SC‘LUTEONS C)?" E'MGENEERING PROBLEMS
EY RELAXATION 0F UMEAR MATRECES
wm THE QIGITAL COMPUTER

Thesis for the 03mm as! M. S.
WCHtGAN $TA?E COLLEGE
Eimm mués $9 $55:
”$955

This is to certify that the
thesis entitled

Solutions of Engineering Problems by
Relaxation of Linear Matrices with the
Digital Computer

presented by

Elmer Louis LeBay

has been accepted towards fulfillment
of the requirements for

Ll. S 0 degree in Ms E.

Ammfiwm

Major professor

Date May 279 195.5

0—169

 

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SOLUTIONS OF ENGIEEERING PROBLEMS BY RELAXATION

OF LINEAR HATRICES WITH THE DIGITAL COMPUTER

By
ELHER LOUIS LE BAY

AN ABSTRACT

Submitted to the School of Graduate Studies of’chhigan
State College of Agriculture and Applied Science
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Mechanical Engineering
1955

Many'problems in the fields of enqineerinq and the
applied sciences find their most direct and natural expression
as a system of linear equations. Thus, there is a perpetual
search for better and faster methods of solving them, and
the combination of Southwell's Relaxation method with the
I M Calculating Punch Type 602-A appears to be a practical
approach with many advantages. The relaxation method is
inherently accurate, offers much in the ray of flexibility
and opportunity for computational skill, and is easily checked.
The IBM 602-A Calculating Punch is a versatile machine, requir-
ing a reasonable amount of set-up time, and is available at a
moderate cost.

Southwell's Relaxation method is an iterative, successive
approximation process that works especially'well with diagonal

.
matrices (diagonal terms large compared to the other coef-
ficients). Fortunately, most physical problems take the form
of an incomplete, diagonal matrix which is a type well adapted
to solution by the computing machine.

The IBM 602—A Calculating Punch is one unit of a wide
variety of electrically operated.machinery designed for use
with punched cards. It is operated by a plug—in wired elect-
rical control panel, and can read data punched in cards, auto-
matically perform one or more calculations (addition, subtraction,
multiplication, division), and automatically punch the results.

It appears that the breakheven point, when machine relax-

ation is compared with hand relaxation, and with some other

methods of solution with the aid of an ordinary desk calculator,
would be somewhere in the neighborhood of matrices of order 10.
Skill at machine relaxation will do much to hasten convergence
of the solutions, and there is considerable opportunity for the
individual relaxer to employ technique and ingenuity to speed
convergence.

Several improvements have been devised during the develop-
ment of this process, one of which is the use of two sixypole
l2—position gang switches to progressively shift the various
reading and punching fields on the punched cards. It is hoped
that this is the beginning of a better method for solving large

linear matrices.

SOLUTIONS OF ENGINEERII‘JG PROBIEMS BY REIAXATION

OF LINEAR MATRICES WITH THE DIGITAL COMPUTER

By
ELuER LOUIS LE BA!

A THESIS
Submitted to the School of Graduate Studies of Michigan
State College of Agriculture and Applied Science
in partial fillfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Iiechanical Engineering

1955

AC KN O‘JIIEDGET .‘EN TS

 

The author wishes to express appreciation to Professor
John W; Donnell fer his inspiration, to his major professor,
Dr. James T. Anderson for his enthusiasm and his fine suggest-
ions, to Mr. Philip Hoffman fer his helpfulness with the
computer, to Professor Leonard C. Price fer his generosity;
and to Dr. Ralph M; Rotty for his encouragement and advice.

He also wishes to express his gratitude to the many

others who have helped him.to attain this goal.

TABLE 93 cammms

INTRODUCTION . . . . . . . . . .
REfiJLnLUQ'L‘J-j I OI\I . O O O O O O O O C 0

BASIC REIAXATION
OPuL‘thIOi-FS TABLE
l{ELAXJ'ITIO}.T TABLE
EXAICrLE OF REIJDCI'I‘ION
CHECKII'IG AND ERRORS
COI‘IVERGE‘ECE
OWE-RELAXATION

IBM PUIICItim CARD ILACHIICES . . . . . .

BIC CARDS

KEEPER-ICE

REPE’ODUC DIG PUNCH
CALCUIATIIEG PUNCH TYPE OZ—A
OPEILITION OF CALCULATOR
SORTER MID COLLATOR

MACHINE REDUCATIC‘l-l . . . . . . . .

CARD PIAUHEING

CMiD FILES

COI-E’UTEEI CUE‘ITRUL [DANIEL

LEECILAI‘IICS CF ll'IACl'lII‘EE RELAXATION
DISCUSSIOI‘I o o c c c o o o o c o
C C'l l {J LUS I Ul‘IS o c o c o o o o o o 0

REM: LEN C ES 0 c o o c o o o o o o

03 momrrww

12
13

16
18
2h
26

INTRODUCTION

 

many problems in the fields of the engineering sciences
find their most direct and natural expression as systems of
linear equations. Among these are plate stresses, structural
frameworks, heat conduction networks, and.mass spectrography.

Large systems of linear equations (order ten and greater)
present a serious challenge to the engineer and the mathematician.
Often, the accuracy of the final results is questionable due
to the growth of rounding error and other forms of inaccuracy;
Thus, there is a perpetual search for better and faster methods
of solving these matrices.

Automatic computing machines offer new approaches to the
solution of these problems, and.many methods of solving linear
matrices have been developed. The combination of the digital
computer and Sir Richard Southwell's Relaxation.method seems
to provide an inherently accurate method, and one which is

easily checked for errors.

RELAXATION

 

The first application of relaxation methods was made in
1935 by Sr. Richard Southwell, F.R.S., then Professor of Engin-
eering Science in the University of Oxford. Since then he has
been energetically and almost continuously devoted to its growth
as a powerful computational weapon for the solution of engineer-
ing and physical problems.

Originally, it was developed as an iteration process to
be applied directly to the solution of pin-connected and stiff-
jointed structural framework problems, heat conduction networks,
and many other types of physical problems. However, it was
found to be a very versatile device in the realm of mathematical
science.

Perhaps the most Widespread use of the relaxation method
has been to obtain particular solutions of partial differential
equations in two dimensions. It is, also, important in the
solution of a wide variety of eigenvalue problems, and linear
simultaneous equations.

As applied to linear simultaneous equations relaxation is
an iterative successive approximation process. A linear system
of equations may be written:

Rl'allxl'l'alZXZ'l’ ’ ' ° ° ° +aijxk+01
R2-a21X14a22X24 ° ° ° - - 4a2jIk+CZ

Rn = ailxl + a:L2X2 + t aijxk * Cn

where i,j,k,n - l,2,3,. . . . .
The functions have calculable values, called residuals RIRZ: . .
Rn, for any values of the variables Xk. To solve the matrix
it is necessary to find the particular values of the variables
which.make all of the residuals zero or approximately so.

Relaxation is begun by selecting an initial set of values
for the variables. Ordinarily, these values are taken to be
zero, residuals equal to the constant terms On, unless there is
some definite reason otherwise. For instance, if all constant
terms were exactly zero, one might well begin with the variables
equal to one, residuals equal to the summation of the coeff-
icients aij.

BASIC RELAXATION

 

The procedure is then to successively reduce the currently
largest residual to zero, by successive relaxation of the
variables. It can immediately be seen that if variable X1 is
relaxed.by one, it would changeiresidual one R1 by 811’ residual
two R2 by'aZl, and the last residual Rn'hy ail' Thus, an

pperations table may be constructed for the unit relaxation of

 

each.variab1e using the coefficients of the variables.

OPERATIONS TABLE

 

operation R R R
X1 I 1 all alg . . . alj
X2 " 1 £121 322 ' ° ° 323'

Relaxation proceeds by taking increments of the variables
called operators, which are multiples of the respective unit
operations; progressively liquidating all of the residuals to
approximately zero.

The relaxation process is accomplished by'means of a
relaxation table which is a running account of the residuals

and the increments of the variables.

RELAXATION TABLE

 

operators R1 R2 Rn

X1°°Xn=0 Cl G2 On
X1 = m mall + Cl male + CZ malj + CD
Xn=h

A summation of the operators gives the desired solutions.

ElfiULPLE 93 RELAX. TIOE-I

 

Solve to two decimal places the pair of equations
- 9X1 + hxg 4 7h 3 0
3X1 - 8X2 + 129 = O

The operations table is

Xl-l -9 3

Xg-l LL -8

The relaxation table is

121 R2
xn = 0 7h 129
x2 = 16 138 1
X1 = 15 3 ms
x2 : 5 23 6
X1 = 2 5 12
X2 = 1 9 h
x1 = 1 0 7
x2 = .8 3.2 .6
X1 = .3 .5 1.5
X2 = .2 1.3 - .1
x1 = .1 .h .2
x1 = .01; .01; .32
x2 = .01. .20 0
x1 = .02 .02 l .06
x2 = .01 - .06 - .02
X1 = .01 .. .03 .01

X1 I 18.h7

X2 = 23.05 -. .03 .01

The last step is the checking operation.

CHECKING AND ERRORS

 

The solutions are checked very simply and easily'hy sub-
stituting into the equations. If theresiduals found in this

Inanner differ from those resulting from the last Operation,

there is an error in the relaxation process.

One of the important advantages of the relaxation method
is that it is not critically dependent upon arithmetical
accuracy in the computation. It is unnecessary to discover a
mistake. Merely accept the correct residuals, enter them into
the relaxation table and continue the liquidation process.

CONVERGENCE

 

Convergence is the rate at which the surmnation of the
increments approaches the solutions. In general, matrices
throughout which the magnitudes of the coefficients are well
scattered are slow to converge. Therefore, it is highly desir-
able that the matrix, and thus the Operations table, be sym—
metrical about the leading diagonal. That is, the coefficients
811’ a22, a33, . . . . . are larger than the other terms in

the matrix.

It is fortunate, indeed, that most engineering problems
and most physical problems are of this form, however, there

are acceptable methods of converting an unsymmetrical matrix.

OVERRELAXATI ON

 

Often the unit operations are such that when used to re-
duce one residual, another is automatically increased. Much
work can be saved in this situation by anticipating the reaction,
since this property is a hindrance to convergence. Knowing that
a later step is going to automatically increase the magnitude

of a residual, it is apparent that there would be advantage to

be gained.from over-shooting zero.

This device is known as overrelaxation. As a general rule,

the overrelaxation should not exceed one and one-half times the

relaxation.required to reduce the residual to zero. Hewever, a

judicious use of overrelaxation.may be used to advantage in

speeding convergence.

Repeating the sample problem using overrelaxation demon-

strates its utility in reducing the number of operational steps.

The relaxation table is

X2 = .06
X1 3 "' .03
X2 = - .01

X1 3 lBob?
X2 ' 23.05

R1 R
7b 129
15h - 31
- 8 23
h — 1
.. ,5
- .26
.01 _
_ .03
— .03

2

.01

There are other variations of the relaxation.method which

do not appear to be applicable here, one of which would be use-

ful in the eventuality of a divergent matrix. However, divergent

matrices are rarely, if ever, found in practical problems.

IBM PUNCHED CARD MACHINES

 

The International Business Machines Corporation manuf-
actures a wide variety of machinery which operates with standard-
ized punched cards. Among these are the keypunch, the reproduc-
ing punch, and the 602A computer. These machines were originally
designed for accounting work, but lend themselves readily to

many other types of determinations.

£1311 CARDS

 

IBM cards measure 3-1/1; inches by 7-3/8 inches with the
upper surface divided into 80 columns and 12 rows. The twelve
rows are designated 0 down through 9 at the bottom of the card,
and the X or 11 row with the 12 row above it at the t0p of the
card. Ordinarily the 12 row is not used, the X row is for
code or minus sign, and the others are for digits, one punch to
a column, although an X punch may occur in a column with a
digit punch.

Adjacent columns are often grouped into _f_i§_l_c_1_s of several
columns. To punch a five-digit number requires a field of five
columns with an X punch somewhere in the field if it is a neg-

ative value .

KEYPUHCH
The keypunch is an electrically operated machine for the
punching of IBM cards. It has a keyboard similar to that of a

typewriter, and automatically feeds the cards through as they

are punched. A program.card.is used to control the passage of
the blank cards through the punching position. It is mounted
on a rotating cylinder, and punching position is synchronized
with the rotation of the program‘card.past the reading bruShes.
An zfpunch in the program.card triggers skipping which continues
by means of lgrpunches until an unpunched column.is reached
which halts skipping for punching to begin. After punching
takes place, skipping may be repeated or used to skip the card

out of the machine.

REPRODUCING PUNCH

 

The reproducing punch, commonly known as a reproducer,
automatically reproduces a punched card in its entirety or in
part into any of the columns ofthe blank<:ard. A plug—in
wired control panel controls the operation of the machine.

Cards to be reproduced are fed from.one hopper, and'blank cards
fromzanother. The machine stops automatically when the last card
is fed in, requiring the start key to be depressed until it has

been.run out.
CALCULATING PUNCH TYPE 602A

The 602A calculating punchlreads data from.punched cards,
automatically performs one or more calculations, and.punches the
results in the last data card or cards that follow. This
machine performs multiplications, divisions, additions and sub-
tractions singly or in various combinations. Some of the

operations may occur simultaneously as for example, calculating

INTERNATIONAL BUSINESS MACHINES CORPORATION

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[BM AUTOMATIC REPRODUCING PUNCH, TYPE 513—514 CONTROL PANEL ._ .
........ FOR SUMMARY PUNCHING—ALPHABETIC ACCOUNTING MACHINE ;.1‘.‘;7~§.?.;.U‘.".
5 REPRODUCING BRUSHES 15 - 20
o o o 0—4 0 o o o o o o o o o o o o o o
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o o o o o O o O O o o o o o O o
15 20 0.9
O o o o o o o o o o O o o o o o
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o o o o o o o o o o o o O o o 0
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o o o o—L o o O o o o o o o o
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M---- ----- : ---------- J ---------- ' ---------------- J—--M
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SBHDLIMS

 

 

 

USE

 

10
and punching.
The machine also has a memory system permitting it to
store data for later operations Or group calculations. Thus,
factors read from the first of a group of cards may be used
for calculations involving each of the following cards. Pos-
itive and negative nmnber conventions are maintained throughout

any calculation.

OPERATION 93': CALCULATOR

 

The 60211 calculator has an electrically wired control
panel through which all calculations and operations are trigger-
ed and sequenced. Some experience is necessary in programming
the machine Operations and wiring of the control panel, although
it is not difficult to wire a control panel from the wiring
diagram. The calculator is wired internally for X pickup and
has a. limited capacity of 20 positions which may be connected
for any 20 of the 80 columns.

Punching placement is a) ntrolled by an adjustable 1191.2
133.31. It contains 80 slots, one for each card column, in which
a small insert is placed wherever punching is to begin.

After the cards are placed in the feed hopper the machine
is started by depressing the start key. The stop key halts the
machine instantly even when in the middle of a calculation, how-
ever the calculation may be résmed without interruption by
starting the machine again. If at anytime the calculations are
unable to continue due to incorrect card punching or control

panel wiring, the machine stops automatically, and the comparing

11
light flashes on. The machine will start again after the reset
key has been depressed. Cards held within the machine at any
time may be run out by opening the control panel cover and start-

ing the machine.

SORTER AND COLLATOR

 

Additional machinery which is often used with IBM punched
cards is the sorter and collator. The sorter separates a.pack
of cards according to the punchings a column at a time. By
this method it may be used to arrange a pack of cards in numer-
ical order if they are serially numbered in one field. The
column representing the lowest numerical.magnitude is sorted
first, followed by sorting on the other columns in their order,
stacking them in ascending order from top to bottom after each
sorting operation.

The sorter may also be used to find the largest value of
a group in a single field by sorting the column of greatest
numerical magnitude first, and resorting only the pack with the
highest numerical punches each time.

The collator combines two packs of cards in any specified
order. This machine is operated.by an electrically wired control
panel the same size the reproducer panel which is half the size
of the 602A calculator control panel. It would be useful only

for the larger matrices.

12

IZACHINE RELAXATION

 

Considerable planning is necessary before a matrix can be
solved with the IBM Calculating Punch type 602-11. The use of
the limited card space must be planned, and the use of the
limited number of X pickups must be arranged. Then the control
panel planning chart should be filled out, and the wiring
arrangement sketched in on the control panel diagram. After
the control panel has been wired and the cards have been punched
relaxation can be carried out on the machine.

The calculation may be formulated as follows:

GA) x (1B) (:0) _-, (iD) where,

A is the Operator (group multiplier) code: X in col. 1

B is the operation " X in col. 3
C is the current residual " X. in col. 5
D is the new residual " X in col. 5

CARD PLANNING

Card planning is an important part of puttirg any com-
putation on the 602-A calculator especially if the entire card
is to be utilized. In this instance the operators, operations
table and the residuals must be converted to punched cards. On
the planning card five columns have been assigned to code and
serial numbering, while the remainder of the card is divided
into fields of five columns for punching five digit values. Of
course, only the necessary code and numbering columns, and the

necessary numerical fields would be used on any one card.

----~ :1
'1

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X" 2 .
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If? residual a; 1 g

i value $-
: fie” :1

 

 

13
The columns have been designated as follows:
Col. 1 X fOr operator code

001. 2 - 3 X in 3 for operations code
operations serial numbers

Col. h - S X in 5 fOr residual code
residual serial numbers

Col. 6 - 10 Operator or operations values
X in 8 negative

Col. 11 - 15 Residual field number 1
X in 15 negative

C01. 16 - 80 Residual fields 2 through it
X in last column of field negative

The internal wiring for X pickup would be for columns 1, 3,

S, 8’ 15’ 20, 25, 30’ 35’ o o o o o , 800

CARD FILE

A standard file of Operator cards, both positive and neg—
ative, is adequate for most relaxation problems. The file
would contain the values:

.001, .002, .003, . . . . . . . . .009

.01, .02, .03, . . . . . . . . . .09

.10, .15, .20, .25, . . . . . . . .95

1, 1.5, 2, 2.5, 3, . . . . . . . 9.5

11, 12, 13, 1h, . . . . . . . .20
Other operator values, if needed, can.be punched up as required.

The file of operations cards contains (n) packs with (n)
cards in each pack; that is, one pack for each Operation, and
within each pack one card to relax each residual. These cards

must be serially numbered and kept in their proper order.

1h

There is one pack Of (n) residual cards, and these too
must be kept in their proper order. it he outset, the initial
residual will appear in residual field number 1, columns 11 - IS,
and the new residuals will be punched progressively across the

card until the last field is reached, and the card is reproduced.

COI IPUTER CONTROL PAI-IEL

 

Considerable knowledge of the calculating machine and the
operations controls section of the control panel is necessary to
understand the functioning of the machine. However, the machine
operator need only be familiar with the reading and punching of
residual values fer the relaxation.process.

In reading, five wires are used to readpin the value of the
residual, and one for the X (negative) pickup. FOr the first
step the five are plugged into readigg hubs (holes) 11 - 15, and

the X pickup is in control readipg hub 6. As relaxation.pro-
gresses the wires for reading must be moved, between calculations,
from.the reading hubs for the current residual field to the reads
ing hubs for the next residual field. The negative pickup wire
must also be moved along from control reading hub 6 fOr residual
field 1 to the corresponding hub fbr the position of the read-
ing wires. This wire is moved along one hub at a time.

Five punching wires must also be indexed along between
calculations, followed by'a skip control wire. For the first
step the punching wires would be plugged into punchigg hubs l6 -

20 for residual field number 2, followed by the skip wire in

hub 21. The skip wire is necessary for thezresidual card to

 

 

PROBLEM ‘ , IBM PLANNING CHART: TYPE 602A CALCULATING PUNCH
' WITH 8 COUNTERS & STORAGE UNITS

 

TRADE MARK

COUNTERS STORAGE UNITS

STORAGE UNIT puNCH UNITS

DIVR.—MULT. DIVIDEND *UNITs POSITION MUST as WIRED 10
III I 6R 1‘ 7L ' 7R 8L

- OPERATION

l

I

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Le «~—+-~-e-+

'3 l_L
1,. 1 +__;+
1 A “I I

.

X—l s o outlgo to
x_3 II II I 11 II
I
ilot selectqr l, ' up g X col. 1 I
in opera r card from CD rol brushes
uplby X col. 3 ,

selectqr 2, ,'
II
sclec 3, up'by X col. 5 I
I i resi card |
selectqr h, uplby X f negatiye
. 11
1’1 a0
a X f eg *
opera value rom re brushes

select r S, _ up

Pilo

selectdr 6, .

up'by X f

negatiI
II

 

 

 

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PRINTED IN U.S.A

 

\wmeO‘a-nnvmm g-IOv—vWW ‘1

TYPE 602 A CONTROL PANEL.

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WITH 8 COUNTERS 8. STORAGE UNITS

CALCULATING PUNCH

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INTERNATIONAL BUSINESS MACHINES CORPORATION

 

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5 —CONTROLREADINGWWW—45'”
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15

be skipped out of the machine and the next calculation to
'begin. These wires are moved along until theydreach field
number 1h where the skip wire is plugged into punching hub
number 1, and field number 1h must be reproduced into field
number 1 in a new pack in order to continue.

Although reading and punching is ordinarily done in
adjacent fields, this is not essential to the operation of the
machine. If for any reason a field is incorrect, it may'be by-
passed.by leaving the reading wires where they are, and moving
the punching wires to the next field. Of course, the reading
wires must also byhpass the incorrect field for the following
step.

If the reading and punching circuits are wired through
two six pole gang switches, moving from.field to field can be
accomplished by'a flick of the switches. Six pole, 12 posit-
ion radio gang switches have been feund to be ideal for this
purpose. Field 1h is not used. The reading switch is wired for
fields 1 through 12, and the punching switch is wired for fields
2 through 12. The punching hubs that are also used for skipping
must be double wired to the gang switch.

The wiring arraxgement and programming used here is simple
and functionally satisfactory, however it is not singularly
unique. hany of the control circuits could be arranged diff-
erently, but a skilled operator would.be required to make the
changes. Sometimes, due to malfunctioning of certain controls
‘Within the machine it is convenient to make minor changes in

'bhe control panel wiring, and use other control circuits.

HECHAHLCS C MAGiINE RELAXATION

 

Relaxation as put to the computer is in a sense an imit-
ation of the hand process, and is not fUlly automatic. The
largest residual must be discovered in some manner whether by
inspection or by sorting. The relaxer would then select the
proper operation and operator, and collate the packs of cards
representing these quantities by hand or by machine. After
the cards have been run through the calculator they must be
separated again by hand or by machine. This process would be
repeated until the variables were determined to the desired
accuracy keeping a running account of the relaxation accompl-
ished in each step.

The reading and punching wiring must also be changed
between steps, and the skip stop in the skip bar must be
inserted in the slot corresponding to the column where punch-
ing is to begin. Two skip bars may be used for convenience,
since they must be removed from the machine to move the skip
stop.

A relatively efficient sequence of events for one step in
the machine relaxation process is:

Find the largest residual

Choose operator value

Enter data into log

Collate operator and residual cards
Place operator card at front of pack

Insert pack into feed hopper

17

Change reading and.punching circuits
Change skip stop or skip bar

Begin calculation, and.move skip stop in the
extra skip bar if one is being used

Remove operator card from pack after calculation

Separate residual and operation packs

Check computation on the largest residual
When all residual fields have been punched, the residual pack
must be reproduced in order to carry on with the relaxation
process.

For the larger matrices, visual scanning of the cards for
the largest residual, and hand collating and separating of the
operations and.residual packs will become too tedious and time
consuming. Depending upon the character of the matrix, the skill
of the relaxer, and the time required to do these operations by
machinery, there will be an optimum size matrix or point of
division between the two methods.

As the cards have been set up, the values of the various
factors are limited to five digits, however, this should be
adequate for most problems in view of the fact that the accuracy
of the coefficients in the original matrix will generally not
exceed five places. The calculator has no sense of decimal
location allowing the relaxer to assign it anywhere within or
outside of the fields as long as all fields are alike in Oper-

ator, operations and residual cards.

18

DISCUSSION

 

Methods of solving simultaneous linear equations may be
conveniently divided into two groupings: (1) direct methods, and
(2) indirect methods. Direct methods are usually lengthy but
yield results of considerable accuracy if a sufficiently large
number of significant figures is retained throughout. Solution
by determinants, and solution by elimination as in the Gauss
process, Cholesky scheme and the Jordan process are examples of
the direct method. Indirect methods are successive approximat-
ion in nature, and the approximate solutions become progressively
more accurate as the process is continued. The Gauss—Seidel
Iteration process and the Southvrell Relaxation process are two
of the better knom indirect methods.

The solution of a series of algebraic linear equations of
order 3}, when Ll is small, is usually done by one of the direct
methods. However, as _r_1 increased the time consumed by these
methods increases roughly with n3.(S). Thus, as the order n of a
matrix approaches 10, the time and labor required become pro-
hibitive, and solution impractical by these methods.

For certain types of matrices in which the diagonal terms
are large compared to the other coefficients in each equation
(diagonal matrix) the successive approximations methods can be
employed. Fortunately, these systems of linear equations are
’ among the most important and most frequently encountered problems
in applied mathematics. Although the time required for successive
approximation methods is also considerable, it increases roughly

with r_1_2.(5).‘ It can be seen immediately that the time required

19

for relaxation solution of algebraic simultaneous linear equat-
ions as compared with direct methods varies inversely with n.
Thus, relaxation appears to have a considerable advantage over
the direct methods.
The main virtue of the relaxation method.is its freedom
from rigid routine and it would seem to have much more flexibility
and opportunity for computational skill than any of the other
successive approximation methods. The mathematical skill and the
physical intuition of the computer may be used in many ways to
accelerate convergence, perhaps reducing the proportional time
for solution a preciably'belOW'nZ.
The time required for setting-up a problem to be solved
with the punched card computer is somewhat more than for hand
solutions with or without the aid of the ordinary desk calculat—
ing machine. It can easily be seen that to convert the operations

2 operations

table to punched cards requires the punching of n
cards. A pack of n residual cards must also be punched out.
However, a Skilled keypunch operator can punch from 10 to 30

2 4 n or

cards per minute. Since,the set up time varies as n
approximately‘n2 with the higher values of n, there will be a
value of n above which it would be impractical to use this
application of the relaxation method.

Fortunately, most physical problems take the form of an
incomplete matrix in which the more distant terms from the diagonal
are zero. Since. it is unnecessary to have a card for zero

operations values, the theoretical number n2 of operations

cards may be reduced to the order of 1/2 n2. This also reduces

20

the number of cards to be collated and separated which, in effect,
raises the value of n for which the use of the collator and
sorter becomes expedient.

It is suggested that the abbreviated pack of operations be
made continuous. That is, a card with the code X in column 3
should.be inserted for any zero values which occur within the
group of terms making up the abbreviated operations pack. This
will simplify collating and aid in preventing errors. of course,
all of the residual cards, including those for which there are no
operations cards, must be contained in the pack run through the
calculator in the order that the residuals in these cards be
reproduced into the new field. When computing is to be carried
out with the abbreviated operations packs, the calculator should
be cleared previous to any computations. This need only be done
once, and may be accomplished by running the clear card through
separately or including it as the first card of the first pack.
The clear card should have X punches in columns 1, 3 and S.

The time required.by the calculator to make the computations
is about 3 seconds fer each operation. Thus, the time elapsed
during one pass of the cards through the machine would be some-

hing less than 3n seconds or about 1 minute for each twenty
residuals. However, the machine time is a small portion of the
total time required to solve a series of equations. Therefore,
a more rapidly cycling machine would not appreciably reduce the
total time necessary to work out acceptable solutions.

A.matrix in its initial state is often not in a good form

to be converted into the operations table. It is sometimes

21

convenient to adjust the diagonal terms to the same relative
magnitude or convert them all to a value of one. This, of course,
can'be accomplished by multiplying or dividing through any of

the equations of the series by'a required constant. Reducing the
diagonal terms to a value of one will mcet likely result in some
of the other coefficients'becoming continuous decimals which
must be rounded off to five digits. The errors thus introduced
can be eliminated at any time by computing the residuals with the
accomplished approximations and the original coefficients; and
proceeding with the relaxation process by operating on the
corrected residuals mentioned above.

In the instance of a diagonal matrix (diagonal terms large
compared to the other coefficients) the terms farthest removed
from.the diagonal terms may be quite small in comparison. In the
operations table these small values have little effect on the
relaxation process, and it may prove expedient to assume them to
be zero. Thus, the number of cards which.must be punched for the
abbreviated operations pack is further reduced, reflecting a
corresponding reduction in the number of cards to be collated and
separated. Errors introduced in this manner can be eliminated
at any time by'checkigg, and.replacing the erroneous residuals
by the correct ones as was done in eliminating rounding off
errors.

Perhaps the greatest advantage of relaxation as carried out
with the automatic computer is its overall relative freedom from
arithmetical error. Relaxation is fundamentally an iteration

process, and when carried out with pencil and paper involves

22

columns and rows of digits which appear to the human.mind as an
endless stream of unrelated numbers. It may be monotonous and
hypnotic to the extent that the human computer becomes quite
susceptible to arithmetical errors, and although the errors can
be discovered.it would.take a series of steps to correct for
them. Machine computation should contribute nothing in the way
of arithmetic errors, excepting for the case where erroneous
data has been submitted to the calculator. In general, errors
in machine solution can be detected by checking the relaxation
performed on the largest residual as a regular part of each step
in the relaxation process.

Due to the great variance of convergence rates among
linear matrices, it is very difficalt to make any concrete
comparison of the solution times required fer various sizes of
matrices. An artificial, perfectly symmetrical series of matrices
was devised in.which the symmetrically corresponding coefficient;
and the constant terms were identical in each equation. The
nonszero terms were l,3,7,3,l and the constant term had a value
of l. The solutions of each proved to be symmetrical as did the
relaxation tables, however, the convergence rates do not appear
to be the same, and thus they could not be used as a basis for
comparison.

Some comparison of the times required for machine solution,
and for pencil and paper solution.with the aid of a slide rule
can.be made. A symmetrical matrix of order 8 representing a
stress network was solved. It took approximately 6—1/2 hours to

solve with the computer, and approximately h hours to solve with

23
hand relaxation. In general, it appears that the breakueven
point between hand relaxation solution and machine solution
would be somewhere in the neighborhood of matrices of order 10.

The same matrix was also solved by the Gauss method with
the aid of a desk calculator. This method took approximately
6-1/2 hours which is the same time as for solution with the
automatic computer (relaxation method), however, set up time
would increase the total time for the punched card computer
method. In general, it appears that the break—even point
between these two methods would also be in the neighborhood

of matrices of order 10.

CONCLUSIONS

 

The application of the IBM 602A Calculating Punch to
relaxation methods for solving algebraic simultaneous linear
equations appears to have several distinct advantages. This
machine is the "workahorse" of automatic digital computing
machinery, and despite the development of faster and more complds—
ely automatic computers, it is yet the most versatile and easily
adapted machine. It has become relatively available at a moderate
cost as compared to the ultra high speed computers, and therefore
is no longer out of reach of the lesser financially endowed
institutions. As a general purpose machine in the fields of
accounting and miscellaneous calculations, the cost of the machine
may easily be justified by the saving of labor costs it can
produce.

Of all the automatic digital computing machinery available
today, the IBM 602A Calculating Punch probably requires the least
skill and knowledge of its operation. much of the later com-
puting machinery is impractical for many'types of problems due to
the prohibitive set-up time required. Even then, cost and avail-
ability put them beyond the reach of all but the wealthiest groups.

Relaxation solutions of engineering and other physical
problems need be carried no further than would be indicated by
the accuracy of the data from which the matrices have been con-
structed. Skill at machine relaxation will do much to hasten
convergence of the solutions to the required accuracy, and there

is considerable opportunity for the individual relaxer to employ

25

every technique he has discovered to speed convergence. This
method offers a practical solution to problems which have been
heretofore shunned by mathematicians and engineers.

Considering the availability at moderate cost of the IBM
602A calculator, the inherent accuracy of relaxation methods,
and the opportunity for the individual ingenuity of the operator,

here is perhaps the beginning of a better method of solving

large linear matrices.

(1)

(2)

(3)

(h)

(S)

(7)

(8)

(9)

(10)

(ll)

 

Allen, .3. de 6., Relaxation Ifle Ilods, KcGraweHill, New Ybrk,

19Sh.

 

Bell, Hilliam D., Punched Card Bechnigues for the Solution 23
Simultaneous Equations and Other Iatrix Operations, Pro-
ceedings of the Scientific Computation Forum, International
Business Kgchines Corporation, New York, l9h8.

Fbrsythe, G.E., Solving Linear .5lgebraic Eauations Can Be
Interesting, Bulletin of the American E;athematicalm
Society, vol. 59 (1953) pp. 299-329.

 

International Business I achines Corporation, Principles of
Operation, IBM Electric Punched Card Accounting Lachines,
Calculating Punch Tyne_602—A, NOJ 1ork, 19:4.

 

 

filne, W.E., Numerical Sol lutions of Dif1°c ential Equations
John Viley and Sons, hen York, 1993.

 

National Bureau of Standards, Simultaneous Linear Equations
and the Determination ggiEicenvalues Aoolied mathematics
Series, vol. 29 (1953), U.S. overnment Printing Office,
hashington.

 

National Bureau of Standards, Contributions to t he Solution
of S*m tens of Linear Equations and the Determination of
bl”9nvelueo, tpolied nauuenatics Series, vol. 39 (19553',
U. 8. Government Printing Office, ”as Elington.

 

 

 

 

Salvadori, M.G., and Baron, M.L., Numerical Methods in
Engineering, Prentice—Hall, New York, 1932.

Shaw, F.S., Relaxation Methods, Dover Publications, New
York, 1953.

 

Southwell, Richard V., Relaxation Methods in Engineering
Science, Oxford University Press, London,l l9h9.

 

Southwell, Richard V., Relaxation Methods In Theoretical.

Physics, Oxford University Press, London, l9L6.

 

 

MICHIGAN STATE UNIVERSITY LIB

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