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This is to certify that the

dissertation entitled

Rigorous Nunerical Analysis
With High—Order Taylor Models

presented by

Jens Hoefkens

has been accepted towards fulﬁllment
of the requirements for

Ph . D degree in Physics /
Mathemat ics

 

it Mu }W

Major professor

M. Berz

Date jl'll ! Ol

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6/01 cJCIRC/DateOuepes-pJS

 

 

RIGOROUS NUMERICAL ANALYSIS WITH HIGH-ORDER TAYLOR MODELS
By

Jens Hoefkens

AN ABSTRACT OF A DISSERTATION

Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics
and
Department of Physics and Astronomy

2001

Professor Martin Berz

 

 

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ABSTRACT
RIGOROUS NUMERICAL ANALYSIS WITH HIGH-ORDER TAYLOR MODELS
By

Jens Hoeﬂtens

Interval techniques have been utilized for rigorous numerical analysis since the
19608. However, their use has been limited by the dimensionality curse and the
dependency problem. The recently developed Taylor model approach alleviates these
problems and allows the use of validated numerics in a wide range of applications. To
broaden the applicability of the Taylor model method, we introduce new algorithms
for the inversion of functional relations and the integration of differential algebraic

equations.

First we present a new method for computing veriﬁed enclosures of the inverses
of given functions over large domains. The approach utilizes Taylor models and the
sharpness of the enclosures scales with a high order of the domain. An integral part
of the new method is the rigorous determination of invertibility of high dimensional
functions over large domains, which is reduced to a veriﬁed linear algebra problem in-
volving only ﬁrst derivatives of the function of interest. Several examples highlighting

various aspects of the methods are discussed.

Differential algebraic equations (DAEs) describe important problems in mechani-
cal and chemical engineering. Existing algorithms for the integration of DAE initial
Value problems have traditionally been restricted to low-index systems and until re-

cently, no practical scheme for the veriﬁed integration of DAEs existed. Recognizing

 

the antiderivation as a natural operation on Taylor models yields a method that treats
DAEs within a fully differential algebraic context as implicit equations made of con-
ventional functions and the antiderivation. The resulting integration scheme can be
applied to high-index problems and allows the computation of guaranteed enclosures

of ﬁnal coordinates from large initial regions.

To demonstrate the general applicability of the Taylor model approach, we present
results from veriﬁed asteroid orbit integrations and the theory of Hamiltonian systems.
We show that the newly developed methods are practical and can indeed outperform
conventional interval methods in a wide class of problems. Finally, we discuss some
details of implementing interval libraries on general purpose computers and present
the concept of language independent software development, which has been used
for the design and implementation of the C++ and Fortran 90 interfaces to COSY

Inﬁnity.

 

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ACKNOWLEDGMENTS

Most of all, I would like to thank my parents and my sister for their unconditional
love. I am especially thankful to them for supporting me in my decision to leave
Germany for the United States. I want them to know that I understand and appreciate
their efforts and sacriﬁces. I also would like to thank Soo Chang for her never ending
love, faith, and support over the last four years. Whenever life seems dull, she opens

my eyes to the beauty of the world.

The ﬁeld of interval analysis is small but has a close and friendly community of
great scientists. I would like to thank the following colleagues for helpful advice and
fruitful discussions over the years: Christian Bischof, George Corliss, Paul Hovland,
Ken Jackson, Vladik Kreinovich, Rudolf Lohner, Ned Nedialkov, John Pryce, and
Bill Walster. Special thanks goes to Ramon Moore for his ground breaking work on
intervals. Working with him on the asteroid problem was one of the best experiences
I had during my time in graduate school. However, my deepest thanks and gratitude
80130 my thesis advisor Martin Berz. He is a constant source of guidance and wisdom.
He 0Pened my eyes to the rigorous analysis of Taylor models and deserves my thanks
for giVing me the opportunity to study at MSU.

I also would like to thank those professors who have kindly served on my guid-

ance committee: John Masterson, Jerry Nolen, Bradley Sherrill, Daniel Stump, and

Clifford Weil. I have also enjoyed many discussions with friends and fellow students

 

over the course of the last ﬁve years. In particular I would like to mention Jens
von Bergmann, Sen Cheng, Jennifer Church, Lars Diening, Bela Erdélyi, Holger Har-
reis, Michael Lindemann, Kyoko Makino, Declan Mulhall, Cristian Opazo, Khodr
Shamseddine, and Ralf Tiinjes.

For ﬁnancial support of my research I owe thanks to the Studienstiftung des
deutschen Volkes, the US Department of Energy, the Alfred P. Sloan Foundation,
and the National Science Foundation. I am grateful for their continuous support of

young researchers and basic science.

Finally, special thanks goes to my father for making me stay in school when I was
ready to graduate with a Masters degree. Without him, this dissertation would have

never been written.

vi

 

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Contents

LIST OF TABLES

 

x
LIST OF FIGURES xii
LIST OF ALGORITHMS xv
1 Introduction 1
2 Background Information 4
2.1 The Differential Algebra "D1, ....................... 5
2.1.1 Elementary Operations ...................... 6

2.1.2 Contracting Operators and Fixed Points ............ 7

2.1.3 Functions and Antiderivation .................. 8

2.1.4 Summary ............................. 10

2.2 Interval Arithmetic ............................ 11
2.2.1 Elementary Operations and Functions ............. 13

2.2.2 Interval Arithmetic and Set Theory ............... 16

2.2.3 Applications of Interval Arithmetic ............... 18

2.2.4 Summary ............................. 22

2.3 Taylor Models ............................... 22
2.3.1 Arithmetic and Operations .................... 26

2.3.2 Accuracy of Taylor Models .................... 29

2.3.3 Applications of Taylor Models .................. 32

2.3.4 Summary ............................. 34

3 Veriﬁed High-Order Inversion 35

3.1 Rigorous Invertibility Analysis ...................... 36

3.1.1 lnvertibility from First Derivatives ............... 37

3.1.2 Verifying Invertibility with Taylor Models ........... 41

vii

 

3.1.3 Comparison of Invertibility Criteria ............... 46

3.2 Guaranteed Enclosures of Inverse Functions .............. 54
3.2.1 Polynomial Inverses ........................ 54

3.2.2 . Inverse Taylor Models ...................... 56

3.3 Examples of Taylor Model Inversion ................... 59
3.3.1 One-Dimensional Function .................... 60
3.3.2 Six-Dimensional Function .................... 61

3.4 A Superconvergent Newton Method ................... 63
3.4.1 Examples ............................. 66

3.5 Implicit Equations ............................ 71
3.5.1 Curves, Surfaces, Manifolds ................... 73

3.6 Analysis of Inversion Algorithms ..................... 75
3.6.1 Space and Time Requirements .................. 76
3.6.2 Order Dependence ........................ 77

3.7 Summary ................................. 78
Differential Algebraic Equations 81
4.1 Background and Motivation ....................... 82
4.1.1 First Order Differential Algebraic Equations .......... 83
4.1.2 General Differential Equations .................. 87

4.2 Veriﬁed Integration of Implicit ODEs .................. 88
4.2.1 High Order Taylor Model Solutions ............... 90

4.2.2 Mathematical Background .................... 92

4.2.3 Discussion of the Algorithm ................... 97

4.2.4 A Second Order Example .................... 101

4.3 Veriﬁed Integration of DAEs ....................... 103
4.3.1 Structural Analysis ........................ 105

4.3.2 Veriﬁed Index Analysis ...................... 109

4.4 Examples and Applications ....................... 110
4.4.1 Kirchhoff’s Law and Electric Circuits .............. 110

4.4.2 Constrained Mechanical Systems ................ 113

4.4.3 Optimal Prescribed Path Control ................ 118

4.5 Summary ................................. 120

5 Applications of High Order Taylor Model Methods 122
5.1 Solar System Dynamics and Asteroid Orbits .............. 123
5.1.1 Physical Background ....................... 124

viii

Drb.

Judi

5.1.2
5.1.3
5.1.4
5.1.5
5.1.6

 

Geometric Description of Kepler Orbits .............
Veriﬁed Orbit Integration ....................
Results and Discussion ......................
Comparison with AWA ......................

Summary .............................

5.2 Existence of Generating Functions ....................

5.2.1
5.2.2
5.2.3
5.2.4
5.2.5
5.2.6

Introduction ............................
Theory of Extended Generating Functions ...........
Enclosing Derivatives of Flows ..................
Examples .............................
Rigorous Analysis of Symplectiﬁcation Errors .........

Summary .............................

6 Implementational Details
6.1 Implementation of Portable Interval Libraries .............

6.1.1
6.1.2
6.1.3
6.1.4
6.1.5

Floating Point Numbers .....................
Directed Rounding ........................
Implementation of Interval Operations .............
Benchmarks and Results .....................

Summary .............................

6.2 Language Independent Programming ..................

6.2.1
6.2.2
6.2.3

6.2.4
6.2.5

The Least Common Denominator Approach ..........
The C++ Interface to COSY Inﬁnity ..............
The Fortran 90/95 Interface to COSY Inﬁnity .........
Performance Analysis .......................
Summary .............................

A Orbital Elements of Major Planets

B The COSY C++ Interface
B.1 Available COSY Functions ........................
B.2 Available COSY Procedures .......................

C The COSY Fortran 90 / 95 Interface

BIBLIOGRAPHY

131
136
143
153
158
158
159
160
162
164

175
176
176
179
185
187
189
190
190
193
202
210
211

213

215
215
216

219

223

 

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‘ 1.1.4
All.)

in“.

List of Tables

3.1

3.2

3.3

3.4

3.5

3.6

4.1

5.1
5.2
5.3
5.4

5.5
5.6

Taylor model of one-dimensional sine function computed with COSY

Inﬁnity. Shown are Taylor coefﬁcients, reference point, domain infor-
mation, and remainder bound .......................

Left-inverse Taylor model for the Taylor model shown in Table 3.1.

Shown are Taylor coeﬂicients, reference point, domain information,
and remainder bound.

Remainder Bounds of the left-inverse Taylor model for six-dimensional
exponential function given by Eqns. (3.31) and (3.33) ..........

Comparison of different Newton methods for the determination of the
ﬁxed point do of Eqn. (3.35) ........................

Enclosures of 7r, obtained by the interval Newton and Taylor model
methods ...................................

Enclosures of the zero for the six-dimensional exponential function,

computed with an interval Newton method and the high order Taylor
model approach ...............................

Taylor model for the solution of the implicit second order ODE initial
value problem given by Eqn. (4.44) ....................

SI units for length, time, and mass. ...................

Astronomical units and their conversions to SI units.

Units of the derived quantities velocity, acceleration, and force. . . . .

Masses, mean distances, and periods of the major planets. Masses are
given as fractions of the Sun mass M, mean distances in astronomical
units, and times in Julian years [125] ...................

Geometric shape of conic sections, depending on the eccentricity 5.

Dates and minimal distances of the predicted closest Earth approach

distances for the near-Earth asteroid 1997 XFll between the years
2000 and 2040 [73]. ............................

5.7 Taylor model describing ﬁnal 2: positions as a function of the initial

conditions 9:0, 00, yo, b0 for the accelerator cell of Sec. 5.2.4. .

6.1 Average execution times for the three interval test cases in pure For-

tran 77 environments ............................

60

61

63

67

70

71

104

125
126
126

129
132

148

171

188

 

1.3

 

6.2 Average execution times for the three interval test cases in COSY In-
ﬁnity language environments. ......................

6.3 Execution times in minutes and seconds for a typical mix of DA oper-
ations on different platforms with different interfaces to COSY Inﬁnity. 210

Al Orbital Elements of the major planets (angles in degree, distances in

astronomical units) at the epoch J 2000 .................. 213
A2 Daily rates of change of the orbital elements {2, z', and w of the nine

major planets . .............................. 214
A3 Daily rates of change of the orbital elements a, a, and M of the nine

major planets ................................ 214
0.1 Deﬁned combinations of COSY objects and standard data types for

the addition ................................. 219
C2 Deﬁned combinations of COSY objects and standard data types for

the subtraction ............................... 219
C3 Deﬁned combinations of COSY objects and standard data types for

the multiplication. ............................ 220
0.4 Deﬁned combinations of COSY objects and standard data types for

the division ................................. 220
C5 Deﬁned combinations of COSY objects and standard data types for

the power operation. ........................... 220
0.6 Deﬁned combinations of COSY objects and standard data types for

the comparison .LT ............................. 220
C7 Deﬁned combinations of COSY objects and standard data types for

the comparison .GT ............................. 221
C8 Deﬁned combinations of COSY objects and standard data types for

the comparison .EQ ............................. 221
C9 Deﬁned combinations of COSY objects and standard data types for

221

the comparison .NE .............................

C.10 Deﬁned combinations of COSY objects and standard data types for
the COSY concatenation .UN .. ..................... 221

H

In.

List of Figures

2.1
2.2
2.3

2.4

2.5

2.6

2.7
2.8
2.9
2.10

3.1

3.2

3.3

3.4

3.5

3.6
3.7

3.8

Commuting diagram for the elementary operations on "D” ....... 7
Commuting diagram for the elementary operations on intervals. . . . 14
Illustration of the wrapping effect caused by a rotation of 45 degrees

in the plane ................................. 17
Illustration of how the wrapping effect can lead to arbitrary large over-
estimations. ................................ 18
Taylor models of order one and ﬁve for the sine function over the
domain interval [—1.5, 1.5] ......................... 25
Left: Enclosing a function by a Taylor model of order eight; right:
Interval bounding of the same function with 50 sub—intervals ...... 25
Commuting diagram for elementary operations on Taylor models. . . 27
Commuting diagram for sine function on Taylor models. ....... 28
Commuting diagram for antiderivation of Taylor models. ....... 29
Interval bounding of a two-dimensional function by Taylor models of
orders seven, eight, nine, and ten. .................... 31
Percentage of random functions that can be shown to be invertible as

a function of dimensionality ........................ 47
Percentage of random functions that can be shown to be invertible as

a function of domain size. ........................ 49
Percentage of functions that can be shown to be invertible as a function

of the non-linearity 5 ............................ 50
Percentage of maximal domain size over which invertibility can be
proven as a function of complexity and non-linearity in the partial
derivatives. ................................ 52
Percentage of random functions that can be shown to be invertible as

a function of functional complexity. ................... 53

Illustration of the domains in the deﬁnitions of inverse Taylor models. 59
Difference between the arcsine function and the reference polynomial

of the 19—th order Taylor model shown in Table 3.2 ........... 62
Transformed Fixed-Point problem: the ﬁxed point an of Eqn. (3.35) is
the intersection point of the two graphs. ................ 67

xii

3.9 Taylor polynomial approximation of order 25 of the sine function over

the interval [0, 12].

IIIIIIIIIIIIIIIIIIIIIIIIIIII

3.10 Charts for the cartesian coordinates x and y of the two-parameter

3.11

3.12

4.1

4.2

4.3

4.4

4.5

5.1

5.2

5.3
5.4

5.5

5.6

5.7

5.8

description Eqns. (3.44) for hyperbolas. Remainder bounds are in the
order of 10‘s. ...............................

Illustration of veriﬁed inversion with intervals. The sharpness of the

75

enclosures scales linearly with the magnitude of the domain sub-intervals. 77

Double-logarithmic plot of the accuracy of the ﬁrst Taylor model N ew-
ton step as a function of initial domain size; results for 6—th order
Taylor models. ..............................

Illustration of an electric circuit described by the index-1 DAE (4.69).
Ub is the operating voltage and U8 is the driving voltage of the circuit.

Illustration of a double pendulum with masses m1 and m2 connected
by massless rods of lengths l1 and I2, respectively. ...........

Coordinates 2:1 and y1 of the double pendulum for 901(0) = 902(0) =
5°,30°,90°, 273(0) = 235(0) = 311(0) = 315(0) = 0 and 0 S t _<_ 100
(integrated in order 6, h = 0.1, g = 1, l1 =12 =1,m1= m2 =1). . . .

.732- and yz-phase space projections for the double pendulum with for
901(0) = ¢2(0) = 5°,30°,45°, 373(0) = $610) = 34(0) = 215(0) = 0 and
0 S t g 100 (integrated in order 6, h = 0.1, g = 1, 11 = l2 2 1,
m1 = m2 = 1) ................................

Coordinates 2:1 and y1 of the double pendulum for 901(0) = 0, (p2(0) =
45°, cc’1(0) = 333(0) 2 y’1(0) = y§(0) = 0 and 0 S t S 100 (integrated in
order 6, h = 0.1, g = 1,l1=l2 =1,m1=100, m2 =1). ........

Orbits of the inner planets Mercury, Venus, Earth, and Mars around
the Sun. ..................................

Orbits of the outer planets Jupiter, Saturn, Uranus, Neptune, and
Pluto around the Sun. ..........................

Illustration of the eccentric anomaly E .................

Residuals of Jupiter; two Very Long Baseline Interferometry observa-
tions from the Galileo spacecraft; courtesy of [130]. ..........

Cartesian positions (in AU) and velocities (in AU /TU) of 1997 XF11
during the ten year integration interval ..................

Logarithmic plot of the diameters of enclosures for the positions (in
AU) and velocities (in AU/TU) of 1997 XF11 during the ten year
integration interval .............................

Evolution of the step size and one-step integration errors during the
ten year integration of 1997 XF11. ...................

Differences in the computed positions and velocities of the asteroid
1997 XF11 between the ten year integrations with and without rela-
tivistic corrections. ............................

xiii

78

111

114

116

117

117

128

129
134

139

146

147

5.9 Illustration of the shrink wrapping method used in the veriﬁed inte-
gration of asteroid orbits ..........................

5.10 One-step and total shrink factors obtained during the ten year integra-
tion of the asteroid 1997 XF11. .....................

5.11 Logarithmic plot of the diameters of the enclosures for positions ob—
tained by AWA (upper) and the Taylor model based integrator.

5.12 Logarithmic plot of the diameters of the enclosures for positions ob-
tained by AWA for an initial box reduced by a factor of 3.275 in each

direction. .................................
5.13 Logarithmic plot of the diameters of the enclosures for positions ob—
tained by AWA for initial condition enclosures of 10‘15 .........

5.14 ’Iiacking picture of the cubic two—dimensional symplectic map, and the
box of guaranteed invertibility of the generator associated with S = 0.

5.15 Illustration of the Fermi-Pasta-Ulam system with three masses and

four springs .................................
5.16 (q1,p1) and (q1,q2) tracking pictures of the half period map of the
Fermi-Pasta-Ulam system for a particle launched along the ql axis.
The box of guaranteed invertibility of the generator associated with
S = 0 extends to at least [—1, 1} in every direction. ..........

5.17 Illustration of the accelerator cell studied in Sec. 5.2.4 .........

5.18 (x, a) and (y, b) tracking pictures of the one-turn map of the accelerator
cell for particles launched along the spatial :r and y axes, respectively.
The box of guaranteed invertibility of the generator associated with
S = 0 extends to at least [—0.1, 0.1] in the spatial variables. .....

xiv

152

153

155

156

157

166

167

168
170

List of Algorithms

6.1 Interval addition with hardware support to set the rounding mode. . 180
6.2 Directed rounding of the two interval endpoints of [$1,132]. ...... 184
6.3 Portable determination of the machine constant 5 at run time. . . . . 184
6.4 Portable determination of the machine constant p at run time ..... 184
6.5 Reliable interval extension of the sine function .............. 186

XV

H.

 

 

 

 

Chapter 1

Introduction

Over the last four decades, the use of computer programs has gained widespread
acceptance in almost all scientiﬁc and engineering disciplines. An implicit trust in
the computed results is based on the assumptions that computer programs are imple-
mented correctly and perform as intended. However, the validity of these assumptions

is questionable, to say the least.

Considering the fallibility of humans, it is presumptuous to believe that software
could be produced free of errors. Moreover, since verifying the correctness of imple-
mentations is equivalent to the halting problem; it is generally impossible to prove
that a given computer program does not contain bugs [117]. While elaborate testing
procedures can reduce the number of errors, even the most important and extensively
tested software systems can fail. As an example, consider the explosion of an Ariane

5 rocket in 1996 which has been traced to a very simple programming error [77].

However, even assuming that software could be written free of bugs, other sources
of errors still exist. Most of these errors are caused by the transition from the perfect
world of numerical analysis to the limited world of ﬁnite state machines. While
numerical analysis often assumes the existence of inﬁnitely many real numbers and

inﬁnitely precise computations, computers have only ﬁnite memory and ﬂoating point

numbers are stored with only ﬁnitely many digits, resulting in computations that are

only approximations to the mathematically correct results.

Most modern computers implement double precision ﬂoating point numbers with
approximately 16 signiﬁcant digits. While this is sufficiently accurate in most ap-
plications, the resulting rounding and truncation errors can accumulate and lead to
arbitrary large deviations from the mathematically correct results. Moreover, while
humans prefer decimal numbers, computers usually store information in binary repre-
sentations [48]. The conversion from decimal numbers to binary numbers, combined
with the limited accuracy of the number representations, can be a signiﬁcant source
of truncation errors. To illustrate this, consider the inﬁnite binary representation of

0.1:

0.110 = 0.001 10012. (1.1)

On a computer with 24 bit ﬂoating point numbers, the error caused by the truncation
amounts to an absolute error of approximately 9.5 - 10'8. While this seems insigniﬁ-
cant, the high failure rates of the Patriot missile system during the Gulf War in 1991

have been traced to exactly this truncation error [141].

It is important to note that the problem of roundoff and truncation errors is a
fundamental characteristic of ﬁnite state computers. While intimate knowledge about
the underlying architecture and signiﬁcant efforts in the implementation of computer
programs could reduce the effects of these errors, such errors can never be completely
eliminated. Moreover, the extra effort necessary would require a software development
model contradicting the idea of using portable high-level programming languages that

help the programmer focus on the correctness of the implementation.

Thus, on the one hand we desire the portable programming of general purpose,

limited precision binary architectures, on the other hand we know from theory and

 

 

 

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practice that this combination can lead to signiﬁcant computational errors. Nev-
ertheless, computers are here to stay and their use is likely to increase. Moreover,
computational results are increasingly used as the basis of important decisions, rang-
ing from the operation of nuclear power plants to the world ﬁnancial markets. In other

words, potentially erroneous results are used to control systems of global importance.

Somewhat contrary to naive intuition, the main problem with these numerical
inaccuracies is that most computed results are actually sufﬁciently good approxima—
tions of the mathematically correct results and only a very small number of cases
exhibit signiﬁcant errors that require further investigation. It is the goal of validated
methods to solve this dilemma by providing the users with bounds that are guaran-
teed to contain the mathematically correct result. While validated methods do not
magically increase the accuracy of computations, they provide a self-validating ap-
proach that computes both an approximation to the mathematically correct result
and a rigorous upper bound on the error of the approximation. In most situations
these are relatively tight bounds and only in a few cases will they be large, indicating

that the computed results may be rather bad approximations of the correct results.

While the fundamental ideas behind validated methods have already been intro—
duced in the dissertation of R. Moore in 1962 [91], early methods have often computed
overly large bounds. Only recently, starting with work by R. Lohner in 1987 [78], have
validated methods been able to produce sharper and more usable bounds. In this dis-
sertation we present new results from the ﬁeld of interval methods [92], which lie at
the core of the validated approach. The results are based on the theory and the ap-
plication of the recently developed Taylor models [83, 82], which combine high order
Taylor polynomials with intervals for validation. Taylor models can in some situations
increase the sharpness of the computed bounds by several orders of magnitude over

conventional methods, thereby improving the applicability of validated methods.

Chapter 2

Background Information

In this chapter we introduce the notational conventions that will be used throughout
this dissertation. We also summarize the mathematical and computational theories
that form the backbone of the material developed in this dissertation: the differential

algebra ﬂD,“ interval analysis, and Taylor models.

Throughout this dissertations N denotes the set of positive integers, Q is the
set of rational numbers, IR stands for the set of real numbers, and C is the set of
complex numbers. For notational convenience, we will always assume that n, v, w E N.
Following the usual conventions, IR” denotes the set of v-dimensional vectors with real

entries; elements of R” are written as

:1: = ($1,...,:1:.,), (2.1)

with real numbers 2:,- 6 IR for z' = 1,. . .,v. If necessary, we denote vectors by if to
distinguish them from real numbers 2:. However, in most cases, such a distinction will
not be necessary. Finally, unless stated otherwise, all functions are assumed to be at

least (n + 1)-times continuously differentiable over their domain.

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2.1 The Differential Algebra n.0,,

Many of the results presented in the following chapters rest on the availability of
accurate descriptions of high-order Taylor polynomials of sufﬁciently smooth functions
on computers. To avoid the need for storing an inﬁnite number of Taylor coefﬁcients,
it is convenient to consider the equivalence classes of functions with the same n—th
order Taylor polynomials. Since these are ﬁnite objects, the resulting structures can
be implemented on computers, and they can be used for the efficient and accurate

modeling and representation of complicated functions.

Let U C R” be an open set containing the origin and consider the space C "+1 (U, R”)
of (n +1)-times continuously differentiable functions that map U into IR”. We deﬁne

the relation of equality up to order n as follows.

Deﬁnition 2.1. For f,g E C"+1(U,R"’) we say that f equals 9 up to order n if
f(0) = 9(0), and all partial derivatives of orders up to n agree at the origin. If f

equals 9 up to order n, we denote that by f =n g.

It is easy to see that equality up to order n establishes an equivalence relation on
the space C"+1(U,R“’) [17]. The resulting equivalence classes are called DA vectors,
and the class containing the function f E C"+1(U,R"’) is denoted by [f],,. The

collection of these equivalence classes is called ”B”. More details on this structure

are given in [12, 17].
PropOSItion 2.1. For f E C"+I(U,R"’), the n-th order Taylor polynomial Tn( f) of

f is contained in [f]n.

This assertion follows easily from the basic deﬁnition of the equivalence classes.

However, the fact that the n—th order Taylor polynomial of f can be used as a rep-

9

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resentative for the class [f],, opens the door for a computer implementation of the

structure "D, by storing and manipulating the coefﬁcients Of Taylor polynomials.

2.1.1 Elementary Operations

Elementary operations like “+” and “x” can be lifted from C”+1(U, R”) in the usual

way, and extend to the corresponding Operations “EB” and “(8” on "D, [12, 13].

Deﬁnition 2.2. Let f, g E C"+1(U,R‘”) be two functions. Then the sum of the DA

vectors [ﬂu and [g]n is given by

lfln 69 [9]" = if + gln- (2.2)

The product of the two DA vectors is deﬁned by
lfln ® [gin = if X Slim (23)

Together with the scalar multiplication r - [f],, = [r - f],,, this deﬁnition of the
elementary operations makes “D, an algebra [17]. Moreover, the diagram in Fig. 2.1
is closed and commuting. In other words, the extension of the elementary operations
to "D, is transparent to the equivalent classes of DA vectors; i.e., knowledge of the
values and derivatives of f and g at the origin is sufﬁcient to obtain Taylor polynomials
of their sums and products. Moreover, in [12, 13] the available Operations on "D,
have been extended to include subtraction and, for a limited class of DA vectors,
even the multiplicative inversion. From now on we omit the distinction between the
Operations on C"+1(U,1R"’) and "D, and will always use the same symbols “+” and

“X” for operations between numbers, functions, and DA vectors.

It has been shown that the derivative Operation (derivation) can be extended
from C"+1(U,R“’) to the algebra "D, in such a way that ”D, becomes a diﬁerential
algebra [12, 17]. While we will not use this intrinsic structure Of "D”, we will make

frequent use of the antiderivation of DA vectors to be presented later.

11‘)

 

fig 4 lfln’lgl"

(+v ) (63, 6)

 

 

ii i
f(+. -)g a [flue ©>lgln

 

Figure 2.1: Commuting diagram for the elementary Operations on "D”.

2.1.2 Contracting Operators and Fixed Points

In the previous section we have shown how elementary operations on the function
space C"+1 (U, R‘” ) can be extended to the differential algebra ”Du. Here we take a look
at the more general concept of operators on “D, and summarize an important ﬁxed
point theorem. The availability of this powerful ﬁxed point theorem for operators
on nDu allows the use of the differential algebra ”D, in a large class of numerical

applications, ranging from the analysis of dynamical systems [15, 23, 27, 18] to global

optimization [68, 88].

Deﬁnition 2.3. For [f],, 6 "D”, the depth A ([f],,) is deﬁned to be the order of the

ﬁrst non-vanishing derivative of f if [f],, 76 0, and n + 1 otherwise.

By deﬁnition of the equivalence classes, this deﬁnition is independent of the choice
of f E [f]. We note that any a E “D, with A(a) 2 1 satisﬁes the condition a"+1 = 0
and is therefore called nilpotent. Using the straightforward deﬁnition of the depth,

contracting operatOrs on ”D, are deﬁned as follows.

Deﬁnition 2.4. Let 0 be an operator deﬁned on M C "D”. O is contracting on M,

’31]

'l

‘v
,I.‘l

'1.

h

I"..

‘. 5

|
,I‘IP
I .

iffor any two [f],,, [g]n E M,

’\(O(ifln) '- 0([gln)) 2 ’\(lfln — iglnl (2'4)

with equality if and only if f 2,, g.

This deﬁnition has a striking similarity to the corresponding deﬁnitions on stan-
dard function spaces. Even more so, a theorem that resembles the Banach Fixed
Point Theorem can be established on "D”. However, unlike in the case of the Banach
Fixed Point Theorem, in “D, the sequence of iterates is guaranteed to converge in at

most n + 1 steps [17].

Theorem 2.1 (DA Fixed Point Theorem). Let 0 be a contracting operator and
self-map deﬁned on M C "D”. Then 0 has a unique ﬁxed point a E M. Moreover,

for any a0 6 M it is 00‘“) (a0) = a.

A proof and further discussion of the DA Fixed Point Theorem can be found
in [17]. Here we just mention that, since )1(a + b) 2 min()\(a), A(b)), it follows easily
that the sum and composition of two contracting Operators 01 and 02 deﬁned on M

is also a contracting Operator.
2.1.3 Functions and Antiderivation

To fully utilize the differential algebra "Du, especially in numerical analysis and com-
puter environments, it is necessary to not only deﬁne the elementary operations on
an but also the standard mathematical functions commonly available on computers:

square root, exponential, logarithm, trigonometric and hyperbolic functions.

At a fundamental level, the basic functions on ”D" are simply deﬁned in terms of

the corresponding operations on the function space [17].

Deﬁnition 2.5. For f,g E C"+1(U,R"’) we deﬁne 9(lfln) = [9(fllrv

 

Y

 

 

. . r. .l. 1 X. L . l. Tru \ .
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While this deﬁnition is straightforward and of great theoretical value, the actual
computation of functions on "D” is based on addition theorems and the Taylor series
of these functions. Although the full details of this procedure are beyond the scope of
this summary, the following example illustrates the general approach. Let [f ]n 6 "D,

be a DA vector and write [f],, = a0 + b with a0 = f (0) Then it is

en) (lfln) = exp(ao + b) = exp(ao) exp(b)- (2-5)

Using the deﬁnition of the exponential function as a power series and the fact that b
is nilpotent, in fact

mm b"
exp ([fln) = exp(ao)' Z 75'" (2-6)
k=0 ’

Further details on the implementation of functions of DA vectors can be found in [12,

13, 17].

The approach outlined above allows us for any function f, for which we have a
code list or algorithm consisting of ﬁnitely many intrinsic functions and elementary
Operations, to obtain the n—th order Taylor polynomial of f around the origin. By
starting the evaluation with the identity DA vectors [id],,, we obtain Tn( f ) by evalu-
ating the code list of f with the argument [id],,. This gives a convenient and powerful
method of computing derivatives Of functions described by computer programs. Un-
like conventional automatic differentiation [111, 52], which is often limited to ﬁrst
and second order, the Taylor series approach does not pose any arbitrary limits on

the maximum order of derivatives that can be computed.

We conclude this section on functions on "D, with an example of an operator
that is unusual but, considering the structure of the differential algebra "D”, actually
quite natural. The antiderivation; i.e., the integration with respect to any Of the v

variables, turns out to be a contracting Operation on all;-

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Proposition 2.2 (Antiderivation is Contracting). For k e {1, . . .,v}, the an-

tiderivation 6,:1 : nD.,—>,,D,, is a contracting operator on "D”.

The proof of this important result is based on the fact that if a, b E "D, agree
up to order l, the ﬁrst non-vanishing derivative of 6,:1(a - b) is Of order l + 1 [17,
59, 60]. It is important to realize that in the DA framework of "Du there is no
fundamental difference between any of the standard mathematical functions like the
sine and exponential functions on "D” and antiderivation. In fact, fully embracing
antiderivation as a normal operation on DA vectors will enable us to develop a new
and powerful method for the veriﬁed integration of ordinary differential equations

(ODEs) and differential algebraic equations (DAES) in Chap. 4.

2.1.4 Summary

For functions that are given by ﬁnitely many intrinsic functions and elementary Op-
erations, the DA approach is equivalent to evaluating the code list with an n-th order
automatic differentiation tool. However, the ease with which the DA approach com-
putes Taylor polynomials, and therefore derivatives, of any given algorithm up to
machine precision and virtually arbitrary order makes the differential algebra “D” an
important computational tool for numerical analysis. Together with the powerful DA
Fixed Point Theorem that guarantees convergence in at most n+ 1 steps, the method
lies at the core of the map approach that has been used successfully in the analysis
and the design of particle accelerators [10, 12, 14, 15, 17]. The differential algebra
”D, has been implemented in the arbitrary order code COSY Inﬁnity [13, 23, 84].
And by storing only non-vanishing coefﬁcients, the implementation can handle even

high-dimensional problems to very high orders.

By combining the symbolic nature Of operations on truncated polynomials with the

10

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numerical operations on ﬂoating point coefﬁcients, differential algebra based methods
offer a unique combination of exact symbolic Operations with the efficiency of ﬂoating
point computations. Moreover, the DA approach avoids the pitfalls of conventional
computer algebra, since it does not suffer from the dramatic increase in storage that
often plagues symbolic computer algebra tools in high-order applications. At the
same time, propagating high order descriptions of computer code avoids many of
the problems associated with traditional numerical methods and opens the door for

rigorous high-order sensitivity analysis of dynamical systems [104].
2.2 Interval Arithmetic

The use of interval methods in computational sciences was started by R. Moore [91],

who observed that,

“Computers carry only a limited number of signiﬁcant digits. Repeating
a computer calculation with more signiﬁcant digits does not necessarily

increase the number of signiﬁcant digits in the result.”
To illustrate this problem, consider the sequence of numbers deﬁned recursively by
1130 = 1 — 1.0—21 and 513".“ = 1:712- (27)

If we compute this sequence on a computer with 10, 16, or even 20 digit accuracy, it

is
$o=$1=---=$75=1, (2.8)

while in fact .1375 < 10““). Since the question in numerical computations is often
how the result compares to some ﬁxed number, this is actually an important problem
and computational errors like the above are relatively frequent. These computational

errors have many sources: ﬂoating point errors, rounding errors, truncation, and

11

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initial errors. And since they are hard to identify at the time of actual computation,

they are often undetected and can have dire consequences.

The previous example illustrates the main problem of naively using computers
to implement results of numerical analysis. Due to the fact the computers can only
represent a small subset of the rational numbers Q accurately, the results obtained by
computer operations are generally only approximations of the mathematically correct
results. While traditional methods of numerical analysis are very powerful and provide
rigorous results in an ideal world, their transformations to computer systems suffer

from the intrinsic limitations of ﬁnite state machines.

However, modern interval techniques use knowledge of these limitations to com-
pute results that are both accurate and reliable. In this section we summarize the
fundamentals of conventional interval methods. The development of powerful new

interval techniques is the main focus Of this dissertation.

Interval Notations

Before presenting details of interval analysis, we introduce notation and conventions
that will be used in connection with intervals throughout the text. For two numbers

a, b 6 R, the interval [a, b] is deﬁned by
[a,b]={a:€lR]a_<_:ch}. (2.9)

Unless explicitly stated otherwise, intervals will always be assumed to be closed and
bounded and therefore compact. Whenever we talk about intervals in general, we use
boldface to distinguish them from regular numbers: X is a point, while X denotes a

compact interval.

To express containment of points and intervals within intervals, we use the stan-

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dard notation

xE[a,b]¢)anSb (2.10)

[a,b]C[c,d]®cgagbgd (2.11)
The union and intersection of two intervals are deﬁned by

[a, b] U [c, d] = {as E IRlz E [a, b] or :1: E [c,d]} (2.12)

[a, b] (I [c, d] = {:12 E Rlx E [a, b] and a: E [c,d]} (2.13)

The union of two intervals will generally not be an interval. The intersection of two
intervals on the other hand is either an interval or empty. Lastly, we denote the

midpoint and the width of an interval by m ([a, b]) and w ([a, b]), respectively.

In the case Of IR” with v > 1, we deﬁne interval boxes to be vectors of intervals and
all previously mentioned operations and relations are deﬁned componentwise. The
width w of an interval box is called the magnitude and is deﬁned as the maximum of

the widths of its components.

2.2.1 Elementary Operations and Functions

We deﬁne elementary Operations on intervals set-wise such that they satisfy the fun-

damental inclusion requirement: for * E {+, —, x, /} we demand that
[a,b]*[c,dl={x*ylas:v_<_b,c5ysd} (2.14)

In other words, the sum, difference, product, or quotient Of two intervals is the set of
sums, differences, products, or quotients of pairs of real numbers, one from each of

the two intervals. All elementary operations can be deﬁned in such a way that the

13

 

 

 

 

any ‘ =- I,,,,,Iy
(+,><, ) (as...)
ii 1)
$(+a X, )y (- 4 Ix(®1®t )Iy

Figure 2.2: Commuting diagram for the elementary operations on intervals.

results of Operations between intervals are again intervals:

Ia: G Iy :: [$1 + 91,552 + y2] (2'15)
Ian 9 I1, = [171— 312,332 — 91] (2-16)

Iz®Iy = [min{$lyly$1y27$2y17$2y2}ymax‘f-lela-lebeI/la3729/2“ (2-17)

If the interval 1,, does not contain 0, the multiplicative inverse is given by

If = [1/y2,1/yll (2.18)

and the quotient Of two intervals is deﬁned in terms of the multiplicative inverse and
multiplication:

hegzhegi (mm

With these deﬁnitions, the diagram in Fig. 2.2 closes and commutes under the inclu-
sion relationship. From now on, we will denote the elementary Operations on intervals

and real numbers by the same symbols.

One problem of interval arithmetic is the dependency problem. It is best illustrated

with the interval I = [-1, 1] and the computation
I - I = [—2,2]. (2.20)

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:1

a“

 

l
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- l
\‘ .,
L"
l"
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Thus, the width of I — I is twice as large as the width of the original interval I.
This problem, known as the cancellation effect, is a manifestation of the more general
problem that conventional interval arithmetic has no provision to distinguish between
independent and dependent variables. As far as interval analysis is concerned, the
two intervals on the left hand side of Eqn. (2.20) are two different entities with no
relation between them, and the fact that they have the same endpoints is seen as
a mere coincidence. The dependency problem especially affects large computations,
since not all occurrences of the dependency problem are as easy tO spot as in this
example. We note that work is under way to include techniques for resolving the
dependency problem within Fortran and C++ compilers [136, 137], where the Opti-
mizers already perform a dependency analysis that is capable of also reducing the
dependency problem. We mention that even conventional ﬂoating point arithmetic is

beset by cancellation effects: on a computer with less than twenty digits of accuracy
(1020 +1) — 1020 = 0. (2.21)

This exempliﬁes a general problem of ﬂoating point arithmetic: computing the dif-
ference of two similar numbers Often results in large relative errors. And although
interval computations always enclose the correct results, the cancellation effect can

lead to signiﬁcant overestimations.

Similar to the way the elementary operations are deﬁned on intervals to satisfy
Eqn. (2.14), the standard mathematical functions for intervals are deﬁned such that
the basic requirement

f (la, bl) = {f(rr) la S cc S. b} (2.22)

is always satisﬁed. In many cases the computation of interval extensions of mathe

matical functions is straightforward and details will be discussed in Sec. 6.1.

15

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2.2.2 Interval Arithmetic and Set Theory

Since interval arithmetic has its roots in the desire to compute rigorous results with
limited precision ﬂoating point hardware, it has originally been used to model small
intervals. However, by realizing that the interval [a, b] is actually the set of real num-
bers between a and b, where a and b can be rational numbers with exact ﬂoating point
representations, computer programs become able to accurately handle real numbers.
Thus, by using intervals to represent sets of real numbers, intervals bring set theory

and the ability to represent any real number to computers.

Utilizing the elementary operations and functions deﬁned in the previous section,

we can Obtain the Fundamental Theorem of Interval Analysis [94].

Theorem 2.2 (Fundamental Theorem of Interval Analysis). Let I = [a, b] be
a compact real interval. If f : I —> IR is a continuous function and f is its inclusion

monotone interval extension, then

17 6 [a,b] => f(x) 6 f(I)- (2.23)

The Fundamental Theorem of Interval Analysis allows the use of computers, de-

spite their limitation, to rigorously answer the question:

Given a function f described by a computer program, consisting of ﬁnitely
many elementary Operations and intrinsic functions, and an interval I, ﬁnd
a set of real numbers that is guaranteed to contain the image of I under

the function f.

By allowing the computation of intervals that are guaranteed to contain the range
1' (I) of f over I, the use of interval analysis allows the use Of computers with ﬁnite

precision to obtain rigorous and trustworthy results.

16

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Some of the most powerful aspects of interval computations are tied to the Brouwer
Fixed Point Theorem, which is the ﬁnite-dimensional version of the Schauder Fixed

Point Theorem:

Theorem 2.3 (Brouwer Fixed Point Theorem). Let D be a non—empty compact
convex subset of IR” and suppose f is a continuous mapping such that the range

1‘ (D) C D. Then f has a ﬁxed point in D, i.e. there is :1: E D such that f(at) : 1:.

Since interval boxes satisfy the requirements on D, the Brouwer Fixed Point
Theorem can be combined with the Fundamental Theorem of Interval Analysis to
allow computer based proofs of the existence of solutions to linear and non-linear
systems. One particular important application of this, the basic interval Newton

method, will be discussed in Sec. 2.2.3.

The Wrapping Effect

Much like the dependency problem, the wrapping effect can lead to substantial over-
estimations in interval computations. It is caused by the need to wrap the results of
interval computations in interval boxes that are parallel to the axes. The effect is
demonstrated in Fig. 2.3: a rotation by 45 degrees leads to an overestimation Of the

magnitude of the result by a factor of \/2.

 

 

 

 

 

 

Figure 2.3: Illustration of the wrapping effect caused by a rotation of 45 degrees in
the plane.

While efforts have been made to ﬁght the wrapping effect by allowing rotated and

even skewed interval boxes [78, 79], the fundamental problem remains and Fig. 2.4

17

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c

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illustrates how the fact that the edges of interval boxes are always given by straight

lines can lead to arbitrary large overestimations of the image sets.

 

 

 

 

 

 

Figure 2.4: Illustration of how the wrapping effect can lead to arbitrary large overes-
timations.

2.2.3 Applications of Interval Arithmetic

By utilizing numerical analysis tools that have been developed speciﬁcally for interval
analysis, interval methods have been used successfully in a variety of applications over
the last decades. In this section we take a closer look at some of the most important
of these algorithms and applications: interval Newton methods, optimization, and

integration of ODEs.

Interval Newton Method

Interval Newton methods are among the most fundamental and important interval
algorithms. They are, in one form or another, part of almost any other interval
algorithm. Since they utilize most of the concepts Of other interval methods, e.g.
ﬁxed point formulations, iteration, and intersections, they are outstanding examples

of the more general interval analysis tools.

Let f : [a, b] —-> IR be a C1 function such that either f’ > 0 or f’ < 0 on [a, b] and
assume that 51:27 E [a, b] such that f (a) = 0. The goal of an interval Newton method
is to compute a small interval enclosure Of :11. According to the Mean Value Theorem,

for any y E [a, b] there is some 5 E [a, b] such that
0 = f(x) = f (y) + f’(€)(:r - y)- (2-24)

18

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l

,[.]o

 

After choosing y = m([a, b]) and solving for 2:, it is

M, , _muuz
— (1 Jill f’(§) . (2.25)

with some unknown 6 E [a, b]. However, if we use intervals to evaluate the interval
extension of the derivative f’, we can guarantee that f’ (6) E f’ ([a, 5]). Assuming

that 0 E f’ ([a, b]), it follows that

f (m ([a, b1»
"'f'(1a.b'1) ' (2'26)

While f (m([a,b])) and m([a,b]) in the last equation are points, 1'" ([a,b]) is an

2: E m([a,b]) ——

interval and the right hand side is therefore itself an interval. This last equation leads

to the following deﬁnition Of the Newton operator N for an interval I C [a, b]:

N (I) = m (I) — M (2.27)

f’ (I)
With the Newton operator, the foundation Of the standard interval Newton method
is given by Thm. 2.4; more details of interval Newton methods can be found in

[92, 94, 3, 57].
Theorem 2.4. For X 0 = I, deﬁne a sequence of intervals by
Xn-l-l : N (Xn) n Xn- (2.28)

If I contains a zero of f, X n will contain the zero for any n. Moreover, if for some

n the intersection N (X n) n X n is empty, I did not contain a zero of f.

It is important to note that intersecting the result of the Newton operator with
the previous enclosure ensures that the elements of the sequence never increase in
size. Thus, unlike in the regular Newton method where the sequence of iterates
might diverge away from a zero, the sequence of magnitudes w (X n) is monotonically

decreasing.

19

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Optimization

Interval algorithms for constrained and unconstrained optimization are based on an
exhaustive search of the search initial region. They usually follow the general scheme
for Lipschitz optimization [106], but use interval evaluations Of the Objective function

instead of estimations based on Lipschitz constants.

The general problem of constrained and unconstrained optimization is given by

minimize: ¢(2:) (2.29a)
such that: C(23) = 0 (2.29b)
9(2) 3 0. (2.29c)

ct : D C IR" —+ IR is the objective function; c : D —+ IR’c and g : D —> IRl are
constraint conditions. If l 2 2, condition (2.29c) is understood to apply to each of

the components of g. If k = l = 0, the problem is considered to be unconstrained.

The basic algorithm for the exhaustive search starts by dividing the initial search
region D into sub-domains and maintaining a list of boxes where the minimum might
be. The boxes in the list are repeatedly tested and split further if necessary. While this
by itself leads to an exponential growth Of the number of boxes, the main idea of the
algorithm is to speed up the search by rejecting boxes from the list; i.e., to rigorously
determine that the minimum cannot be inside a particular box. Interval methods
play an important part in this algorithm. They are used in storing information on
possible regions in the list and, more importantly, they are used to compute veriﬁed
range enclosures Of the objective function to possibly reject boxes from the list of
candidate boxes or to verify that the box contains the global minimum. More details

of such algorithms can be found in [114, 58, 56]-

Sophisticated extensions of the basic branch and bound algorithms exist that use

20

 

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interval methods to improve the decision process Of rejecting boxes [115, 34]. However,
while global optimization with intervals has found successful applications [37], it Often
struggles due to the exponential growth in the number of candidate boxes. This is
especially true for complicated multidimensional objective functions [21]. More details
on constrained and unconstrained optimization can be found in [140, 29, 41, 55, 63,
95, 38]. We would like to highlight the GlObSol package [67, 68, 46] as one particular

sophisticated implementation of global Optimization algorithms with intervals.
ODE Initial Value Problems
Consider the general explicit ﬁrst order ODE initial value problem
22' :2 f(t,:r), 23(t0) = 2:0 (2.30)

with a sufﬁciently smooth function f deﬁned on a suitable subset of IR x R”. Inter-
val methods for the integration Of ODE initial value problems provide enclosures for
the ﬁnal positions, such that at every time step t,- the actual solution to the initial
value problem is contained in the solution interval. To achieve that goal, the integra-
tion methods take extended initial conditions, mathematical truncation, and roundoff

errors into account.

However, mostly due to the wrapping effect discussed in Sec. 2.2.2, methods for the
integration of ODE initial value problems are among the most demanding problems
for designers of interval algorithms [98]. While the ﬁrst attempts at interval based
integration tools [92] have struggled with the prOpagation of large initial regions even
over small time intervals, newer methods like AWA use changes of variables, ellipsoid
interval boxes and other techniques [79, 101] to ﬁght the wrapping effect with varying
degrees of success. But only recently have interval-based techniques been developed

that alleviate the problem of the wrapping effect in the integration of ODE initial

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re "C

u‘ ‘.;,_

 

\. ‘
] gm)
~.‘

ad.

5"»...
"Mid

 

value problems [22, 82]. These methods will be summarized in Sec. 2.3.3 and their
application to solar system dynamics will be presented in Sec. 5.1. By successfully
avoiding the wrapping effect to high orders, the new approach allows the long term

propagation of large initial condition sets without substantial overestimations.

2.2.4 Summary

Interval arithmetic is arithmetic deﬁned on sets of numbers instead of single numbers
and its results have been connected to modern computing machinery and numerical
analysis in 1962 by R. Moore [91]. The use of interval arithmetic allows rigorous
computations Of veriﬁed results on computers by accounting for the effects of ﬁnite
precision and roundoff errors, mathematical truncation, and extended sets Of starting
values. As such, interval computations overcome the general limitations of ﬁnite state
machines and bring the strict mathematical rigor of numerical analysis and set theory
to the computational sciences. Good introductions and overviews to interval analysis

can be found in [92, 94, 3].

Interval methods have been used successfully in a wide range Of applications:
chemical and electrical engineering [102, 5], computer graphics [96], dynamical sys-
tems and chaos [50, 124], computer assisted proofs [74, 75], and expert systems [70]
to name only a few. However, the wrapping effect, the dependency problem, and
the dimensionality curse prevent interval methods from widespread acceptance in the

computational sciences.

2.3 Taylor Models

Intervals are well suited to model enclosures Of real numbers. However, since they

only propagate information about function values and neglect any further knowledge

22

 

w- ‘1‘."-

._.'. c.

r‘ ‘\
“trig“A

”a.

., .
a“

 

about derivatives, they are not well suited for the rigorous enclosure of functions. In
fact, modeling functions with intervals usually requires splitting the domain D into a
collection {D,-} of sufﬁciently small subdomains and evaluating the functions on each
of the D,s. Then the “evaluation” of f consists of ﬁnding the appropriate interval

enclosure 1' (D;) of the range of f over the subdomain D,-.

While splitting a one-dimensional interval is usually acceptable, splitting higher
dimensional interval boxes results in an exponential increase of the number of subdo-
mains. This phenomenon is known as the dimensionality curse. It frequently plagues
conventional interval techniques and limits their applicability to large, both in terms

of dimensionality and domain size, problems.

Taylor models Offer a remedy to this problem by combining high-order Taylor
polynomials with real number coefﬁcients and intervals for veriﬁcation. Taylor mod—
els allow the rigorous modeling of complicated multidimensional functions over do-
mains that are usually several orders of magnitude larger than the ones over which

conventional interval methods can work with the same sharpness.

Deﬁnition 2.6 (Taylor Model). Let D C IR” be a box with 230 E D. Let P : D —+
IR‘” be a polynomial of order n and R C R” be an non-empty convex compact set.

Then (P, 2:0,D,R) is called a Taylor model of order n with expansion point 2:0 over

D.

Following these notations, P is called the reference polynomial, 2:0 is the expansion
or reference point, and R is called the remainder bound of the Taylor model. For the
rigorous modeling of functions on computers, we usually view Taylor models as subsets

of function spaces by virtue of the following deﬁnition.

Deﬁnition 2.7 (Taylor Models as Sets Of Functions). Let T = (P, 2:0, D, R) be
an order n Taylor model. Then, identify T with the set of functions f E C"+1(D, R‘”)

23

O—w

1'5 3,,
If” r'
- fl .1
§

.
3"". r.

, a:

l
"1»

It".
Ar-

 

such that f (2:) —P(£L‘) E R for all 2: E D, and the n-th order Taylor series of f around
2:0 equals P. Furthermore, if a C"+1 function f is contained in a Taylor model T, we

call T a Taylor model for f.

To illustrate the concept of Taylor models, consider two Taylor models T1 and T5
of orders one and ﬁve given by
T1 = (2:, 0, [—1.5,1.5] , (—1.122182, 1.122182» (2.31)

T5 (2: — 23/31+ 25/51, 0, [—1.5, 1.5] , (~—0.015781,0.015781)) (2.32)

The two Taylor models T1 and T5 have been computed as Taylor models for the sine
function with COSY Inﬁnity [23, 26] and are shown together with their remainder
bounds in Fig. 2.5. While the ﬁrst order Taylor model has remainder bounds that
are similar in size to the domain interval; i.e., the ﬁrst order reference polynomial
is a rather bad approximation Of the sine function over the domain, the ﬁfth order
Taylor model encloses the sine function with a sharpness that is of the order of
one percent of the domain size. We mention that the requirements on the Taylor
expansion of the functions contained in a Taylor model are sometimes omitted, and
Taylor models are then simply viewed as sets Of 6"“ functions that are R-close to the
reference polynomial. However, these situations are rare and will always be mentioned

explicitly.

Similar to the way intervals allow the rigorous implementation of set theory and
numerical analysis on computers, Taylor models allow the veriﬁed representation of
arbitrary smooth functions within computer programs. As such, Taylor models com-
bine the mathematically rigorous results of functional analysis with the approximative
ﬂoating point representations on computers to obtain methods to rigorously compute

analytical results.

For purposes of illustration, Figure 2.6 compares the enclosure of a function by

24

 

 

 

 

 

 

 

 

 

Figure 2.5: Taylor models of order one and ﬁve for the sine function over the domain
interval [—1.5,1.5].

Taylor models and regular intervals. In this case, the virtue Of the method lies in the
fact that a single Taylor model over a relatively large domain box guarantees a sharp-
ness that interval bounding cannot achieve, even with multiple smaller domains. This
helps signiﬁcantly in ﬁghting the dimensionality curse inherent in interval bounding
where it is of the utmost importance to avoid subdivisions Of the domain boxes, due

to the otherwise exponential growth in the number of subdomains.

 

 

 

 

 

 

 

 

Figure 2.6: Left: Enclosing a function by a Taylor model Of order eight; right: Interval
bounding of the same function with 50 sub—intervals.

We note that the actual implementation of Taylor models and related Operations in
COSY Inﬁnity is mostly due to K. Makino [82]. Since the deﬁnition of Taylor models
is so intimately related to Taylor polynomials, the implementation of Taylor models

rests on the foundations laid by the differential algebra package and the interval

.25

1

 

«-
[I

1'rl'.
ind" u
.

 

1.,
‘ 1”,]
!
riff],
I“ ‘1',
.
Ll'gu'c
l

I
. l
I'M ,
V4

implementation to be discussed in Sec. 6.1.

2.3.1 Arithmetic and Operations

Methods have been developed for arithmetic Operations on Taylor models that pre-
serve the deﬁning inclusion relationships. Hence these methods allow the computation
of Taylor models for any sufﬁciently smooth computer function. In the following, let
P and D be as above and denote by B(P, D) a guaranteed enclosure of the range of
the polynomial function P over the box D. Moreover, for k E N, P0,) and P(k+) are
the parts of P of orders up to k and greater than k, respectively. Using this notation,

the next theorem summarizes results presented in [82] and shows how elementary

operations can be deﬁned on Taylor models.

Theorem 2.5. Let T1 = (P1,2:0,D,R1) and T2 = (P2,2:O,D,R2) be two Taylor

models of order n and deﬁne
RP = R1 ' R2 + R1 ' B(P2, D) + B(Pla D) ' R2 + B((P1° P2)(n+)v D) (233)
Obtain new Taylor models T5 and Tp by

TS : (Pl + P2: $0,131 R1 + R2) 9 (2.34)

TP ((PI ' P2)(n),$0, D1 RP) ° (2.35)

Then, T; and Tp are Taylor models for the sum T5 and product Tp of T1 and T 2. In

particular, for two functions f1 E T1 and f2 E T2, it is

(f1 + f2) 6 Ts and (f1 - f2) 6 Tp. (2.36)

Proof. If we deﬁne C"‘H functions 61 = f1 - P1 and 52 = f2 — P2, then 51(3) 6 R1

and 52(2)) E R2 for any 2: E D. Thus, for a given a E D
((1. + f2) — (P1 + 122)) (2:) = 3, (2:) + 62(2) 6 R1 + R2 .-_- Rs. (2.37)

26

in I

Eb . nub
.

7‘

how

".' N]

a...“

Since the n-th order Taylor expansion of the sum f1+ f2 equals the sum of the Taylor

series, T5 is indeed a Taylor model for the sum T1 + T2. Also, over the domain D

((f1'f2)—(P1'P2)(n)) = ((P1+61)(P2+62))—((Pl-P2)—(P1'P2)(n+))

= P1'52+51'P2+51'52+(P1'P2)(n+)- (2-38)

Moreover, since the n—th order Taylor polynomial of f1 - f2 equals the polynomial

product (P1 - P2)(n), T,» is a Taylor model for the product T1 -T2. E]

The general theory of arithmetic on Taylor models has been developed in [83,
22, 82] and all elementary operations have been deﬁned such that the basic inclusion
properties are maintained throughout the arithmetic. In particular, all operations are

inclusion monotone and lead to closed commuting diagrams as shown in Fig. 2.7.

 

 

 

 

f) g (- T va T9
[*3 X] [39, ‘3]
l, v
fl+1']g r —’ Tfler ®ng

Figure 2.7: Commuting diagram for elementary operations on Taylor models.

As with the elementary operations, it is possible to extend the standard mathemat-
ical functions to Taylor models such that the fundamental inclusions are maintained;
the effect of this is illustrated by the closed commuting diagram for the sine func-
tion shown in Fig. 2.8. The actual implementation of the intrinsic functions uses an
approach resembling the way functions are deﬁned on "D” to obtain Taylor models

for the standard mathematical functions available on computers. More details on the

27

deﬁnition and implementation of intrinsic functions on Taylor models can be found

in [83, 22, 82].

 

 

 

f 9 4' T1
[Sin( ' )1 [Sin( - )l
l l
sin(f) ‘ ﬁ- sin(Tf)

 

Figure 2.8: Commuting diagram for sine function on Taylor models.

Last but not least, we deﬁne the antiderivation of Taylor models, which is a
particularly powerful operation with important applications in veriﬁed Taylor model

computations.

Deﬁnition 2.8. For an n-th order Taylor model T = (P, .730, D, R) and k = 1, . . . , 1),
let
an.
Qk 2/ P(n-1) (551,”-a$k—l,€k,$k+1a---,$v)d€k- (2.39)
o

The antiderivative 6,:1 of T is deﬁned by

61:1(T) : (QM-730,1): (B(P(n) _ Phi-1), D) + R) ' B(xk: D» ' (2°40)

Since (2;, is of order n, the deﬁnition assures that for a n-th order Taylor model
T, the antiderivative 6;1(T) is again a n-th order Taylor model. Moreover, since all
terms of P of exact order n are bound into the new remainder, this deﬁnition of the

antiderivation is inclusion monotone and lets the diagram shown in Fig. 2.9 commute.

As in the case of the differential algebra "D”, it is noteworthy that the antideriva-

tion does not fundamentally differ from other intrinsic functions on Taylor mod-

28

al‘ H
.3 ll}
’l’sgart.
the .A I
'F‘wll
Lil;

L1“
‘l

T

 

to“ :31

 

 

l l
o“ f(€)d€k ‘ : 3"1(Tf)

 

Figure 2.9: Commuting diagram for antiderivation of Taylor models.

els. Moreover, since the corresponding DA operation is contracting and smoothness-
preserving, it has desirable properties for computational applications. Finally, it
should also be noted that the antiderivation of Taylor models is compatible with the

corresponding Operation on the differential algebra ”DU.

2.3.2 Accuracy of Taylor Models

As indicated earlier, one of the main advantages of Taylor models over conventional
interval methods lies in the fact the propagating high-order information on the func-
tions allows the accuracy of the enclosures to increase. In fact, the accuracy of the
enclosures obtained by Taylor models scales with a high order of the domain size. This
makes Taylor models particularly well suited for high-dimensional problems over large
domains, since it reduces the number of necessary domain splittings. The following
theorem states that in most cases the width of the remainder bounds scales in fact

with the (n + 1)-st order of the domain size.

Theorem 2.6. Let f be a function that is represented by a ﬁnite number of elemen-
tary operations and intrinsic functions, and assume that B is an inclusion monotone '

palynomial bounder that scales at least linear with the magnitude of the domain. If

29

LJ‘
.1».

T = (P, 2:0,D, R) is a Taylor model of order n obtained by evaluating the code list
representing f, then the remainder bound R scales with the (n + 1)-st order of the

magnitude of D.

Proof. The proof follows by induction over the elementary operations and intrinsic
functions that make up the code list for f. Firstly, the assertion is correct for the
constant Taylor models and for the identity function, since the respective reference

polynomials can be bound with an arbitrary precision.

Since for two Taylor models A and B, R4“; = RA + By, the assertion also holds
for the sum of Taylor models. As shown earlier, the remainder bound of the product

A - B is given by
RA-B = RA ° R3 + RA ° B (PB) + B (PA) R3 + B ((PA ° PB)(n+)) . (2.41)

Since each of the terms scales with at least order (n + 1), so does their sum. Thus,
the assertion holds for all elementary operations. (Note that division is deﬁned and
implemented in terms of the multiplicative inverse, which is conceptually an intrinsic

function.)

Finally, the computation of remainder bounds of intrinsic functions is exempliﬁed

by the exponential function. According to [82],

RexpM) = (3 (PA — PA(0)) + RA)"+1 ' exp ([0,1] ' (3 ((PA * PA(0))) + RA))- (2-42)
By inclusion monotonicity, the second term never increases with a decreasing domain
size and thus, the product scales with the (n + 1)-st order of the domain size.

Using the complete deﬁnitions [83, 82], similar arguments can be made for all the
intrinsic functions of Taylor models, including the computation of the multiplicative
inverse. Thus, the remainder bound of any ﬁnite Taylor model computation does

indeed scale with the (n + 1)-st order of the domain. Cl

30

Figure 2.10 illustrates how the fact that the sharpness of Taylor models scales with
the (n + 1)-st order of the domain allows us to obtain sharp bounds quickly, even
in the multidimensional case. While the Taylor model of order seven has remainder
bounds that are of the same order as the domain, the size of the remainder bounds
quickly decreases with increasing order of the Taylor models. Finally, a Taylor model
of order ten encloses the function with a remainder bound that is only a fraction of

the underlying domain.

 

Figure 2.10: Interval bounding of a two-dimensional function by Taylor models of
orders seven, eight, nine, and ten.

31

2.3.3

3 'Fdilt

if i'u'.’

 

la

m

m

I w

"i‘ ,
M ,1.
tan},-

2.3.3 Applications of Taylor Models

Since their ﬁrst development in 1996 [83], Taylor model methods have been used for
a variety of applications. While recent results will be discussed in later chapters,
we summarize some of the fundamental Taylor model algorithms in this section to

illustrate the general applicability of the approach.

Bounding Ranges of Functions

One of the most challenging problems in global optimization is the problem of deter-
mining bounds on the ranges of functions over given domain boxes. While interval
arithmetic can answer that question rigorously [113], the cancellation and dependency

problems often lead to range bounds that are too pessimistic to be of practical value.

Taylor models reduce the problem of bounding the range of arbitrary functions to
the problem of bounding the range of polynomials. While this is still a hard problem,
it is already substantially easier to solve than the more general problem [129]. These
Taylor model based techniques have been used in several situations, including veriﬁed

bounding of highly complex functions [21, 16] and global optimization [88].
Rigorous Quadrature

Quadrature, the numerical computation of integrals, is an important area of numeri-
cal analysis. Extensive theories provide formulas for computing approximative values
of the integral and rigorous upper bounds for the integration errors. While these
methods often converge with high-orders, they usually also require the computation
of bounds of higher order derivatives. Although conventional interval methods and
automatic differentiation can in principle be used to compute these bounds, the com-

putational overhead for accurate results is often tremendous [92, 131] and the com-

32

 

n,‘
1
‘.

3“} Pl

 

putations suffer from the well known problems of interval arithmetic and automatic

differentiation: exponential inﬂation of code and dependency problems.

Using the antiderivation of Taylor models, the computation of integrals becomes
a mere application of the antiderivation. Simply modeling the function by a Taylor
model over the region of interest and using the antiderivation yields, according to
Def. 2.8, a rigorous enclosure of the primitive. Hence, subsequently computing a
rigorous bound of the range of the Taylor model for the primitive gives the desired
result. Details on using this approach for high-dimensional veriﬁed quadrature are

given in [25].
Integration of Ordinary Differential Equations

While conventional interval methods have long been used for the integration of
ODEs [92, 66, 35], the wrapping effect and the dependency problem have prevented
these methods from successfully integrating complicated systems over large time in-
tervals. On the other hand, one of the most important applications of Taylor model
methods lies in the computation of solution of ordinary differential equations (ODES)
under avoidance of the wrapping effect for practical purposes [24]. To illustrate the

approach, we consider the initial value problem

22' = f(t,:r), m(to) = :60. (2.43)

It is a well established fact that the solution to Eqn. (2.43) can be obtained as the

ﬁxed point of the Picard operator ’P given by

’P(x) = 2:0 +/t f (T,:L')dr. (2.44)

Using the antiderivation for Taylor models, the operator 17 can be extended to Taylor
models and yields a method that allows the computation of veriﬁed enclosures of

ﬂows of Eqn. (2.43). This approach has been discussed in [24, 82] and has recently

33

 

 

been used successfully in a variety of applications ranging from solar system dynamics
(of. Sec. 5.1 and [27]) to beam physics (c.f. Sec 5.2 and [81, 82]). Unlike methods that
use regular interval computations to enclose the ﬁnal conditions, the Taylor model
approach avoids the wrapping effect to very high order and is therefore capable of

propagating extended initial regions over large integration intervals.
2.3.4 Summary

At the time of writing, Taylor models are a relatively new concept combining high or—
der Taylor polynomials with ﬂoating point coefficients with intervals for veriﬁcation.
They allow veriﬁed computations while avoiding some of the difficulties inherent in
normal interval arithmetic. The Taylor model approach guarantees inclusion of func-
tional dependencies with an accuracy that scales with the (n + 1)-st order of the
domain over which the functions are evaluated. In particular, as shown in [85], this
method can often substantially alleviate the following problems inherent in conven-

tional interval arithmetic:

o Sharpness of the Result
0 Dependency Problem

0 Dimensionality Curse

The remainder of this dissertation focuses on the development of new algorithms

and computational methods utilizing the Taylor model approach to obtain guaranteed

results.

34

 

Ch

 

Chapter 3

Veriﬁed High-Order Inversion

In this chapter we derive Taylor model methods to prove the existence of inverse
functions and to compute Taylor models for them if they exist. The fundamental

problem can be paraphrased as follows.

Given a function f : D C R” —> R” (known only up to some accuracy)
deﬁned over a box D, is the function invertible over its range f (D)? And

if so, ﬁnd a representation of the inverse as accurately as possible.

In real analysis, as opposed to interval analysis, it is usually sufficient to determine
invertibility in the neighborhood of a point, and the Inverse Function Theorem es-

tablishes a sufﬁcient condition for this:

Theorem 3.1 (Inverse Function Theorem). Let U C R” be open and assume

that f : U ——> R” is of class C"+1 with n 2 0. If

mﬂmyw4w an

is a linear isomorphism, then there is a neighborhood V C U of $0 such that f : V -—>

R” has a 6"“ inverse g : f (V) —> V such that
g o f = idv and f 09 == idf(V)- (3.2)

35

 

 

While the Inverse Function Theorem is one of the most powerful theorems in
conventional point analysis, its main limitations in practical situations are that it
does not give any means of computing the neighborhood V over which the function
f has an inverse, nor does it provide a method to actually determine that inverse
function. This is usually not a problem in point analysis where the Inverse Function
Theorem is used with great success. However, in numerical analysis and interval
analysis we are often interested in both computing the inverse function and deciding

the question of invertibility over extended regions.

In this chapter we take a fresh look at the problem of computing rigorous enclo-
sures of inverse functions. We ﬁrst derive a new Taylor model based criterion for
invertibility that requires only knowledge about ﬁrst derivatives. While the method
has important applications in studying dynamical systems [133], we will use it as a
ﬁrst step in computing Taylor models for inverse functions. In a second step, we show
in Sec 3.2 how Taylor models for inverse functions can be computed from the origi-
nal Taylor models. By studying several examples, we demonstrate that the method
outperforms conventional interval approaches both in terms of accuracy and usable

domain size.

3.1 Rigorous Invertibility Analysis

The ﬁrst step in the process of computing inverse functions is the determination
of invertibility over the domain of interest. While the Inverse Function Theorem
guarantees invertibility over neighborhoods of points, it gives no estimates on the sizes
of these regions of invertibility. In this section we derive Taylor model based methods
to rigorously prove invertibility of functions over given domains and demonstrate the

applicability of the approach by comparing its performance with several conventional

36

 

interval methods.
3.1.1 Invertibility from First Derivatives

There are a variety of ways to decide invertibility for a given function f, some of which
rely on second and higher order derivatives. These methods are often computationally
expensive since they require bounding complicated functions like the operator norm
of the second derivative map over the domain of interest [2]. In this section we present
a method that requires only bounds on ﬁrst partial derivatives to decide the question
of invertibility. Furthermore, and perhaps more importantly, the method exhibits
some important structure regarding the points at which the derivatives actually have
to be evaluated. We will later capitalize on this structure to signiﬁcantly reduce
cancellation problems in the necessary veriﬁed linear algebra. It should be noted
that this method is not local in nature, but can indeed guarantee global invertibility

everywhere in the given domain.

The following theorem enables us to decide whether a given function is invertible
over a domain by just evaluating its ﬁrst derivatives. Hence it greatly reduces the
computational overhead necessary. It seems to originate in works by K. Kovalevsky
from the beginning of the twentieth century but has been “rediscovered” by many

others (e.g. [103, E 5.3-4]).

Theorem 3.2 (Invertibility from First Derivatives). Let D C IR” be a boa: and

f I D —) IR” a C1 function. Assume that the matrix

3%(X1) 35:00)
M: 3 a (3.3)
%,,L‘;(x.) 35x.)

is invertible for every choice of X1. . - - aXv 6 D ° Then f has a Cl-inverse deﬁned on

f (D); where f (D) denotes the range of f over D.

37

 

Proof. It sufﬁces to show that f is injective over D. So assume that 3 y E f (D) and

331,2.) E D such that

ff-Tl) = f(xz) = 31- (34)
For t E [0,1] and i = 1, . . . , v, deﬁne auxiliary functions
h.(t) = f.- (331 ' (1 - t) + $2 - t)- (3-5)

Since h,(0) = h,(1) for all i = 1,. . . , v, the Mean Value Theorem asserts that for each

i there exists a t,- E [0, 1] such that
dh,
EU»:(Vf)($1'(l-ti)+x2'ti)'(x2—$l)=0 (3.6)
Then for i = 1,. . .,v, deﬁne
Xi = $1 ' (1 — t,) + $2 ' ti- (3.7)
By the convexity of D, x,- E D for all i = 1, . . . , v and moreover,
(Vf,)-(:r2—$1)=0foralli=1,...,v. (3.8)

However, the last equation is equivalent to M - (11:2 — $1) = 0. But by assumption M
is regular and the kernel of the linear map associated with M is therefore just {0}.
We conclude that $1 = 3:2 and therefore that f is one-to-one over the domain D.
As such it has an inverse, which, by the Inverse Function Theorem, is also of class

C1. [:1

Regularity of the Jacobian at each point in the domain is a necessary but insuffi-
cient condition for invertibility. Thus, while Thm. 3.2 resembles the Inverse Function
Theorem, it is in fact stronger in the requirements on f. To see that it is impossible
to deduce invertibility from the fact that the Jacobian is non—singular at each point

within the domain, consider the function

x5 - 1011:3112 + 5503/4

f ( Z ) = ( y5 _ 10:32:13 +5134?! ) ’ (3.9)

38

rich 1

which is related to the complex function f : C —> C, f(2) = 25 via the identiﬁcation

z = x + iy. The derivative of f at any point (2:, y) E IR2 is given by

"4 _ 2 2 4 3 _ 3
53: 301: 31 +53; 20:1: y 20:03; ) (310)

M (x’ y) = ( 202:3/3 — 20x33; 511:4 — somzy2 + 53,4
The determinant det (M) : R2 —> R of M is a polynomial deﬁned on the whole space
R2 and

det (M (:r, y)) = (5:1:4 — 305102;;2 + 5y4)2. (3.11)

Hence, over the domain D = [01,1] x [01,1], the derivative of f is regular at every

point. However, for (p1 = 1r/15 and 902 = 77r/15

f<:::::)=f(:::::)
And since cos (p1 z 0.978, sin (pl :3 0.208, cos (pg z 0.105, and sin (pg 7:: 0.995, f
satisﬁes the requirements of the Inverse Function Theorem at each point in the domain
D but fails to be an isomorphism. This shows that a naive application of the Inverse
Function Theorem is not sufﬁcient to establish invertibility over extended domains.
However, Thm. 3.2 shows that if we view the Jacobian of f as a function of v2

variables, instead of just v variables, its regularity at each point in the extended

domain D” gives a sufﬁcient criterion to guarantee invertibility of f.

An immediate corollary of Thm. 3.2 is the following interval formulation of sufﬁ-

cient conditions for the existence of the inverse function, which in this version is very

well known [100, 119].

Corollary 3.1. Let f and D be as in Thm. 3.2. For i,j = 1,...,v let pm C R be

compact intervals such that

0f.-
6231'

 

(:13) E pm- Va: E D. (3.13)
If the interval matrix P = (Pm) is regular, then f has a Cl-inverse deﬁned on f (D).

39

 

rise W

“1‘
, _ .
ﬂint“.
l.

3:;» a

,.

u‘. "t

'1‘ am:
L 112V»

2%? .31.

A practical algorithm based on Thm. 3.2 and Cor. 3.1 requires efﬁcient and accu-
rate methods to model partial derivatives and depends on the availability of efﬁcient
methods for the determination of regularity of an interval matrix with potentially
large entries. Both these problems can become challenging for naive interval methods
which have difﬁculties modeling complicated functions: they often give signiﬁcant
overestimations in bounding the derivatives of the functions, and suffer from the well
known problems that come with an increasing number of variables and an increase in

domain sizes.

Moreover, interval methods are not particularly well suited for practical applica-
tions of Thm. 3.2'via Cor. 3.1 because of difﬁculties in handling the fact that the
individual rows of the Jacobian can each be evaluated at one point. A new Taylor
model based method that circumvents this problem will be presented in the next sec-
tion. But nevertheless, once the interval matrix P of Cor. 3.1 has been determined
as accurately as possible, the next question is how to establish regularity of it. While
in lower dimensions it may often be sufﬁcient to compute the interval determinant of

the matrix, more sophisticated methods are needed for higher dimensional problems.

It has been shown that regularity of a matrix is equivalent to the componentwise
distance to the next singular matrix being greater than one [122]. In the following
paragraphs we present two advanced interval based methods to prove the regularity
of the interval Jacobian P. In Sec. 3.1.3 we will compare these interval methods for

the determination of invertibility with a new approach based on Taylor models.

The ﬁrst method of verifying the regularity of an interval matrix A = 431+ [—A, A]
is based on the singular value decompositions of both the midpoint matrix ll and the
A-matrix. If 0(M, i) denotes the i-th singular value of a matrix M (sorted in non-

increasing order), then the following assertion holds [120, 121]:

40

 

ner“"’]
[Lit A
f

L»: If!)

mgr:
-) ,

”'1'":
v_ k!

l
'l; ,_
m, U.
o;_
.5] ‘

 

 

Theorem 3.3. Given A = A + [-—A,A]. Then 0(A,1) < 0(A,n) implies that A is

non-singular.

The success of this method depends mostly on the sharpness of the models of the
partial derivatives, and as such this method has difﬁculties scaling with dimensionality

and complexity of the functions of interest.

The next theorem has been presented in [100], and it is one of the most powerful

approaches to proving the regularity of a given interval matrix A.

Theorem 3.4. Given A = A + [—A,A]. If A is regular, let S be an approximate
inverse of A. If the spectral radius p(I — S - A) is less than 1, then any matrix in A

is non-singular.

This method uses a preconditioning of the matrix A, which allows the method
to work very well for medium sized problems, but as we will see below, in higher
dimensions this method still suffers from the fact that the entries of the product
matrix S - A are computed from addition and subtraction of intervals and as such

are subject to signiﬁcant overestimations due to cancellation and other dependency

problems.

We note that the most practical method to establish the contraction of an interval
matrix A is to start out with an arbitrary non-empty interval vector 130 containing
the origin and iterate n+1 = B - w,c with B = I — S - A. If for any k E N we can show
that n+1 C (ck, we have proved that B is indeed contracting and therefore p(B) < 1

as required by Thm. 3.4.

3.1.2 Verifying Invertibility with Taylor Models

We now come back to the question whether a given function f, deﬁned over some

interval box D, is invertible over its image. We Will combine the results 0f Thm. 3'2

41

 

with the regularity criterion T hm. 3.4 and Taylor model based techniques to derive
a new algorithm to rigorously answer this question. Compared to the corresponding

interval version of Cor. 3.1, we will be able to avoid an overly pessimistic behavior.

The proof of Thm. 3.2 is based on proving regularity of the matrix

gfi-(xl) 35m)
3,93%) 3530a)
with points X1, . . . , Xv E D. Thus, any combination of entries from the same row of

M can be evaluated using the same set of domain variables. While interval arithmetic
cannot capitalize on this intrinsic structure, as we shall see in the following, Taylor
models are particularly well suited for this task. Instead of modeling each partial
derivative of the functions f,- by intervals, we now model each of the gradients by one
single Taylor model, leaving us with the task of proving regularity of a matrix-valued

Taylor model, which we will be doing using a method derived from Thm. 3.4.

A naive application of Thm. 3.2 leads to matrix valued Taylor models depending
on the v2 variables in D”. However, since the size of n-th order Taylor models in v

variables is approximately

(n+v)____(_m_)_!, (3.15)

v v! n!
a dependence on v2 variables would restrict the method to low orders even for rela-

tively small problems.

In the remainder of this section we present an algorithm that uses Thms. 3.2
and 3.4 to prove invertibility of functions modeled by Taylor models over extended
domains and requires only Taylor models depending on v variables. Thus, the method
avoids an inﬂation of dimensionality beyond the v variables that are required anyway

to model the functions of interest.

42

 

 

Algorithm
(1) For i, j = 1,. . . , v let Tm- be Taylor models for the partial derivatives of f; i.e.,

aft ‘
5;; E Tm. (3.16)

(2) Let T be an interval enclosure of the range of the Taylor models TM; i.e.,

g E Tm- => g(:v) E TM for all a: E D.
(3) Let T be the midpoint matrix of T and similarly for the transpose T” of TT.
(4) Let S be an approximate ﬂoating point inverse of TT.
(5) Compute a new Taylor model N by
Ni,j = 6i,j —' Z Si,kTIZ:j = 61d — ZSUCEJC, (3.17)
k=1 k=1
where 6“ denotes the Kronecker Delta.

(6) For i = 1, . . . , v compute Taylor models r,- with the Taylor models e,,+j that are

deﬁned over new domains, independent of those of the N,,J-’s:

Ti = 2N3]: ' €u+j = ZN“ ° ev+j- (3.18)
j=l 3'21

(7) Find interval enclosures I i for the Taylor models r,-.
(8) If Ii C [—1, 1] for all i = 1, . . . , v, the original function f is invertible.
Remarks

In practice, the Taylor models Tm» needed in step (1) can be obtained by evaluating
the ﬁrst order automatic differentiation [111, 112] code belonging to f in Taylor

model arithmetic. Additionally, extensions of the Taylor model approach are being

43

 

.
. m
w.“

developed that allow the determination of the TM directly from an extended Taylor
model of f. Finally, if f is computed as the ﬂow of a differential equation; i.e.,
the function that maps initial conditions to ﬁnal conditions, Taylor models can be

obtained as solutions of an augmented differential equation (c.f. Sec. 5.2).

As outlined earlier, all Taylor models with the same index i; i.e., belonging to the
same row of the matrix, can be evaluated using the same set of domain variables.
The next step now is to compute a suitable preconditioning matrix for the transpose
of T. However, instead of showing that the matrix T is regular, we will show the
equivalent statement that the transpose of T is regular. By working with the transpose
T7 in steps (3—5), the resulting entries of the matrix N are all computed using
linear combinations of Taylor models that can be evaluated over the same set of
domain variables. Hence each entry of N is again a Taylor model in v variables,
and the columns of N are now modeled over the same set of domain variables. As
a consequence, the sum 22:1 8.»chch can be evaluated in Taylor model arithmetic,
and in particular any manifestations of the dependency problem are suppressed, as

in other Taylor model computations.

The ﬁnal stage of the method in steps (6—8) is to show that the spectral radius
of the resulting matrix-valued Taylor model N is less than unity. To establish this,
the image of the unit ball (in the maximum norm) under the map N is considered.
If we succeed in showing that the image is properly contained in the unit ball itself,

invertibility is veriﬁed.

Similar to the above, it proves advantageous not to work with the matrix N, but
rather with its transpose NT. In this way, mixing of Taylor models from different
columns of N is again avoided, which allows suppression of blow up in the necessary

linear algebra. This is justiﬁed since the transposed matrix has the same spectral

44

9131. El

 

radius as the original one. As a ﬁnal application of Taylor models, we model the unit
ball by Taylor models as follows. For any small 6 > 0, the identity function over the
interval [—1, 1] is contained in (117,0, [-1, 1], (—€,e)). Thus, if we denote the i—th such
Taylor model by e,-, a bound on the range of e,- contains the closed unit interval in
z,. It should be noted that due to the special nature of the Taylor models r,-, the
bounding is actually quite simple: each of the v terms of the sum that constitutes the
Taylor models r,- has a different set of variables. Thus, there is no cancellation in the
sum, and hence the sharpest possible bound of the sum equals the sum of the bounds
of the individual terms. Moreover, the latter are just Taylor models multiplied by
the interval [—1,1], and hence their ranges are just the ranges of the Taylor model

multiplied by that interval.

Summary

The presented method utilizes Taylor models at two crucial points. Firstly Taylor
models are used to model the partial derivatives of the function f. As will be seen
later, this helps tremendously in ﬁghting the issues of complexity and cancellation
in modeling the derivatives. Moreover, the Taylor models are used to compute en-
closures for the derivative matrices under consideration, and they are used for all
arithmetic operations on these matrices which minimizes the overestimation due to
cancellation, since the majority of the functional dependence is propagated in the

reference polynomial and therefore not subject to the dependency problem [85].

Finally, we have been able to modify the computation in such a way that all
arithmetic Operations on Taylor models can be performed using only v variables,
which allows for a favorable computational complexity and allows the method to

scale well to high dimensional problems.

45

3.1.3

)5 {rill}
ten 0\'
lill'illll
[:lll Elli
:33 we
infra]

37% 1h

:3 ~05
.U Ml

 

3.1.3 Comparison of Invertibility Criteria

As outlined earlier, we are mainly interested in proving invertibility for a given func-
tion over domains as large as possible. But not only the sheer sizes of the domains of
invertibility matter, but an equally important question is how good we are able to ﬁll
out the maximum region of invertibility from within; i.e., how close to a critical point
can we still prove invertibility. Moreover, in many applications we are likely to be
confronted with high dimensional problems. The following examples will demonstrate
how the new Taylor model based method performs in each of these areas as compared

to standard interval methods.

In each of the following examples we have tested a number of functions for invert-
ibility over various domains. All tests have been performed with each of the methods
listed below, and the results are presented in graphs and discussed. Note that the

determinant test has only been used for problems of up to six variables.

(1) Interval tests based on Cor. 3.1 and an interval determinant
(2) Interval tests based on Cor. 3.1 and Thm. 3.3
(3) Interval tests based on Cor. 3.1 and Thm. 3.4

(4) Taylor model based tests as presented in Sec. 3.1.2

For the interpretation of the results discussed in this section it is important to note
that in all cases the necessary bounds on range enclosures have been computed using
regular interval arithmetic: no dedicated range bounders utilizing domain decompo-
sition have been used either in the interval or Taylor model approaches to keep the

computational overhead within reasonable limits.

46

 

Invertibility as a Function of Dimensionality

The ﬁrst example illustrates how the performance of the presented methods behaves
as a function of dimensionality. Invertibility of one thousand 15-th order polynomials
with uniformly distributed random coefficients in [-1, 1] has been tested with the
methods (1) to (4). The domain of the random polynomials was [-0.005,0.005]”
for v = 1,. . . , 10. Fig. 3.1 shows the percentage of these random polynomials that
could be veriﬁed to be invertible as a function of dimensionality. To simulate realistic
conditions, the Taylor models for these order 15 polynomials have been of order ten

and the resulting remainder bounds are in the order of 10‘”.

 

 

 

 

100 ﬁ— r 9 (r3 l r j—
§ u
.. Z 0
g 80 _ a: o ‘]
Q)
I; 60 - x D l
t I
2
5.: x
”5
a, 40 - ‘ l
0)
£9 . x
c u
8 + Interval Determmant
213 20 l x Singular Value Decomposition ' «
‘1 x Interval Spectral Radius . x
a Taylor Model Based Spectral Radius i
O ‘ ‘ 1 4V— l 4 E
0 2 4 6 8 10

Dimensionality v

Figure 3.1: Percentage of random functions that can be shown to be invertible as a
function of dimensionality.

As expected, for increasing dimensionality the number of successfully established
invertible polynomials is decreasing. But it is important to note that the Taylor model
based method performs much better in establishing invertibility and suffers much less

from an increase in the number of variables than the interval based methods do.

47

Unfortunately there is no good way to assess what fraction of the original functions
are truly invertible; but it is to be expected that this fraction decreases with the

dimensionality, accounting for the drop of predicted invertibility in all approaches.
Invertibility as a Function of Domain Size

For the next example we have investigated how the presented methods of establishing
invertibility behave as functions of the domain size. We have restricted ourselves
to the medium sized problem of 10—th order polynomials in six variables. For each
polynomial we have applied the methods (1) to (4) over domains of increasing size.
All domain boxes have been placed symmetrically around the origin and the abscissa
in Fig. 3.2 shows the magnitude of the domains. For each of the different domain
sizes we have plotted the percentage of polynomials for which invertibility could be
proven using the four different schemes. As in the previous example, the computation
was performed with 1000 random polynomials and the Taylor models were extended
with artiﬁcial remainder bounds in the order of 10’”. The results of this experiment

are illustrated in Fig. 3.2.

In Thm. 2.6 we have shown that the accuracy of Taylor models scales with the
(n + 1)-st order of the domain size, while interval arithmetic scales at most quadrat-
ically with the domain sizes. That explains why the Taylor model based method can
assert invertibility over domains that are much larger than the ones interval based
methods can handle. Moreover, it should also be noted that the number of invertible
polynomials naturally decreases with an increasing domain size, since the probability

of singular points being contained in the domain D increases with the volume of D.

48

 

 

 

 

 

100 G F l l l— 1 I— *1 T I
g a + Interval Determinant
. a a x Singular Value Decomposition

a, x ‘ E, n Interval Spectral Radius
_5 80 - D u D Taylor Model Based Spectral Radius.
Q ~ “’0
a U D
Q X T D
g 60 ' x l' D 0 ~
11 D
2 . x " 0
.s x . D a
"6 "I U

40 - x u ‘
8, X I G D
“'2' X I 0
q, + x n
O I
5 20 " x ‘ ..
o. + x x

+ ' . § x x
‘ X
0 4 4* +‘: .%£ a—t4: ﬁL—‘r :1:+T .Lg‘sé"
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05
Domain size

Figure 3.2: Percentage of random functions that can be shown to be invertible as a
function of domain size.

Invertibility as a Function of Non-Linearity

This example demonstrates how the performance of the discussed methods changes
with the non-linearity of the functions of interest. To that end we have considered
functions f :2 (f1,...,f6) : [~1, 1]6 C R6 —+ R6 given by

6
_ a,- (I)
2;“ i" "2 (3.19)
(I. — Zk=1bikak) + E2
with 5 ER and 6 x 6 matrices A 2 (age) and B = (bi,k) With coefﬁcients in [“1:1l-

fi($1‘l ' ' ' 12:6) :

For small values of e and appropriate coefﬁcients a”, and biJca this may give ill-deﬁned

functions because of vanishing denominators. However, in these cases we count the
functions as not invertible.

We have generated 500 functions by choosing the coefﬁcients of the matrices A
and B randomly in the interval [—1,1], and the presented methods have been used
to prove invertibility over [—1,1]6. Fig. 3.3 shows the percentage of functions that

can be shown to be invertible depending on the non-linearity 5. All Taylor model

49

 

ﬁjrﬁlplll

teal.

..

LIEU.

. 'l
g fI-a‘!

computations have been performed in order 6 and the value of 5 has been increased

linearly from 0 to 250.

 

 

 

 

 

100
(D
.5. so-
‘5
C
.2
G)
3.5 60 '
t
9
.E
‘5
a, 40 -
O)
5!
C
8 x
5 20 - f + Interval Determinant ]
o. _f + x Singular Value Decomposition
x f u Interval Spectral Radius
,3 ,, +3». 0 Taylor Model Based Spectral Radius
0 ""13.” 4 n 1 1 1
O 50 100 150 200 250

Non-linearity s

Figure 3.3: Percentage of functions that can be shown to be invertible as a function
of the non-linearity e.

The non-linearity of these functions is mostly determined by the quantity 5, such
that for 5 >> 1, the functions are almost linear and their invertibility depends mostly
on the invertibility of A. Since almost all computer generated random matrices are
invertible, it is to be expected that all methods succeed in proving invertibility for

sufﬁciently large 5. Fig. 3.3 indicates that this is indeed the case.

For 5 ~ 1 on the other hand, the resulting functions show a large non-linearity and
all methods fail in proving invertibility. It is likely that due to the size of the domain

box and the high degree of non-linearity, these functions are truly not invertible.

Between these two extremes, with increasing 8, the different methods become more
successful in establishing invertibility. However, the Taylor model based method starts

succeeding in proving invertibility very suddenly for e t: 20, while the success of the

50

Invert

.,_,

other methods sets in only for larger 5, and increases at a much slower rate.
Invertibility in the Vicinity of a Critical Point

As another example of how the presented methods scale to larger domain sizes and
to illustrate how they behave in the neighborhood of a singular point, consider the

function f = (fl, . . . , f6) : R6 ——> R6 with components f,- given by

fi($1, - - - ,176) = (171' — 370)2 + ACOSHQR — $0)'($1r(i) — 550))

—A(:c,,(,-) — x0) sin(:r,- -— mg) (3.20)

whererr(i)=i+1fori=1,...,5 and 1r(6)=1.

At the point (2:0, . . . ,xo) the Jacobian of f vanishes and hence for symmetric
domain boxes centered at the origin, 2:50 is an upper bound for the magnitude of
the domain of invertibility. Fig. 3.4 shows the percentage of this maximal domain
diameter 22:0 (for $0 = 0.1) over which the different methods can prove invertibility
as a function of the parameter A, which is gradually increased from 0 to 2. The
invertibility tests have been performed for the domains [-p - 0.001, p - 0.001]6 with
p = 1,. . . , 100, and the ﬁgure shows the maximal value of p for which the various
methods can determine invertibility as a function of A. The results indicate that the
Taylor model based test can guarantee invertibility over a much larger region and is

less sensitive to the perturbation A.

It is important to note that all interval based methods perform equally poorly for
increasing A. This indicates that the real problem of establishing invertibility in this
example is with the use of conventional intervals for the modeling of the derivatives
and not so much with the different methods to establish regularity of the interval
matrix of derivatives. This example illustrates how Taylor models can enclose even

complicated functions extremely accurate: the remainder bounds of the Taylor models

51

la:

 

 

 

 

100 We 0 m I F r— f r *1 w
l D D a
g 1‘ I! D a D o
'5 80 " "‘ G U o D T
.— I D
(U n D D
S ' .. ° 0 a
'U T n D D D D
F: 60 - T n D D _]
.g Ii " I! D U D n D
6 ‘ " l 0 D 0 El
E " I . ﬁ ‘
B 40 - ! " x ,. -
0 n T " l I
O) . If n T " ll
#3 + Interval Determmant
3 x Singular Value Decomposition
8 20 - n Interval Spectral Radius -
o. D Taylor Model Based Spectral Radius
0 __1 1_ l l 1 L A __l_ 1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Complexity and non-linearity A

Figure 3.4: Percentage of maximal domain size over which invertibility can be proven

as a function of complexity and non-linearity in the partial derivatives.

for the partial derivatives are all in the order of 10‘”.

Invertibility as a Function of Functional Complexity

In this last example we study how the presented methods behave as a function of

computational complexity of the original function f. To that end, we have emulated

functional complexity using the following method.

For the previously introduced random polynomials of order ten in six variables

with coefﬁcients in [—1, 1] we model computational complexity c by

1 c
f(rv) — ﬁgs-(x)

(3.21)

where each of the f,- is a random polynomial as before and the scaling factor has been

introduced to maintain the standard deviation of the coefﬁcients of the resulting

DOIynomials. All tests were performed over the [-—0.005, 0.005]6 domain box and the

results are shown in Fig. 3.5.

52

It is important to note that the functions have been evaluated by adding the
results of the individual polynomial evaluations. This is a realistic model of practical
applications where the supplied functions often come as black boxes that do not permit

any further simpliﬁcations to control cancellations.

 

100 I I 1— ﬁ— I f I f I
a
a o
E] a D
80 ["3 D a D “
n
60-} ~

.o. l

Interval Determinant

Singular Value Decomposition

Interval Spectral Radius ~
Taylor Model Based Spectral Radius

Percentage of invertible functions

I
DIX-O-

20

IX

 

 

* . i i
o s . . . _
o 5 1o 15 20 25 so 35 4o 45 so

Complexity c

.I. .I. I. .I. .I.

 

Figure 3.5: Percentage of random functions that can be shown to be invertible as a
function of functional complexity.

The ﬁrst observation worth noting here is that the Taylor model based method
does not suffer from an increase in complexity, but rather even seems to improve
with it. This is caused by the way we simulate computational complexity: while the
coefﬁcients in the original polynomials are uniformly distributed in [—1, 1], the coefﬁ-
cients of the resulting “complex” one are not uniform anymore, and this distributional

change leads to a simpler behavior with improved invertibility.

As outlined in Sec. 2.3, Taylor models are particularly well suited to deal with the
cancellation problems that plague conventional interval arithmetic, since the bigger

Part of that cancellation takes place in the polynomial coefﬁcient real number arith-

53

n‘

 

:-
‘1

use i

metic and only a small fraction of it contributes directly to the remainder bounds.
(This is in contrast to normal interval arithmetic where all the functional depen-
dence is propagated in the remainder bound.) As such, Taylor models are expectedly
much better in modeling the derivatives themselves and hence the Taylor model based
methods can succeed in proving invertibility for computationally complex functions

that cannot be properly modeled by intervals.

3.2 Guaranteed Enclosures of Inverse Functions

Once the existence of an inverse function has been established for a function f con-
tained in a Taylor model, the question of actually computing a Taylor model for the
inverse function f ‘1 arises naturally. In the following we develop methods for com-
puting inclusions of inverses of general functions in Taylor models. We show how
Taylor model techniques can be combined with the methods for the veriﬁcation of
invertibility presented earlier to derive a new method that allows the computation of
Taylor models containing the inverses of given invertible functions enclosed in Taylor

models.

3.2.1 Polynomial Inverses

If we start out with a Taylor model T for an invertible function f, the ﬁrst step in
computing a Taylor model S for f ‘1 is the computation of the reference polynomial
of S. According to Def. 2.6, this requires the computation of the n—th order Taylor
Polynomial of f ‘1. In this section we utilize the DA Fixed Point Theorem to calculate

this n-th order Taylor polynomial of the inverse.

Assume that f E C"+1(]R”,1R”) is an origin-preserving map; i.e., f (0) = 0. If the

derivative M of f at 0 is a linear isomorphism, then the Inverse Function Theorem

54

guarantees that there is a neighborhood V of 0, such that f ‘1 exists and is also of class
C“1 on f (V) Once the linearization of f has been shown to be invertible, within
the DA framework it is possible to compute a representative of the equivalence class
of [f 4],, from a single representative M of the equivalence class [ f1". Split the map

Mas
M=M+NM (3.22)

where NM denotes the purely non-linear part of M. Composing from the right with

the locally existing inverse results in

I = MOM_1=M0M—1+NMOM_1 (3.23)
=> M-1 = M—1 o (I — NM 0 M“) = 0(M-1) (3.24)

where I is the identity map. The last equation is actually a ﬁxed point relation for
the map M'l. Viewing this equation as a relation on equivalence classes, it turns

out that 0 is contracting because
(0 ([Aln) — 0 (Wall = M“ 0 (WMln ° [Bln - WMln ° [Alnl- (325)

Thus, with [A],, and [8],, agreeing up to order k, the nilpotency of N implies that

[NMln 0 [B]n and [NM],, 0 [4],, agree up to order k + 1.

According to the DA Fixed Point Theorem, this assures the existence of a unique
ﬁxed point of the operator 0, which can be reached in at most n + 1 iterations.
Moreover, this method generates all derivatives of up to order n of f"1 at the origin
in ﬁnitely many steps through mere iteration. Because it only requires the iteration
of a rather simple operator, this approach is particularly useful for computational

applications. More details on this method can be found in [17].

While the presented method by itself allows only the computation of inverse poly-

nomials for origin-preserving maps, we will show at the end of the next section how

55

m

(””1“
.

I‘leLt'J

fir

3an

[ll ilil‘l

 

it can nevertheless be used for the computation of Taylor models for general inverse

functions.

3.2.2 Inverse Taylor Models

The computation of veriﬁed inverses starts by modeling the functional dependence of
interest by a Taylor model and the a priori determination of invertibility. With the
knowledge that the function contained in the Taylor model is actually invertible, the

computation of an enclosure of the inverse follows the steps outlined in this section.

First, it should be clariﬁed what exactly constitutes an inverse Taylor model, since
compatibility requirements of the underlying domains force us to consider two kinds
of inverse Taylor models as given by the next deﬁnition. We denote by B (T) an
inclusion of the range of all functions in the Taylor model T and by I a Taylor model
(as, 0, D, (-—e,e)) of the identity function over the domain D (with some arbitrary
small 8 > 0). Finally, for a polynomial P and a Taylor model T, P(T) denotes the
result of the formal evaluation of the polynomial P with the argument T. This is
well deﬁned since it requires only additions and multiplications of Taylor models with

Taylor models and real numbers.

Deﬁnition 3.1 (Inverse Taylor Model). Let T = (Pm 1130, D, R) and S = (Gm yo,

A,Q) be two Taylor models. S is called a left-inverse Taylor model for T if

1. GnOPnan,
2'Pn($0):y0;
3. fET=>f(D)CA,

4. B(G. (T) — I) g o.

56

 

5' _‘_l .
.lkil ll

ii'ﬂldl

 

Similarly, S is called a right-inverse Taylor model for T if T is a left-inverse Taylor

model for S.

It turns out that the rather general deﬁnition of left-inverse Taylor models en-
sures the veriﬁed enclosure of left-inverses for all invertible functions contained in T.
Moreover, the computation of left-inverse Taylor models is usually sufﬁcient for prac-
tical applications, their computation does not require a modiﬁcation of the original

domains, and their domains are easily computed.

Theorem 3.5. Let T = (szro, D, R) and S = (Ga, yo, A, Q) be given Taylor models
such that S is a left-inverse Taylor model for T. Assume that f E T is invertible over

D. Then there is g E S such that g = f"1 on f (D).

Proof. Deﬁne 6, : D —+ R” by (if (as) :2 Gn( f (cc)) — :12. Note that the Taylor expansion
of 6; around 2:0 E D vanishes up to order n since the expansion of f around 2:0 agrees

with R, up to order n and by assumption Gn(P,,(:r)) =n 3:.

Since f is invertible over D, Vy E f (D), there is a unique 23,, E D such that

f(IIJy) = y. Then, for y E f“(D) deﬁne the function
{3(9) = (My) — 5r($y) = 07.01) — (Hf—101))- (3.26)
Then, for any a: E D we have

s (f (19)) = Gn (f (33)) - 5r (f‘1 (f (It‘ll) (3.27a)

= a: + a, (x) — 6, (r‘ (f (z))) = e. (3271))

Thus 57 is the unique inverse to f on f (D)- Since lily) - Gn(y)l = l5f(f"l(y))| E 9,
it follows that g is contained in the Taylor model 5; := (Gmyo, f (D) ,Q).

As the continuous image of a compact set, 1’ (D) is itself compact and hence the

function Q can be extended (n + 1)-times continuously differentiable to a function g

57

deﬁned on the whole set A. Moreover, since (2 is a proper superset of g (f (D)), it
is possible to satisfy (g — G") (A) C Q. Since the n-th order Taylor expansion of g
around yo = f (.720) E D equals the n—th order expansion of g, which in turn equals

G", it follows that g E S. Finally, for :r E D, g(f(:r)) == g(f(:r)) = 2:. E]

The ﬁrst step in translating the previous theorem into a practicable algorithm
lies in the efﬁcient computation of the inverse polynomial C". To this end, let T 2
(Pa, 2:0, D, R) be a given Taylor model and assume that P" is locally invertible around

2:0. Then consider the polynomial
P(x) = P(a: + 2:0) — P(rro). (3.28)

Apparently P(O) = 0, and hence, according to Sec. 3.2.1, [P‘l] can be calculated in

at most n + 1 steps. With yo = P(ro) we obtain 012(9) as

(My) = are + 13‘1 (y - 310) (3.29)

Once the reference polynomial of the inverse Taylor model has been computed,
the next step is the determination of an appropriate domain A such that either f E
T => f (D) C A (left-inverse Taylor model) or g E S => g(A) C D (right-inverse
Taylor model). Fig. 3.6 illustrates the various domains involved in the computation
of inverse Taylor models and their relationships

X=Uf(D) andY=nf(D)a (3-30)

feT feT
and hence X and Y are the smallest and largest sets that could be used as domains
for left- and right-inverse Taylor models, respectively. Since a general computation
and representation of X and Y is impossible, a practical application strives to ﬁnd

a small overestimation W for X and a large underestimation Z for Y such that

58

Ii‘vv]

.1.
AAA‘

,
II.

 

.
r} 11'
g)
.
mm.“
«31.4.

[f 'i

\[i‘]

 

 

 

 

‘_——----
'o
I
I
\

.
\
‘s 0’.
'I. ‘0'
.-.-I-’

Figure 3.6: Illustration of the domains in the deﬁnitions of inverse Taylor models.

Z C Y C X C W. Our implementation of this method chooses A = T(D) as
the domain for the left-inverse Taylor model. Computation of the latter consists of
bounding the range of the reference polynomial Pu over D and adding the remainder
bound R. While the bound on the polynomial does not necessarily have to be accurate
to obtain a left-inverse Taylor model, the practical application of left-inverse Taylor

models beneﬁts from overestimations as small as possible.

3.3 Examples of Taylor Model Inversion

In the following we demonstrate the performance of the inversion methods presented
in the previous two sections in examples. All computations have been performed

using the Taylor model objects in the arbitrary order code COSY Infinity [23, 26, 82].

59

 

..-.

3.3.1 One-Dimensional Function

As a ﬁrst example, consider the one—dimensional sine function over the domain D =
[—0.5, 0.5]. The functional dependence is modeled by a 19-th order Taylor model over
that domain, and the result is presented in Table 3.1. The 19—th order Taylor model
encloses the sine function with an accuracy that is close to the machine epsilon of

approximately 10‘”.

 

RDA VARIABLE: NO8 19, NV= 1
I COEFFICIENT ORDER EXPONENTS
1 1.000000000000000 1 1
2 -.1666666666666667 3 3
3 0.8333333333333333E-02 5 5
4 -.1984126984126984E-03 7 7
5 0.2755731922398589E-05 9 9
6 -.2505210838544172E-07 11 11
7 0.1605904383682162E-09 13 13
8 -.7647163731819817E-12 15 15
9 0.2811457254345521E-14 17 17
VAR REFERENCE POINT DOMAIN INTERVAL

1 0.000000000000000 [-.5000000000000000 ,0.5000000000000000]
REHAINDER BOUND INTERVAL

R [-.1085432243394823E-014,0.1085432243394823E-014]
*t**#*****************##tt*********tttttttttttttttttttttt

Table 3.1: Taylor model of one-dimensional sine function computed with COSY Inﬁn-
ity. Shown are Taylor coefﬁcients, reference point, domain information, and remainder
bound.

Invertibility of the sine function is easily veriﬁed using the method presented in
Sec. 3.1.2. The next step is the computation of a polynomial approximation of the
inverse of the reference function. Not surprisingly, the methods of Sec. 3.2.2 obtain
the 19—th order Taylor expansion of the arcsine function up to machine precision.

Proper computation of the remainder bounds and the domain gives the left-inverse

Tiii/let model shown in Tab. 3.2.

Fig. 3.7 shows the difference between the arcsine function and the reference poly-

60

'2
l

“I

RDA VARIABLE: NO= 19, NV= 1

 

I COEFFICIENT ORDER EXPONENTS
1 1.000000000000000 1 1
2 0.1666666666666667 3 3
3 0.7500000000000000E-01 5 5
4 0.4464285714285714E-01 7 7
5 0.3038194444444444E-01 9 9
6 0.2237215909090909E-01 11 11
7 0.1735276442307693E-01 13 13
8 0.1396484375000000E-01 15 15
9 0.1155180089613970E-01 17 17
10 0.9761609529194068E-02 19 19
VAR REFERENCE POINT DOMAIN INTERVAL

1 0.000000000000000 ['.5210953054937487 ,0.5210953054937487]
REMAINDER BOUND INTERVAL
R [-.7707363654262549E-008,0.7707363654262549E-008]

******$*****¥*******************************¥*****¥**#*#*

Table 3.2: Left-inverse Taylor model for the Taylor model shown in Table 3.1. Shown
are Taylor coefﬁcients, reference point, domain information, and remainder bound.

nomial over the interval sin (D) = [-0.479425...,0.479425...]. As expected, the
error stays within the remainder bounds (illustrated by the horizontal lines), and the
remainder bounds represent an overestimation of the true error by less than a factor
of four. It should be noted, however, that the arcsine function as such is not contained
in the resulting left-inverse Taylor model. But, as outlined earlier, the part of the
arcsine function necessary to act as a left-inverse of the sine function over the domain
sin (D) is contained and can be extended such that the resulting function is deﬁned

over the whole domain of the left-inverse Taylor model.

3.3.2 Six-Dimensional Function

The following example involves a six-dimensional exponential function that has a

dense 8—th order Taylor polynomial with 3003 non-vanishing coefﬁcients in each of

the six reference polynomials.

61

le

' r.
‘r. .
limit

DU

with

10-08 1 ﬂ f I

 

 

 

ﬂ“

 

 

 

 

 

-5909 . 3
Error of Arc Sine Approximation
Remainder Bounds -------
_1 908 1 Mr 1 1 1 1 1 1 1
-0.4 03 -0.2 -0.1 0 0.1 0.2 0.3 0.4

Figure 3.7: Difference between the arcsine function and the reference polynomial of
the 19-th order Taylor model shown in Table 3.2.

Let A = (a,,-) be an invertible 6 x6 matrix and consider f = (fl, . . . , f6) : R6 —-> R6

deﬁned by
6 \
f,(:r1, . --,$6) = exp (201347)“ I — 1. (3.31)
i=1 /
If B = (b,,-) is the inverse matrix of A, the inverse function g = (g1, . . . , g6) of f is
given by

6
91(91,- - - 1316) = 2% log (”l/j + 1) -

i=1

Using the matrix

1
-—1
-1
—1

1

1

 

p—tb—iHi—‘HH
p—Lh—lF—‘HI—‘H
HHHHHH

1
1
1
-1
—1
1

1 )
—1

1
-1

 

—1
_1 )

(3.32)

. (3.33)

the 8-th order Taylor model method for veriﬁed inversion has been applied to the

62

firth:

ll'f

w

Ffldlfl

Still 1

L' ~
“All”?

.7? in
Arl 1"“

function f over the domain box D :2 [—-0.01,0.01]6.

The resulting left-inverse Taylor model is deﬁned over the domain [——0.061686,
0.061686]6, and it has been veriﬁed that it contains the true inverse function g. The
remainder bounds of the individual components of the left-inverse Taylor model are
listed in Tab. 3.3. Since the range of the exponential function and its inverse are
in the order of 10“, the relative accuracy of the remainder bounds is about 10‘”).
In Sec. 3.6.1 we will present a comparison of these results with conventional interval

techniques.

 

Component Remainder Bound

1 [-0.4190638646976846E—011, 0.4184087823912867E—011]
[-0.2791908825275360E—011, 0.2791908821988238E—011]
[-0.2791908824574486E—011, 0.2791908821987869E—011]
[~0.1396454411975411E—011, 0.1396454410994258E—011]
[—0.1396454411909750E—011, 0.1396454410994186E—011]
[-0.1396454411225902E—011, 0.1396454410994267E—011]

GWAOJM

 

Table 3.3: Remainder Bounds of the left-inverse Taylor model for six-dimensional
exponential function given by Eqns. (3.31) and (3.33).

3.4 A Superconvergent Newton Method

While the conventional interval Newton method presented in Sec. 2.2.3 is a very
powerful and versatile tool for the rigorous enclosing of zeros, its relatively slow

quadratic convergence often results in the need for domain subdivisions.

We will now apply the methods derived in the previous section to a higher order
version of the interval Newton method. For practical root ﬁnding problems the new
method offers a larger domain of convergence, and is more robust for complicated
functions because of the decrease of the dependency and cancellation problems. Since

the method converges with an order higher than two, it is also called superconvergent.

63

C7

5’1

While the conventional interval Newton methods are essentially based on ﬁrst
order Taylor approximations and inversions, the availability of high order Taylor
models and the corresponding inversion tools enables us to overcome the limitation
of slow convergence of the otherwise powerful interval Newton method. Based on the
idea of approximating the function f by a high order Taylor model and computing a
left inverse Taylor model for it, the Taylor model based computation of zeros is elegant
and straightforward, and can easily be implemented using the COSY Inﬁnity [23, 84]

language environment. The algorithm takes the following form:

(1) Set k = 0 and let D”) be a box that contains a zero of f and assume that f is

invertible over 0(0).

(2) Model the functional dependence of f over DU“) by an n-th order Taylor model

Tl") with the reference point m (D02).
(3) Compute a left-inverse Taylor model S (k) for T (k).

(4) 001111311“? a new domain via DOC“) = D“) F) S (k) (0). If the enclosure DOC“)

of the zero is not accurate enough, replace 1: by k + 1 and repeat from (2).

Obviously, for n = 1 this method is equivalent to the traditional interval Newton
method with centered forms (c.f. Sec. 2.2 and [92, 94, 3]). However, the high order
version of the interval Newton algorithm allows for a fast and accurate determination
0f guaranteed enclosures of the zeros of a given function. Since the accuracy of n-th
order Taylor models scales with the (n + 1)-st order of. the domain size, this method
converges very rapidly once the domain size has become sufﬁciently small. This fact
will be illustrated further in Sec. 3.6.2. Using the notation of the previous algorithmic

description, the following is an important consequence:

64

Theorem 3.6 (Interval Newton for Taylor Models). Given a Taylor model T
for f as before and assume that f is invertible over the domain D. If f has a zero
33" E D then, for 13(0) = D, it is 33" E DU“) for each k E No. On the other hand, if

D“) 0 S0“) (0) = 0 for any k E No, then f has no zeros in D.

Proof. We prove the ﬁrst part by induction. For k = 0 the assertion is true by
assumption, since :5“ E D = 13(0). Now we assume that the assertion is true for
some k E No. Then, since .v“ E DU‘) and f E T”), 0 is contained in the domain
of Sl"). Moreover, since 506) is a left-inverse Taylor model for TV“), El g E S (k) such
that g( f (:r*)) = ac”. However, since f (22") = 0, that implies that x“ E SW (0).

Consequently, the ﬁrst part of the theorem holds for all k E No.

The second part of the assertion follows from the ﬁrst part: if there was a zero in
DU“) then, according to the ﬁrst part, it would also be in DU“) 1') S“) (0). However,

if the intersection is empty, there was no zero to begin with. E]

It is important to note that, just like for the conventional interval Newton method,
it is generally not possible to give rigorous estimates on the speed of convergence of
this method. However, once the requirements of Thm. 3.6 have been satisﬁed, it will
usually converge with the (n + l)-st order of the size of the inclusion D; see also
Sec. 3.6.2 for a further discussion. It should also be pointed out that unlike the ex-
tended Interval Newton method, the presented algorithm cannot handle local extrema
of f within the domain. However, this can easily be circumvented by performing ex-
tended Interval Newton preconditioning steps that split the original domain until the

Taylor models on the smaller boxes satisfy the requirements of Thm. 3.6.

65

3.4.1

vir’tm ‘
{infulf

0190

w -

.2b 933'
me .\e'
a or

¢39L~
4.5.le

, l. ..
with:

3.4.1 Examples

The following examples demonstrate the Taylor model based high-order extension of
the Interval Newton method. In particular, they show that the method converges

extremely fast over relatively large domains.
One-Dimensional Fixed Points

This example compares the presented DA and Taylor model high-order extensions of
the Newton method and the ﬂoating point and interval versions of that algorithm
in a one-dimensional setting. It shows how the convergence rate of the high-order

methods can indeed outperform the traditional algorithms by a wide margin.

The ﬂoating depth h of a cylindrical trunk of density p = 0.66 and radius r in

water is given by

h = r (1 — cos (do/2)), (3.34)

where do = 3.655403079564624 is the unique ﬁxed point of
a = sin(a) + 21rp. (3.35)

This ﬁxed-point relation for do can readily be transformed into a root ﬁnding problem

and the resulting function is shown in Fig. 3.8.

We used several Newton methods to determine the ﬁxed point by solving the
corresponding root ﬁnding problem: a traditional point Newton algorithm, an interval
Newton method, a Taylor model based Newton method as presented in this section,
and a non-veriﬁed point-polynomial version of the latter. The interval [3.3, 4.3] has
been the initial enclosure of the zero, and the midpoint 3.8 has been used as a starting
value for the non-veriﬁed methods. The computations have been performed in order

19 with an accuracy goal of 10‘”. The ﬁnal approximations and enclosures of do are

66

 

 

 

 

 

 

 

4 . .— r r
3 _ sin(x)+1.32n-x
2 _ .
1 _ -
0
-1 .
-2 . ..
-3 . I . .
1 2 3 4 5 6

Figure 3.8: Transformed Fixed-Point problem: the ﬁxed point do of Eqn. (3.35) is
the intersection point of the two graphs.

shown in Tab. 3.4 together with the number of iterations and CPU time required by

the different methods.

 

 

Newton Method Result Steps Time
Floating Point 3.655403079564624 4 000001663
Polynomial 3.655403079564624 0.00162493

 

1
Interval [3.655403079564622, 3.655403079564625] 4 000003908
Taylor model [3.655403079564619, 3.655403079564628] 1 0.00484088

Table 3.4: Comparison of different Newton methods for the determination of the ﬁxed
point do of Eqn. (3.35).

It should be noted that the high-order methods reach the desired accuracy after
just a single step. The traditional algorithms on the other hand need four iterations
to achieve the desired precision for this rather simple function. However, since the
computational complexity of Taylor models exceeds the one of intervals, and since

all methods work without domain splitting, the traditional algorithms are still more

67

 

efﬁcient.

Polynomial Functions

The following one-dimensional example uses the veriﬁed Newton methods to enclose
the value of 7r to almost machine epsilon — in fact the desired accuracy goal has been
set to 10‘”. To that end, the sine function has been approximated by its 25—th order
Taylor polynomial P, and the two Newton methods have been used to determine the
unique zero of P in the interval [1.8, 4]. The order of the approximation has been
chosen in such a way that the value of that unique zero and 7r agree up to machine

epsilon.
$2k+1

12
P(x) = §(-1)km (3.36)

Due to the alternating sign of the terms, the naive implementation and evalua-
tion of P exhibits strong cancellations. While it is clear that more efﬁcient methods
for the approximation of the sine function exist, this particular setup is typical for
applications where the functional dependence is often given by complicated and re-
dundant expressions that exhibit strong cancellation. While interval methods are
known to have problems with this, the Taylor model approach can successfully avoid
these problems. Fig. 3.9 shows the sine function and its 25-th order approximation
between 0 and 12. In the region around 3 that is most interesting for this problem,

the two functions agree up to machine precision.

This example illustrates how Taylor models can successfully control cancellation
and how the high-order methods converge much faster than the regular ﬁrst-order
Newton methods: over the same initial domain, it takes the regular interval Newton
method eight iterations to enclose the zero with the desired accuracy. On the other
hand, Tab. 3.5 shows that the Taylor model method achieves a comparable accuracy

in just a single step. Moreover, the interval Newton method had to resort to extended

68

 

 

 

25"“ order TaWar-Approximation of Sine Function
Sine Function ---------

I— l l

 

 

-1.5 ' 4 A
o 2 4 6 3 1o 12

 

Figure 3.9: Taylor polynomial approximation of order 25 of the sine function over the
interval [0, 12].

interval divisions and subsequent domain splitting in the ﬁrst four of the eight steps.

A Multidimensional Function

This example illustrates the ability of the Taylor model based Newton method to read-
ily extend to higher dimensional problems. The example shows the determination of
the zeros of the six-dimensional exponential function deﬁned in Sec. 3.3.2 with initial
enclosures of the zeros given by [——0.25,0.25]6 for Taylor models and [—0.02,0.02]6
for the conventional interval Newton method. The computation has been performed
with 8-th order Taylor models and the desired accuracy has been set to 10"”. The

enclosures returned after each step of the iteration are listed in Tab. 3.6.

Unlike the conventional interval method, where the computation and inversion
of the interval matrix for the derivative poses a signiﬁcant challenge, the Taylor

model based Newton method generalizes easily from the one-dimensional case to

69

Table I

v] ,
M~L t
1,.

\
“pun!-

'.
n, .
LIV]

 

Method Step

Enclosures of 7r

Interval Newton 0 [1.800000000000000, 4.000000000000000]
[2 .91907 767271 1 509, 4.000000000000001]
[2.919077672711509, 3.427717355896160]
[2.919077672711508, 3.165511666111006]
[3.085798724266786, 3.16551 1666111007]
[3.136629538119056, 3.154844930481172]
[3.141102949424404, 3.141988434766514]
[3.141592414707353, 3.141592894936869]
[3.141592653589790, 3.141592653589801]
[1 .800000000000000, 4.000000000000000]
[3.141592653589765, 3.141592653589826]

Taylor models

i—aoooacﬁo‘ﬁww"

Table 3.5: Enclosures of 7r, obtained by the interval Newton and Taylor model meth-
ods.

higher dimensional problems. Furthermore, the rapid convergence of the method
over a relatively large domain demonstrates the power of the high-order approach.
It converges in only two steps to an extremely accurate enclosure of the zero. But
even more striking, the second step improves the sharpness by about 11 orders of
magnitude. Once again, this is a vivid example of how the accuracy of Taylor models

scales with the (n + 1)-st order of the domain size.

The interval Newton method on the other hand, reached the desired accuracy in
eight steps; since the computational complexity of the Taylor models is about three
orders higher than that of intervals, this improvement does not yet justify the use of
Taylor models. However, the interval Newton method required a much smaller initial
enclosure of the zero to converge. Combining this with the dimensionality of the

problem and assuming that the domain has to be split uniformly in each coordinate
direction, the Taylor model method turns out to be two orders of magnitude more

efﬁcient than the interval Newton. We will revisit this particular aspect of the method

in Sec. 3.6.2.

70

E3“

 

Method Step Magnitude of Enclosures of Zero

Interval Newton 0 [-0.20000000000000E—001, 0.20000000000000E-001]
[0.200000000000003001, 0.20000000000000E-001]
[-0.20000000000000E—001, 0.200000000000003001]
[~0.20000000000000E—001, 0.200000000000003001}
[-0.87410175378563E-002, 0.87410175378563E—002]
[-0.96852367808498E—003, 0.96852367808498E—003]
[-0.54933148362165E-005, 0.54933148362165E-005]
[-0.82475281503554E-010, 0.82475281503554E-010]
[-0.94355649697386E—014, 0.94355649697386E-Ol4]
[0.250000000000003000, 0.250000000000003000]
[-0.47478831445046E—003, 0.47478831445046E—003]
[-0.60171167482408E—014, 0.60171167482408E-014]

 

 

Taylor models

[\DD—‘OCN‘QCDCﬁi-bww'"

 

Table 3.6: Enclosures of the zero for the six-dimensional exponential function, com-
puted with an interval Newton method and the high order Taylor model approach.

3.5 Implicit Equations

In this section we develop a method to compute Taylor model enclosures of functions
that are described by implicit relations. This method has interesting applications
in prescribed path control [58] and can even be used to obtain solution enclosures
for implicit ordinary differential equations [59]. While the new approach uses the
methods for computing inverse Taylor models discussed earlier, the method is based

on the Implicit Function Theorem.

Theorem 3.7 (Implicit Function Theorem). Given two open sets U C IR” and
V C R‘" and afunction f E C"+1(U X V,R“’). Write f(iE, y) With 33 E U and y E V

and suppose that (2:0, yo) 6 U x V and that f (270, yo) = 0- If

D(fl({a:o}lew)nW) ($0ay0) 3 Rm “i Rm (3.37)

is a linear isomorphism, then there is an open set U’ C U C IR” containing 1120, an

Open set V’ C V C R” containing yo, and a unique 0"“ function g : U’ —+ V’ such

71

that for

By

tr inc

Elﬂff

iZELUH

sigh]

that for all :1: E U’

9(130) = 310 and f (x, g (3)) = 0- (3-38)

By combining this fundamental result of calculus with the presented methods
for enclosing inverse functions, we are able to compute Taylor models for implicitly
described functions 9 as above. Suppose that f satisﬁes the conditions of the Implicit

Function Theorem. Then we deﬁne a function (I) : U’ x V’ :-—> R‘” by

<1> ( 3 ) = ( “If,” ). (3.39)

Since f satisﬁes the Implicit Function Theorem, it follows that (I) satisﬁes the con-
ditions of the Inverse Function Theorem around (1:0, yo) 6 U’ X V’. Thus, there are
neighborhoods U” of 2:0 and V” of go such that the inverse <1)“1 : U” x V” ——> R'“
exists and is well deﬁned. Then we use <I>"l to compute an explicit expression for the

implicitly described function g:

(;)=¢-‘(3)=(,E‘.))- (3'40)

Thus, in the semi-algebraic framework of Taylor models, we can use an inver-
sion algorithm to obtain rigorous enclosures of implicitly prescribed functions. The
only drawback of this method is that augmenting the system requires additional vari-
ables that might not be available due to memory constraints. However, in many
important applications like prescribed path control [58], the variable 3: represents
one-dimensional time. Thus, the augmented system is often enlarged by only a single

variable, which is usually within the workable range of the Taylor model methods.

We also mention that the technique of augmenting implicit systems has been
applied extensively to ﬁrst order in bifurcation theory [72] and to higher order for

symplectic integration [11]. Other combinations of the Implicit Function Theorem

72

and ﬁr

are dis

‘
_‘ tr ‘4
. (11.1,

i .
ll [l
EUHj

0 Eli

and ﬁrst order automatic differentiation [111, 52, 112] in the ﬁeld of control theory

are discussed by Evtushenko [42].

3.5.1 Curves, Surfaces, Manifolds

Hyperbolas are conic sections with an eccentricity e > 1. Denoting the foci of a

hyperbola by f1 and f2. Hyperbola with a semi-major axis of a are described by

where the two different signs denote the two branches of the hyperbola [61]. If we
denote the distance between the origin and the foci by e; i.e., e = [ f1], the eccentricity

e and the semi-major axis a are related by:

a - e = e. (3.42)

Given these relations and notation, the semi-minor axis b and the parameter p are

connected via

b2 = 32 -— a2 and pa 2: b2. (3.43)

We denote the angle between the zit-axis and the vector connecting the focal point
f1 with the point (x, y) on the positive branch of the hyperbola by oi. Then in the
special case of a = e = 1, the cartesian coordinates x and y and the parameters 5 and

o are connected by

 

h, (5,45,33,31) = x2 — 52": 1 — 1 = 0 (3.443)

 

hz (e, d), 3:, y) = (1 — ecos(¢)) - (ﬂit — e)2 + y2 — (52 — 1) =: 0 (3.44b)

In the remainder of this section, we use the approach outlined in Eqns. (3.39)

and (3.40) to compute Taylor models containing charts for the cartesian coordinates

73

l0
[ll Si] :

W‘iw
maul

The 1

mail

1". I:

"y..
JON

:1: and y as functions of the parameters 5 and d. The results show that the method
can indeed be used to solve implicit equations and to compute parameterizations.
Besides its applicability to static problems as discussed here, this method has an im-
portant application in prescribed path control, where the implicit relation connecting

coordinates and control functions is obtained as the result of an ODE integration
step [58].

To compute charts for :1: and y from Eqns. (3.44), we start with initial conditions

of so = 2 and (to = 7r/ 2 and use a standard Newton method to determine consistent

initial conditions 2:0 and go such that
hl (50195011301310) = h2 (50, $0,150, yo) = 0- (3-45)

Once the consistent initial conditions 360 = 2 and yo = 3 have been determined to

machine precision, we model the functions hl and I12 by 19-th order Taylor models

T1 and T2 with reference point (so, (to, $0,110) and domain
D = [199,201] x [1.56079633,1.58079633] x [1.95, 2.05] x [288,312]. (3.46)

The resulting Taylor models have remainder bounds with a width in the order of the
machine precision 10'16 and their reference polynomials have 59 and 4284 coefficients,

respectively.

After augmenting the system and deﬁning the function (1) according to Eqn. (3.39),
we use Taylor model inversion and Eqn. (3.40) to obtain Taylor models for a: and y

as functions of s and (b. The resulting Taylor models are deﬁned over the domain
D’ = [199,201] x [1.56079633,1.58079633], (3.47)

have reference polynomials with 210 and 205 non-vanishing coefﬁcients, and remainder

74

6111115

fig. 3.?

25“

Eli -

138'

bounds

R1 = [—0.635775672959 x 10-8,0.635776786515 x 104], (3.483)

R2 = [—0.537150447777 x 10—3, 0.537151554117 x 1078]. (3.48b)

Fig. 3.10 shows the resulting surfaces 3: (e, (1)) and y (e, 45) over the domain box D’.

Implicit x(e.¢) Implicit y(c.0)

 
     

    

1.56 ’51-99 1'56 15 0052.01
' . 2.005 ' . _ .

.01
Angle 0 Ewentricity e An9'9 4’ Eccentricity a

Figure 3.10: Charts for the cartesian coordinates x and y of the two-parameter de-
scription Eqns. (3.44) for hyperbolas. Remainder bounds are in the order of 10‘s.
This example demonstrates how the Taylor model inversion tools can indeed be
used for the computations of implicit functions and charts of smooth manifolds. While
the Implicit Fimction Theorem can guarantee the existence of local parameteriza-
tions and is therefore of great theoretical value, the presented method allows rigorous
computations of these parameterizations. As illustrated by the example, the result-
ing Taylor models offer a highly accurate description of the charts of the implicitly
described manifolds. The ability to rigorously solve implicit equations with small
overestimation is of great interest to ﬁelds as diverse as prescribed path control [58]

and differential algebraic equations [59].

3.6 Analysis of Inversion Algorithms

Before we close this chapter on Taylor model based inversion methods, we use the

previously presented examples to study some of the complexity and performance

75

312036
niques
ahead

11150

3.6.]

The C
if Elli.
the s:

E

characteristics that distinguish conventional interval methods and Taylor model tech—
niques. While the theoretical limits that govern Taylor model computations have
already been discussed in Sec. 2.3, here we focus on practical results to underline and

illustrate these aspects.

3.6.1 Space and Time Requirements

The computational complexity of individual Taylor model operations exceeds the cost
of the corresponding standard interval arithmetic by a wide margin. For example,
the storage size S of an n-th order Taylor model in 1) variables scales approximately

as

 

S=(n:v)=(n+v)!' (3.49)

v! n!
Moreover, while the cost of simple operations like addition and assignment scales
linear with S, multiplication and intrinsic functions scale with higher orders of S [83,
22, 82]. However, since Taylor models can often avoid domain splitting, their real
strength lies in high-dimensional problems with large domains. As an example, con-
sider enclosing the inverse of a one-dimensional function as illustrated in Fig. 3.11.
Using only intervals, this requires domains of the size of the desired accuracy, since

the number of sub-intervals has to scale with the inverse of the desired accuracy.

As another example, consider the six-dimensional exponential function presented
in Sec. 3.3.2. The computational cost of the 8—th order Taylor models is about 1500
times higher than that of conventional intervals. However, achieving a uniform in-
clusion accuracy of 10‘11 with intervals, as is often needed for root ﬁnding problems,
would require approximately 109 sub-intervals in each dimension: resulting in 1054
separate interval evaluations, which would make it impossible to use conventional in-
terval methods for that task. Thus, the advantages of Taylor models often outweigh

their additional costs over naive interval techniques if enclosures of high precision are

76

 

 

 

l

J

I
L

 

I
l
T
l

 

 

I
_L
f
l

 

ﬂ
1
T
J_

 

 

I
l
I— 7
l

 

 

 

 

 

 

 

 

Figure 3.11: Illustration of veriﬁed inversion with intervals. The sharpness of the
enclosures scales linearly with the magnitude of the domain sub-intervals.

desired.

As a general rule of thumb, if conventional interval methods and Taylor models
compete over the same domain and box-splitting is not required to achieve the neces-
sary sharpness, the simple interval methods will almost always be more efﬁcient. But
if using intervals requires splitting of the domains, as is frequently the case, the use

of Taylor models will generally become favorable.

3.6.2 Order Dependence

This ﬁnal example of the chapter illustrates how the accuracy of Taylor model ap-
proaches indeed scales with the (n + 1)-st order of the domain size as shown in
Thm. 2.6. For the six-dimensional exponential function deﬁned in Sec. 3.3.2, Fig. 3.12
shows the magnitude of DU) after just one iteration of the 6-th order Taylor model

Newton method as a function of 0“” = [—0.25, 025l-

The data illustrate how that accuracy of the method scales with the 7-th order of

77

 

the initial domain 13(0), For purposes of illustration, a line of slope seven has been
ﬁtted to the data points. Deviations from that theoretical line for small domains are
due to limits imposed on the veriﬁed computations by the machine epsilon. For large
initial domains on the other hand, D“) is bigger than the theoretical line would imply,
caused by a decreasing accuracy of the 6-th order Taylor polynomial approximations
over the initial domain. Hence, for large domains the Taylor models operate in a

region where the rules of (n + 1)-st order convergence fail to apply.

 

 

 

 

0 I T I I
+
+ X
‘2 r + :1,” l
,x.
+ t”
.4 r- t 4"”, _]
13;,
,v
—6 ,. yr
l
-8 h ’3’”, ..
,4-
I,
-10 - ,«J «
’2'
(I
-12 )- A» +
.4!"
+’+’+
'14 7 +++++++++jxv -
’ + Accuracy vs. Domainsize

-16 . ----- Linear Approxnmation wuth slope 7 -

_18 " 1 I l 4—
-2.5 -2 -1.5 -1 -0.5 0

Figure 3.12: Double-logarithmic plot of the accuracy of the ﬁrst Taylor model Newton
step as a function of initial domain size; results for 6-th order Taylor models.

3.7 Summary

In this chapter we have presented new Taylor model based methods to rigorously
answer the question of whether a function is invertible over a certain domain of
interest. This method uses only ﬁrst derivatives and as such it can be implemented

with little computational expense.

78

ls
intent

it] an

00:15
01rd)

[161'

9120
I

fill]

lite?

As a next step, we developed a new method for computing rigorous enclosures of
inverse functions of arbitrary complicated multidimensional functions. Its applicabil-
ity and performance has been demonstrated in the computation of veriﬁed enclosures
of the inverse function for given invertible functions. The method combines Tay-
lor models with the method for proving invertibility and efﬁcient differential algebra
algorithms that allow the determination of exact n-th order polynomial inverses in

ﬁnitely many steps.

Using Taylor models, we have been able to overcome some limitations of conven-
tional interval based methods to determine invertibility. Namely we have successfully
shown that Taylor model based methods can model computationally complex func-
tions much more accurately than interval methods. Since the accuracy of Taylor
models scales with the (n -l- 1)-st order of the domain size, we have shown that the
new methods work over signiﬁcantly larger domains and scale better to high dimen-
sional problems than interval based methods. Moreover, the newly developed method
can handle complicated functions by using Taylor models which have been shown to
control the dependency and cancellation problems inherent in regular interval arith-
metic [85]. This is especially important in the establishment of invertibility prior to

the computation of the inverse Taylor model itself.

As an immediate application of the new method, an extension of the conventional
interval Newton method has been presented. It has been demonstrated to converge
much faster than the regular interval-based version of that algorithm. Moreover,
combining the inversion tools with a fresh look at the Implicit Function Theorem
resulted in a new method for the rigorous computation of implicitly described func—
tions. Important applications of the new methods are in the computation of solutions
of differential algebraic equations (c.f., Chap. 4 and [4, 107, 59]), the determination of

existence of generating functions (c.f., Sec. 5.2), and the stability analysis of asteroid

79

(villi:

orbits near the Lagrangian points [133].

80

Chapter 4

Differential Algebraic Equations

Under certain conditions, the solutions of ordinary differential equations (ODEs)
and differential algebraic equations (DAEs) can be expanded in Taylor series in the
independent variable and the initial conditions. In these cases, we can obtain good
approximations of the solutions by computing the respective Taylor series [31, 32].
Moreover, the Taylor model approach often allows us to compute rigorous enclosures

of the solutions of initial value problems [24, 82].

By using structural analysis [105, 108] and differentiation, it is often possible to
transform a given DAE into an equivalent system of implicit ODEs. If the derived
system is described by a Taylor model, representing each derivative by an independent
variable, veriﬁed inversion methods discussed in the previous chapter can be utilized
to solve for the highest derivatives as functions of lower order ones. The resulting
Taylor model forms an enclosure of the right hand side of an explicit ODE initial value
problem that is equivalent to the original DAE. While this explicit system is suitable
for integration with Taylor model solvers [24], the intermediate inversion often requires
a substantial increase in the dimensionality of the problem, limiting the approach
to relatively small systems. An application of this inversion-based integration of

differential equations has been discussed in [59].

81

In this chapter we derive a method for the veriﬁed integration of general implicit
ODEs that is based on the observation that solutions can be obtained as ﬁxed points
of a certain operator containing the antiderivation. We show that this Operator is
particularly well suited for practical applications in Taylor model settings since its
restriction to DA vectors is guaranteed to converge to the exact solution in at most

11 + 1 steps, where n E N is the order of the Taylor models.

While sophisticated methods have been developed for the numerical integration
of DAEs [51, 30, 53, 4], they are usually based on multistep methods that generally
do not provide any means of veriﬁcation and validation. However, since the new
method also allows the computation of the differentiation index Vd of DAEs [4], it
can be utilized for a veriﬁed index analysis. Combining this with a more traditional
structural analysis of DAEs [105, 108], we get a scheme for transforming DAEs into
implicit ODEs, which can then be solved with the new method. Since this combination
uses veriﬁed Taylor models at every stage of the computation, it can be used to
compute Taylor model enclosures of the solutions of DAEs. Additionally, by utilizing
high order Taylor models (n > 20 is not uncommon), the scheme can even be applied
to high-index problems that pose serious challenges to existing non-veriﬁed DAE

integration methods.

4.1 Background and Motivation

The standard ﬁrst order initial value problem is usually associated with an explicit

ordinary differential equation of the form

x’ =: f(t,:L‘). (4.1)
However, a more general ﬁrst order ODE problem is given by the implicit form

F(t, 516,:6’) = 0, (4.2)

82

when

‘5 as
{0 ill

. [1
HIGH
5'10.
Slilll‘

l0 1

where the Jacobian
6F (t, u, v)
01)

(4.3)
is assumed to be non-singular for all arguments in an appropriate domain. According
to the Implicit Function Theorem, it is then possible to turn the problem into an
explicit ﬁrst order system that is as smooth as the original problem. Thus, if F was
sufﬁciently smooth to begin with, the implicit system is guaranteed to have a smooth

solution. However, the solutions of implicit ODEs are not guaranteed to be unique.

To illustrate this, consider the following initial value problem for an implicit ODE

(:6’)2 = 3:2, 12(0) = 1. (4.4)

‘t are both valid solutions to this problem,

Obviously 23+(t) = e‘ and 2:-(t) = e
illustrating that smooth implicit systems may indeed have multiple smooth solutions.
We will later address the implications of this phenomenon by introducing the concept

of consistent points, which will enable us to guarantee the local uniqueness of solutions

of implicit ODEs and DAEs.

4.1.1 First Order Differential Algebraic Equations

ODEs with constraints are another extension of the regular eXplicit ﬁrst order ODE

problem

23’ = f(t, x, z) (4.5a)

0 = g(t, 27, 2)- (4.5b)

Unlike in the case of standard ODEs, m(t) is not only a solution to the differential
Dart (4.5a), but is also forced to satisfy the algebraic constraint condition (4.5b).

While the system (4.5) is given in the so—called semi-explicit form, it can easily be

83

 

rewritten in an implicit form by introducing the variable 5 == (2:, z)T.

I _ x'_f(t1$iz) _. 46
F(t7€a€)_( g(t,$,Z))-O. (')
However, unlike in the case of implicit ODEs, the resulting Jacobian matrix
6F(t,u,v) __ I 0
Ti. .)

is no longer regular.

Generalizing the previous example and the deﬁnition of the general ﬁrst order
implicit ODE (4.2), we write the general ﬁrst order differential algebraic equation
(DAE) as

F(t, :12, :r’) = 0, (4.8)

where F is a sufﬁciently smooth function deﬁned in an appropriate domain. In the
case of DAEs we do not impose any additional restrictions on the regularity of the
Jacobian (4.3). In fact, the rank and the structure of the Jacobian may even depend
on the solution m(t) However, for simplicity we will always assume that the rank and

the structure of the Jacobian along a single solution m(t) are independent of t.

The class of DAEs contains all ODEs and all algebraic equations, as well as a
large number of problems between these two extremes. As such, DAEs can generally
not be solved by simple algebraic means. Moreover, unlike the case of ODEs, even
smooth DAEs do not necessarily have solutions. It is interesting that the ﬁrst re-
sults on the existence and uniqueness of solutions of general DAEs have only recently
appeared [109, 110]. While these results are based on theoretical arguments of dif-
ferential geometry, in Sec. 4.3 we will present results that guarantee the existence of

solutions to a large class of DAEs based on direct computations.

The most common approach of determining the solution of a given DAE is to

differentiate the system until we can pick 1) equations from the enlarged system such

84

that the Jacobian (4.3) of this new system is regular. The resulting implicit ODE
system can often be solved for a solution of the original DAE problem. This basic

idea is formalized by the deﬁnition of the index [4]:

Deﬁnition 4.1 (Differentiation Index). For the general DAE system (4.8), the
differentiation index 11,, along a solution m(t) is the minimum number of differentia-
tions of the system which would be required to solve for 33’ uniquely in terms of the

dependent variable a: and the independent variable t.

Thus, the index Va) is deﬁned in terms of the overdetermined system

F(t,a:,:r’) = 0

F’(t, :6, 27’, :6”) = 0
(4.9)

F(p)(t,:1:, ,m’, . . . ,x(p+1)) : 0

to be the smallest integer p so that (4.9) can be solved for the highest derivatives in
terms of lower order derivatives and t. In the following, the index will indicate the
differentiation index unless we compare to other concepts of indices in which case we

will use the fully qualiﬁed name.

It might be somewhat of a surprise that the deﬁnition of the index explicitly refers
to a solution of (4.8). However, while we will always assume that the index does not
change along a single solution; i.e., is independent of the independent variable t, a
given system can have different indices along different solutions. To illustrate this,

consider the system

0 = 13’ — 2 (4.103)
0:30-9) (4mm
0 =xy+z(1—y)—t (4.10c)

85

Assuming continuous solutions, the second equation permits exactly two solutions:

ya(t) = 0 and ye (t) = 1. Depending on which of the solutions is compatible with the

initial conditions, we get two different scenarios:

a: The system reduces to at’ = z and 0 = z —- t; it has index 1 with the solution

(180 + t2/2, 0, t)T.

6: The system reduces to 56’ = z and 0 = :1: — t; it has index 2 with the solution

(t, 1,1)7'.

While this example shows that the index does indeed depend on the solution of the
DAE, it also illustrates how the solution manifold of a DAE can vary with the index.
While any sufﬁciently smooth ﬁrst order ODE system of size v has exactly v degrees of
freedom, the solution manifold of the DAE system (4.8) can have any dimensionality

between 0 and v.

By permitting only discrete values for y, the previous example highlights how the
constrained nature of DAEs even extends to the initial conditions. While the initial
conditions for ODEs are usually free to vary over large ranges, the initial conditions
of DAE problems have to be consistent with the constraints. However, frequently the
consistent initialization of a DAE problem is not obvious: problems with an index
larger than one have hidden constraints which are only revealed by the intermediate
derivatives in (4.9). Thus, while the highest derivatives are needed to transform the
system into a solvable ODE, all lower order derivatives are describing the constraint

manifold and the consistent initial conditions.

Standard DAE Problems

Short of testing all possible combinations of derivatives, Def. 4.1 provides us with no

efﬁcient algorithm for computing the index of the DAB (4.8). While we will revisit

86

 

this issue in Sec. 4.3, many practical DAEs can be written in standard Hessenberg

forms with known index, separating the ODE part and the algebraic constraints.
Deﬁnition 4.2 (Hessenberg Index-1). The diﬂ'erential algebraic equation
IL" = f(t,a:,z) (4.113)
0 2 g(t, 2:, z), (4.11b)
with the Jacobian matrix function Bg/Bz assumed to be regular for all t, is in Hes-

senberg Index-1 form.

DAEs in Hessenberg Index-1 form are also called semi-explicit index-1 systems.
Since the constraints can in principle be solved for the algebraic variable z, these
systems are closely related to implicit ODE problems. Another important class of

DAEs is given by the Hessenberg Index—2 problems.
Deﬁnition 4.3 (Hessenberg Index-2). The differential algebraic equation
33’ = f(t,:L‘, z) (4.12a)
0 = g(t, 2:), (4.12b)
with the product of the Jacobian matrices (ﬁg/6:6) (6 f / (92) assumed to be regular for

all t, is in Hessenberg Index-2 form.

We note that in many index-2 problems the algebraic quantity 2 plays the role
of a Lagrange multiplier. Thus, many problems of classical mechanics can easily

be written in the form of index-2 DAEs. An example of this will be discussed in

Sec. 4.4.2.

4.1.2 General Differential Equations

Up to this point, we have exclusively dealt with systems of ﬁrst order differential

equations. This is commonly justiﬁed by the observation that any n—th order differ-

87

ential equation in v variables can be transformed into an equivalent ﬁrst order systems
with up to n x v variables. This approach is known as order reduction and has many
advantages for the theoretical analysis of general ODE problems. However, the in-
creased size of the problems often challenges veriﬁed numerical integration methods.
In the remainder of this chapter we will therefore focus on methods for the direct

numerical integration of general high-order differential equations.

For a deﬁnition of the most general DAE problem, we consider the independent
scalar variable t, the v dependent variables 2:, = xj(t), and sufﬁciently smooth func-

tions f,- for i = 1, . . . , v, and form the system

f1(t,x1,...,a:(1£”),...,a:v,...,a:,(,£l”)) = 0
(4.13)

fv (t,:c1,. ..,:1:[£"‘),...,2:,,,...,xffm) = 0.

For a given j, the i-th equation of this system does not necessarily depend on 1:],
the derivatives 33;”) for n < (,5, or even any of its derivatives. However, if there is a
dependence on at least one of the derivatives (including the O-th derivative 3]” = 317,-),

we denote the order of that highest derivative by f,g.

4.2 Veriﬁed Integration of Implicit ODEs

While sophisticated general-purpose methods for the veriﬁed integration of explicit
ﬁrst order ODEs have been developed [79, 78, 82, 24, 97, 99], none of these can
be readily used for the veriﬁed integration of implicit ODEs, let alone differential
algebraic equations. As a ﬁrst step toward the veriﬁed integration of general DAEs,
we consider the problem of integrating the implicit ODE initial value problem (4.2).

Unlike in standard ODE integration methods, we will not limit ourselves to ﬁrst order

88

problems, but will eventually be able to integrate the arbitrary order problem
F (t,:r,:r’,...,x(”)) , (4.14)

where the derivative of F with respect to the highest derivatives in each of the compo-

nents is assumed to be non-singular for all argument values in an appropriate domain.

The new integration method is based on an extended use of antiderivation. To
motivate the combination of high order methods and antiderivation for the veriﬁed
integration of general differential equations, consider the explicit second order ODE

initial value problem
:6” = f (11:,33’, t), m(te) = 1:0, .r’(t0) = 3:2,. (4.15)

While the conventional approach to solving this system is based on order reduction
to a two-dimensional ﬁrst order problem, in the framework of Taylor models we can

use the intrinsic antiderivation and substitute f = :r’; i.e.,

270) = $0 + [toté(r)d'r- (4.16)

After inserting the expanded expression for m(t) into (4.15), we obtain an explicit ﬁrst

order ODE for 5.
t
E’ = F(€,t) = f (2:6 + avast). (4.17)
to
This system can readily be integrated with the existing Taylor model based integration

scheme discussed in Sec. 2.3.3, and the solution m(t) of Eqn. (4.15) can be computed

from the Taylor model for f (t) by a ﬁnal application of antiderivation.

c
:1: = x0 + €(r)dr. (4.18)

to

While this approach seems obvious from a theoretical point of view, numerical
analysis has traditionally avoided explicit references to antiderivation in its algo-

rithms since antiderivation is usually not available in the classical computational

89

frameworks. Only recently, with an increased use of symbolic and semi-symbolic
tools like Mathematica and COSY Inﬁnity, has the explicit use of antiderivation in
numerical computations become feasible. From a practical point of view, this ap-
proach has the advantage of reducing the computational complexity of the right hand

side of the ODE (4.15), often allowing for favorable computations.

4.2.1 High Order Taylor Model Solutions

In this section we present a Taylor model based algorithm for the veriﬁed integration

of the general ﬁrst order ODE initial value problem
F(t, IL‘, 113’) = O, (B(to) 2 $0. . (4.19)

While we will later extend the algorithm to higher order ODEs, for now we assume
that the problem is stated as an implicit ﬁrst order system with an (n + 2)-times

continuously differentiable v-dimensional function F and regular Jacobian matrix

6F(t, u, v)

6v (4.20)

in appropriate domains.

The method discussed in this section uses techniques and tools based on the dif.
ferential algebra "D, and the Taylor model approach presented in Chap. 2. Utilizing
DA vectors and Taylor model methods for the veriﬁed integration of initial value
problems allows the propagation of initial conditions by not only expanding the so-
lution in time, but also in the transverse variables [24]. By representing the initial
conditions as additional DA variables, their dependence can be propagated through
the integration process, allowing Taylor model based integration schemes to reduce
the wrapping effect to high orders [86]. Moreover, in the context of the new algorithm,

expanding the consistent initial derivative in the transverse variables further reduces

90

 

the wrapping effect and allows the system to be rewritten in a derivative-free, origin

preserving form suitable for veriﬁed integration with Taylor models.

Algorithm: Taylor Model Integration of Implicit ODEs

For notational convenience, we write the initial condition of the implicit ODE prob-
lem (4.19) as are, and denote functional dependence on the initial condition 2:0 in a
suitable domain by y. With these conventions, a single n-th order integration step of

the implicit ﬁrst order ODE (4.19) consists of the following sub-steps:

1. Using a suitable numerical method like the Newton method discussed in Sec. 3.4,
solve the implicit system

F(th y’xl) = 0 (421)
for a consistent initial condition :c’(t0, y) = 2:6(y).
2. Utilizing antiderivation, rewrite the original problem in a derivative-free form
t
<I>(t.y,€) = F (t,v+/ €(T.y)dr,€) = 0, (4.22)
to
where 5 = g (t, y) = :r’ (t, y) has been substituted for the derivative of 2:.

3. Translate the problem into an origin-preserving form by substituting C (t, y) =

E (t, y) — :1:[,(y) to obtain a new function
‘1’(t. y. C) = <1) (t. y. (0. y) + $6(y))- (4.23)

4. Since \Il(to, are, 0) = 0, within the differential algebraic framework it is possible

to write the ﬁrst order truncation of ‘1! without the constant part as

‘1’“, y, C) :1 LC(<) + LR“) g), (4.24)

where L( and L R denote the linear parts in C and (t, y), respectively.

91

. If LC is regular, transform the previous expression into an equivalent ﬁxed point

formulation for C.
((t. y) = MC) = -L;1(4(t,y,g)— 14(4)). (4.25)

. Using the operator ”H, deﬁne a sequence (av) of DA vectors in "DH, by a0 = 0
and

au+1 = 71(au). (4.26)

Then deﬁne the polynomial P(t, y) = on“.

. Construct a Taylor model T with the reference polynomial P over an appropriate

domain T x D C R x R” containing the reference point (to, 2:0) such that

’H(T) c T. (4.27)

. Compute a Taylor model X from T by using the relation

may) = y + f (<14. 4) ”some. (4.28)

to

As we will see in the remainder of this section, the ﬁnal Taylor model X is in

fact a Taylor model for the ﬂow x(t,y) of the original ODE problem (4.19) for the

particular choice of the initial derivative.

4.2.2 Mathematical Background

The algorithm presented in the previous section rests on several non-trivial assertions

that will be proven here. We provide its mathematical foundation and establish the

basis for the discussion in the next subsection. The proofs can be split into two

groups: the differential algebraic part of the algorithm and the Taylor model results.

Implementation and user interface issues of the algorithm will be discussed in the

next section.

92

 

Differential Algebraic Results

In this section we discuss the differential algebraic results needed to justify the pre-
sented algorithm for the veriﬁed integration of implicit ODEs. First, we show that
the operator ’H introduced above is well deﬁned and DA-contracting. We then show
that up to an additive constant, its unique ﬁxed point lies in the same equivalence

class as the derivative of the ﬂow of the original ODE problem.

Lemma 4.1. The operator ”H given by Eqn. 4.25 is a contracting operator and self

map on the set M = {a 6 "DH, [ Ma) > 0}.

Proof. To ﬁrst order regularity of L4 is equivalent to the assumed regularity of the

Jacobian 8F(t, u, v) /0v.

g(teme = g(tmyoﬂ‘l‘ $6) =1 aa—iaoﬂmo (429)

Thus by assumption, the linear map LC is regular in a neighborhood of the initial

conditions and the operator ’H is therefore well deﬁned on all of "DH”.

To show that ’H is contracting on the subset M of nilpotent DA vectors, let
a,b E M be given and assume that a and b agree up to order k. Since L21 is

invertible, it suffices to show that
A ((‘I’ftoa Elma) — Lc(a))—(‘1’(to,yo,bl— Lc(b))) > k- (4.30)
Since ‘I’(te, 51:0, 0) = 0, the map \Il(te, 2:0, C) is origin-preserving and can be written as
‘I’(t0, 5130, C) = L((C) + LR(t03$O) +N(t0, 53010, (4.31)

where N is a purely non-linear function. Thus, it sufﬁces to show that N is contract-
ing. However, if a and b agree up to order 16, their images N(to, are, a) and N(to, :60, b)
trivially agree up to order k + 1. Finally, since \II is origin-preserving, ’H is indeed a

self map of M, and therefore a contracting operator on M. E]

93

While this lemma guarantees the existence of a unique ﬁxed point of the operator
’H, the next theorem summarizes the main result of the DA part of the presented

algorithm.

Theorem 4.1. If we denote the flow of the implicit ﬁrst order ODE initial value

problem (4.19) by m(t, y), then the ﬁxed point of ’H is a representative for

[540431) - xtfylln- (43?)

around the expansion point (to, 9:0).

Proof. In principle, this assertion follows from the construction of the operator 71.
However, in the following we summarize observations on existence, uniqueness, and

smoothness of the solutions to the IVP (4.19) that are required for a full justiﬁcation.

Since the original function F is of class 6"” and its Jacobian matrix is assumed
to be non-singular over a suitable region containing (to, 2:0, 363), at least one solution
to the initial value problem exists. Moreover, once a consistent initial derivative 276
has been ﬁxed, the Inverse Function Theorem guarantees the existence of a unique
solution m(t, y) in a neighborhood of (to, 1:0, 3:3). The solution is the unique solution
to an explicit ﬁrst order system, which can in principle be derived from the original
implicit problem. Since the Inverse Function Theorem guarantees that the explicit
system is also of class 6"” in a neighborhood of the consistent initial conditions,
Thm. 5.2 ensures that the ﬂow m(t, y) is a C"+2 function of its variables. Thus, the
equivalence class of its derivative is well deﬁned in C’“r 1, and since the solution C is

unique, the ﬁxed point of ’H is indeed a representative for that class. Cl

Taylor Model Results

In this section we prove the main Taylor model result needed for the presented al-

gorithm: the self-inclusion of the Taylor model in (4.27) is a sufﬁcient condition to

94

 

guarantee the enclosure of a ﬁxed point of Eqn. (4.25).

Deﬁnition 4.4 (L-Taylor Model). Let T = (P, 11:0, D, R) be an n-th order Taylor
model and let L > 0 be a Lipschitz constant for P over D. Then the L-Taylor model

TL is the set of all functions f E C0 (D,R“’) such that
1. f(x) — P(x) 6 R for all a: E D,
2. f(xo) = P(IBO),

3- [f(rv) - f(y)l S le - ill for all 23.9 E D-

First, it should be noted that this gives a well deﬁned and non—empty set of
functions, since P E T 1,. And while there is an obvious connection to normal Taylor

models, L—Taylor models contain a different set of functions:

1. Taylor models are “more restrictive” than the corresponding L-Taylor models,

since the latter may contain non-differentiable functions.

2. Taylor models are also “less restrictive” than the corresponding L—Taylor mod-

els, since they do not pose any limits on the Lipschitz constant of its members.

Finally, unlike standard Taylor models, L-Taylor models are subsets of a Banach

space, namely C0 (D, R”)-

Lemma 4.2. The L-Taylor model TL deﬁned as above is a convex subset of the

Banach space C0 (D, R”) [82]-

Proof. Given two functions f0 and f1 in TL and t 6 [0,1], deﬁne ft = t- f1 + (1 —t) - f0.
Since the sum of continuous functions is continuous, ft 6 C0 (D, R”) for any t E [0, 1].

Moreover, by convexity of R
Mas) — P(x) = (t - 11(4) + (1 - t) - 3(4)) - P(x) 6 R (4.33)

95

 

for any t E [0, 1]. Since

UM?) " ft(y)l = lt' f1($) + (1 ‘tl 'f2(17) —t°f1(f/) — (1 “ t) ' 12(9)]
St: |f1($) -f1(y)|+(1—t) ' Ifo(1‘) -fo(y)| (4-34)
gt-L+(1—t)-L=L,

TL is indeed a convex subset of the Banach space C0 (D, R”). E]

Lemma 4.3. The L-Taylor model TL deﬁned as above is a compact subset of the
Banach space C0 (D, lR‘”) [82].

Proof. It sufﬁces to show that every sequence (fu) in TL has at least one limit point in
TL. According to the Arzela—Ascoli Theorem it suffices to show that (fu) is uniformly

bounded and equicontinuous.

We will ﬁrst show that the sequence is uniformly bounded. Since P is a polynomial
and therefore continuous on D, [P] assumes its ﬁnite maximum M1 over D at some
point 5::

M1: [P(zit)[= max{|P(:r)| [z E D}. (4.35)

Since R is bounded, there is a constant M2 > 0 such that
y E R => |y| 3 M2. (4.36)

After deﬁning 5,, = f,, — P, 6,,(x) E R for all a: E D. Thus for any a: E D and any
:2 e N

|f,,(:r)| _<_ [P(cc)| + |6,,(:r)[ < M1 + M2 < oo. (4.37)
Hence, the sequence is indeed uniformly bounded. Moreover, since by deﬁnition the
sequence is also Lipschitzian with uniform Lipschitz constant L that is independent

is u, it follows that it also equicontinuous.

Thus, the L-Taylor model TL is a compact subset of the Banach space C0 (D, R'”). C]

96

The main result of this section is that the self-inclusion given in Eqn. (4.27) in
indeed sufﬁcient to guarantee that the ﬁxed point of ’H is contained in the Taylor
model T. The proof of this assertion is based on the next theorem, which is an

almost immediate consequence of the previous two lemmas.

Theorem 4.2. Let T and D be interval domains containing the points to and 51:0

respectively. Let L > 0 be a Lipschitz constant for P and deﬁne the L-Taylor model
TL = (Pa “0.580) ,T X D4Rl- (438)

If 7-l(TL) C T L, then the L-Taylor model TL contains a ﬁxed point of ’H.

Proof. Since ”H is a continuous operator on C0 (T x D, R”), the assertion follows from

the last two lemmas and the Schauder Fixed Point Theorem. El

According to this theorem, a ﬁxed point of H is contained in the L-Taylor model
TL. However, as indicated earlier, one of the ﬁxed points is in fact equal to $’(t, y) —
r[,(y) and is therefore of class 6"“. Moreover, according to Thm. 4.1, the n-th order
Taylor expansion of the ﬁxed point around (t0,:1:0) equals P. Thus, if [H] < 1, the

ﬁxed point is also contained in the regular Taylor model

T = (P, (40,420) ,T x D, R). (4.39)

We also note that since the reference polynomial P is already a very good ap-
proximation of the mathematically correct ﬁxed point. In practice the self-inclusion
of the image can almost always be achieved by choosing sufﬁciently small domains T

and D around the reference points to and 2:0.

4.2.3 Discussion of the Algorithm

After having established the mathematical foundations of the algorithm in the previ-

ous subsection, we will now comment on the individual steps of the basic method. In

97

the following discussion we focus on how each stage can be performed automatically

in a computer environment, with only limited need for manual user interventions.

1. In the integration of explicit ODEs, the initial derivative 11:6 2 23’ (to) is uniquely
determined by the systems initial conditions. In the context of implicit ODE
on the other hand, the solution to an initial value problem is not guaranteed to
be unique. Thus, the consistent initial condition 2:6 has to be obtained during a
pre—analysis step. And since the consistent initial condition may not be unique,
veriﬁed methods have to be used for an exhaustive global search. To simplify

this, the user should be able to supply initial search regions for 326.

In addition to the example given in Eqn. (4.4), the following example illustrates
how the solutions of implicit ODE initial value problems are not necessarily
unique.

(45(4))2 + (4.2111(4))2 =1 and 4(0) = 0 (4.40)

Obviously, s:._(t) = — sin(t) and 23+ (t) = +sin(t) are two distinct smooth solu-

tions of the problem with different initial derivatives -1 and +1.

Since the following stages of the algorithm require an expansion of the consistent
initial velocity in the transverse variables, the use of high-order Newton methods
or the inversion schemes discussed in Sec. 3.5 are warranted at this point. This
will compute both a Taylor model and a suitable DA vector for the following

stages of the algorithm.

As part of the ﬁrst step, the regularity of the J acobian matrix (4.20) should also
be veriﬁed, and rigorous bounds on the sizes of the domains of regularity have
to be established for t, 2:, and z’. Naturally, all subsequent steps then have to
ensure, by possibly shrinking the domains for the independent variable t and the

initial conditions, that all computed quantities stay within these upper bounds.

98

2. From a user’s perspective, it is important that by combining a suitable user
interface with a dynamically typed runtime environment like COSY Inﬁnity,
the substitution of the variables with antiderivatives can be performed auto-
matically, and there is no need for the user to rewrite any of the equations by

hand.

3. By shifting to coordinates that are relative to the consistent initial condition 2:3,
the solution space is restricted to the set M = {a E "D, : /\(a) 2 1} of nilpotent
DA vectors. In step 6, this allows the deﬁnition of a DA—contracting operator,
and the application of the DA Fixed Point Theorem. Again, this coordinate
shift can be performed automatically within the semi-algebraic DA framework

of COSY Inﬁnity.

4. As in the previous two steps, the semi-symbolic nature of the DA framework
allows the linear part L; to be extracted accurately and automatically, since
the linearizations are computed automatically and are readily available in DA
and Taylor model based methods. It should also be noted that the existence of
the inverse over the domains of interest has already been established in the ﬁrst

step, since the regularity of LC follows from the regularity of the Jacobian (4.20).

5. Frequently the solution of a system can be speciﬁed as the ﬁxed point of a
sufﬁciently smooth operator. Here we have the derivative of the solution of an
implicit ODE initial value problem given as the ﬁxed point of such an Operator.
Moreover, according to Thm. 4.1, the ﬁxed point does indeed represent the

desired solution.

6. According to Lem. 4.1, ’H is well deﬁned and DA-contracting on the set M of
nilpotent DA vectors. Hence the DA Fixed Point Theorem guarantees that the

sequence (a,,) of iterates converges in at most n + 1 steps to the n-th order

99

solution polynomial.

an+1 = P(t, v) =4 C(t, v). (4.41)

It should however be noted that within a computer environment the implemen-
tation and iteration of ’H ﬁnds only a ﬂoating point polynomial which is a ﬁxed
point of the floating point operator ’H. While the coefﬁcients of the polynomial
might differ from the mathematically exact n—th order expansion by the ma-
chine epsilon, it is in fact sufﬁcient to ﬁnd a ﬁxed point of ’H only to machine
precision, since deviations from the exact result will be accounted for in the

remainder bound of the Taylor models.

. It has been shown that for explicit ODEs and the Picard operator ’P deﬁned
in Eqn. 2.44, inclusion is guaranteed if the solution Taylor model T satisﬁes
P(T) C T. Although ’H differs from ’P, based on the proof sketched in [82], we

have been able to establish a similar result in Thm. 4.2.

While all previous steps are guaranteed to succeed whenever at least one consis-
tent 2:[, can be found for which the linear part is regular, practical applications
of this step of the algorithm can fail if no suitable Taylor model can be con-
structed, although decreasing the size of the domains will generally lead to a
successful inclusion. Details on suitable strategies for the construction of the
Taylor model T can be found in [82, 24]. Lastly, it should be noted that this
step of the algorithm requires a guaranteed computation of ’H, using Taylor

model enclosures for 2:3 and L4-

. This ﬁnal step computes an enclosure of the solution to the original problem
from the computed Taylor model containing the derivative of the actual solu-
tion, and it relies on antiderivation being inclusion-preserving. To maintain full

veriﬁcation, the Taylor model enclosure of 2:3 has to be added to the result of

100

the integration.

Similar to the example presented in (4.15), the new method also allows the direct
integration of higher order implicit ODE initial value problems without explicit order
reduction. While this approach does not reduce the dimensionality of the Taylor
models if dependence on initial conditions is desired, it can however improve the
sharpness of the ﬁnal remainder bounds. Since the magnitude of the time domain T
is usually smaller than one, Def. 2.8 shows that the remainder bounds of the actual

solution will often be smaller than the ones of the computed highest derivative.

To illustrate how the algorithm can be adapted to higher order ODEs, consider

the general second order implicit ODE initial value problem
0(1), 2:, 29,42") = 0, 23(to) = 2:0, 2:’(t0) = 2:2,. (4.42)

If we assume that the Jacobian matrix 8G (t, u, v, w) / 0w is non-singular in a suitable
domain, this can be written as
t r t
(I)(t,§) = G (we +/to (2:6 + to {(o)do) dr,2:[, + to £(r)dr,§,t) = 0, (4.43)
and the algorithm works with only minor adjustments. Similar arguments can be
made for more general higher order ODEs. The performance of this approach will be

illustrated in the next section with the direct integration of a second order system.

4.2.4 A Second Order Example

To illustrate how the new algorithm can be used for the direct integration of higher

order problems and to demonstrate how the method works in practice, consider the

101

implicit second order ODE initial value problem

ex" + 2:” + :r = 0, (4.44a)
2:’(0) = 2:3 2 0. (4.44c)

While the demonstration in this section uses explicit algebraic transformations for
illustrative purposes, it is important to note that the actual implementation uses the
DA framework and does not rely on such explicit manipulations. We also mention
that for purposes of illustration, this example does not expand the solution in the

transverse variables.

1. Compute a consistent initial value for 233’ = ”(0) such that e30 + x3 ’+ 2:0— — 0.
A simple interval Newton method, with a starting value of 0, ﬁnds an enclosure

of the unique solution 23’ = -1.278464542761074 in just a few steps.

2. Rewrite the original ODE in a derivative-free form by substituting C = 2:”.
t r
@(C, t) = 35“) + C(t) + (2:0 +f (2:3 +/ C(o)do) dr) 2 O. (4.45)
0 o

3. Deﬁne the new dependent variable C as the relative distance of C to its consistent

initial value and substitute C = C — 2:3 in <1> to obtain the new function \II given

by
n t r
‘IJ(C, t=) C+ 2:3 ’+ eerC+1+ 1:2th +/ / C(o)dodr = 0. (4.46)
o o

4. The linear part L((C) of ‘II is 1+ 633 where the 1 is the constant coefﬁcient and

e33 results from the linear part of the exponential function eC.

5. With LC from the previous step, the solution C is a ﬁxed point of the DA-

contracting operator ’H deﬁned by
1

1 + 6%, (e353 (( — (3C) — 2:3’— —$23 -—t2-— [0 /0( C o() dodr). (4.47)

102

 

71(4):

. Starting with an initial value of C (0) = 0, the n—th order expansion P of C is

obtained in exactly n steps.
0"“) = a (0“) . (4.43)
. The result is veriﬁed by constructing a Taylor model T with the computed
reference polynomial P such that ’H(T) C T. With the Taylor model
T = (P, 0, [0, 0.5], [—10-14,10—14]), (4.49)
’H(T) = (P, 0, [0, 0.5], [—0.659807722 . 10‘”, 0.659857319 - 10-14]) . (4.50)

Since P is a ﬁxed point of H, the inclusion ’H(T) C T can be checked by simply
comparing the remainder bounds of T and ’H(T); the inclusion requirement is

satisﬁed for the constructed T.

. A Taylor model for 2: is obtained by using the antiderivation of Taylor models

according to

t r
2: (t) = 2:0 +/ (233 +/ (163’ + C (0)) do) dr. (4.51)
o o
The listing in Tab. 4.1 shows the resulting Taylor model of order 25 computed

by COSY Inﬁnity

This example demonstrates how the new method can be used for the veriﬁed

integration of implicit ODE initial value problems to high accuracy. In this context

it is important to note that the width of the ﬁnal enclosure of the solution is in the

order of 10‘14 for a relatively large time step of h = 0.5.

4.3 Veriﬁed Integration of DAEs

In this section we revisit the question of how a given DAE initial value problem can

be transformed into an equivalent system of implicit ODEs. Since no general purpose

103

RDA VARIABLE: N03 25, NV= 1
I COEFFICIENT ORDER
1.000000000000000
-.6392322713805370
.4166666666666668E-01
.1993921404777223E-02
.6314945441169959E-04
.2635524930464548E-05 1
.4411105791086625E-06 12
.1533094467519992E-07 14
0.8104707776528831E-08 16
10 -.3384116382961162E-09 18
11 -.1389729003787960E-09 20
12 0.1981078695604361E-10 22
13 0.1549987273495670E-11 24

IOOIO
OGGPN’O

1
2
3
4
5
6
7
8
9

 

VAR REFERENCE POINT DOMAIN INTERVAL
1 0.000000000000000 [0.00000, 0.50000]
REMAINDER BOUND INTERVAL
R [-.2500253775762034E-014,0.2500000000000003E-014]
*ttttttastestastiestattests*ttttastes*ttttttttttttttttttt

Table 4.1: Taylor model for the solution of the implicit second order ODE initial
value problem given by Eqn. (4.44).

methods for the direct integration of DAEs exist, this transformation is in general
the only available method of computing solutions to the initial value problems of

differential algebraic equations.

We ﬁrst summarize the main result of the E-method by J. D. Pryce [107, 108],
which uses a structural analysis of DAEs to establish the existence of solutions. Ad-
ditionally, the method ﬁnds an upper bound on the problem’s differentiation index
Vd- Moreover, and more importantly, it also determines an explicit scheme for the

transformation of the DAE system into a solvable system of implicit ODEs.

However, within the previously presented algorithm for the integration of ODEs,
the regularity of L4 already offers a sufﬁcient criterion for the solvability of the derived

ODEs. While the linear map LC will generally be singular, by repeatedly differenti-

104

ating the individual equations of the DAB, we will eventually obtain a regular linear
map LC- Additionally, once the consistent initial derivative is computed, the minimum

number of differentiations gives the differentiation index 114 of the DAE.

4.3.1 Structural Analysis

Here we summarize the main results of the structural analysis of DAEs deveIOped
by J. D. Pryce [105, 107, 108]. The signature method, or E-method, is used to
decide whether a given DAE possesses a unique solution and to transform it into an

equivalent system of implicit ODEs. The method considers the general DAE problem

f1(t,2:1,...,2r[E“),...,:r,,,...,rff‘”) = 0
(4.52)

f.) (t,2:1, . . . ,xﬁf'”), . . . ,2:,,,...,2:,(f””)) = 0.

We recall that for a given j, the i-th equation of this system does not necessarily
depend on 2:], the derivatives 2)") for 77 < 5,], or even any of its derivatives. However,
if there is a dependence on at least one of the derivatives, including the zeroth order

derivative 56].") = 22,-, we denote the order of that highest derivative by Cij. We formally
deﬁne the v x v matrix 2 = (0.3:) by

—oo if the j-th variable doesn’t occur in f: (4.53)
01'2“ — 5,.)- otherwise.

The matrix 2 is called the signature matrix of the problem (4.52) and it forms the
foundation of the structural analysis.
We denote the set of permutations of length v by P” and consider the assignment

Problem [9] of ﬁnding a maximal transversal T E P, of the signature matrix 2.

Maximize ”T” = 23:10:2(1) With 0‘1',T(i) Z 0- (4-54)

105

Generally, an assignment problem is the task of matching v workers with v jobs. The
entries of the assignment matrix 2 measure the effectiveness of the person i for the job
j. An effectiveness of —00 indicates the inability of performing a particular task. By
requiring the entries of the transversal T to be ﬁnite, we effectively limit the problem

to the sparsity pattern

S={@ﬂbe>—my (4%)

It should be noted that this analysis of the sparsity structure can be ill-posed and the
assignment problem does not have any feasible solutions. More importantly, it has
been shown by Pantelides that the assignment problem might be ill-posed even for
solvable DAE problems. While this is one of the largest drawbacks of the structural
analysis, the method works for many important DAE problems, including DAEs of

index 0 and DAEs in Hessenberg form [108].

If a maximal transversal exists, we can consider the linear programming problem

in the variables (a), (dj) 6 Z” deﬁned by

Minimize Z = Ed,- — Z c,-, (4.56)
j=l i=1
subject to the restrictions
dj-q20ijV(i,j)€S, c,20‘v’i=1,...,v. (4.57)

While this problem does not have a unique solution, there is a uniquely determined
smallest solution [108], and the smallest (Ci), (dj) 6 Z” are called the oﬂsets of the
problem.

Once the offsets have been determined, we form the v x v system Jacobian matrix

J = (J,,-) with entries
as 458
Jr = '————.'— ( ' )

J

106

where J,, = 0 if the derivative is not present or if d,- < cg. We will later see that in the
framework of structural analysis the system Jacobian plays the role of the Jacobian

matrix (4.20) in conventional index analysis of DAEs.

Next we consider the collection of equations obtained by taking derivatives of the

f, with respect to the independent variable t:

fl“) = fl” = =- ff“) = o.

(4.59)

fl”) = fl” = ... = fl“) = 0.

By deﬁnition of 0,,- and Eqn. (4.57), the derivatives of the 2:,- that appear in the

equations (4.59) are all in the set
{25(10),2:[1),.. ”2:90,. ..,2:S,0),2:§,1),. ..,2:(1d”)} . (4.60)

If we write the set (4.60) as a vector X and collect all the equations (4.59) in the

function F, we can write the derived system as
F (t,X) = 0, (4.61)

where F and X have M = v + Zc, and N = v + Z d,- components, respectively.

Within the framework of structural analysis of DAEs, a consistent point of (4.52)
is deﬁned as a scalar to and a set of scalars C), indexed over a ﬁnite set I of indices
(3', I) such that there is a unique solution of (4.52) near t = to such that (cg-(Rte) = 35,,
However, the set I is not required to be a minimal index set, and it has been shown
that it generally will not be minimal [116]. The concept of consistent points allows
a simple success check of the structural analysis. If we view (4.61) as M algebraic
equations in N variables and assume that the system has a solution (t‘, X *), then if

the system Jacobian (4.58) is regular at the point (t‘, X ‘), it is a consistent point and

107

the system (4.52) has a unique solution near t = t“. Moreover, in a neighborhood of
(t*, X ‘) the system has D degrees of freedom, where D is the common optimal value

of (4.54) and (4.57),

U

D = urn = 2 = Zd, — 26,. (4.62)
j=1

i=1

The structural analysis also provides an upper bound for the differentiation index Vd

of the system.

0 ifalldj>0

1 otherwise. (4'63)

udgmaxq+{

While it has been suggested that the index 14, might equal the given expression,

examples in [116] show that (4.63) provides only be an upper bound.

To utilize structural analysis for the numerical integration of the DAE (4.52) we
use the subset of equations belonging to the highest derivatives in (4.59) and solve
it as an implicit ODE initial value problem. Thus, we solve the v-dimensional ODE

problem

1“" = --- = fl“) = 0 (4.64)

to ﬁnd the unique solution of the original DAE problem. Moreover, all the interme-

diate equations

(0) _ = f1(°1_1)=0

1

(4.65)

is”) = = fir-1E 0

express the obvious and hidden constraints. These equations can in principle be used
to verify initial conditions and make a posteriori adjustments to the computed ﬁnal

coordinates.

Finally, it should be noted that the structural analysis is not infallible, since it

relies on the presence of a substantial sparsity pattern. To illustrate this, consider

108

some DAE system F = 0 for which the structural analysis succeeds and the c,- are
not all zero. If we then premultiply the system by a random non-singular matrix, the
resulting DAE is equivalent to the original one. However, the signature matrix of the
new system will generally be constant in each column, since the system is unlikely to
exhibit any signiﬁcant sparsity patterns. Thus, all the offsets will be c,- = 0 and the

system Jacobian is identically singular.

4.3.2 Veriﬁed Index Analysis

The applicability of the structural analysis summarized in the previous section is of-
ten limited by its inherent limitation; namely, the E-method fails if the system has no
obvious sparsity pattern. Moreover, it has been shown that index one problems can
have arbitrary large structural and Taylor indices, rendering the method unpractical
in these cases [116]. More formally, the fact that the index set I is not necessarily

minimal, may lead to arbitrarily large indices and unnecessary computational over-

head.

Within the DA framework on the other hand, the regularity of LC in the neigh-
borhood of a consistent point is a sufﬁcient criterion for the existence and uniqueness
of solutions. Moreover, since the regularity of LC is equivalent to the regularity of
the Jacobian (4.20), this approach allows a rigorous determination of the differen-
tiation index 11.1. In practical terms, this approach is simpliﬁed by the availability
of the differentiation in the DA framework of COSY Inﬁnity. After determination
of a consistent initial velocity, a conventional DA analysis can easily determine the
differentiation index of the system, provided it is in fact smaller than the order of
the DA vectors. High order Taylor model methods can then be used to rigorously
guarantee regularity of the resulting system Jacobian and determine the existence of

a consistent point for the initial conditions.

109

Clearly this approach beneﬁts from the availability of high order derivatives in the
DA framework. However, to fully automate this method and avoid the need for hand-
coded derivatives, it requires the additional availability of the derivative operation on
Taylor models. Its utilization can completely eliminate the need for manual user
intervention from the whole algorithm, which has repeatedly been recognized as one
of the most important aspects of successful user interface design for veriﬁed integration

schemes [135, 34, 65].

4.4 Examples and Applications

To further illustrate the concept of DAEs and to show the wide range of applications
in which they arise naturally we discuss two important representatives of general
DAE problems in this section. Moreover, as an interesting application of DAEs, we
will demonstrate how prescribed path control problems can be formulated as DAE
problems. In light of the Taylor model approach the method has distinct advantages
over other existing methods for veriﬁed feedforward control problems; namely, a re-
duction in computational complexity and a broader range of applications, since it is

not limited to explicit ﬁrst order systems.

4.4.1 Kirchhoff ’3 Law and Electric Circuits

Differential algebraic equations are a natural way of modeling systems that obey
intrinsic conservation laws. Frequently these systems can be written in either Hes-
senberg Index-1 or Index-2 form. While we will encounter an index two problem in
the next section, here we illustrate the concept of DAEs with an index one example
from of electrical engineering. Consider the electric circuit shown in Fig. 4.1, which

Combines resistors, capacitors, and transistors.

110

 

 

 

 

 

 

 

 

 

 

 

Figure 4.1: Illustration of an electric circuit described by the index-1 DAE (4.69). Ub
is the operating voltage and U, is the driving voltage of the circuit.

Ohm’s Law describes the simple linear relationship between the voltage U and the

current I at a resistor of resistance R by

U = R1. (4.66)

At a transistor on the other hand, the voltage and the currents at base, emitter, and

collector are connected by nonlinear relationships of the form

IE = f(U) = 5 (BU/UP — 1) (4.67a)
IC = —GIE (4.67b)
I3 = (0l — 1) IE, (4.67c)

where U p, 02, and H are characteristic quantities of the transistor. Across a capacitor C
the connection between voltage and current is given by a simple ﬁrst order differential
equation

I = CU’. (4.68)

If we apply Kirchhoff ’s Law, which formalizes the conservation of current, to the

111

ﬁve points marked in Fig. 4.1, we get the following set of equations:

U1-—Ue

 

 

 

 

 

 

0 = — (a — 1) f (U1 —- U3) (4.6931)

R1
C(U; — Uj) = UbgzUz + “AU? — af(U1— U3) (4.6%)
0 =.- U3 ’ U0 —- f (Ul — U3) — f (U, — U3) (4.69C)

R3
0 (U1 — 04) = (a — 1) f (U. — U3) — U4 1:12 (4.69d)
0 2 U4 — Uh + Of (U4 — U3). (4.698)

R5

Thus, the response of the system to the driving voltage Ue(t) is determined by a
differential algebraic equation with index one. The system consists of four algebraic
constraints. Since the two differential equations are linearly dependent, they amount
to one differential and one algebraic equation. It is easy to see that the problem has

only one degree of freedom, since only the difference U2(0) — U4(0) can be chosen
freely.

This circuit is a typical example for a large class of engineering problems that
have conservation laws at their core. Most, if not all of these dynamical systems are

naturally described by differential algebraic equations, providing an almost inﬁnite

number of interesting and important DAE problems, many of which can and should

be solved with veriﬁed methods.

112

4.4.2 Constrained Mechanical Systems

Constrained mechanical systems can often be written as differential algebraic equa-

tions and are typically of the form

M (2:) -2:” := f (t,2:,x’) + G (2:) /\ (4.70a)

5(4) = 0 (47%)

where 2: E R” is the state vector, A 6 R‘” is the Lagrange multiplier, and <I>’ =
BQ/aq = G”. We will generally assume that M is positive-deﬁnite symmetric; so
its diagonal is non-zero. Moreover we will also assume that <I>’ has full row rank,

implying that v 2 w.

Extensive theories have been developed that give cookbook recipes for solving these
problems by using the constraint conditions (4.70b) to introduce new variables that
reduce the problem’s dimensionality [49, 61]. However, these schemes usually rely, to
a certain degree, on the user’s intuition and often require substantial arithmetic to
reorganize and simplify the resulting ODEs into explicit ﬁrst order systems. Moreover,
while the use of the constraints (4.70b) does lead to simpliﬁed ODEs, it generally does

not reveal the hidden constraints of the system.

Within the context of differential algebraic equations, the regular and the hidden
constraints are easily accessible. Moreover, the availability of automated integration
schemes for DAE initial value problems removes the need for intuition and physical
insight into the problems and allows the automated integration of these important

Systems without user intervention.

Example: Double Pendulum

As a prototypical example for constrained mechanical systems, consider a planar pair

of connected pendulums in a frictionless environment. Assume that the pendulums

113

 

 

l,

Figure 4.2: Illustration of a double pendulum with masses m1 and me connected by
massless rods of lengths l1 and [2, respectively.

are massless and inextensible, with point masses on the ends, as illustrated in Fig. 4.2.
If we denote the tensions in the strings by A1 and A2, they take the roles of the

Lagrange multipliers in this problem and the equations of motion, expressed in the

Cartesian coordinates 2:1, y1,2:2, ye, are given by

771133,; + MIDI/l1 — 42(332 " 370/12 = 0
7712in + 4191/51 - 42(92 — yll/l2 — mlg = 0
7112.72"; + /\2(.’132 — 330/12 2 0
(4.71)
mzvé' + 42(92 - y1)/lz - 0229 = 0
2:? + y? — l? = 0

(2:2 — 11:1)2 + (92 — yl)2 — 13 = 0.
While it is common practice in classical mechanics to reduce this problem to a two
dimensional ODE in the variables (t, and 452, here the focus is on treating this problem

as a differential algebraic initial value problem in the six variables 2:1, y1,2:2,y2 and

A1, Ag.

The 2 matrix of the system (4.71), with the entries forming a maximal transversal

114

in bold face, is given by

2—10—10 0(

—12—1 0 0 0

0—1 2—1—10

—-1 0 -1 2 —1 0 ' (4'72)
—1
-1)

O O —1 -I -—1
Further analysis gives the offsets c = (0, 0, 0, 0,2, 2) and d = (2,2,2,2,0,0) and

 

 

(0000—1

reveals that the system has four degrees of freedom, a structural index of three and
a differentiation index Vd of two.
Based on this analysis, the implicit ODE problem that can be derived from the
DAE (4.71) is given by
mir’,’ + Alan/l1 — A2(2:2 — 2:1)/12 = 0
”123/’1’ + Mill/l1 — 42(112 — fill/l2 — 77119 = 0
mgx’z’ + /\2(2:2 —- 2:1)/lg = 0
N (4.73)
7712312 + 42(92 — yll/lz — ng = 0
2 (2’12 + 2:12:3’ + y’12 + yiyi’) = 0

2 ((4; -- xi)’ + (y; — yl)’ + (42 — 2:1)(23’2’~— 43') + (42 — y1)(y3’-—- 3,9) = 0,

Additionally, the complete set of obvious and hidden constraint conditions for this
system is given by
2:? + y? — if = 0
($2 - 51302 + (312 — 3102 - lg = 0
(4.74)
2($13314- 3131i) = 0

2 ((422 — 2,) (2’2 — 2’1)+(yz — 91) (vi _ 91)): 0
The ﬁrst two constraints express that the lengths l1 and l2 are constant, and the

115

 

 

-o.4 ' Y I . l
-o.2~ -. ” l

.i I: :2 5‘. I~ - ..., -‘

.. .. . -. .- :. .3

.. .. .. .. .. _. "_ ‘.

.. ,. .. .. .. ;. ,4 1;; (~;

..1l .. >1 .. ,n .- , - -,
.. 1- a .. ., .. _. .. _.

. ,, ., .1.l ., .. .. .- ,:'-‘.>‘

.... 41.. :. .. _. 1; '...._‘

.. .1- .. . .. .. . '..:.,,V
., ., .~ .. .. .. _. .. '..

. ,. . .. .r .u - W!

:3.:.-:... .. 4.. ~.

a 5‘ h .. A. '.~ ‘. ~04 l

004
-v-1-‘.- 4. .. .. :..~. .4
...,;:,: .. .:.. ,,:
fifj.v'-11 l-?. 'l'. :'.
.v'. .. v ; ’ '. '-

., .‘ . ~ a.

....:.,»:. _,‘. .--. ,.‘- .,.~
.... ... _, _,‘. . . .. .,.-
111.... .-- U. ..._ _.:. .,.-
. ..,..~-.--. ,'. ‘vv.. _... .,.v
...-.~-.--. .'. -.8.. 4H -.'
..,.,.-....-.<, -, ... '....'..
.,,,,..‘..,-.-, ~, ..,4,.._-..
..,...-....;,-.

.....--,.... ~. '~ - .:.«.. ,-.

1 .. - : ' -

 

 

 

 

 

 

 

 

 

_A__P_A_ ;_L_L _A_—l g L4

0102030405060708090100 0102030405060708090100

Figure 4.3: Coordinates 2:1 and y1 of the double pendulum for 221(0) = rp2(0) =
5°,30°, 90°, 2’, (0) = 23(0) = y[(0) = y3(0) = 0 and 0 S t _<_ 100 (integrated in order
6,h=0.1,g=1, l1=12=1, m1=m2=1).

hidden constraints reveal that the velocities of the masses are tangent to the circles

on which the masses move.

To illustrate the motion of the double pendulum, Figs. 4.3 to 4.5 show results
obtained from various integrations of the double pendulum with different initial val-
ues. All plots highlight the oscillatory motion of the system with an energy transfer
between the two masses. Fig. 4.3 shows the coordinates of the upper mass ml for
various initial energies. While the system follows an almost periodic path for small

energies, it shows signs of chaotic motion for high energies.

The phase space projections of the second mass m2 shown in Fig. 4.4 indicate that
the system moves on almost periodic orbits on its oscillating orbit. Finally, Fig. 4.5
shows the time evolution of the coordinates 2:1 and y1 for the case where the upper
mass is 100 times heavier than the lower one. In this case, we see slow oscillations
superimposed with the small and fast oscillations of the lower mass. Moreover, the

evolution of y1 indicates slow but noticeable energy transfers between the two masses.

116

 

 

 

 

 

 

 

 

 

 

X2092 Yz’Y'z
2 ~ 1 ’ l
1 l 0.5 r .......... ..
.“_...:-.;;‘7$;”#"”' ii“, .3 52),”:
° ‘ or (91/
"l§?‘<;\«;;\ ' unit“:
-1 h '0.5 L ‘‘‘‘‘‘‘ “3. 2:1...”
.2 . l .1 _
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

Figure 4.4: my and yg-phase space projections for the double pendulum with for
901(0) = 092(0) = 5°,30°,45°, 221(0) = 103(0) = y{(0) = y§(0) = 0 and 0 5 t g 100
(integrated in order 6, h = 0.1, g 2 1, l1 =12 =1, m1 = m2 = 1).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

X1 Y1
0.015 f f . 0.9999 . . . . . . . n
0.01 lv 1 0.99992 U U U n n n U n
°-°°5 0.99994 1
° ' 0.99996 l ,
-0.
005 l 0.99998 + .
-0.01 l I U U U U w
1

0.015 g a 4_- e 4

 

 

0102030405060708090100

0102030405060708090100

Figure 4.5: Coordinates $1 and yl 0f the double pendulum for 901(0) = 0’ “22(0) = 450’
51(0) = 332(0) —_— y'1(0) _—_ 315(0) = O and 0 _<_ t g 100 (integrated in order 6, h = 0.1,
9:1,llzl221,m1=100,m2=1)

117

4.4.3 Optimal Prescribed Path Control

Prescribed path control problems for dynamical systems are traditionally solved by
using feedback control methods. Feedback control is based on the idea that we can let
the system operate with a pre—determined control function A for a small time interval
h and adjust the control function at the end of the time interval, depending on the
deviation of the system from its prescribed path. While feedback control is usually
easy to implement, its applicability is often limited to non—stiff problems that vary

only slowly over time.

However, it is also possible to handle prescribed path problems in feedforward
mode, where the appropriate control function is determined according to the desired
future behavior of the system. Feedforward control tends to be more accurate in
following the prescribed path with larger time steps h. Thus, feedforward control can
often result in more accurate modeling with less computational cost. If we denote the
independent variable by t E [t,-, t,+1], the state of the system by m(t) and the control

input by A(t), the general prescribed path problem can be written as
F(t,x,x', A) = 0 Vt E [t,-,t,~+1]. (4.75)
Using the Taylor model approach for the integration of implicit ODES, problems
of this form can be solved for the system’s state as a function of both the independent

variable t and the control function A. Thus, we can obtain a Taylor model with an

explicit dependence on the control function A,
m(t, A). (4.76)

Then, given a. prescribed path Mt), we deﬁne a new function, A = A(t, A), as the

difference of the solution m(t, A) and the prescribed path 112(15).
Mt, A) = m(t, A) — ¢(t)- (4.77)

118

The function A can be viewed as an implicit description of the control function A(t)
such that the system follows the prescribed path. We may assume that m(t,) = 2/2(t,-)

and that we have an initial control A(t,~) such that

(éég—Al) (t,, A(t,-)). (4.78)

is regular. Then, the Implicit Function Theorem guarantees the existence of a function

A

A(t) such that
A(t, 5\) = 0 (4.79)

in a neighborhood of ti. Since the partial inversion discussed in Sec. 3.5 allows the de-
termination of a Taylor model for the optimal control function A from this expression,
at this point the problem of veriﬁed prescribed path control can be considered solved.
However, the potentially large number of additional variables required in solving the
implicit equation (4.79) often renders this method unfeasible. With the availability
of general DAE integration tools on the other hand, we can view (4.75) as a DAE
problem and use methods for the veriﬁed integration of high order DAE initial value

problems to compute appropriate control functions for the system (4.75).

In comparison to traditional methods of prescribed path control, the virtue of
this method lies in the fact it requires only a single ODE integration for a rather
long time step. Existing methods for prescribed path control optimize the control
function through a series of repeated integrations or use extremely short time steps
with the assumption that the control function stays constant over a single integration
interval. The DAE approach on the other hand, allows the determination of a smooth
and explicit expression for the control function to very high order. Moreover, by
utilizing Taylor models at each step of the algorithm, the new method even allows the
computation of rigorous error bounds on the computed control functions, providing

the operators of the system with guaranteed and trustworthy results.

119

Example: Two-Link Robot Motion

To illustrate the formulation of prescribed path control problems as DAE initial value
problems, the following system describes a slightly simpliﬁed version of the equations
for the prescribed path control of a two-link robot arm [107]. With sufﬁciently smooth
and highly non-linear functions f1, f2, and f3, the system has a differentiation index

of 5 and is given by

x’l' = fl (t, 2:2, :33, 3’1, 223,111,112) (4.80a)
233' = f2(t,:c2,:r3,:r'1,zg,u1, U2) (4.80b)
17;: = f3(t, $2, $53,333, $3,111,112) (4.80c)
0 = cos(a:1) + cos(:l:1 + 11:3) — cos(e‘ — 1) — cos(t — 1) (4.80d)
0 = sin(xl) + sin(scl + x3) -— sin(l — e‘) — sin(l - t). (4.80e)

The prescribed path is expressed in the constraint conditions (4.80d) and (4.80c) and

requires valid solutions 3:1 and $3 to be of the form

271(1) = 1 — et ' (4.81a)

x3(t) = e‘ —t (4.81b)

4.5 Summary

We have developed a method for the veriﬁed integration of general ODE initial value
problems. It is based on a combination of high order Taylor models and the an-
tiderivation, and it allows the automatic computation of tight solution enclosures for
a large class of solvable initial value problems. The new approach avoids the wrap-
ping effect to very high orders by expanding the solution not only in the independent

variable t, but also in the transverse variables. Thus, the method computes accurate

120

enclosures of the ﬂow of ODE systems. Additionally, by using the method, or alter-
natively a more traditional structural analysis, for the index-analysis of differential
algebraic equations, it can be extended to the integration of the large and important

class of DAE initial value problems.

Finally, we point out that although we deliberately restricted the discussion to
DAE problems with constant indices, many interesting practical problems have in
fact non-constant index: valves in chemical processes may open and close, surfaces in
mechanical devices may touch. However, a change in index is usually also associated
with a non—smooth change in the equations, which is something the Taylor model
approach cannot handle well. But the method can still be used in these cases by
dynamically changing the equations between time steps and using the Taylor model
describing the ﬁnal coordinates at the end of the previous step as the initial condition

for the next step.

121

Chapter 5

Applications of High Order Taylor
Model Methods

In this chapter we present applications in which the availability of Taylor model meth-
ods allows the use of computers to obtain rigorous results. These applications require
the high order Taylor model approach for two reasons: standard ﬂoating point meth-
ods fail to provide the required mathematical rigor, and conventional interval tech-
niques fail to adequately address the problems of wrapping and dependency. Since the
systems discussed in this chapter are not necessarily available for direct experiments,

the Taylor model approach is the only method to study them rigorously.

First we study the veriﬁed integration of near-earth asteroid orbits. This problem
has long been seen as a signiﬁcant challenge for interval techniques, since it involves
large initial sets, long integration intervals, and highly non-linear equations of motion,
resulting in signiﬁcant wrapping and dependency problems. However, we will show
that the Taylor model approach can overcome all of these obstacles and return prac-
tical results. Unlike in the case of conventional ﬂoating point methods, the computed
results are guaranteed enclosures of future positions of the asteroids, accounting for

the effects of uncertain initial conditions, measured data, and mathematical trunca-

tion.

122

In a second application we use high order Taylor models to prove the existence of
generating functions for Hamiltonian systems over extended domains. The approach
combines the mathematical rigor of the invertibility tests discussed in Sec. 3.1 with
Taylor model based integration of initial value problems. This novel combination pro-
vides the rigorous justiﬁcation for the method of symplectic tracking of Hamiltonian
systems, which has important applications in the study and the design of particle

accelerators.

5.1 Solar System Dynamics and Asteroid Orbits

Spurred by recent media attention, the possibility and implications of asteroid im-
pacts on Earth have recently come to the public’s attention. Indeed, the potentially
devastating consequences of such events justify research in preventing them, including
possibly by altering the orbits of near-earth asteroids on collision courses. But pro-
tective measures cannot be taken unless near-earth asteroids are detected and their
orbits are predicted. Moreover, for an increased response time it is essential to detect
hazardous asteroids as early as possible and perform long-term integration of orbits
from known observational data. Since the results of these predictions will potentially
justify substantial preventive efforts, it is of utmost importance to actually get them
right. Thus, the integration of asteroid orbits from initial measurements is an obvious
application of veriﬁed ordinary differential equation (ODE) integration methods. Due
to the computationally challenging models and the large uncertainties, however, so
far all such computations have been performed using conventional high order ﬂoating

point integration schemes with subsequent non-veriﬁed error analysis [73].

The integration of the motion of asteroids in the solar system poses several chal-

lenges to veriﬁed integration methods, mostly due to the large uncertainties in the

123

initial conditions, measured physical constants (e.g. the gravitational constant is one
the least accurately known constants [62]), and strong non-linearity in the mathemat-
ical models. Therefore, veriﬁed tools have not found widespread application for these
problems. However, due to the potential seriousness of the outcome of the integration,

they are important applications and test cases of veriﬁed methods [90, 135].

The major problem of integrating solar system dynamics lies in the relatively large
uncertainties in the positions and velocities of the objects involved. This problem
particularly hurts conventional veriﬁed integration schemes that are prone to the
wrapping effect discussed in Sec. 2.2.2. It leads to a substantial overestimation on
the range of ﬁnal positions and velocities, and this overestimation often results in
situations that do not permit reasonable conclusions, since the bounds are overly
pessimistic. The other main obstacle to veriﬁed integration of solar system dynamics
lies in the strong non-linearity of the equations of motion (5.3). Together with the
uncertainties in the positions and velocities, this lets conventional veriﬁed integration
suffer from cancellation problems and makes it almost impossible to obtain usable

long-term integration results.

We will demonstrate, however, that the use of Taylor model based ODE integration
methods [83, 85] can successfully alleviate these problems. In fact, the Taylor model
approach can indeed be used to perform veriﬁed long-term integration of asteroid
orbits in the solar system. We will demonstrate the performance of the method with
several examples obtained from orbit integrations of the asteroid 1997 XF11, which is

predicted to have several nearby passages of Earth between the years 2000 and 2040.

5.1.1 Physical Background

Before we discuss any details of rigorous asteroid integrations, in this section we

summarize the physical principles that govern the motion of planets and asteroids in

124

the solar system. We give an overview of the solar system and present an accurate
formulation of the equations of motion for a point mass in it. Some of the notation
introduced deviate from the standard conventions used in the International System

of Units (SI) and will be used throughout the text.
Physical Units

The basic physical quantities relevant for the mathematical description of planetary
motion are length, time, and mass. They are usually expressed in terms of the SI

units Meter, Second, and Kilogram.

 

Quantity SI Unit Symbol

Length Meter In
Time Second 5
Mass Kilogram kg

 

 

 

Table 5.1: SI units for length, time, and mass.

For practical calculations it is sometimes convenient to use different units than
the ones listed in Tab. 5.1, mostly to avoid excessively large or small values of phys-
ical quantities. The following list gives a short description of the commonly used

astronomical units of time, mass, and length:

0 The astronomical unit of time is the time interval of one Earth-day; an interval

of 36525 days is called a Julian century.
0 The astronomical unit of mass is the mass of the Sun.

0 The astronomical unit of length (AU) is the radius of a circular orbit in which a
body of negligible mass, and free of perturbations, would revolve around the Sun
in 365.25692 days, and its value is almost equal to the mean distance between

the Earth and the Sun [125].

125

The rules required for a conversion of these units into the SI system are given in
Tab. 5.2. Several other physical quantities, namely the velocity, the acceleration,
and the force), are expressed in derived units; Tab. 5.3 lists them in terms of the

fundamental quantities length, mass, and time.

 

 

Quantity Unit Symbols Conversion
Time Day d 1 d = 86400 3
Julian Year J 1 J = 365.25 (1
Mass Sun Mass M 1 M = 1.9891 x 1030 kg
Length Astronomical Unit AU 1 AU 2 1.496 x 1011 m

 

Table 5.2: Astronomical units and their conversions to SI units.

 

Quantity Unit

Velocity Length/ Time
Acceleration Length/Time2

Force Length x Mass/Time2

 

 

 

Table 5.3: Units of the derived quantities velocity, acceleration, and force.

Astronomical calculations also make frequent use of angles, which are commonly
measured in degrees (360 deg is equivalent to a full circle) or radians (27r rad E
360 deg). Finally, practical computations often require the use of ad hoc units to
satisfy requirements of the algorithms used, and these units will be introduced as

needed.
Gravitation

Gravitation is an attractive force between masses, and for the purpose of gravitational
interactions in the solar system, all masses can be treated as point masses; i.e., as if
their mass was concentrated in the center of mass. Classically, the force inﬂicted by

a point mass mg on a mass m1 acts along the line connecting the two points and is

126

given by

 

(5.1)

where G denotes the gravitational constant and F12 is the vector connecting ml with
m2. Eqn. (5.1) is due to Newton, who derived it from observational data of planetary
motion and Kepler’s Laws. Since E12 2 -—F‘21, gravitation satisﬁes Newton’s Third

Law: actio = reactio.

Predicting the motion of two point masses m1 and m2 under their mutual New-
tonian gravitational inﬂuence, neglecting all other forces and interactions, is known
as the two-body problem and has been studied extensively. It can be shown that the
barycenter of the two masses moves with constant velocity and that the two masses
move on one of the conic sections circles, ellipses, parabolas, or hyperbolas with the
common barycenter in the focus [49, 61]. Consequently, in a barycentric coordinate
system, the two masses move on planar conic sections around the origin. Details of

the motion on conic sections will be discussed in Sec. 5.1.2.

Within the framework of classical celestial mechanics, the superposition of the
gravitational force is linear. Hence, the total force ET on a mass M induced by N

other masses m,- is merely the vector sum of the individual forces 13,-z

N
_‘i M 3 -0
FT = 21:14}: 2:1|:G|TTUL—3T. (5.2)

A more general and more complete model for the force on a point mass under the

inﬂuence of other masses will be discussed in Sec. 5.1.1.

The Solar System

The solar system is the dynamical system consisting of the objects that are conﬁned

to orbits around the Sun. The largest members of the solar system are the Sun,

127

the nine major planets, some of which are orbited by moons, and a large number of

asteroids and comets.

Fig. 5.1 shows the orbits of the four innermost planets: Mercury, Venus, Earth,
and Mars. As shown in Tab. A.1, the orbital ellipse of Mercury has a noticeable ec-

centricity. Fig. 5.2 shows the orbits of all nine major planets; the m/y-plane coincides
Y

a ,
ye

Figure 5.1: Orbits of the inner planets Mercury, Venus, Earth, and Mars around the
Sun.

 

with the ecliptic; i.e., the plane of Earth’s orbit, of the year 2000. The up-to-scale
inclusion of the orbits of the inner planets illustrates the size of the solar system:
Pluto’s perihelion is almost 50 astronomical units away from the Sun, and it takes

the light from the Sun almost seven hours to reach Pluto.

The Sun is by far the heaviest and largest object in the solar system, as its mass
of approximately 1.9891 x 1030 kg exceeds the masses of the other planets by at
least three orders of magnitude. As an important consequence of this dominance, the
maximal distance between the solar system barycenter and the Sun’s center of mass is
only 1.5 x 109 m, or about twice the radius of the Sun. For most practical purposes it
is therefore sufﬁcient to assume that all planets orbit the ﬁxed Sun. Tab. 5.4 lists the
masses of the nine major planets with their moons as fractions of the Sun mass, their

mean distance to the Sun, and the average times needed for one revolution around

128

\t ...)

 

Figure 5.2: Orbits of the outer planets Jupiter, Saturn, Uranus, Neptune, and Pluto
around the Sun.

the Sun. It should be noted that, according to Kepler’s Third Law, the period T and

the mean distances a are related via T = am.

 

Planet Mass (m) Dist. a a3/2 Time T

Mercury 1 / 6023600 0.39 0.2436 0.240
Venus 1/408523.5 0.72 0.6109 0.615
Earth 1/328900.5 1.00 1.0000 0.999
Mars 1 / 3098710 1.52 1.8739 1.881
Jupiter 1/ 1047.355 5.20 11.858 11.856
Saturn 1/ 3498.5 9.54 29.466 29.424
Uranus 1 / 22869 19.19 84.064 83.747
Neptune 1 / 19314 30.07 164.89 163.723
Pluto 1/ 3000000 39.48 248.07 248.021

 

 

 

Table 5.4: Masses, mean distances, and periods of the major planets. Masses are
given as fractions of the Sun mass M, mean distances in astronomical units, and
times in Julian years [125].

In addition to the Sun, the nine major planets and their moons, the solar system
contains numerous other small objects, mostly asteroids and comets, the largest of
which are commonly referred to as minor planets [134]. The sixteen largest asteroids

in the solar-system have diameters between 240 km and 1000 km and are part of

129

the main asteroid belt, which consists of several thousand objects of non-negligible
mass and size and is located between the planets Earth and Mars. During past space
craft missions, the asteroids Ceres, Pallas, Vesta, Iris, and Bamberga, all of which
are orbiting the Sun in the main asteroid belt, have been found to have a noticeable
gravitational effect on the motion of objects in the Earth-Mars range [125]. Hence,
an accurate formulation of the equations of motion of objects on near-earth orbits has
to include their gravitational effects. We will revisit this when we discuss an accurate
mathematical model for the motion of near-earth asteroids in Sec. 5.1.1. Finally, it
should be noted that the term asteroid belt is misleading, since it is by no means a
dense object. In fact, several spacecrafts have passed it undamaged and the mean
distance between non-negligible objects in the main asteroid belt is several million

kilometers.
The Equations of Motion

In the previous sections we have given an overview of the solar system and the classical
description of gravitation. Now we present an accurate mathematical model [139, 125,
44, 76] describing the motion of point masses in a solar system barycentric coordinate
system by a second order differential equation. The model includes Newtonian effects
of the Sun and the major and minor planets and relativistic corrections to the classical

Newtonian description of gravitation of the Sun and the major planets.

r}, 73,-, and if,- denote the solar-system barycentric position, velocity, and accel-
eration of the j-th major planet, where j = 0 denotes the Sun. 7 and B are the
post-Newtonian parameters measuring space curvature produced by the unit rest
mass and nonlinearity in the superposition of gravity; in accordance with general
relativity, both are assumed to be one. The masses have been combined with the

gravitational constant into the coeﬂicients u,- = G'mj, and the position of the point

130

mass is given by 7'". The distances between point mass and major planets are expressed
by r,» = Hr —- 773]]2, the respective velocities are denoted by v, = H153”2 and v =2 “F”,
N is the number of minor planets that are included in the computation and c is the

velocity of light.

 

9 N
3 4 u .1:—
+ + 7 afﬁxing? (5.3)

This differential equation shows that the force on a point mass is essentially given
by the Newton force inﬂicted by all other major objects in the solar system and
some correction terms in the order of 1/c2. Although Eqn. (5.3) gives only a partial
description of the acceleration exerted on a point mass in the solar system, it provides
a highly accurate description of the motion, and it has been used for the preparation

of the most accurate ephemeris system so far [125].

5.1.2 Geometric Description of Kepler Orbits

As mentioned earlier, the Kepler problem of predicting the motion of two point masses
under their mutual Newtonian gravitation permits only conic sections as orbits, with
the barycenter being the focus. The orbits lie in planes that are perpendicular to the

SYstem’s angular momentum and pass through the barycenter. In general, the shape

131

of conic sections is determined by the polar equation, which relates distance to the

barycenter r and the angle in the plane of motion «p via
r (1 + scos((,0)) = a (1 -— 52). (5.4)

Depending on the value of the eccentricity e, the orbit is given by one of the following

geometric ﬁgures:

 

Eccentricity Orbit

 

 

s = 0 Circle
0 < s < 1 Ellipse
e = 1 Parabola

s > 1 Hyperbola

 

Table 5.5: Geometric shape of conic sections, depending on the eccentricity 5.

In this section we summarize how Kepler ellipses can be described in terms of
orbital elements and how this description can be used to compute the positions and
velocities of planets in a barycentric cartesian coordinate system whose :r/y-plane

coincides with the ecliptic of the Earth.
Orbital Elements

The shape and orientation of Kepler ellipses in the solar system can be described un—
ambiguously by six parameters, the so-called orbital elements. While several equiva-
lent choices for these parameters exist, the following list shows one particular selection

that signiﬁcantly simpliﬁes the subsequent computations:

O Longitude of ascending node (2: The angle, along the ecliptic, from the vernal
point to the ascending node, which is the intersection between the orbit and

the ecliptic, where the planet moves from south of to north of the ecliptic; i.e.,

from negative to positive latitudes.

132

o Inclination i: The tilt of the orbit plane relative to the ecliptic.

Argument of perihelion w: The angle from the ascending node to the perihelion,

along the orbit.

Semi-major axis a: The mean distance to the Sun.

Eccentricity e: The parameter describing the shape of the orbit (c.f. Tab. 5.5).

Mean anomaly M: The product of the mean motion of the orbiting body and

the time since the body passed the pericenter at some speciﬁc point in time.

The mean anomaly is best described if one imagines a small circle within the orbit,
and centered on the focus f. Then M is the angle between the perihelion P, the focus
f, and the position the planet would occupy if it was traveling at a constant angular
speed. While the last three elements describe the orbit ellipse themselves, (2, i, and
w describe the orientation of this ellipse in space. Other equivalent choices of the
orbital elements include the mean longitude L = M + w + Q, the time at perihelion,

and the mean daily motion 17.. For reference, Appendix A lists the orbital elements

and daily rates of change of the major planets.

Cartesian Coordinates

Since the equations of motion (5.3) describing the acceleration of a point mass in the
solar system are only valid in a cartesian system of inertia, for practical computations
it is usually necessary to convert orbital elements into cartesian coordinates. This
transformation is based on the eccentric anomaly E, which is due to Kepler and

connects the cartesian coordinates of a planet with its orbital elements.

Consider a large circle of radius equal to the semi-major axis of the orbit. Let the

center of this circle be at c. Now, on the diameter of the circle which passes through

133

the long axis of the ellipse, raise a perpendicular which meets the planet at r on the
ellipse. Continue this perpendicular until it crosses the large circle and call the point
where the perpendicular crosses the large circle Q. Then the angle E is called the

eccentric anomaly (c.f. Fig. 5.3). The mean and eccentric anomalies are related by

 

 

Figure 5.3: Illustration of the eccentric anomaly E

the implicit equation

MzE—esmm. am

If the eccentricity 5 is small, a Newton method can easily ﬁnd E for given values
of M and 6. Once E has been determined, the cartesian positions F = (51:, y, z) and

velocities i" = (is, g, z") are computed from the following equations [118, 93]:

G(M+m) 1

n = a (l — scos(E)) (5.6a)

 

a = a (cos(E) — 5) (5.6b)
ﬂ = a\/1 — 52 sin(E) (5.6c)
fy = —nsin(E) (5.6d)

6 = nx/l — 52 cos(E) (5-59)

134

p. = cos(w) cosm) —- sin(a) sin(w) cos(i)
p, = cos(w) sin(Q) + cos(fl) sin(w) cos(i)
p, = sin(w) sin(i)

q. = — sin(w) cosm) — sinm) WM 0080‘)
q, = — sin(w) sin(a) + cos(fl) cos(w) cos(i)

Q, = cos(w) sin(i)

x=a~px+59x
y=a-py+6-qy
z=a-pz+ﬁ-qz
i=7-pz+6-qx
UZV‘Py'l’a'Qy

2:7'pz'l'6‘Qz

(5.7a)
(5.71.)
(5.7c)
(5.7a)
(5.7e)

(5.70

(5.8a)
(5.8b)
(5.8c)
(5.8a)
(5.8e)

(5.8f)

The origin of this coordinate system coincides with the barycenter of the combined

two-body system of the Sun and planet, but the coordinates can easily be transformed

into a. solar system barycentric coordinate system by translation.

Planetary Motion and Perturbations

For the two-body problem, the six orbital elements are constants and are determined

uniquely by the six initial conditions of the position i" and the velocity i". Due to the

overwhelming mass of the Sun and the large distances between the planets, even in

the presence of other planets, the orbits of the planets are very close to being ellipses.

Minor corrections to the two-body Kepler ellipses of the planets are due to:

o inﬂuences from planets other than the Sun,

135

o deviations of the force from the simple inverse square law (relativistic correc-

tions) .

It is convenient to treat the orbits as instantaneous ellipses whose parameters are
deﬁned by the instantaneous values of the position and velocity vectors, since for small
perturbations the orbits are approximately ellipses. However, these perturbations
cause the six formerly constant parameters to vary slowly, and the instantaneous
perturbed orbits are called osculating ellipses: the osculating ellipses are the elliptical
orbits that would be assumed by the bodies if all the perturbing forces and corrections

were suddenly turned off.

This perturbation theoretical approach of describing planet orbits relative to
slowly varying elliptical reference orbits has proven itself extremely successful in parti-
cle optical systems [17, 87], and based on the arbitrary order approach of COSY Inﬁn-
ity [23, 26], sufficiently accurate smooth description of planet orbits can be computed
even for long-term integrations. Moreover, in light of the Taylor model approach, this
strategy has the particular advantage that the orbital elements of planets are almost
linear functions of time, which eases the description by Taylor models and increases

the accuracy of the enclosures.

5.1.3 Veriﬁed Orbit Integration

In the previous sections we summarized the concepts that govern the motion of masses
in the solar system and presented the principle of describing nearly closed orbits by
perturbations to osculating ellipses. Here we show how these concepts can be used for
the veriﬁed integration of asteroid orbits. Since this involves highly non-linear equa-
tions, large sets of initial conditions, and substantial uncertainties in the mathematical

description, we have found conventional veriﬁed integration schemes [79, 99] unable

136

to handle the problem of creating a guaranteed ephemeris for asteroids in the solar
system. As indicated earlier however, Taylor model based integration schemes [83, 85]
can offer accurate long-term veriﬁcation even under these challenging conditions. To
illustrate this, we have based the integration on Taylor model methods, Eqn. (5.3),

and the ephemeris DE405 [130].

The basic idea of the integration is that the presence of a relatively small asteroid
will not change the orbits described by the DE405 ephemeris. While this is clearly
only an approximation, due to the mass ratio between the planets and asteroids and
due to the relatively large distances separating them, the deviations from this approx-
imation are in fact extremely small and will be accounted for through general error
bounds on the planets’ positions and velocities. Thus, instead of integrating the full
(60 + 3 x N )-dimensional system involving the Sun, the nine major planets, and the
N minor planets as Eqn. (5.3) would suggest, we use osculating ellipses together with
the corresponding errors of the ephemeris to obtain Taylor models for the positions
and velocities of the planets. These Taylor models describing positions and velocities
of the planets and interval enclosures for effects not described by Eqn. (5.3) are then
used for a Taylor model based integration of the six-dimensional equations of motions

for the asteroid alone.

This particular approach allows a substantial reduction of the problem’s dimen-
sionality and makes its veriﬁed integration feasible. Utilizing high-order Taylor mod-
els limits the wrapping effect and enables us to obtain long-term predictions, even for
large sets of initial conditions. Moreover, the approach does not lead to a dramatic
inﬂation in the size of enclosures of the ﬁnal coordinates. In the remainder of this
section we present a detailed discussion of how the veriﬁed integration of Eqn. (5.3)

can be achieved and results will be presented in Sec. 5.1.4.

137

Accuracy of the Ephemeris

The mathematical models and concepts presented so far allow the computation of
accurate planet orbits. For the computation of the ephemeris DE405 [125, 130],
which has been used to compute the motion of asteroids, rockets, and satellites in the

solar system [142], the following effects have also been included:

0 Newtonian gravitation of the Sun and the major planets
o relativistic effects of orders up to 1 / 62

o Newtonian effects of the 300 largest minor planets

Since planets are not rigid spheres with a uniform mass density, an accurate inte-
gration includes the deviations from the simple point-mass concept. The following

factors have been included in the preparation of the DE405/LE405 ephemeris:

o the difference between geometric centers and barycenters (ﬁgure effects)
0 Earth tide effects

0 physical librations of moons

Finally, the results of the numerical integration have been adjusted to past measure-

ments. Measurements are, among other things, based on the following techniques:

0 meridian transits (i.e., optical triangulation),
0 satellite astronomy,
0 radar ranging to planet surfaces (including surface models),

0 radar ranging to spacecrafts,

138

0 laser ranging to lunar reﬂectors.

Residuals of the ephemeris DE405/LE405 have been presented in [130] and it has
been shown that the ephemeris DE405 is indeed suitable to accurately predict the
positions and velocities of massive objects in the solar system [142]. Since Jupiter has
the second largest mass of all objects in the solar system, its positional errors translate
into potentially large errors in the integration results of asteroids and its residuals with
respect to DE405 are therefore of great importance. Fig. 5.4 shows these residuals
with respect to DE405, obtained via Very Long Baseline Interferometry observations

from the Galileo spacecraft.

VLBI residuals of Jupiter w.r.t. DE405

0.05

 

 

 

 

 

r *— ﬁ— *ﬁ 1— f v— r“ 1 mm f ﬁ—ﬁ— ﬁ— ﬁ— ﬁ ﬁr —'
.. J, l
'3 ] [ ] l l
g o Mﬁgﬁ f_‘L1 M] l r; rd j1_ hf Ir— L[‘ll ﬂ M4 f LL
L i l T l [ U1 11 l ’
. 1
8+ _L l M M l M l M 1* l
‘? 96 1996.5 1997 1997.5 1998

Figure 5.4: Residuals of Jupiter; two Very Long Baseline Interferometry observations
from the Galileo spacecraft; courtesy of [130].

The residuals shown in Fig. 5.4 translate into positional uncertainties of only
about 100 km. Combining this with even higher accuracies for the inner planets
(the ephemeris system DE405 is believed to be accurate to 0.001 arc seconds [130]),
it is possible to predict the positions and velocities of the Sun and the major and
minor planets quite accurately for the next 50 to 100 years. As a result, the data
of DE405 can be used to obtain time-dependent descriptions of the orbital elements.
These can then be used to compute Taylor model descriptions of the barycentric
cartesian coordinates of the planets that are sufﬁciently accurate for veriﬁed long-

term integrations. Consequently, Taylor model integration schemes can be used to

139

obtain veriﬁed long-term results for the motion of small objects in the solar system.

In addition to the effects included in Eqn. (5.3), our integration also considers the
acceleration due to the rotation of the solar system barycenter around the galaxy;
i.e., the deviation of the solar system from an inertial frame of reference. While this
contribution amounts only to about 5-10"8 [7], it is in the same order as the relativistic
corrections and can therefore not be neglected. By determining the direction of this
acceleration, we include it in the accurate model of determining the acceleration on

a point mass in the solar system given by Eqn. (5.3).

The center of our galaxy is believed to coincide with Sgr A * and its ecliptic latitude

ﬂ and longitude A at the year 2000 are given by [127]:

s = 266.85 deg, (5.9)

A = -5.61 deg. (5.10)

Translating these into cartesian coordinates allows the determination of a vector de-
scribing the acceleration of objects in the solar system due to the rotation of the solar
system barycenter around the center of the galaxy. Since this vector varies over time,
and since its direction and length are based on measurements, it is necessary to in-
clude error bounds on it. However, since these errors are several orders of magnitude

smaller than other errors, they do not challenge the rigorous integration.

Additional Effects

Traditional numerical integration schemes, as used in the preparation of the ephemeris
DE405, can only include quantiﬁable eﬂects. Moreover, in the framework of tradi-
tional ﬂoating point integration schemes it is useless to account for quantities that
have smaller effects on the result than the known uncertainties in other more in-

ﬂuential parameters. However, some of these omitted effects are actually indirectly

140

accounted for, since the results of the ephemeris have been ﬁtted to observational
data by modifying the underlying physics [125]. The most important of the omitted

forces in the preparation of the ephemeris DE405 and Eqn. (5.3) are:

relativistic effects of orders higher than 1/62,

relativistic corrections for the minor planets,

non-gravitational forces (e. g. friction),

static background forces.

Veriﬁed integration methods, however, cannot ignore these contributions, regardless of
their actual inﬂuence. The easiest, and usually sufﬁciently accurate, way of including
them in the mathematical models is to add error intervals to the right hand side of

the equations of motion (5.3).

Taylor Models for Orbits

To obtain an enclosure for the acceleration of a point mass in the presence of the Sun
and the planets, Eqn. (5.3) requires veriﬁed descriptions of the positions, velocities,
and accelerations of the Sun and planets. As outlined before, we obtain these Taylor
models from the time dependent descriptions of the osculating ellipses. The orbital
elements of the planets are slowly time-dependent functions with a dominating linear
part. As such they can easily be modeled within the differential algebraic framework
presented in Sec. 2.1. Within the differential algebraic framework of COSY Inﬁnity,
these polynomials can be carried through Equations (5.6 — 5.8) to obtain time de-
pendent polynomials describing the positions and velocities of the individual planets

and the Sun. Taylor models for the cartesian coordinates are then constructed by

141

utilizing bounds obtained from the residual analysis of the DE405 ephemeris [130]

and an accuracy analysis of the polynomial descriptions.

It is important to underscore that we currently do not have long-term data for the
orbital elements for which the described procedure yields polynomial descriptions that
are equivalent in accuracy to DE405. However, the system provides the framework
necessary to include time dependent descriptions of the orbital elements that allow
the computation of such orbits. Currently, the implementation assumes a linear time-
dependence of the orbital elements as shown in Appendix A. Thus, while the results
presented in Sec. 5.1.4 are not veriﬁed with respect to the actual solar system, they
are consistent and accurate within the given description of the orbits. Moreover, our
results indicate that a more accurate description of the orbital elements will neither

degrade the performance nor the sharpness of our integration method.

Initial Conditions and Observational Data

A major challenge for veriﬁed integration of asteroid orbits is the problem of obtain-
ing guaranteed sets of initial conditions for the asteroid at some time. Since actual
measurements are the main source for initial conditions, they do not have rigorous
error bounds, but come with conﬁdence intervals instead. This is based on the as-
sumption that, in the absence of systematic errors, the measurements have a Gaussian
distribution around the correct value. Hence, the quantity that describes the width
of the conﬁdence interval most naturally is a, the parameter describing the width of
the Gaussian curve. While this description leaves the possibility of undetected sys-
tematic errors, measurements are frequently conﬁrmed using independent methods.
Thus, it is usually appropriate to assume that initial conditions obtained from a 30

neighborhood are good enclosures of the actual initial conditions.

Due to the numerous measurements of the inner planets, their positions and ve-

142

locities are known to a very high degree of accuracy. For asteroids on the other hand,
often only a few measurements are available. However, if the asteroid ever comes close
to Earth, high accuracy measurements using radar Doppler effects and radar delays

will yield accurate results with small uncertainties for the initial conditions [142].

In the computation of the results shown in Sec. 5.1.4, we assumed that the ini-
tial positions and velocities of the asteroid of interest can be enclosed in cartesian
boxes of approximately i,75 km and 21:2 . 10‘5 km/s. Since current sigma position
uncertainties for near Earth asteroids tend to be between 10 km and 103 km [142],
obtaining enclosures this sharp poses a signiﬁcant challenge for astronomers. How-
ever, the actual results shown in Fig. 5.6 indicate that the Taylor model methods
can successfully handle initial condition sets as wide as 10"5 AU, or approximately
1500 km. Thus, for the performance of the actual integration with algorithms based

on Taylor models, the width of the enclosure boxes is not a signiﬁcant factor.

5.1.4 Results and Discussion

After combining the theory of Taylor model based ODE integration with the detailed
model of the underlying physical concepts presented earlier, we have performed several
veriﬁed integrations of asteroid orbits using Taylor model methods. Here, we show
results from these integrations to illustrate the performance and applicability of these

integration tools.

All computations presented in this section are based on special units for the dis-

tance and time that aid the integration process:

0 Time is measured in time units (TU), where one time unit equals 365.25/27r z

58.131 days.

0 Distance is measured in astronomical units (AU).

143

Since the Earth moves on an almost circular orbit, it travels approximately 2n astro-
nomical units in 271’ time units. Hence, in these units the velocity of the Earth on its
orbit is approximately v = (21r AU) / (21r TU) 2: 1 AU / TU. Since in these units both
the distance of the Earth to the coordinate center and its orbital velocity are close
to unity, all components of the right hand side of the differential equation describ-
ing the asteroid’s orbit are of similar magnitude. This simpliﬁes the error analysis
of the integration tools, since it removes the necessity of weighting the errors of the
individual components against each other. Moreover, this aids the step size control
of the integrators, since it turns the potentially stiff system, with different step sizes

for different components of the system, into a non-stiff one.
Integration Results for 1997 XF11

In this section we present results obtained from integrations of the near-Earth aster-
oid 1997 XF11, which has a high eccentricity of e = 0.48, low-inclination orbit with
i = 4.10, and a diameter of several kilometers. It orbits the Sun with a period of
approximately 1.7 years. Since one of the two intersection points of its orbit with the
ecliptic is in the vicinity of the Earth’s orbit ellipse, there is a non-vanishing probabil-
ity for close encounters between the asteroid and Earth. Tab. 5.6 lists predictions for
the close encounters between the years 2000 and 2040: the closest approach distance
is predicted to be 0.006 astronomical units in the year 2028. While this is about
one quarter of the distance between the Earth and the Moon, it is close enough to
warrant further research into the future orbit of the asteroid 1997 XF11. In fact,
1997 XF11 is an important test case for veriﬁed solar system integrations, since a

successful integration would be able to verify the non-impact in the year 2028.

The integrations discussed in this section are based on initial conditions obtained

from the HORIZONS system [128]. As indicated in Sec. 5.1.3, we assumed the initial

144

conditions to be in boxes of 10“7 and 10“6 for the positions and velocities. The exact

position r?) and velocity r?) at the initial time to are given by:

to = JD2450465.5 z January 17,1997 (5.11)
as e (—1.772691...,+0.14s722...,—0.0792s4...)4.0.5-(lo-7,104,104)

r0 6 (+0.237203...,—0.612525...,+0.045832...)$0.5-(104,104,104).

The cartesian positions and velocities of the asteroid 1997 XF11 during the integration
interval starting at to are shown in Fig. 5.5 and were obtained from the relativistic
integration discussed in Sec. 5.1.4. Since this problem is dominated by the two-body
interaction between the Sun and the asteroid, the evolution of the coordinates exhibits

the periodic behavior of elliptical orbits with high eccentricity.

In the remainder of this section we present results obtained from two different
integrations of 1997 XFll’s orbit. The ﬁrst integration uses Eqn. (5.3), while the
mathematical model of the second integration does not include the relativistic cor-
rections to the classical Newton gravitation. As such, the two integrations allow us
to measure the actual inﬂuence of the correction terms on the asteroid’s ﬁnal coordi-
nates, which gives an estimate on the computational cost and ﬁnal beneﬁt associated

with these corrections.

Integration with Relativistic Corrections

In this section we present results from a ten year integration of the asteroid 1997 XF11,
using Eqn. (5.3) with relativistic corrections and initial conditions boxes given by
Eqn. (5.11). The asteroid’s positions obtained from this integration are shown in
Fig. 5.5. As indicated earlier, the results demonstrate that the asteroid is moving on
an almost elliptical orbit around the Sun. More interestingly though, Fig. 5.6 gives

the size of the enclosures for the coordinate a: and the velocity i2. It is important

145

 

 

 

Position x E
Position Y
2 _ -------------- Fashion 2
"I, “\,\ :/'\\ l, “A: "I,
1 . .' ' ' l f ’- "f
. 1‘. .‘
l‘ [

o .1 .................... .7 ......... .......... . . ’

:1 ,
-1 l " I

 

 

 

1 997 1 999 2001 2003 2005 2007

 

Velocity v?
.......... Velocity v

15 ~ -------------- Velocity v;

 

 

 

 

     

 

 

MM

-1 4 . .
1 997 1 999 2001 2003 2005 2007

 

Figure 5.5: Cartesian positions (in AU) and velocities (in AU/TU) of 1997 XF11
during the ten year integration interval.

to note that the sizes of these enclosures do not just grow monotonically with time,
but even decreases at certain points. Moreover, the enclosures have a very small
average growth rate, which is the main reason for the ability of the Taylor model

based integrator to propagate initial conditions over large time intervals.

Tab. 5.6 lists the closest approach distance between 1997 XF11 and the Earth in
the year 2028 as 0.006 AU. By extrapolating the enclosures that can be obtained from
the Taylor model integration on the other hand, we observe that the enclosures at
that point will likely be smaller than 0.006 AU. Since this is sufﬁciently accurate to

guarantee a non-impact, it shows that the Taylor model approach can indeed be used

146

 

 

 

 

 

0'001 Enclosure 01x
19-04 : ‘1
16-05 r 1 1
l l
1e-06 ~
l 1
16-07 4 t 4

1997 1999 2001 2003 2005 2007

 

 

 

 

 

 

 

 

0-001 3' Enclosure?! v, f r u
l
l
l
16-04 g l
l l
l 1
l 1
16'05 .- ']
E i
r l
l l
1606 + + g a *

 

 

 

1997 1999 2001 2003 2005 2007

Figure 5.6: Logarithmic plot of the diameters of enclosures for the positions (in AU)
and velocities (in AU/TU) of 1997 XF11 during the ten year integration interval.

to obtain meaningful results from veriﬁed long-term integrations of real world systems.
In Sec. 5.1.5 we will compare this with similar results obtained from conventional

veriﬁed integration tools.

To illustrate the technical aspects of the integration method, Fig. 5.7 shows how
the automatic error and step size control of the Taylor model integration method
work in practice. The maximally allowed local error was set to 10‘9 and Fig. 5.7
shows that the integrator has been able to meet this goal by reducing the step sizes

accordingly. A comparison between Figs. 5.7 and 5.5 shows a strong correlation

147

 

Date Min. Dist. (AU)

 

October 31, 2002 0.064
June 10, 2016 0.180
October 26, 2028 0.006
June 8, 2035 0.174
November 4, 2040 0.150

 

Table 5.6: Dates and minimal distances of the predicted closest Earth approach
distances for the near-Earth asteroid 1997 XF11 between the years 2000 and 2040 [73].
between the local error and the current location of the asteroid on its orbit: the local
error tends to increase, with a corresponding reduction in the step size, whenever
the asteroid is closest to the Sun. In these regions its acceleration is at a maximum
and the asteroid moves with its largest orbital velocity. It should be noted that a
signiﬁcant contribution to the local errors stems from the uncertainties of measured
quantities and the planets’ orbits. Thus, the main source for local error can therefore
not be subject to veriﬁed error control. Consequently, the desired local error is close

to the optimum for this particular problem.

Integration without Relativistic Corrections

Here we present results of a ten year integration similar to the one discussed in
the previous section. However, this integration is based on the classical Newton
gravitation without any of the relativistic corrections included in Eqn (5.3). This
allows us to quantify the actual effects of the relativistic corrections on the computed
orbit of 1997 XF11. Hence, the analysis determines the importance of a complete
model including the relativistic corrections given by Eqn. 5.1 and puts a measure on

the computational overhead of these corrections.

For this integration, we used the same initial conditions as in the previous example.

Moreover, the automatic step size control of the Taylor model integrator chooses

148

 

r

h —— Stepsize mgr time

 

 

 

o 1 _1 1_ 1
1 997 1 999 2001 2003 2005 2007

 

1,2 . . a ﬁ
6‘09 -—-— One-step integration error

16-09 [-
86-10 F i

6e-10 - J

4e-10 l

 

26-10 [

 

 

M

1997 1999 2001 2003 2005 2007

 

Figure 5.7: Evolution of the step size and one-step integration errors during the ten
year integration of 1997 XF11.

the same step sizes as in the previous computation, indicating that the relativistic

corrections have only a small inﬂuence on the ﬁnal result.

Firstly, it should be noted that the integration without the relativistic corrections
requires only about one tenth of the CPU time necessary for the accurate integration
discussed before. Secondly, the enclosures of the computed ﬁnal positions and veloc-
ities are virtually indistinguishable from the ones shown in Fig. 5.6. The differences
between the positions and velocities computed by the corrected and the uncorrected
integrations are shown in Fig. 5.8. Since the differences are much smaller than the

sizes of the enclosure boxes, the performance of veriﬁed solar system integration could

149

39-05

 

28-05 7

19-05 -

o .

-1 e-05

1

-29-05 >

 

——-— Difference—Ax
......... -. Difference Ay
.............. Difference A2

1*

- .' - -2 J .
w::::.‘:_". I. .2- -

    

L

 

 

-3e-05

1997

39-05
2.56—05
26-05
1 .56-05
1 e-OS
56-06

0 .

-56-06
-16-05

1 997

Figure 5.8: Differences in the computed positions and velocities of the asteroid
1997 XF11 between the ten year integrations with and without relativistic correc-

tions.

be improved signiﬁcantly by ﬁnding guaranteed enclosures of the relativistic correc-
tions in Eqn. (5.3) that are easier to compute than the corrections themselves and
are not overly pessimistic. The net effect of this approach would be a relatively small
tradeoff in accuracy for a substantial gain in integration speed. However, such rigor-
ous estimates on the size of the relativistic corrections are not known and it is not

clear if they could be rigorously computed with an effort that is signiﬁcantly smaller

1999 2001 2003

2005

2007

 

r Difference sz

-—— 'DiffererTce Av,

........... Difference Avy

r

l 1 J

 

  

 

 
 

 

 

 

1999 2001 2003

2005

than the one needed to compute the corrections proper.

150

Shrink Wrapping

An important aspect of the Taylor model based integration method is the ability to
dynamically shrink the size of the remainder bounds during the integration. This is
extremely useful since it delays the exponential inﬂation of errors by several orders

in time. This dynamic reduction of errors is called shrink wrapping [28].

The main idea behind shrink wrapping is illustrated in Fig. 5.9: at certain points
during the integration, the linear part of the Taylor model M propagating the initial
conditions is enlarged, resulting in a new Taylor model M5. The polynomial part
of this new Taylor model maps the set of initial conditions to a set that properly
contains the original enclosure of the ﬁnal coordinates plus the remainder bounds. At
this point, the remainder bounds of MS can be shrunk to zero width, since the poly-
nomial map of M 5 already ensures containment of the image of the initial conditions
in the ﬁnal enclosure. However, this parts with the Taylor model approach viewing
the reference polynomial as the Taylor expansion of the ﬂow of the differential equa-
tion. On the other hand, shrink wrapping enables Taylor model based integration
methods to obtain long-term results without an overly pessimistic enclosure of the
ﬁnal positions and velocities. Since the errors of the right hand side of Eqn. (5.3) are

relatively large, shrink wrapping has been performed after each integration step in

the computation of the results presented in Sec. 5.1.4.

The shrink factor is the factor by which the linear parts of the Taylor models’

reference polynomials are enlarged. We distinguish two kinds of shrink factors:

0 The one-step shrink factor determines the amount of scaling in a single shrink

wrap Operation.

0 The accumulated shrink factor gives the amount by which the linear parts have

151

 

 

 

 

 

Figure 5.9: Illustration of the shrink wrapping method used in the veriﬁed integration
of asteroid orbits.

been scaled over the whole integration at this point.

Fig. 5.10 showsvthe values of both shrink factors, computed during the ten year
integration with relativistic corrections presented in Sec. 5.1.4. The ﬁgures illustrate
two important and somewhat surprising aspects of the shrink wrapping method: on
average, the one-step shrink factor decreases exponentially over time and the total
shrink factor grows only linearly and not exponentially. This behavior is particularly
helpful for the computation of long term results, since it prevents an exponential
inﬂation of the enclosures of the ﬁnal positions and velocities over the integration
interval. Comparisons between integrations with and without shrink wrapping show
that the availability of the method is a key ingredient of an overall strategy geared

towards veriﬁed long-term integration results [86].

152

1.0010
1.0009 ~
1.0008 ~
1.0007 -
1.0006 -
1.0005 -
1.0004
1.0003
1.0002

1 .0001 ‘ ‘ ‘
1997 1999 2001 2003 2005 2007

 

 

One-step shrink fafztor

 

 

 

 

 

3.5 .

 

Total shrink factor

 

3.0 -

2.5 r

2.0 t

1.5 *

 

 

 

_L J _1

1 .0 ‘ L ‘
1 997 1 999 2001 2003 2005 2007

 

Figure 5.10: One-step and total shrink factors obtained during the ten year integration
of the asteroid 1997 XF11.

5.1.5 Comparison with AWA

During the last decade, AWA [79, 80] has become a standard benchmark for veriﬁed
ODE integration methods. While it tends to be overly pessimistic for complicated
problems, the method works reasonably well for converging linear problems. In this
section we compare the performance of AWA with the Taylor model based integra-
tion schemes used before, and we will demonstrate that the Taylor model approach

performs favorably when compared with the conventional interval techniques used in

153

AWA.

Since AWA lacks advanced control structures which would aid the implementation
of the right hand side as given by Eqn. (5.3), and cannot handle uncertainties in the
right hand side of the differential equations very well, we have only used it for the

integration of the two-body problem of the Sun and the asteroid 1997 XF11.
2°. ’7 ..
r = ———-—-r. (5.12)
HT”3

'7 =2 0.9986 is the value of the product of gravitational constant C and the Sun mass M
in the units that have been used for all previous integrations. It should be noted that
Eqn. (5.12) represents a much simpler problem than the one previously discussed,
since it contains only one planet orbiting the Sun, no relativistic corrections, no
corrections accounting for the solar system being a rotating coordinate system, and

no uncertainties for the positions and velocities of the Sun.

As in the previous cases, the initial conditions are given by Eqn. (5.11), and AWA
has been set to integrate the system in order 18 with an initial step size of 0.0001. For
comparison, the same integration has been performed with Taylor models of order
ten and an initial step size of 0.1. In both cases the maximum local errors were set
to 10‘“. It should be noted that the model discussed in this section is an extremely
simpliﬁed version of the real problem described by Eqn. (5.3) and has therefore no
relevance for the veriﬁed integration of solar system dynamics. Also, while we would
have liked to perform the integrations over time intervals spanning more than one
year, AWA was unable to integrate the system further, since the wrapping effect led
to an exponential inﬂation of the enclosures. (The time evolution of the position and

velocity enclosures is shown in Fig. 5.11).

The results given in Fig. 5.11 show that the Taylor model approach allows the com-

putation of much sharper bounds on the ﬁnal coordinates than conventional interval

154

 

 

 

 

0.1 , Y— T ﬁ 1 '—
' + Enclosure of x , .
. x Enclosure 01y ,3 .
0.01 r - Enclosure of z #3.. ,
. , x . 1
[ +/M xx I
0.001 1' +3. x .-
* If“, I
l ’ ﬁ
16-04 E + ,th /
, + + x I
++' x '
1 6'05 1' W. x ..
l #yl" X I 1
‘4'” Mix...
l r I [
1e-06 - £39593" ,
. i ,
' l
1 6’07 :— 9 J; 1* g

1997 1997.2 1997.4 1997.6 1997.8 1998 1998.2

 

 

 

0'1 l . ' Enclosﬁre ofx‘ # (
.- : 52333329: 3
0.001 [ l
1e-04 [- [
16'05 I Minamzﬂll i
1e-06 “‘1“. i
1e-07 [ ‘I ]
1e-08 ’ + A 4

 

1997 1997.2 1997.4 1997.6 1997.8 1998 1998.2

Figure 5.11: Logarithmic plot of the diameters of the enclosures for positions obtained
by AWA (upper) and the Taylor model based integrator.

techniques, represented by AWA. At the point where AWA terminates the integration,
the Taylor model enclosures are almost four orders of magnitude sharper than the
corresponding results obtained by AWA. Moreover, at the ﬁnal point of integration,
the enclosures obtained by AWA exhibit a large exponential growth while the Taylor
model results show a much smaller growth rate and even decrease at certain times.
This smaller rate of increase gives the Taylor model methods the ability to integrate

over much larger time intervals than conventional interval based tools.

155

 

0.1

‘1 I F—

+ Enclosuré of x J
x Enclosure of y , .1! J
0-01 F - Enclosure of z :" ‘

0.001 L Wt: _' 3

. . .

18-04 7 3' i
. xi ’
. ++X

 

 

 

1e-05 :- .. [
. ”W” x 'l 1
19-06 :- .+*w W 1‘"- 1
I 5.1%}: l
1e-07 :- _‘ 1
. ' l
1e-08 ‘ + L E

1997 1997.2 1997.4 1997.6 1997.8 1998 1998.2

Figure 5.12: Logarithmic plot of the diameters of the enclosures for positions obtained
by AWA for an initial box reduced by a factor of 3.275 in each direction.

Since the Taylor model approach is much more computationally complex than
regular interval arithmetic, it is natural to ask if we could split the original problem
for AWA in such a way that the total CPU time used is the same for both AWA and
the Taylor model integration. Using AWA, the computation of the results shown in
Fig. 5.11 took 4.5 seconds of CPU time. On the same machine, the Taylor model

based integration took more than 92 minutes, or about 1234 times longer. Since
(3.275)6 z 1234,

we then split the boxes of initial conditions into boxes that are 3.275 times smaller in
each coordinate direction and ran AWA with initial conditions given by these smaller
boxes. Fig. 5.12 shows the resulting enclosures of the ﬁnal positions for the same
integration period as before. It is striking that even if AWA uses the same amount
of CPU time as the Taylor model based integrator, there is hardly any change in the
resulting size of the enclosures. Moreover, the sharpness of the results given by AWA

is still not comparable with the one computed from Taylor models.

As a ﬁnal test, we tried to determine the size of initial condition boxes for which

156

AWA can compute a ﬁnal enclosure comparable in sharpness to the Taylor model
results. However, even the smallest possible initial sets; i.e., boxes with a magnitude
of the machine epsilon of 10‘15 in each of the six coordinate directions, do not allow
the computation of ﬁnal enclosures comparable to the Taylor model results. Details
of this integration are shown Fig. 5.13 and it is important to note that even with
these tiny initial boxes, the ﬁnal enclosures are virtually indistinguishable from the
ones obtained with larger initial sets. This indicates that the main challenge for the
veriﬁed integration of this system lies in the wrapping effect, which is known to limit

the applicability of conventional interval techniques.

 

0.01 w— r ﬂ
r + Enc[osure o: x z , s
x no osureo y ,.;:. .
19'04 ' - Enclosure ofz ,W..-‘ J
l 9.13“?" 1
16-06 F “,5“? 5
' W;: :1 II. [
9"” .-
16-08 ' WW II" ~
l K .
1e-10 [ ,. l
. Ix ]
1e-12 l :i' l
. . [
19-14 l t‘ (
[ f
1e—16 " .

 

 

 

1997 1997.2 1997.4 1997.6 1997.8 1998 1998.2

Figure 5.13: Logarithmic plot of the diameters of the enclosures for positions obtained
by AWA for initial condition enclosures of 10‘”.

While normal interval techniques will almost always outperform Taylor model
methods for problems that do not require any domain splitting, it has been shown
in Sec. 3.6.1 that if domain splitting becomes necessary, the additional expense of
using Taylor models is more than compensated for by the increased accuracy. Here
we even have a problem for which the normal interval methods, represented by AWA,
cannot compete with Taylor models in terms of sharpness regardless of the amount of

domain splitting. Lastly we mention that the integration of a two—body system could

157

have been implemented more favorably for AWA, since the problem can be reduced
to four dimensions. However, the main purpose of these computations was to gauge
the applicability of conventional interval techniques to the six-dimensional problem

of solar system dynamics and not to fabricate a simple test case.

5.1.6 Summary

The results presented in Sec. 5.1.4 and Sec. 5.1.5 show that the Taylor model approach
can indeed be used for the veriﬁed integration of solar system dynamics. And it has
been shown that the Taylor model approach is capable of propagating large sets of
initial conditions of large time spans without falling prey to the wrapping effect that
often leads to signiﬁcant overestimations of the ﬁnal enclosures in veriﬁed integrations

of ODE initial value problems.

Moreover, by comparing the results from the Taylor model based integration
method with similar results obtained from tools utilizing conventional interval tech-
niques, it has to be concluded that the Taylor model approach is the only method
that allows the veriﬁed long-term integration of asteroid orbits and other real world
problems. Compared to conventional interval techniques, the Taylor model approach
gives several orders of magnitude sharper results and can propagate initial conditions

over orders of magnitude larger time spans.

5.2 Existence of Generating Functions

In this section we discuss a particularly interesting application of Taylor model meth-
ods to the theory of generating functions of canonical transformations. We show that
using high-order Taylor models enables us to rigorously establish lower bounds on

the size of the domains of deﬁnition of any type of generating function. Moreover,

158

we will see that the computed domains often enclose the dynamic apertures of the
examples studied. The results of this analysis have far reaching implications in the

study of particle accelerators and symplectic tracking.

5.2. 1 Introduction

It is well known that coordinate transformations of Hamiltonian systems that keep
the Hamiltonian structure intact are called canonical transformations, or, in more
modern terminology, symplectic maps [17, 49, 123]; the time evolutions of Hamilto-
nian systems are also symplectic. Any symplectic map can be represented in terms of
a single scalar function, called the generating function. Until recently, only the four
conventional Goldstein—type generators were well known [49]. However, following the
introduction of extended generating functions, it has been shown that to each sym-
plectic map, inﬁnitely many generating function types can be constructed [40]. In
certain applications, such as long term simulation of accelerators and other Hamilto-
nian systems, it is important to maintain symplecticity during tracking [47, 1, 40]; one
available method to achieve this is the generating function based symplectic track-

ing[14,40,138]

In principle, any of the valid generators could be used for tracking of a given
Hamiltonian system. Although it can be shown that for any globally deﬁned sym-
plectic map, global generators can always be found [40], their construction is however
usually not straightforward. Indeed, as shown in [17, 40], the representations of sym-
plectic maps through generators commonly available in practice are often only locally
valid. Frequently, the purpose of tracking simulations is to estimate the region of
space where stable particle orbits can be found, the so—called non-wandering set, or
dynamic, aperture in the beam dynamics terminology. Hence, a sizable phase space

region must be covered by tracking, and if the generating function method is used,

159

the generating function must be deﬁned at least in that region. However, so far
nothing has been known about the size of the domains of deﬁnition of generating
functions, and the necessity to study this interesting problem has been recognized

in [14, 138, 33, 40].
5.2.2 Theory of Extended Generating Functions
Following the exposition of [40], we regard every map as a column vector. Let

a = ( Z: ) (5.13)

be a diffeomorphism from the Open set U C IR” onto its image; a; and (12 are
ﬁrst and second 2w components of 0:, respectively. In other words, for i = 1,2,

a,:U-—>V,~C1R2“’. L613

A B
Jac(a) = ( C D ) (5.14)
be the 420 x 4w Jacobian of a, split into 2w x 220 blocks and deﬁne
” _ J2w 0211)
A,” ), (.15
where \
0w 1,,
= ( -1. 0w ). (5....

In the last equation, Iw denotes the unit matrix of appropriate dimension. We call a

map a conformal symplectic if its Jacobian satisﬁes
(Jac (0))T ' J4w ' Jac (a) = p ' j4w, (5.17)

with some non-zero real constant p [6]. Finally, a map M is called symplectic if its

J acobian M satisﬁes the symplectic condition [39]

MT-J-Mz J. (5.18)

160

Henceforth, we will consider only symplectic maps that are origin preserving;
i.e., maps around ﬁxed points. We call a map a gradient map if it has a symmetric
J acobian N, and it is well known that, at least over simply connected subsets, gradient
maps can be written as the gradient of a function, which is called the potential of
the map [17]. With these preliminaries, the main result of the extended generating

function theory is best expressed by the following theorem [40].

Theorem 5.1 (Existence of Extended Generating Functions). Let M be a
symplectic map and denote the identity map by I. Then for every point z in the
domain of M, there is a neighborhood of z such that M can be represented by functions

F via the relation

W..-(..(A;».(..(g»a

where a = (0:1, a2)T is any conformal symplectic map such that
det (C (M (z) ,z) - M2 + D (M (2) , 2)) 7t 0. (5.20)
Conversely, for any (32 function F the map .M deﬁned by
M = (NC — A)“1 (B — ND) (5.21)
is symplectic, where the matrices A, B, C, D, M, and N are deﬁned as above.

The function F is called the generating function of type a of M, and denoted by
Fam. Theorem 5.1 says that, once the generator type a is ﬁxed, locally there is a
one-to-one correspondence between symplectic maps and scalar functions, which are
unique up to an additive constant. The constant can be normalized to zero Without
loss of generality. Due to the fact that there exist uncountably many maps of the

form (5.17), to each symplectic map one can construct inﬁnitely many generating

161

function types. But in any case, according to Eqn. (5.19), for a given M the domain

of deﬁnition of the generator of type a equals the domain of invertibility of

(mm)-

Thus, ﬁnding the domain of deﬁnition of a generator is equivalent to ﬁnding the
domain of invertibility of (5.22). A large class of practically used generator types are
the generators associated with linear maps oz. These can be organized into equivalence

classes [S], associated with

—J L"1 J )
a = _ , (5.23)
(§(I+JS)Ll §(I—JS)
and represented by arbitrary symmetric matrices S [40] where the matrix L denotes

the linear part of M.

Finally, we note that the size of the domains of deﬁnition of the extended generat-
ing functions is directly related to the problem of optimal symplectic approximation
of Hamiltonian flows [40]. A very general theoretical argumentation based on Hofer’s
metric states that in general the generating function associated with S = 0 gives
the Optimal approximation. In the examples shown in Sec. 5.2.4 we will therefore

concentrate on this particular choice of S.

5.2.3 Enclosing Derivatives of Flows

In order to prove the existence of a generating function over extended domains
with Taylor model methods, we need to obtain Taylor models for the derivative of
Eqn. (5.22). In the case of linear maps oz, this reduces to the computation of Taylor
models for the J acobian of the map M. While simple systems may allow the explicit
determination of the symplectic map and its derivative, in the more common case of

practical problems, the exactly known information are the Hamiltonian functions, or

162

equivalently, the corresponding ordinary differential equations. Since closed form so-
lutions of these are generally not available, numerical integration schemes are utilized
to compute approximations to the mathematically correct solutions. If we use the
Taylor model integration method discusses in Sec. 2.3.3, we can enclose the mathe-
matically correct solution by a Taylor model. However, since the derivative operation
does not extend to Taylor models, we cannot use this Taylor model to compute Tay-
lor models for the derivates. But the following result from the theory of ordinary
differential equations gives us the means to compute Taylor models for the J acobians

of flows [54].

Theorem 5.2 (Smooth Dependence on Initial Conditions). Suppose that f (t, 11:)
is a 6"“ function deﬁned on an open set 52 C R x IR". Let p(t, u, y) be the unique

solution to the initial value problem
17' = f(t,x), m(u) = y. (5.24)

Then the function ¢(t, u, y) is a 6"“ function of the variables (t, u, y) and the partial
derivative of the solution 239% with respect to the spatial initial conditions satisﬁes the

matrix initial value problem

I” = (3;: (t,¢(t,u,y))) -Y, Y(u) = id. (5.25)

Thus, by using Taylor model based integration techniques to rigorously enclose

the solution of the enlarged ODE initial value problem

23' = f(t, 1:), (5.2%)
Y’ = (g (m) ~Y, (5.26b)
m(to) = x0, (526(2)
mo) = I, (5.26d)

163

we can compute Taylor models containing not only the map M(tf, to, .710) at the ﬁnal
time tf, but also the Jacobian g—ﬁfﬂbtmxo) of the ﬂow. In light of Eqn. (5.26),
the symplectic condition (5.18) loses its differential algebraic nature and becomes
a purely algebraic constraint. Thus, similar to energy conservation, the symplectic
condition expresses in fact a conservation law, which can open the door for symplectic

integration in the framework of as differential algebraic equations.

It should be noted that the augmented ODE system (5.26) is less smooth than
the original problem, resulting in solutions that are also less smooth than before.
However, since the systems of interest are commonly analytic over their domains, this
is usually not a problem. Finally, it is important to note that while the augmented
system is v(v + 1) dimensional, there is no need to propagate any dependency on the
additional initial conditions in the augmented system. Thus, the integration of the
augmented system can still be done with Taylor model in v variables, resulting in a

system that is as manageable as the original problem.

5.2.4 Examples

To illustrate how the existence of generating functions can indeed be proved with full
mathematical rigor, we will now discuss several examples of symplectic systems. We
show that the Taylor model based invertibility tests can often guarantee invertibility

of the maps over regions that properly enclose the dynamic aperture.
An Exactly Symplectic Map

As a ﬁrst example we consider a two dimensional cubic polynomial, which is exactly
symplectic and given by
M = N 0 £, (5.27)

164

where

__ cos§ sin§
Lﬂ(—sin§ cosg)’ (5'28)
and
—3((1+I))3
N(q)=(q ). 5.29
p P+3(Q+P)3 ( )

The main advantage of this relatively simple example over the more complicated
examples discussed below is it allows an analytical treatment. According to Thm. 3.2,
extended generating function exist over extended regions D C IR2 if the following
polynomial expression for the determinant in the four variables ql, p1, qg and p2 has

no zeros in D x D C R4:

[(1--\/§)ql+(1+~/®101]2
1 + 9 8(1+\/§)2 [(2+\/3)(312+1)+322] (5.30)

- 1 x/— 1
+ 9l(1 @fl:%; 3)p]2 [(2+\/-)(312-1)+ (7+4\/§)311],

 

 

where

S = ( S“ 312 ) (5.31)
812 822

is any symmetric matrix with real entries. It is easy to see that Thm. 3.2 guarantees

the existence of globally deﬁned inverses if the entries of S satisfy:
’322< (2 + f) (312 + 1) (5.32a)

2+\/§
—311$
7+ 4f

 

(312 — 1) (5.325)

Thus, we can analytically prove the existence of a large class of. globally deﬁned
generators for this symplectic map. However, while global domains of deﬁnition are of
great theoretical value, in practical applications we are interested mainly in proving
invertibility in ﬁnite regions that enclose the dynamic aperture. For these cases, the

Taylor model based methods are well suited. For example, employing the Taylor

165

 

 

 

 

 

Figure 5.14: Tracking picture of the cubic two-dimensional symplectic map, and the
box of guaranteed invertibility of the generator associated with S = 0.

method for S = 0, the tracking picture shown in Fig. 5.14 was obtained. The box
surrounding the tracking is the region for which invertibility of the corresponding
generator can be proved using the Taylor model approach. Clearly, it extends well

beyond the dynamic aperture, which is approximately 0.2.

The Fermi-Pasta-Ulam System

Since the map of the previous system was given by an exactly symplectic polynomial,
we were able to derive an analytic expression for the derivative of the map and
compute a Taylor model for the Jacobian. This Taylor model allowed us to apply
the invertibility test to prove the existence of the generating function over extended
domains. Here and in the next example we study problems in which the maps are
obtained by integrating an ODE initial value problem, requiring the application of

Thm. 5.2 to obtain Taylor models for the Jacobians of the ﬂows.

The Fermi-Pasta—Ulam system [45] is an extensively studied Hamiltonian system
which is known to exhibit many interesting features and has stimulated important

developments in nonlinear dynamics. The Fermi-Pasta-Ulam system consists of a

166

ﬁnite series of Springs; regular springs obeying a linear force law alternating with stiff

non-linear springs. The general Hamiltonian function of the system is given by

d 1 n k n n
H ((1,15) = E 2 (13327—1 + 1033') + 5 Z (921' " Q2i—1)2 + 2012:“ "' (1204- (5-33)
i=1 i=1 ':0

For this example we considered the problem with n = 2, k = 5000, and q4 = q5 :-

p4 = 0, resulting in the system shown in Fig. 5.15.

V \3
ql q2 q3
Otttouattttauo

Figure 5.15: Illustration of the Fermi-Pasta—Ulam system with three masses and four
springs.

By using the Taylor model based integration scheme, we obtained a Taylor model
for the half-period map of the ﬂow of (5.33), and then tracked several particles by
iterating this map. Typical phase space plots are shown in Fig. 5.15. The domain
of the generator associated with S = 0 extends to at least [—1, 1] in every direction
of the phase space. Although the concept of dynamic aperture does not apply in
this example, since the system is bounded, the size of the resulting domains are

comfortably large.

A Particle Accelerator Cell

We conclude the section with an example of practical interest in accelerator physics.
The Hamiltonian representing an accelerator magnet with the arc length 3 as inde-

pendent variable and on-energy particles is given by [17]

H(:r,y,a,b)=—(1+%)m—%(1+%)As(x,y). (5,34)

167

 

 

 

 

 

 

 

 

Figure 5.16: ((11, p1) and (q1, Q2) tracking pictures of the half period map of the Fermi—
Pasta—Ulam system for a particle launched along the q] axis. The box of guaranteed
invertibility of the generator associated with S = 0 extends to at least [—1, 1] in every
direction.

Here p is the curvature radius of the magnet, A, is the 3 component of the vector
potential, and ((m, a) ; (y, b)) are two pairs of canonically conjugate variables. For the
sake of computational simplicity, we assume that the magnetic ﬁelds are piecewise
independent of the arc length, in particular we use the sharp cutoff approximation

for fringe ﬁelds, as is commonly done in beam physics.

To ﬁnd the Hamiltonian functions for a speciﬁc magnetic element or a ﬁeld free

regions of space, all that is necessary are the speciﬁc values of p and A,. In ﬁeld free

168

space is obviously A, = 0 and p 2 00. For a homogeneous dipole magnet, the vector

potential and curvature radius are connected via

B
A. = «,9 (a: + p), (5.35)

where qBo/Po = 1/ p. For a quadrupole magnet on the other hand p z co and the

vector potential is given by

A, = -—-;E (2:2 — yz) , (5.36)

where k is the quadruple strength. The case It > 0 denotes a magnet that is focusing
in the :r. direction, and correspondingly k < 0 indicates a magnet that is defocusing
in the a: direction and focusing in the y direction. Lastly, the Hamiltonian function

of a sextupole magnet is determined by p z co and

A8 = —g (a:3 —- myz) , (5.37)

where h is the sextupole strength.

Combining these elements, we set up a cell consisting of the following sequence of
elements: drift, defocusing quadrupole superimposed with a sextupole, drift, bend-
ing dipole, drift, defocusing quadrupole superimposed with a sextupole, and drift.
The defocusing quadrupoles have the same strength k = —0.0085 and the sextupole
strength is h = 0.06. The lengths are 1 meter for the drifts and 0.5 meter for the
magnets; the dipole’s curvature radius is p = 2.5 meter and the reference particle is a

proton with an energy of 1 MeV. The resulting arrangement is illustrated in Fig. 5.17.

Integration of the system yielded a Taylor model containing the true solution of
the corresponding differential equations. For illustration, Tab. 5.7 shows the ﬁrst
ten terms of the Taylor model containing the x-component of the ﬂow, which is the
function mapping the initial conditions :ro, a0, yo, and b0 to the at position at the

end of the accelerator cell. The full Taylor model is of order 17 and its reference

169

 

 

 

 

E:g
<N
\
_MJ D

 

 

 

Figure 5.17: Illustration of the accelerator cell studied in Sec. 5.2.4

polynomial has 1933 non-vanishing coefﬁcients. It should be noted that the Taylor
model encloses the flow with a relative overestimation of only 10’1”, showing yet
again that the Taylor model approach avoids the wrapping effect to very high orders

and allows the computation of extremely tight solution enclosures.

According to the tracking results shown in Fig. 5.18, the dynamic aperture of
this accelerator cell can be estimated to be (:r, y) = (0.03,0.045). Using the Taylor
model based invertibility tests with the results of the integration, we have been able
to prove the existence of the generating function for S = 0 over domains that extend
as far as i0.1 and therefore comfortably enclose the dynamic aperture of this system.
Moreover, since ten centimeters corresponds to a typical accelerator magnet aperture,

the results even bear practical relevance.

5.2.5 Rigorous Analysis of Symplectiﬁcation Errors

Based on the results of summarized in Sec. 5.2.2, generating function based symplectic
tracking has been deve10ped for COSY Inﬁnity by B. Erdélyi [40]. The basic ideas of

the actual implementation are summarized below. With an arbitrary initial condition

170

RDA VARIABLE: N08 17, NV8 4

 

I COEFFICIENT ORDER EXPONENTS

1 0.1398389113940111 1 1 0 0 0

2 0.1038317686361456 1 0 1 O 0

3 -.2447944264979660E-01 2 2 O 0 0

4 -.1183394850192213E-01 2 1 1 0 0

5 -.2119694344941219E-02 2 0 2 0 O

6 0.2361409770673162E-01 2 0 0 2 0

7 0.12127660974103088-01 2 0 0 1 1

8 0.2185093364668458E-02 2 0 0 0 2

9 0.99448307635409458-03 3 3 0 0 O
10 0.8715481705011164E-03 3 2 1 0 0

VAR REFERENCE POINT DOMAIN INTERVAL
1 0.000000000000000 [-1.00000000 , 1.00000000 ]
2 0.000000000000000 ['1.00000000 , 1.00000000 ]
3 0.000000000000000 [-1.00000000 , 1.00000000 ]
4 0.000000000000000 [*1.00000000 , 1.00000000 1
REMAINDER BOUND INTERVAL

R [-.2157121190249145E-012,0.2178948979195422E-012]

*It***********litti*********************¥#**************#*#

Table 5.7: Taylor model describing ﬁnal :5 positions as a function of the initial con-
ditions .170, an, yo, b0 for the accelerator cell of Sec. 5.2.4.

2 in the domain of the generating function, we can write Eqn. (5.19) as
é—M-zzM-Jv(VF)T(C-(23—M-z)+2), (5.38)
where 2? = M[S](z) is the image of 2 under the formally symplectiﬁed map M5], and
C=%(I+J-S)-M“. (5.39)
After deﬁning 2 — M - 2, we write this as
w=M~J-(VF)T(C.w+z). (5.40)

The last equation is in fact an implicit relation for w = w(z). Once Eqn. 5.40 has

been solved for w, the ﬁnal result is then computed via
i=w+M-z. (5.41)

171

 

 

 

 

 

 

 

 

 

Figure 5.18: (:r, a) and (y, b) tracking pictures of the one-turn map of the accelerator
cell for particles launched along the spatial a: and y axes, respectively. The box of
guaranteed invertibility of the generator associated with S = 0 extends to at least
[—0.1, 0.1] in the spatial variables.

This result is then, up to machine precision, the image of 2 under the exactly sym-
plectic map MIS], which is close to the actual map M of interest. By repeating this

procedure, it can be used for the tracking of particles with an exactly symplectic map

that is close to the original map of interest.

While this algorithm has been shown to give outstanding results [40], it does not
provide any rigorous measures on how close the actual map M and the symplectiﬁed

map MIS] are, since the method does not provide any bounds on the error of the

172

image 2 with respect to the mathematically correct result. However, according to the
presented analysis, all quantities determining the implicit relation (5.40) are readily
available in the form of Taylor models. By using the Taylor model approach to solving
implicit equations presented in Sec. 3.5, we can therefore compute a Taylor model
for w = w(z) from (5.40). Using this Taylor model, Eqn. (5.41) describes in fact a
Taylor model for 2 in terms of the initial condition 2. If we denote this Taylor model
by T2 and the Taylor model for M computed by integration of the ODEs by T1, then
the Taylor model A = T1 — T2 gives a rigorous enclosure for the deviation of the
symplectiﬁed map M[S] from the mathematically correct map M. The Taylor model
A can then be used to compute a rigorous upper bound for the error introduced by

the symplectiﬁcation.

Since the Taylor model approach allows the computation of tight enclosures even
after extended arithmetic operations, the computed bound on the symplectiﬁcation
error is generally very small in comparison to the estimated dynamic aperture of the
Hamiltonian system. Hence, the error analysis can provide a rigorous a posteriori

justiﬁcation for the generating function based symplectic tracking.

5.2.6 Summary

We have shown that the Taylor model approach can indeed be used to rigorously prove
that the assumptions of theorems are satisﬁed. This is of great interest in a variety of
settings of theoretical physics and numerical analysis, where sophisticated theorems
are often used without any means for an automated analysis of the applicability.
Moreover, while a priori or a posteriori analysis can sometimes be used to justify
the applicability of methods and theorems, this approach often fails if the input data
are obtained as results of previous computations. Here we have seen how the Taylor

model approach can be used to simultaneously compute the results and provide a

173

rigorous justiﬁcation for the applicability of the theory.

In the particular examples studied here, we have proved the existence of generating
functions for Hamiltonian systems. Since these generating functions can be used for
the symplectic tracking [14, 40, 138] of Hamiltonian systems, the method is of great
practical importance in the study of general dynamical systems and especially of

particle accelerators.

174

Chapter 6

Implementational Details

In this ﬁnal chapter we focus on some of the practical aspects of developing veriﬁed nu-
merical methods for real world computer environments. In particular, we discuss the
ideas and principles that govern the implementation of the portable interval library
in COSY Inﬁnity [26, 20]. The design of this library has been guided by demands
for efﬁciency, portability, and reliability. To that end, the package has been imple-
mented in standard Fortran 77 and does not rely on hardware support for the directed
rounding. However, great care has been taken to reduce the unavoidable performance

overhead imposed by the additional measures to ensure rigorous enclosures.

Finally, we close with a summary of the least common denominator approach
towards language independent software development, and we discuss the C++ and
Fortran 90/95 interfaces to COSY Inﬁnity that have been developed based on this
paradigm. With the availability of these interfaces, it is now possible to use Taylor
model methods within the frameworks of these modern object oriented languages and

to interface Taylor model algorithms with existing codes and applications.

175

6.1 Implementation of Portable Interval Libraries

In the eyes of most computational scientists, ﬂoating point numbers are somewhat of
a mystery. While ﬂoating point support is clearly required for most modern hardware
platforms, it was not until 1987 that the IEEE has formally agreed on a standard for

the implementation of ﬂoating point support [64, 48]

However, to properly implement interval arithmetic on computers, some knowl—
edge about the implementation and the basic layout of ﬂoating point numbers is
essential. Moreover, to implement portable interval arithmetic, a common standard
for ﬂoating point numbers is required. Still, most of the complications in imple—
menting interval libraries stem from the desire to design packages that are portable

between a wide range of hardware and software systems.

6.1.1 Floating Point Numbers

In the following we will assume that ﬂoating point numbers are stored in a T-digit,
base-B format that complies with the IEEE Standard 754 for ﬂoating point num-
bers [64]:

(s)- (2 cc,- - B“) - (BE). (6.1)

The number s is an internal representation for the sign of the ﬂoating point number,
the number E satisﬁes the condition Emin g E g Emax with machine dependent
integers Emin, Emu, and T. It should be noted that Emin < 0, while Emax > 0. With
the exception of the number 0, all numbers are normalized such that the leading digit
2:1 is non-zero. While the IEEE standard 754 requires B to be two and allows T = 24
and T = 53, the newer standard IEEE 854 also permits B = 10. However, in the
remainder of this section we will always assume that all ﬂoating point numbers are

represented in a ﬁxed binary format; i.e., we assume that B = 2 and that the integers

176

T, Emin, and Emax are ﬁxed.

In the following small letters at, y, and 2 denote real numbers, and their corre-
sponding representations on the computer are denoted by 2:, y and z. The elementary
operations +, —, x, and / on the set of ﬂoating point numbers are denoted by EB, 69,

®, and Q, respectively.

The most fundamental and most limiting property of ﬂoating point numbers is
that only a ﬁnite subset of rational numbers has an exact ﬂoating point representation.

We denote this set of ﬂoating point numbers by

T
f={x€Qi(Z$12-i) 2EaEminSESEmaXazlzl} (6'2)
i=1

It should be noted that within any given ﬂoating point system .73, some important
real numbers have exact ﬂoating point representations: —2, —1, 0, 1, 2 E .7. The next
proposition highlights another two important floating point numbers which will be

used throughout this section.

Proposition 6.1. For a given binary representation of ﬂoating point numbers, deﬁne

the numbers 6 and ,u by

e = 21‘T (6.3a)

p = 23min“. (6.3b)
Then 5, u E .7: and the two numbers can be characterized by

e = min{x|$6}'and1€Bm>1}, (6.4a)

p = min{x|z€fand:c>0}. (6.4b)

Proof. To prove that the two numbers have exact ﬂoating point representations, we

write

5 = 21‘T _-: 21 ~2‘T. (6.5)

177

Thus, according to Eqn. (6.2), it is indeed e = 6:. Similarly, it is
’1’ : 28min+1 : 21 . 23min (6.6)

Hence, the number it also has an exact ﬂoating point representation. Next we show

that e is the smallest ﬂoating point satisfying the condition 1 ED :1: > 1:

1 69 e (21 - 2-1) 99 (21 . 2") = 21 ® (2-1 + 2-T) (6.7a)

II

= 21-(2—‘+2-T)=1+e>1=1. (6.7b)

On the other hand if we assume that there is y E F such that y < e and y 69 1 > 1,

then 3; is of the form
T
y = 21‘T - 23 . 2.2.2“ (6.8)
i=1

with Emin + T — 1 g E S 0 and 2:1 2 1. Consequently it would be

T
1+3; = (21.2-1)ea(21434-2334) (6.9a)

i=1
T
= 21- (2‘1 EB 2244“?) (6.95)
i=1

= 21-2*1=1= 1, (6.9c)

in contradiction to the assumption that y ED 1 > 1. Thus, the ﬂoating point number

5 is indeed characterized by (6.4a).

The last part of the assertion follows from the deﬁnition of p and the basic require-
ments of ﬂoating point representations: clearly ,u > 0, and any positive ﬂoating point
number smaller than p would have either E < Emin or $1 = 0, both violating the
deﬁnition of the set .7: of ﬂoating point numbers. Thus, it can indeed be characterized

by (6.4b), making it the smallest positive ﬂoating point number. (:1

178

6.1.2 Directed Rounding
In Sec. 2.2 we showed that the sum of two intervals is given by
[131,332] + [91,312] = [331 + 132,311+ 312] (6-10)

However, if we naively implement this interval addition on a computer with ﬂoating
point numbers satisfying T = 53, the result of the following addition would violate

the basic inclusion property of intervals:
[—1,1]+ [264,264]. (6.11)

While all interval endpoints have exact ﬂoating point representations, the ﬂoating

point sums needed for the interval addition would lead to truncation:

—169264 = 264, (6.12a)

169264 = 264. (6.125)

Clearly, this situation is not acceptable in an interval library used for rigorous com-
putations and the necessity of ﬁnding a solution to this problem has already been
recognized in [92]. The easiest way of alleviating this problem is by using directed
rounding for the result of interval computations: at the end of each elementary inter-
val computation, the lower bound of the resulting interval is replaced by a number
that is, by an appropriate amount, smaller than the originally computed ﬂoating point
number. Similarly, the computed result for the upper bound is shifted by an appro-
priate amount towards +00. Since ﬂoating point numbers are not uniformly spaced,
the results cannot be computed by simply adding ﬁxed numbers to the endpoints.

Instead, directed rounding towards —oo and +00 is used.

The most accurate and most efﬁcient way of implementing directed rounding in an

interval library would be to rely on intrinsic routines built into IEEE 754 compliant

179

ﬂoating point hardware: the standard requires the hardware to provide at least the

following three rounding modes:

1. rounding towards —00,
2. rounding towards +00,

3. rounding to the nearest ﬂoating point number.

With the ability to set the rounding mode, Alg. 6.1 could be used for the addition of

two intervals [11:1, 12:2].

 

Algorithm 6.1 Interval addition with hardware support to set the rounding mode.

save current rounding mode
set rounding towards —0o

21 4: 101 EB 91

set rounding towards +00

22 ¢ $2 + 312

restore saved rounding mode

 

However, this strategy deﬁes the goal of writing a portable interval library, since it
would require hardware-speciﬁc code to change the rounding modes. Moreover, even
on the same platform, different compilers use different conventions of embedding hard-
ware instructions, leading to an unacceptable platform dependence in the resulting
code. Finally, it should also be noted that on many current hardware platforms a
change in the rounding mode results in a severe performance penalty, since it leads to
pipeline ﬂushes that render this approach unacceptably slow [65, 135]. Only recently
have interval researchers been able to convince hardware manufacturers of the need
for more efﬁcient rounding mode switches, and the resulting interval implementations

have shown substantial increases in performance [136, 137].

In order to implement directed rounding in software, it is necessary to know by

how many Units in the Last Place (ULPs) the computed result of any of the ﬂoating

180

point Operations differs from the mathematically correct result. We will denote this
number by N and note that most Of the current computer manufacturers guarantee
N to be at most one. Based on the number N, the following theorem allows the use

of directed rounding for veriﬁed interval results:

Theorem 6.1. Leta: E .77 be given and assume that it is sufﬁciently large ( c. f. Cor. 6.1
for a deﬁnition of how large a: has to be). Provided that none of the following opera-

tions leads to a ﬂoating point overﬂow, the number

(1 EBsgn (m)®N®e) ®x (6.13)
is by at least N ULPs bigger than :6. Similarly the number

(1 esgn($)®N®6) 82: (6-14)

is by at least N ULPs smaller than 2:.

Proof. The assertions will be proved for N = 1 and for positive numbers 2:. Similar

arguments hold for higher values of N and negative numbers at. It is

(1 + e)a:— — 21 (2‘ + 2”) 2E-(: av.- 2") (6-159)
= 21+E (2 +2”) (22.2 “) (6.15b)
T T
= E. (Z x12"‘+Z:r,-2‘(‘+T‘l)) (5-150)
T- 1
=]ZE ((iaﬂ” + (xr+:r1) )2 T+Z$ '2 (”T 1)))J (6115(1)

= E. (: +($T + $1)2_T) (6.15e)

181

Since $1 =- 1, the result is indeed by at least one ULP bigger than the original number

(a. A similar computation shows that the second multiplication does indeed round
towards —00:

(1 - e)m = a: 9 ea: (6.16a)

T T
= 23. (211.2“) — 21*T - 253- (21,24) (6.165)
i=1 i=1

T T
= 23 - (211,2-i — Z ass-(”F”) (6.16c)
i=1 i=1

T—l T
2 ]2E ' (2 “322—1. + (331" " $1)2_T + Z$i2_(i+Ti3))] (616d)
i=1 i=2

Since 131 = 1, the result is indeed by at least one ULP smaller than the original

number m, which ﬁnishes the proof. [:1

A Portable Algorithm for Directed Rounding

Based on the results derived in the previous section, we can now present an algorithm
that, under mild assumptions on the accuracy of the hardware and system software,
allows the computations of veriﬁed interval Operations with directed rounding. The

next two corollaries are immediate consequences of Thm. 6.1 and they provide the

mathematical foundation for software implemented directed rounding:

Corollary 6.1. The number 6 = p/ (1 — Ne) is the smallest positive floating point

number that can be rounded towards zero without remaining ﬁxed or resulting in an

underﬂow result of 0.

Corollary 6.2. Let x,y E R be such that min(|x|,ly|) 2 6 and assume that the

following multiplications do not result in a ﬂoating point overﬂows Then the following

182

inclusion holds:

[23, y] C [(1 — sgn(:c)Ns) - m, (1 + sgn(a:)Ne) - m]. (6.17)

Corollary 6.2 excludes two special cases that require exception handling in an im-

plementation of an interval library that uses software implemented directed rounding:

o The multiplications might result in overﬂow errors and since this cannot be

handled in a general and portable way, the exception handling is left tO the

runtime environment.

0 Rounding Of ﬂoating point numbers a: with 0 _<_ [2:] < 6 need special attention

because a multiplication may result in an underﬂow error or a ﬁxed point.

From now on we will assume that N = 1, and we deﬁne the following two numbers:

M1, 2 1+5, (6.18a)

Md 2 1—8. (6.18b)

With these numbers and the previously deﬁned 6, Alg. 6.2 describes portable software

simulated directed rounding: if [$1,232] is a ﬂoating point interval, then Alg. 6.2

performs the necessary rounding of that interval.

On average, Alg. 6.2 is too pessimistic since it assumes that all ﬂoating point
Operations have systematic errors, even the ones with exact ﬂoating point Operands
and results. Moreover, both endpoints of the intervals are always rounded outwards,
although on average half the computed endpoints are already properly rounded. How-
ever, considering the result Of Thm. 6.1, the presented algorithm is optimal for soft-
ware simulated directed rounding. In order tO implement Alg. 6.2, it is necessary

to know the machine dependent numbers )1 and e, which can be calculated with the

algorithms 6.3 and 6.4.

183

 

 

Algorithm 6.2 Directed rounding of the two interval endpoints Of [2:1, 272].
if 2:1 2 6 then
$1 4: $1 ® Md
else if (121 5 —6 then
$1 <= $1 ® Mu
else if x1 > 0 then
$1 4: 0
else
$1 <= —0
end if

if $2 2 0 then
$2 ¢ $2 ® Mu

else if x2 3 —6 then
$2 4: $2 ® Md

else if x2 _>_ 0 then
$2 4: (5

else
$2 <= 0

end if

 

Algorithm 6.3 Portable determination Of the machine constant e at run time.
8 <= 1
while 1 + 6/2 > 1 do
8 <= 5 / 2
end while

 

A_lgorithm 6.4 Portable determination Of the machine constant p at run time.
u 4: 1
while u/2 > 0 do
H <= M2
_end while

184

Thus, by combining Alg. 6.2 with the two algorithms 6.3 and 6.4 we can indeed

implement a truly portable interval library with fully rigorous directed rounding. The

performance Of the combined algorithms will be discussed in Sec. 6.1.4.
6.1.3 Implementation of Interval Operations

In Sec. 2.2 we noted that the intrinsic functions can be extended to intervals in such

a way that the basic inclusion property is always maintained:

f([a1bll = {f(ib‘) I a S a: S b}- (6-19)

Here we discuss for some Of the intrinsic functions how the extension to intervals
can be deﬁned such that inclusions are indeed propagated. The interval extensions
of the standard mathematical functions are not based on their respective Taylor
expansions, but rely on domain decompositions such that the functions are monotonic
over the subdomains. The following two proposition show how the monotonicity Of

certain functions can be used to compute simple and accurate interval extensions for

monotone functions:

Proposition 6.2. For any interval [a, b] C R, the interval extension of the exponen-

tial function is deﬁned as

exp ([0131]) = [exp(a), exp(b)l (5-20)

and satisﬁes the basic requirement (6.19).

Proposition 6.3. If we assume that a > 0, then the interval extension of the logo-

rithm is deﬁned as

10909.0) = [10800110800] (6.21)

and satisﬁes the basic requirement (6.19).

185

Alg. 6.5 shows an algorithm for extending the sine function to interval arguments
[a1,a2]. Special attention is given to arguments that are too large for an accurate
determination of the sine function. However, since the interval [—1, 1] is always a

correct, albeit overly pessimistic enclosure for the correct result, even this case can

be handled rigorously.

 

Algorithm 6.5 Reliable interval extension of the sine function.

n2 <= 2n
if max(|a1|, [a2|) < 106 then
perform outward rounding on the interval [a1, a2] and store the result in [b1, b2]
r1 4: nearest integer to ((b1 ® W2) 6 1/ 4) 89 7T 2
if ()1 > T1 then
7‘1 <= 71 EB 7T2
end if
r2 4: nearest integer to ((bg ® ﬁg) 03 1/4) (8 W2
if b2 < r2 then
7‘2 4': T2 8 7T2
end if
if b1 3 r2 and b2 2 r1 then
C1 4: —1
C2 <= +1
else if b1 5 r2 then
c1 4: min (sin(bl), sin(b2))
C2 <= +1
round cl towards —00
else if b2 2 r1 then
C1 <= —1
02 4: max (sin(bl), sin(b2))
round c2 towards +00
else
c1 <= min (sin(bl), sin(bzl)
c2 42 max (sin(bl),sin(b2))
perform outward rounding on the interval [61. C2]
end if
else
C1 ¢ —1
C2 4: +1
end if
return the interval [c1, C2]

186

6.1.4 Benchmarks and Results

Here we present several benchmark results Of the interval implementation in the
COSY Inﬁnity [26] language environment, which is based on the previously presented
algorithms. The results show that the implementation is very efﬁcient and introduces

only a small overhead in the interval Operations Of COSY Inﬁnity.

We chose the multiplication Of intervals as the benchmark Operation, since its
computational cost is similar to the one Of the presented software-based rounding
algorithm. For the actual implementation, we have measured the execution times Of
9 - 106 interval multiplications that test all nine cases of the interval multiplication.
Thus, the results allow a very good comparison between the execution times and the
imposed overheads. For additional comparison, we also compared the new code tO
the previous implementation Of the interval routines in COSY Inﬁnity, which did not
provide any rounding and. were therefore not fully veriﬁed. Altogether, we tested the

following three cases:

A: The code has been executed with an unmodiﬁed version Of the original interval

routines Of COSY Inﬁnity.

B: The code has been executed with the new interval routines, but rounding was

disabled.

C: The code has been executed with the new interval routines and outward round-

ing was enabled.

The results Of A and B allow a comparison between the new and the Old interval
libraries in COSY Inﬁnity. A comparison between B and C on the other hand, puts

a measure on the performance overhead imposed by the software rounding.

187

First we tested the new routines in a pure Fortran 77 program, to eliminate the
overhead Of the COSY Inﬁnity runtime environment. The program had tO be compiled
without any optimization, thus the execution times should be viewed relative to each

other, rather than in absolute numbers.

 

 

 

Platform, Compiler Test A Test B Test C A /B C/B
VMS/Alpha, Digital F77 9.16 s 4.63 8 13.39 s 1.98 2.89
Linux/Intel, G77 9.75 S 8.44 8 14.77 S 1.16 1.75
Solaris/Sparc, Forte F77 8.29 s 6.22 s 11.03 s 1.33 1.77

Digital Unix/Alpha, Digital F77 2.80 s 1.40 s 3.10 s 2.00 2.21

Table 6.1: Average execution times for the three interval test cases in pure Fortran 77
environments.

Next we computed the interval products in the COSY Inﬁnity language environ-
ment and the corresponding execution times are given in the next table. This time
we used the highest Optimization levels available. Thus the numbers should not be

compared to the previous benchmarks.

 

 

 

Platform, Compiler Test A Test B Test C A /B C/B
VMS/Alpha, Digital F77 28.97 8 25.55 S 58.28 S 1.13 2.28
Linux/Intel, G77 24.71 3 26.24 8 31.72 s 0.94 1.21
Solaris/Sparc, Forte F77 17.27 8 18.32 8 19.81 s 0.94 1.08

Digital Unix/Alpha, Digital F77 7.24 s 7.63 s 8.77 s 0.95 1.15

 

Table 6.2: Average execution times for the three interval test cases in COSY Inﬁnity
language environments.

It should be noted that the new interval routines have a different calling scheme in
the COSY Inﬁnity language environment than the Old interval routines: compared to
the case A, B requires one additional subroutine call, while C requires two additional
calls. The plain Fortran programs on the other hand have only one additional call in
the case Of C and the same number Of subroutine calls in the cases of A and B. Since

the new call graph leads to an improved maintainability Of the code, the additional

188

overhead is however acceptable. It should also be noted that the actually computed
interval results Of A and B are always equal, but unveriﬁed. The case C on the other
hand produces correctly rounded enclosures for the mathematically correct results in

all situations.

6.1.5 Summary

In this section we have presented the basic ideas behind rigorous interval arithmetic
implementations. In particular, we introduced the concept Of directed rounding,
which can be used to eliminate truncation and underﬂow errors from conventional
ﬂoating point Operations. We also outlined how hardware support can in principle be
utilized for outward rounding. However, the main focus of the discussion has been
the presentation of a portable algorithm for the rigorous implementation Of directed

rounding.

To illustrate how the use Of directed rounding can avoid the problem illustrated

in (6.11), consider the following similar example with directed rounding:

([264, 264] + [_1,1]) _ [264, 264] z [264 J" 264 T] _ [264, 264] (622a)
: [264 _ 2111 264 + 211] _ [264,264] (622b)

= [—211 — 2'42, 211 + 2‘42] . (6.22c)

Although the last interval is a rather pessimistic enclosure Of the mathematically
correct result of [—1,1], it is nevertheless a guaranteed enclosure. And it is easy
to see that the result is in fact the best enclosure one can Obtain for this problem
on a computer with T = 53. Generall, overly large ﬁnal interval enclosures Often
indicate that the program, although mathematically correct, is stated in a numerically

unstable form and should be reformulated.

189

6.2 Language Independent Programming

Scientiﬁc software is commonly deve10ped in high-level programming languages like
C [69], C++ [132], and Fortran 90/ 95 [89]. More recently the use of Java has been
discussed for the use in scientiﬁc computing. Despite its performance problems, the
main advantage of Java lies in its combination of the object oriented programming
paradigm with the promise of portability across a wide range of platforms; while
almost all other languages have undergone standardization efforts, none has ever

fully achieved true portability between systems.

Although all compiled programming languages with support for ﬂoating point
numbers are more or less equivalent, for the purposes of computational mathematics,
the portability of software is an issue of great concern and often guides important
design decisions: compilers are not necessarily available all platforms and legacy
applications need to be extended. Thus, it is natural to ask for a language independent
programming model. The obvious approach to this problem is to write software in a
meta-language and use code generators to derive native language programs from this
high-level description. While we are not aware of any large scale system developed in
this model, we have been able to ad0pt a similar approach for the development of the
C++ and Fortran 90/95 interfaces to COSY Inﬁnity. The basic ideas, results, and

experiences with this approach will be discussed in this section.

6.2.1 The Least Common Denominator Approach

The COSY Inﬁnity language environment offers an object oriented approach to ad-
vanced numerical data types. However, access to these data types has traditionally
required using the COSY Inﬁnity programming and run time environment. This

restriction has often made it difﬁcult to interface COSY Inﬁnity’s data types and

190

algorithms with existing software packages, which are likely to be written in compiled
languages like C++ and Fortran 90/95. The C++ and Fortran 90/ 95 interfaces to
COSY Inﬁnity offer a solution to this problem: the ﬂexibility of a modern object-
oriented language combined with the power of the high performance data types and

algorithms of COSY Inﬁnity.

While it can be argued that Fortran 77 this is not the most appropriate language
for the deveIOpment of a large-scale software package — version 8.1 of COSY Inﬁnity
consists of more than 35,000 lines of highly optimized source code — the use of For-
tran 77 also has undeniable advantages over the deployment of more recent languages

like C++, Java, or Fortran 90/ 95:

Availability Together with C, Fortran 77 is one of the most widely available pro-
gramming languages. Virtually all computer platforms have a Fortran 77 com-

piler.

Performance Fortran 77 is a very simple language with low runtime overhead, re-

sulting in fast execution times and outstanding resource utilizations.

Legacy Many existing and well tested software packages are written in Fortran 77,

providing programmers with a large pool of reusable software components.

Fortran 77 is in fact the least common denominator of most modern programming
languages, since it has no support for operator overloading, dynamic memory allo-
cation, and low level hardware operations like bit manipulation and pointers. Thus,
any program written in Fortran 77 can, at least in principle, be mapped one-to-one to
languages like C, C++, and Fortran 90/ 95. Since the Fortran family of programming
languages is backwards compatible, this mapping is trivial for the Fortran 90 / 95 en-

vironments. Moreover, with the availability of the F2C converter [43], we also found

191

the appropriate tool for a conversion from Fortran 77 to C and C++ source code.

COSY Inﬁnity is a programming environment that uses a byte-compiler approach
similar to Java: source code is translated into an internal byte code, which is then exe-
cuted by an interpreter. Thus, the COSY Inﬁnity system consists of three main parts:
the compiler, the interpreter, and the support libraries implementing all the available
language features. Not much would be gained by simply converting Fortran 77 code
to C++ source or renaming the Fortran 77 source ﬁles into Fortran 90/ 95 source ﬁles.
However, both C++ and Fortran 90/ 95 offer object oriented language features, which

allow the deﬁnition of new data types, and have been used for the implementations

of the interfaces to COSY Inﬁnity. This approach offers:
0 native language support via classes (C++) and Modules (Fortran 90/ 95),

a thin and lightweight object oriented wrappers with low overhead,
0 embedding of the COSY interpreter into Fortran 90/ 95 and C++ programs,

0 operator overloading for transparent transition from built-in numerical data

types to COSY objects.

The interfaces disregard the byte-compiler part of COSY Inﬁnity and embed the
interpreter to gain access to the support libraries of the system. Since all advanced
data types of COSY Inﬁnity are implemented as part of the support libraries, they

are immediately available to C++ and Fortran 90/ 95 programmers.

We also mention that the system has been implemented in a way that completely
removes the need for any manual code manipulations in any language other than
Fortran 77. Thus, at the core of the least common denominator (LCD) approach lies

a set of Fortran 77 source ﬁles that implement all algorithms and data structures and

all code for the interfaces is generated from this uniform code base in an automated

192

way. Therefore, the interfaces immediately beneﬁt from any enhancements in the
COSY Inﬁnity system. In the remainder of this section we discuss the C++ and the
Fortran 90/ 95 in further details and present their application programming interfaces

(APIs).
6.2.2 The C++ Interface to COSY Inﬁnity

The C++ interface is implemented through the Cosy class, which offers access from
within C++ to the core of COSY Inﬁnity. This interfacing is achieved by embedding
the COSY Inﬁnity execution engine into a C++ class. Since the glue that holds
the two systems together is a very thin wrapper of C++ code, the performance of
the resulting class is comparable with the performance of COSY Inﬁnity itself and

exceeds that of other approaches (c.f. Sec. 6.2.4 for further details).

The COSY Inﬁnity programming language [20] uses an object-oriented approach
to programming which centers around the idea of dynamic typing: all Cosy objects
have an internal type, which may be real, string, logical, etc., and the exact meaning

of operations on Cosy objects is determined at runtime and not at compile time.

The Cosy class attempts to be compatible with the C++ double precision data
type. In most cases, it should be possible to convert an existing numerical program
to a Cosy-based one by simply replacing the string “double” with the string “Cosy”
in the source. However, using this approach would underutilize the Cosy class, which
shows its real strengths if the advanced data types like intervals, DA vectors, or Taylor
models are used. For example, replacing the double precision numbers in an existing
program with Cosy objects that are initialized to DA vectors would allow high-order
sensitivity analysis of the original program. Other beneﬁts lie in the automatic veri-

ﬁcation of existing programs by using intervals or Taylor models.

193

Memory Management

The Cosy class manages its own internal memory and does not use dynamic allocation
of memory by either malloc or new. To a large extent, this is the reason for the

performance advantage that COSY Inﬁnity has over languages like C and C++.

As a consequence of this, every Cosy object requires a small portion of space in
some non-dynamic memory region. While this is never an issue with global and local
variables, this becomes an issue when Cosy objects are created dynamically by using
new or new []. Consequently, dynamic allocation of Cosy objects should be avoided
whenever possible. If Cosy objects really have to be created dynamically, care should
be taken to delete the objects as soon as possible, or the COSY system will exhaust

its internal memory.

Constructors

To allow an easy conversion of existing code from the double data type to the Cosy
data type, several constructors have been deﬁned that should accommodate this
through a variety of implicit constructions. Together with the built-in type con-
versions of C++, this mechanism should be able to handle almost any situation

correctly.
Cosy( );

The default constructor creates a Cosy object with enough internal space to store one
number or character. The object’s type is initialized to RE (real) and its value is set

to zero.

Cosy( const double val, int len = 1 );

Create a Cosy object with enough internal space to hold len numbers or characters.

The parameter 1en is Optional and defaults to 1. The object’s type is initialized to

194

RE and its value is set to val.
Cosy( const int val, int len = 1 );

Create a Cosy object with enough internal space to store len numbers or characters.
The parameter 1en is optional and defaults to 1. The type of the object is initialized
to RE, since COSY Inﬁnity does not have a dedicated data type for integers; its value

is set to val.
Cosy( const bool f );

Create a Cosy object with enough internal space to store one number or character.

The object’s type is initialized to L0 (logical or boolean) and its value is set to the

boolean value f.
Cosy( const char *str );

Create a Cosy object from a C string str. The object’s type is set to ST (string)
and enough internal memory locations are allocated to hold the string (without the
terminating NULL character, which is not needed in COSY). The object is initialized

with the string str.
Cosy( const Cosy& src );

Create a new Cosy object from an existing one. The new object is initialized with a

deep copy of src.
Cosy( integer 1en, const int 11, const int dimI] );

This special constructor creates a Cosy object that represents a Cosy array of dimen-
sionality n. The length of each of the dimensions is given in the array dim. And each
entry of the array has internal space for len numbers and is initialized to zero with

tYpe RE. For further details on Cosy arrays, refer to Sec. 6.2.2.

195

Assignment Operators

The Cosy class supports all assignment operations available in C++. Moreover, all
the assignment operations that are commonly used with ﬂoating point numbers are
implemented in a way compatible with the standard C++ deﬁnitions for ﬂoating

point data types.

Cosy& operator =(const Cosy& rhs)

Assign a deep c0py of rhs to the object and return a reference to it.

Cosy& operator+=(const Cosy& rhs)

Add rhs to the object and return a reference to it; equivalent to :1: = a: + rhs.
Cosy& operator-=(const Cosy& rhs)

Subtract rhs from the object and return a reference to it; equivalent to a: = :c — rhs.
Cosy& operator*=(const Cosyk rhs)

Multiply the object with rhs and return a reference to it; equivalent to a: = a: a: rhs.
Cosy& operator/=(const Cosy& rhs)

Divide the object by rhs and return a reference to it; equivalent to a: = 112/rhs.
Cosy& operator&=(const Cosy& rhs)

Unite the object with rhs and return a reference to it. For numerical Cosy objects,
the result of a union is usually a vector. It should be noted that this implementation

of this operator is not compatible with the default behavior of this operator in C++.

Unary Mathematical Operators

The Cosy class supports all unary operators available in C++. The operators are

compatible with the default implementations for ﬂoating point variables.

196

Cosy operator+()

Return the positive of the object. This is in fact an identity operation and is included

only for completeness.

Cosy operator-()

Return the negative of the object without modifying it.

Cosy operator++()

Add one to the object and return the result.

Cosy Operator--()

Subtract one from the object and return the result.

Cosy operator++(int)

Add one to the object and return a c0py of the object before the operation.
Cosy operator--(int)

Subtract one from the object and return a copy of the object before the operation.
Array Access

Cosy get(const int coeffI], const int n)
Obtain a copy of an array element. The element is described by the n-dimensional

array coeff. More details on Cosy arrays are provided in Sec. 6.2.2.
void set(const Cosyh arg, const int coeff [] ,const int n)

COpy the Cosy object arg into an array. The target element is described by the

n-dimensional array coeﬂ'. More details on Cosy arrays are provided in Sec. 6.2.2.

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Printing, IO, and Streams

As indicated earlier, the code for the Cosy class is automatically derived from For-
tran 77 code by using the F 2C [43] converter. Consequently, the IO handling of the
underlying C code is conceptually closer to the printf-type ideas of C than it is to the

streams of C++.

However, by using temporary ﬁles, the Cosy class has partial support for the
stream based IO of C++. This mechanism uses the ﬁle COSY.TMP in the current
working directory as a translation buffer. This allows the Cosy class to be compatible

with output streams.
friend ostreamh operator<<(ostream& 3, const Cosy& src)

Print a representation of the object src onto the ostream s. The printing uses the

formats speciﬁed in [20].
Type Conversion

While the implicit type conversion mechanisms of C++ allow a transparent transition
from the default C++ data types to Cosy objects. The conversion of Cosy objects into

standard C++ data types on the other hand requires use of the dedicated conversion
functions listed below.

friend double toDouble(const Cosy& arg)

Return a double precision variable that represents the result of calling the function
CONS on the Cosy object arg.

friend bool toBool (const Cosy& arg)

Return a boolean variable that contains the boolean value of the Cosy object arg. If

arg is not of type LO, the return value is undeﬁned.

198

friend string toString(const Cosytz arg)

Return a C++ string object that contains the string contained in the Cosy object

arg. If arg is not of type ST, the result is undeﬁned.

Elementary Operations and Functions

The COSY Inﬁnity environment has a large number of operators and functions built
into its language. The C++ interface to COSY Inﬁnity aims to give transparent
access to these functions by trying to be compatible with both the notations of C++
and of COSY Inﬁnity. To that end, the Operators are compatible with the C++ no-
tations, and the elementary functions are compatible with the standard C++ naming
conventions, and almost all functions deﬁned in math.h for double precision ﬂoating

point numbers are supported for Cosy objects.

As a general rule, all functions in C++ are named with the lower case version of
their corresponding COSY Inﬁnity identiﬁer. However, whenever COSY Inﬁnity uses
a name for a function that does not exist in C++, e.g.,the absolute value function is
called obs in COSY Inﬁnity, while it should be called fabs in C++, both names are
made available. Whenever the name of a COSY function clashes with reserved words
of C++, the ﬁrst letter of that function’s name is capitalized: e.g., the COSY Inﬁnity

function REAL is called Real in the C++ interface.

For the operators deﬁned in COSY Inﬁnity, the following deviations from these

general rules exist:

0 While the exponentiation is an operation in COSY Inﬁnity, C++ uses the func-

tion pow(. . .) for this.

0 The operator # of COSY Inﬁnity is not deﬁned in C++ and has been replaced

with the standard C++ operator !=.

199

o The operator & does not follow the standard C++ conventions and computes
the union Of two Cosy objects. However, since the Cosy class is meant to be used
for the development of new programs, or as a replacement for double variables,

overloading this operator is unlikely to cause any problems.

The signatures Of all Operators and functions derived from the COSY operators are:

Cosy Operator+

Cosy Operator-

Cosy operator*

Cosy operator/

Cosy pow

bool operator<

bool Operator>

bool operator==
bool Operator!=
Cosy operatort

bool Operator<=
bool operator):

The standard functions deﬁned for the Cosy class are listed in Appendix 8.1. These

functions are also referred to as intrinsic functions for Cosy Objects.

COSY Procedures

The COSY Inﬁnity language environment has several procedures built into its lan-
guage. These procedures range from diagnostic tOOls (e.g., MEMFRE) over ﬁle han-
dling tO complex tasks (e.g., POLVAL). For a complete interface from C++ to COSY
Inﬁnity it was necessary to make these procedures available as void functions. The

C++ interfaces to the procedures all have a standardized signature:
void <name> (...);

All procedures take at least one argument, and all arguments are either Of type
Cosy at or const Cosy &. The list in Appendix B.2 shows the names of the COSY

Inﬁnity procedures in the ﬁrst column, and the declaration Of the corresponding C++

200

functions. Further details on the corresponding COSY Inﬁnity procedures are given

in [20].
Cosy Arrays vs. Arrays Of Cosy Objects

In the COSY Inﬁnity language environment, arrays are collections of Objects that may
or may not have the same internal type. Thus, within COSY Inﬁnity, it is conceivable
to have an array with entries representing strings, intervals, and real numbers. In that
sense, the notion of arrays in COSY Inﬁnity is quite similar to the notion of arrays

Of Cosy Objects in C++.

However, there is a fundamental difference between the two concepts: a C++
array Of Cosy Objects is not a Cosy Object. Due to this difference, the C++ interface
does not use C++ arrays Of Cosy objects (although the user obviously has the freedom
to declare and use them). As a consequence, the interface provides two different (and
slightly incompatible) notions of arrays. Arrays of Cosy Objects are C++ arrays
and they can be used wherever C++ permits the use of arrays. Cosy Arrays, on
the other hand, are individual Cosy Objects which themselves contain Cosy Objects.
Since several important procedures Of COSY Inﬁnity assume their arguments to be
Cosy arrays, Cosy arrays are quite important in the context of COSY Inﬁnity and its

C++ interface.

Since the C++ interface to Cosy does not use the [] Operator for the access to

elements, users should use the utility functions

Cosy get(const int coeffI], const int n)

and

void set(const Cosy& arg, const int coeffI], const int n)

described in Sec. 6.2.2 to access the elements Of a Cosy array. To simplify the access

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to individual array elements, we suggest that users use inheritance or external utility
functions for convenient access to the elements Of Cosy arrays. For two-dimensional

arrays, such a function could be imlemented as:

Cosy get(Cosy ta, int 1, int j) {
int c[2] = {1+1. j+1};
return a.get(c, 2);

Since Cosy arrays start at one, (as Opposed to C++ arrays that start at 0), these
utility functions could also be used to mask this implementational detail from the
user. However, since the user’s requirements on the dimensionality Of Cosy arrays
vary widely, the distribution Of the C++ interface does not provide any of these

convenience functions.

Finally, we point out that the two different concepts of arrays lead to the possibility
Of having C++ arrays Of Cosy arrays — although it might be challenging to maintain a

clear distinction between the various indices needed to access the individual elements.

6.2.3 The Fortran 90/ 95 Interface to COSY Inﬁnity

The Fortran 90/ 95 interface to COSY Inﬁnity gives Fortran 90/ 95 programmers easy
access tO the sophisticated data types Of COSY Inﬁnity. The interface has been
implemented in the form of a Fortran 90/ 95 module, which has recently been used as

part of the GlObSOl [46] system for constrained global Optimization.
Special Utility Routines

The Fortran 90 / 95 interface tO COSY Inﬁnity uses a small number Of utility routines
for low-level access to the internals. In this section we describe these routines in

detail.

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SUBROUTINE COSYJNIT [<N'I'EMP>] [<NSCR.>] [<MEMDBG>]

Initialize the COSY system. This subroutine has to be called before any COSY

objects are used.

NTEMP sets the size Of the pool Of temporary Objects and defaults to 20. This pool of
variables is used for the allocation of temporary COSY Objects. Since Fortran 90/ 95
does not support automatic destruction of Objects, it is necessary to allocate all
temporary Objects beforehand and never deallocate them during the execution of the
program. The pool is organized as a circular list; and in the absence of automatic
destruction Of Objects, if the number of actually used temporary variables ever exceeds
NTEMP, memory corruption will occur. It is the responsibility Of the user to set the

size appropriately.

NSCR defaults to 5000 and sets the size Of the variables in the pool. Additionally,
the subroutine SCRLEN is called to set the size of COSY’s internal temp variables.
MEMDBG may be either 0 (no debug output) or 1 (print debug information on memory
usage). It should never be necessary for users Of the Fortran 90/95 module to set

MEMDBG.

Neither the size Of the pool, nor the size Of the variables in the pool can be
changed after this call. More details on the pool Of temporary Objects are provided

in Sec. 6.2.3.
SUBROUTINE COSY-CREATE <ser> [<LEN>] [<VAL>] [<NDIMS>] [<DIMS>]

Create a variable in the cosy core. All COSY Objects have to be created before they
can be used! This routine allocates space for the variable and registers it with the

COSY system. SELF is the COSY variable to be created.

LEN is the desired size Of the variable SELF, which determines how many DOU-

BLE PRECISION values can be stored in SELF, and defaults to 1. If VAL is given,

203

the variable is initialized to it; otherwise VAL defaults to zero. Independent of the

parameters LEN and VAL, the type of the variable is set to RE.

This routine can also be used for the creation Of COSY arrays, which are discussed
in Sec. 6.2.3. If NDIMS and DIMS are speciﬁed, the variable SELF is initialized to be
an NDIMS-dimensional COSY array with length DIMS(I) in the i-th direction. Each

entry of the array has length LEN and is initialized to VAL with type RE.
SUBROUTINE COSYDESTROY (SELF)

Destruct the COSY Object SELF and free the associated memory. If SELF hasn’t been

initialized with COSY_CREATE, the results are undeﬁned.
SUBROUTINE COSYJRRAYGET <SELF> <NDIMS> <IDXS>

Return a copy of an element of the array SELF. NDIMS speciﬁes the dimensionality
Of the array and IDXS is an array containing the index Of the desired element. More

details on COSY arrays are provided in Sec. 6.2.3.
SUBROUTINE CUSYJRRAYSET (SELF) <NDIMS> <IDXS> <ARG>

Copy the COSY Object ARC into an element of the NDIMS-dimensional array SELF.
The target is speciﬁed by the NDIMS—dimensional array IDXS which contains the index

of the target. More details on COSY arrays are provided in Sec. 6.2.3.
SUBROUTINE COSY_GE'I'I'EMP <SELF>

Return the address of the next available temporary Object from the circular buffer
of such Objects. While the value Of the returned variable is undeﬁned, the type is

guaranteed to be RE. Refer to Sec. 6.2.3 for more details.
SUBROUTINE COSYDDUBLE (SELF)

Extracts the DOUBLE PRECISION value from the variable SELF by calling the
function COSY function CONS.

204

SUBROUTINE COSYLLDGICAL (SELF)

Extracts the logical value from the variable SELF. If the type of SELF is not LO, the

result is undeﬁned.
SUBRDUTINE COSYLURITE <SELF> [<IUNIT>]

Writes the COSY variable SELF to the unit IUNI'I‘, which defaults tO 6. This func-
tion uses the same algorithms employed by the COSY procedure WRITE, which is

discussed in [20].
SUBRUUTINE CUSYQTMP <ARG>

Return a temporary COSY Object initialized with the value ARG, which may be either
Of type DOUBLE PRECISION or INTEGER. The main purpose Of this function is
for the temporary conversion Of parameters to COSY procedures. As an example,
consider the following two equivalent code fragments. They illustrate that the use of

the function COSY.'I'MP leads to simpler and less error prone code:

TYPE(CDSY) :: A,B,X
CALL COSY_CREATE(A)
CALL CDSY_CREATE(B)
CALL COSY_CREATE(X,2)
A 8 2

B = 5

CALL INTERV(A,B,X)
CALL COSY_DESTROY(A)
CALL COSY_DESTROY(B)

TYPE(CDSY) :: x

CALL COSY_CREATE(X.2)
CALL INTERV(CDSY_THP(2),COSY_THP(5),X)

Operators

The Fortran 90/ 95 interface to COSY Inﬁnity Offers all Operators that the standard
COSY system offers. For the convenience of of the user, additional support functions

are provided that allow mixed Operations between built-in data types and the COSY

205

Objects and all Operations behave as expected. A complete list of all the deﬁned
operations between COSY Objects and built-in types is given in Appendix C. It

should be noted that all Operations involving COSY Objects return COSY Objects.

Assignment

The Fortran 90 / 95 interface to COSY Inﬁnity provides several assignment Operations
that allow an easy transition between built-in data types and COSY Objects. This
section lists all the deﬁned assignment Operators involving COSY Objects.

COSY LHS = COSY RHS

Copies the COSY object RHS to LHS. If LHS hasn’t been created yet, it will be created
automatically.

DOUBLE PRECISION LHS = COSY RHS

Converts the COSY Object RHS to the DOUBLE PRECISION number LHS by calling
the function COSY_DOUBLE.

LOGICAL LHS = COSY RHS

Converts the COSY Object RHS to the LOGICAL variable LHS by calling the function
COSYLLOGICAL.

COSY LHS = DOUBLE PRECISION RHS

Copies the DOUBLE PRECISION variable RHS to the COSY Object LHS. If LHS hasn’t
been created yet, it will be created automatically. The type Of LHS will be set to RE.
COSY LHS = LOGICAL RHS

Copies the LOGICAL variable RHS to the COSY Object LHS. If LHS hasn’t been created

yet, it will be created automatically. The type Of LHS will be set to L0.

COSY LHS = INTEGER RHS

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Copies the INTEGER variable RHS to the COSY Object LHS. If LHS hasn’t been

created yet, it will be created automatically. The type of LHS will be set to RE.

Functions

The Fortran 90/95 interface to COSY Inﬁnity supports most of the functions sup-
ported by the COSY language environment. The following list shows all the functions
that are supported for COSY objects by the Fortran 90 / 95 interface to COSY Inﬁnity.
The left column shows the Fortran 90/ 95 name of the function and the right column
shows the name Of the function in the COSY Inﬁnity environment. All functions have

the exact same names as the corresponding COSY inﬁnity functions discussed in [20].

Subroutines

All the standard procedures of the COSY Inﬁnity language environment are available
as subroutines from the Fortran 90 / 95 interface to COSY. The names and parameter

lists of the subroutines match the names and parameter lists Of the normal COSY

Inﬁnity procedures.

Automatic argument conversion is not available. That means that all arguments
have to be either previously created COSY Objects or temporary COSY Objects Ob-

tained from calls to COSYJI'MP.
Memory Management

The COSY Fortran 90/ 95 module is based on the standard core functions and algo-
rithms of COSY Inﬁnity. As such, it uses the ﬁxed size memory buffers Of COSY
Inﬁnity for storage of COSY Objects. While this fact is mostly hidden from the user,

understanding this concept helps in writing efﬁcient code.

When a COSY Object is created by using the routine COSY_CREATE, memory is

207

allocate in the internal COSY memory. This memory is not freed until the routine
COSY_DESTROY is called for this object. Moreover, since COSY’s internal memory is
stack based (and not garbage collected), memory occupied by one Object will not be

freed until all Objects that have been created at a later time have also been destroyed.

Since Fortran 90/95 does not have automatic constructors and destructors, all
objects have to be deleted manually. While this is generally acceptable for normal
Objects, this is impossible to guarantee for temporary Objects. TO allow temporary
Objects in the COSY module, a circular buffer Of temp. objects is created when the

COSY system is initialized with COSY-INIT.

As an example on how the pool of temporary Objects should be used, consider
the following fragment Of code that implements a convenience interface to the COSY
procedure RERAN. Internally, the function CRAN obtains one Object from the pool
for its return value. This avoids the obvious memory leak that would result if it was

creating a new COSY Object.

FUNCTION CRAN()
USE COSY_MODULE
IMPLICIT NONE
TYPE(COSY) :: CRAN
CALL COSY_GETTEHP(CRAN)
CALL RERAN(CRAN)

END FUNCTION GRAN

However, it has to be stressed that the ﬁxed size Of the pool Of temporaries bears
a potential problem: there is no check in place for possible exhaustion Of the pool.
In other words, the pool has to be sized large enough to accommodate the maximum
number Of temp. objects at any given time during the execution of the program.
Since this number is easily underestimated, especially for deeply nested expressions,

the buffer should be sized rather generously.

208

COSY Arrays vs. Arrays of COSY Objects

In the COSY Inﬁnity language environment, arrays are collections Of objects that may
or may not have the same internal type. Thus, within COSY Inﬁnity, it is conceivable
to have an array with entries representing strings, intervals, and real numbers. In that
sense, the notion Of arrays in COSY Inﬁnity is quite similar to the notion Of arrays

of COSY Objects in Fortran 90/ 95.

However, there is a fundamental difference between the two concepts: a For-
tran 90/ 95 array of COSY objects is not again a COSY Object. Due to this difference,
the Fortran 90/95 module does not use Fortran arrays Of COSY Objects (although
the user Obviously has the freedom to declare and use them). As a consequence, the
interface provides two different (and slightly incompatible) notions Of arrays. Arrays
of COS Y Objects are Fortran 90/ 95 arrays and they can be used wherever Fortran
permits the use Of arrays. COS Y Arrays on the other hand, are individual COSY
Objects which themselves contain COSY Objects. Since several important procedures
of COSY Inﬁnity assume their arguments to be COSY arrays, COSY arrays are quite

important in the context Of COSY Inﬁnity and its Fortran 90/95 interface modules.
TO access the elements Of COSY arrays, users should use the utility routines
SUBROUTINE COSYJRRAYGET (SELF) <NDIMS> <IDXS>

and

SUBROUTINE COSYJRRAYSET <SELF> <NDIMS> <IDXS> <ARG>

Finally, we point out that the two different concepts of arrays lead to the possi-
bility Of having Fortran 90/95 arrays of COSY arrays — although it would be quite
challenging tO maintain a clear distinction between the various indices needed to

access the individual elements.

209

6.2.4 Performance Analysis

To demonstrate the success Of the LCD approach in the creation of C++ and For-
tran 90 / 95 interfaces to COSY Inﬁnity, Tab. 6.3 lists the execution times Of a mod-
erately complicated DA algorithm in six variables and order ten on various hard-
ware/compiler combinations. We have timed the exact same algorithm with the

different interfaces and it is important to note that:

a) the Fortran 90/ 95 interface is not signiﬁcantly slower than the standard COSY

Inﬁnity system

b) the C++ interface is only about two to three times slower than the correspond-

ing Fortran code

While the C++ interface seems slow, most Of the computational overhead can be
attributed to the implementation of the Cosy class that is Optimized for maintain-
ability and not performance. Moreover, to the best of our knowledge all other DA
package implemented in C++ are at least ten times slower than COSY Inﬁnity. Thus,
the C++ interface to COSY Inﬁnity is in fact the fastest general purpose DA pack-
age available; and gives users from all areas of the computational sciences access to

advanced Taylor model algorithms and high-order veriﬁcation.

 

COSY, C77 C++ COSY,Fort77 Fort90

 

 

 

Alpha/666, Linux 1:26 4:15 1:11 1:09
1:17 4:18 1:09 1:10
Intel PII/333, Linux 5:36 10:31
5:51 10:31
5:54 10:21
Alpha, True64 1:53 1:54
1:53 1:54

 

Table 6.3: Execution times in minutes and seconds for a typical mix Of DA Operations
on different platforms with different interfaces to COSY Inﬁnity.

210

6.2.5 Summary

In this section we have presented the least common denominator (LCD) approach
to language independent software develOpment. We have demonstrated its applica-
bility by summarizing its utilization in the development Of C++ and Fortran 90/ 95
interfaces to COSY Inﬁnity. While we refrain from recommending the LCD approach
with Fortran 77 code in new software projects, our experiences show that the method
can be used for the maintenance Of large existing legacy codes that cannot easily be
rewritten in new programming languages. However, it has proven invaluable in the
design and development Of the C++ and Fortran 90/ 95 interfaces to COSY Inﬁnity,

broadening the applicability and availability of advanced Taylor model methods.

211

APPENDIX

212

Appendix A

Orbital Elements of Major Planets

For the computation Of asteroid orbits presented in Sec. 5.1, the orbital elements of the
nine major planets play an important role: they are the basis of the computation Of
the planets’ positions and velocities. For reference purposes, the following tables list
the orbital elements and their linear time dependences. It should however be noted
that approximating the time dependence Of the orbital elements by linear relations is
not suﬂicient to reproduce the accuracy of the ephemeris DE405 [130].

Tab. A.1 lists the orbital elements Of the nine major planets; Tab. A.2 and A.3
list the daily rates Of change Of the orbital elements. The numerical values are listed
in [125] and are accurate at the epoch J2000. While the number Of signiﬁcant digits
is reduced in the following tables, the computations have been performed with the

full accuracy available.

 

 

 

 

Planet 0 i w a e M
Mercury 48.33167 7.00487 29.12478 0.38710 0.20563 174.79439
Venus 76.68069 3.39471 54.85229 0.72333 0.00677 50.44675
Earth —— 11.26064 0.00005 114.20783 1.00000 0.01671 357.51716
Mars 49.57854 1.85061 286.46230 1.52366 0.09341 19.41248
Jupiter 100.55615 1.30530 —85.80230 5.20336 0.04839 19.65053
Saturn 113.71504 2.48446 —21.28310 9.53707 0.05415 317.51238
Uranus 74.22988 0.76986 96.73436 19.19126 0.04717 142.26794
Neptune 131 .72169 1.76917 —86.75034 30.06896 0.00859 259.90868
Pluto 110.30347 17.14175 113.76329 39.48169 0.24881 14.86205

 

Table A.1: Orbital Elements Of the major planets (angles in degree, distances in

astronomical units) at the epoch J 2000.

213

 

 

 

 

Planet (2' i’ w’
Mercury —0.3394174461 - 10‘5 —0.1787968669- 10'6 0.7756255234- 10'5
Venus —0.7581489085 - 10‘5 —0.2175070345 - 10‘7 0.6754049719- 10‘5
Earth —0.1386284128° 10‘3 —0.3569853221 - 10’6 0.1477415013- 10‘3
Mars —0.7758688874 - 10‘5 —0.1937029431 - 10"6 0.1962864091 - 10‘4
Jupiter 0.9256749562 ' 10'5 —0.3156133568- 10‘7 —0.2868963411 - 10“5
Saturn —0.1210015971 - 10‘4 0.4646741210 - 10‘”7 —0.2721423688 - 10"5
Uranus 0.1278728420- 10‘4 —0.1589474479- 10‘7 —0.2805080227- 10’5
Neptune —0.1150277600 - 10’5 —0.2768271345 - 10‘7 —0.5271731676- 10‘5
Pluto —0.2838999240° 10‘6 0.8418891347- 10‘7 —0.7218799621 - 10"6

 

Table A.2: Daily rates Of change Of the orbital elements (2, i, and w Of the nine major
planets .

 

 

 

Planet a’ e’ M’
Mercury 0.1806982342- 10‘10 0.6918549067~ 10"9 4.092334433949377
Venus 0.2518818487- 10’10 —0.1351950719- 10’8 1.602131301695948
Earth —0.1368904989- 10‘ll —0.1041478438~ 10‘8 0.985599987451460
Mars -0.1977002118- 10‘08 03258590009 10‘8 0.524021165107646
Jupiter 0.1662888405- 10‘07 —0.3526351815- 10'8 0.083080374325026
Saturn —0.8255441486- 10‘07 —0.1006488706- 10"7 0.033485450148293
Uranus 0.4162217593- 10‘07 —0.5242984255o 10‘8 0.011721311354533
Neptune —0.3427679829 - 10‘07 0.6882956878- 10’9 0.005987479199909
Pluto —0.2105736030 - 10'07 0.1770020547 ~ 10’8 0.003976577306242

 

Table A.3: Daily rates of change of the orbital elements a, e, and M Of the nine major

planets.

214

Appendix B

The COSY C++ Interface

In this section we describe the intrinsic functions and procedures of COSY Inﬁnity
that have been exported to the C++ interface. The deﬁnitive reference for these
information is given by [20].

B.1 Available COSY Functions

The following table lists all COSY Inﬁnity functions that are available from the C++
interface. The ﬁrst column lists the name of the COSY function as described in [20]
and the second column shows the complete signature of the corresponding C++
function.

TYPE Cosy type ( const Cosytz x );
LENGTH Cosy 1ength( const Cosytz x );
VARMEM Cosy varmem( const Cosytz x );
VARPOI Cosy varpoi( const Cosytz x );
NOT Cosy not ( const Cosytc x );
EXP Cosy exp ( const Cosy& x );
LOG Cosy log ( const Cosy& x );
SIN Cosy sin ( const Cosytt x );
COS Cosy cos ( const Cosyl’t x );
TAN Cosy tan ( const Cosyh x );
ASIN Cosy asin ( const Cosytt x );
ACOS Cosy acos ( const Cosytc x );
ATAN Cosy atan ( const Cosytr x );
SINH Cosy sinh ( const Cosytz x );
COSH Cosy cosh ( const Cosy& x );
TANH Cosy tanh ( const Cosytr x );
SQRT Cosy sqrt ( const Cosy& x );
ISRT Cosy isrt ( const Cosytc x );
SQR Cosy sqr ( const Cosy& x l;

215

ERF
WERF
ABS
ABS
NORM
CONS

IN
WIDTH
REAL
IMAG
INT
NIN T
NIN T
DA
CMPLX
C ON J

Cosy
Cosy
Cosy
Cosy
Cosy
Cosy
Cosy
Cosy
Cosy
Cosy
Cosy
Cosy
Cosy
Cosy
Cosy
Cosy
Cosy

erf
werf
fabs
abs
norm
cons
re

in
width
Real
Imag
Int
rint
nint
da
cmplx
conj

const
const
const
const
const
const
const
const
const
const
const
const
const
const
const
const
const

Cosyt
Cosy&
Cosy&
Cosyt
Cosy&
Cosyk
Cosy&
Cosyh
Cosyh
Cosy&
Cosyl
Cosy&
Cosy&
Cosyk
Cosy&
Cosy&
Cosy&

NNﬁNﬂNNNNNNNHNNNN

VVVVVVVVVVVVVVVVV

.0 .0 .0 .0 .0 .0

.0 .0

no no .0 90 no no .0

.0

B.2 Available COSY Procedures

The following table lists all COSY Inﬁnity procedures that are available from the
C++ interface. The ﬁrst column lists the name Of the COSY procedure as described
in [20], while the second column shows the complete signature Of the coresponding
C++ function (for readability, c denotes a const Cosy A argument, while v stands

for Cosy & arguments).

MEMALL void memall ( v );

MEMFRE void memfre ( v );

OPENF void openf ( c, c, c );
CLOSEF void closef ( c );

REWF void rewf ( c );

BACKF void backf ( c );

CPUSEC void cpusec ( v );

QUIT void quit ( c );

SCRLEN void scrlen ( c );

DAINI void daini ( c, c, c, v ).
DANOT void danot ( c );

DAEPS void daeps ( c );

DAPEW void dapew ( c, C. c, C )
DAREA void darea ( c, v, c );
DAPRV void daprv (c, c, c, c, c );
DAREV void darev (v, c, c, C. C );
DAFLO void daflo (C C. V. C )3

(SIIFIA)
IEUERUAIJ
[MAFLAIJ
IlAJDIIJ
IlAJJEﬂi
IIAJIQTT
IIAJILIJ
I1AJ?EHE
l)AdPEU\
IlAJPEH?
IlAJNCTVV
CDF2
CHDPJF‘
CHDPIFTlA.
CHDPJFTJS
AdlﬂlEﬂﬂ
IJTQ‘I
IJDEXP
MBLOCK
EHJEMTITL
SONSIEEI
IiEKDS]?
VELSET
’VTELC3EXF
VUEZHERK)
VUELBJACK
VEFILL
IhdLHNITT
LEUILHE
LJWALéHE
IDVFEHIV'
IIJSIENH)
IPIL()
IPJLHP
I‘FVTELC)
I‘fVOELHP
IN/SEKF
IDVCHEIT
OIELEM
CHILI
(TVOELJELJ
()VODV
INTPOL
ILILANHAIL
‘EUD‘DAIL

void
void
void
void
void
void
void
void
void
void
void
void
void
void
void
void
void
void
void
void
void
void
void
void
void
void
void
void
void
void
void
void
void
void
void
void
void
void
void
void
void
void
void
void
void

cdflo
reran
daran
dadiu
dader
daint
daplu
dapee
dapea
dapep
danow
cdf2
cdnf
cdnfda
cdnfds
mtree
linv
1det
mblock
substr
stcre
recst
velset
velget
vezero
velmax
vefill
imunit
ltrue
1false
interv
insrnd
inlo
inup
ivvelo
ivveup
ivset
ivget
Oielem
Oiin
ovelem
oviv
intpol
rdavar
rdvar

AAA/\AAAAAAA/‘AAAAAAAAAAAAAAAAAAAAAAAAAAAAA/\AAA

c, c, v, c );

v );

v, c );

c, v, v );

c, v, v );

c, v, v );

v, c, c, v );

c, c, v );

c, c, c, v );

v, v, v, v );

c, c, v );

v, v, v, v v );
v, v, v, v v, v,
v, v, v, v v, v,
v, v, v, v v, v,
v, v, v, v v, v,
c, v, c, c, v );
c, c, c, v );

c, v, v, c, c ).
c, c, c, v );

c, v );

c, c, v );

v, c, c );

c, c, v );

v, v, v );

c, v );

v, c, c, c, c ),
v );

v );

v );

c, c, v ).

c );

c, v );

c, v );

c, v );

c, v );

v, c, C )i

c, c, V );

c, v );

c, c, v );

c, v );

c, c, v );

v, c )3

c, c, c, c, c, c, v );
c, c, c, v );

217

RDANOT
RDREA
DAEXT
RDRBN D
RDAREF
RDADOM
RDITIG

RDNPNT

RDIN T
RDPRIS
CLEAR
GRMOVE
GRDRAW
GRDOT
GRCURV
GRPROJ
GRCHAR
GRCOLR
GRWDTH
GRMIMA
RKCO
POLVAL
POLSET
FOXDPR
MEMWRT
VARMAX

void
void
void
void
void
void
void
void
void
void
void
void
void
void
void
void
void
void
void
void
void
void
void
void
void
void

rdanot
rdrea

daext

rdrbnd
rdaref
rdadom
rditig
rdnpnt
rdint

rdpris
clear

grmove
grdraw
grdot

grcurv
grproj
grchar
grcolr

grwdth
grmima
rkco

polval
polset
foxdpr
memwrt
varmax

AAA/NAAA/\AAAAAAAAAAAAAAAAAA

. . . . .
VVVVV
no 0.- no no .-

~<<<<<O

D

vv.
.0 n .-

000'-

00040000000000

.

218

<

0

0

V
.0

Appendix C

The COSY Fortran 90 / 95 Interface

In this section we describe the Operations that are deﬁned between COSY Objects and
the standard Fortran 90 / 95 data types. The deﬁnitive reference for these information
is given by [20]. It should be noted that all Operations involving COSY objects return
COSY Objects.

 

 

 

 

Addition +

COSY COSY COSY
DOUBLE PRECISION COSY COSY
COSY DOUBLE PRECISION COSY
INTEGER COSY COSY
COSY INTEGER COSY

 

Table C.1: Deﬁned combinations Of COSY Objects and standard data types for the
addition.

 

 

 

Subtraction -

COSY COSY COSY
DOUBLE PRECISION COSY COSY
COSY DOUBLE PRECISION COSY
INTEGER COSY COSY

COSY INTEGER COSY
- COSY COSY

Table C.2: Deﬁned combinations Of COSY Objects and standard data types for the
subtraction.

219

 

Multiplication *

 

 

COSY COSY COSY
DOUBLE PRECISION COSY COSY
COSY DOUBLE PRECISION COSY
INTEGER COSY COSY
COSY INTEGER COSY

 

Table C.3: Deﬁned combinations of COSY Objects and standard data types for the
multiplication.

 

 

 

Division /

COSY COSY COSY
DOUBLE PRECISION COSY COSY
COSY DOUBLE PRECISION COSY
INTEGER COSY COSY
COSY INTEGER COSY

 

Table C.4: Deﬁned combinations of COSY Objects and standard data types for the
division.

 

 

 

Power **

COSY COSY COSY
DOUBLE PRECISION COSY COSY
COSY DOUBLE PRECISION COSY
INTEGER COSY COSY
COSY INTEGER COSY

 

Table C.5: Deﬁned combinations Of COSY Objects and standard data types for the
power Operation.

 

Comparison . LT .

 

 

COSY COSY COSY
DOUBLE PRECISION COSY COSY
COSY DOUBLE PRECISION COSY
INTEGER COSY COSY
COSY INTEGER COSY

 

Table C.6: Deﬁned combinations Of COSY Objects and standard data types for the
comparison .LT..

220

 

Comparison .GT.

 

 

COSY COSY COSY
DOUBLE PRECISION COSY COSY
COSY DOUBLE PRECISION COSY
INTEGER COSY COSY
COSY INTEGER COSY

 

Table C.7: Deﬁned combinations Of COSY Objects and standard data types for the
comparison .GT..

 

Comparison .EQ.

 

 

COSY COSY COSY
DOUBLE PRECISION COSY COSY
COSY DOUBLE PRECISION COSY
INTEGER COSY COSY
COSY INTEGER COSY

 

Table C.8: Deﬁned combinations of COSY Objects and standard data types for the
comparison .EQ..

 

Comparison . NE .

 

 

 

COSY COSY COSY
DOUBLE PRECISION COSY COSY
COSY DOUBLE PRECISION COSY
INTEGER COSY COSY
COSY INTEGER COSY

 

Table C.9: Deﬁned combinations Of COSY Objects and standard data types for the
comparison .NE..

 

Concatenation .UN.

 

 

COSY COSY COSY
DOUBLE PRECISION COSY COSY
COSY DOUBLE PRECISION COSY
INTEGER COSY COSY
COSY INTEGER COSY
DOUBLE PRECISION DOUBLE PRECISION COSY
DOUBLE PRECISION INTEGER COSY
INTEGER DOUBLE PRECISION COSY
INTEGER INTEGER COSY

 

Table C.10: Deﬁned combinations of COSY Objects and standard data types for the
COSY concatenation ...UN

221

BIBLIOGRAPHY

222

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