120
131
THS

 

CERTAIN SUMMATECN AND
CLL’BATURE ir‘CRML‘LAS

Thesis for the Degree of M. A.
MICHIGAN STATE CCLLEGE
jack I. Northan‘:
3939

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d5}. ' .-

'3‘.

 

CERTAIN SUMMATION AND CUBATURE FORMULAS

by%'
Jack Iffigrgmm

Submitted 1n.partial fulfilment of the requirements
for the degree of Master of Arts in the
Graduate School, Michigan State College,

Department of Mathematics

June, 1939

a

'o. ~u.

V1-

 

'IID

MATH. La,

9"“rwt-3é .‘ '.
~.._
C’\ C)
$ 3

NEQK

 

ACKNOWIEMMENT

To Doctor William.Dowell Baton
whose suggestions and encouragement

have made this thesis possible.

12.1479

 

 

lo
2.
3.
4e
5.
6.

CONTENTS

Introduction

Extension of Lubbock's Formula of the First Type
Extension of Lubbock's Formula of the Second Type
Extension of Lubbock's Formula of the Third Type
Extension of Woolhouse's Formula of the First Type
Extension of Hardy's Formula of Mechanical Quadrature

Bibliography

17
29

40

52

 

pres

~I c:
Q.‘

kl)

CERTAIN SWTION AND CUBATU’RE FORMULAS
I. INTRODUCTION

Certain approximation formulas in one variable have been
presented ° by which a sum is estimated by taking h times the sum of
every h“h tens. The value of these summation formulas of Lubbock and
Ioolhouse is that they contain remainder terms which give upper bounds
for the error in calculation.

The major purpose of this paper is to obtain extensions
into two variables of Lubbock's formulas of the first, second, and
third type, and of Woolhouse's formula of the first type. Approxima-
tims of a double sum are obtained by taking h]: times the sum of every
h17h tern in every kth array of one of the variables. To these approxi-
mations are added certain corrective terms which involve finite differ-
me: or derivatives. Remainder terms are obtained, and examples il-
lustrate the use of these double summation formulas.

Also a formula for approximating the value of a double
integral, called a cubature formula, is obtained. It may be compared
with Hardy's formula of mechanical quadrature in one variable. a re-
mainder term is again available, together with an illustrative example.

He shall make extensive use of the notation of the cal-

eulus of finite differences. For one variable we define

(1) 4.5“” = {(M) - fix),

 

° JJ'. Steffensen, Interpolation, Baltimore, The William and Wilkins
Cupany, 1927, p. 133. Hereafter referred to as Steffensen.

 

 

2.

 

 

(2) CH“ = f(“%) “ f(x-1t);
(“it -—€})

(3) Ehlf(x): f ):)((X .

It follows that

(4) o §£<x> = 4W 3%

The quantities defined in (l)-(4) are called the descend -
ing difference, the central difference, the mean, and the mean central
difference of f(x) reapectively. Generally h = l, in which case we
writeA, 8 ,CI ,and [218.

For two variables we define

(5) A, A..)‘(x31)= Ae [flu-1+!) “f(mfl

_-. A1[J[(X+';1) - HW]
: f(xuflw) — f(K;‘1+’)‘)£(X+‘;1) +)L()‘}‘l)°

The symbols, A, and A1 , are commutative.

Again in one variable we define a displacement symbol Eh
such that when it Operates on f(x) we obtain f(x + h). That is, Ehf(x)
= f(x + h). Hence A f(x) = (E - 1)f(x), showing that the symbol
A is a linear function of a displacement symbol. For descending dif-
ferences of higher order, A“ f(x) = (n - 1)m f(x).

For two variables we define
4n 0| m 4.
A, A1 f(m) = (Em) (6,-0 Hm).
For m = n = l, we obtain (5).

For the central differences we define

(6) 5x Sith) = 81 [{(X‘t—iqfl ‘ f(x‘fisifl
= 8x[}(X;1+-25- Her-H]

3.4.3!” .3. rue). ‘
_

me - JEN Erie _

I. 1

am.

3.

=Hx+' v+t)- f(x but) f‘+‘»‘!‘s’>+7L("‘fi ‘5'

and for the mean central differences
(7) ,
CI 5x C] S, = T[f(x+l;1+l) ~f(x-t;1+z)
sow-l) now]

The treatment of the remainder terms in this paper de-
pends on certain quantities called divided differences. For one vari-
able we define

{Ga/2,) z JV”) ‘ f@L

4-4
0|

7((dn‘u‘u) :— i("' A)‘ f(au‘l .L

.-a

and finally

flee-m) "("“'“"9‘fl"""“~>,

th

where this last expression is an n order divided difference of f(x)

'with respect to the arguments a0, a1, ... , a . It may be shown that

11

‘ “ .f’(‘:)
f 40; ‘1 ' ° ' 4“) — g (4." “(J "' (4’5'a"")(d.“d'dh) --. (4"- ‘4‘) )

so that the value of a divided difference of f(x) of n‘+ 1 arguments

 

depends upon the value of f(x) at the n,+ l arguments.°
Similarly, for two variables,

/(4.. ,, . flW ~f<w

 

 

° Steffensen, p. 15.

4.

. _ f(é.,cu.j,.)_ f(40.¢,"/(rl/3
f(0m14” 1:0, 44,) _ 1'. - A,

 

z f(<.:1r.)- flashy f(e.;1r,)+ {(4,4)

(‘.",} (la-1v.)

or, in general ,

 

f(aha,” avg.) = i{&-— - (”l-,4.) - f(a'mdn' 4.)

60—4,.

and

I

-, _ m~ «mt-v4...)-few-«.-.;s-«xni
fl“- “WA” 1") (drama-IR)

+ settle—1r.-.)+/<e~-«..,z,,~x,,)
* (e,-o,)(1r,-1r,)

 

A formula of major importance0 for the evaluation of a

divided difference is
f(X,Q,--~a.;1,fie-~Iz~') : (“03”“), i+bh,,(§31t) )

where a,£—féd..a~4( Leas/n, ifxandylieinthesame

 

closed intervals as 5 and “L . The formula indicates that the order
of the partial derivative for each variable is one less than the number
of arguments for that same variable in the divided difference.

In the formulas that will be used, the following factor-
ial notation will appear frequently.
(8) xM = x(x - l) ... (x - v + 1),

 

° Steffensen, p. 205.

5.

 

[as
(9) x = x2(x2 - 1)(x2 - 4) [x2 - (v- 1%] ,
DAMP?“ 1 1 2 3'
00> —-— x<*-é)(x-—i~)--' [x - —————‘“;"],
fail“ Eu]
(11) x .2 X )
X
awn-v run
(12) x =_______.x" .
X

Function (8) is the descending factorial. Funotions
(9)-(12) are central factorials. It is seen that (9) and (12) are even
functions, while (10) and (11) are odd functions. Each factorial is a
polynomial of degree indicated by the exponents. For V = 0, (8), (9),
and (12 ) are assigned the value of l, and hence (10) is equal to x and
(11) is equal to l / x.

The following Theorem of Mean Value for sums will be
used repeatedly.
81 ) f(x) is continuous in the closed interval (a,b)/
32 ) ¢(x) does not change sign in (c,d),
H3) (excel, as ,5!“ (Leia-1,,

1‘

c1) Z mom) = m) :wa ,

(2!
Proof: Since f(x) is continuous in the closed interval (a,b) it as-
sumes a maximum, l, and a minimum, m, in that interval. Then, if ¢ (x)

is positive or zero (otherwise we take - ¢(x)),

a inn» 2 Zen-Hm e M 2de ,

which would not be true except for H2. Hence there exists an N where

méflél such that

6.

N :55“) = Z1: ¢(x) fag) .

But f(x), being continuous, equals N for some x = f in (a,b). C1 is
obtained by replacing N in the above equation by 11? ).

The Theorem of Mean Value may be extended to double
sums. Its use will be illustrated in the following important example.
Suppose we have the double sum '

(P!

i Z ' (3-) (—2)”;
see tee “f" k k ’,t({5;«*)°

The factorials, (-3-) , (T) , and hence their product, do not change sign

 

within the limits of the summation for s and t. It is assumed through-
out this paper that the function under consideration has continuous
partial derivatives of sufficiently high order in x and y. Then the
Theorem of Mean Value applies, and we have

“" "" (P) m

E: ;,,;, (t) (-E) f (amzozwf’yga),

Pit

 

 

where
L-l h-l (f) (t)

091; : Z; Z: Fifi—(’2’) (”3) I

and where min.{£3 ifs max.{f,§ and min. {41,} s 11. a max.(1t.}. These
minimum and maximum limits for f and K may be used because the end
points of the closed intervals in El of the Theorem may be chosen as

the minimum and maximum values of f5 ( and “t ).

2.

tvovariables
. s.
(15) ;(‘*z,

here the rem

1an and divi

7.
2. EXTENSION OF LUBBOCK'S FORMULA OF TEE FIRST TYPE

The interpolation formula in descending differences for
two variables may be written

<11'J>'j((“1.‘~:7*1r ::~{-AA1 1)+R,

where the remainder term may be expressed in terms of descending factor-

 

ials and divided differences as follows:

(14) R 2 (—Z—)m7{(x+i,x 114-1-, x+p~1-7+T+)
Wet/(11711111 ----- 7111—1)

(
1—1‘”({-)"f(x+1‘.~1w 111111111117” " 111-1)-

The remainder term may also be written 0

(15) R 2 Thaw?” (£’1+$—-) + T,- (SE) firfiuifiuf)

“7(7)“ )(Ly‘ ‘1I7fl’t(f"1‘1)*

where “alga?! x t+p -l, y_11,',,11 g y +q-l. {51”1', 11‘4”. each
dependon x+s/h and y-t—t/k.

For p :_ q = 3, the interpolation formula (13) becomes
f(“ivWfl = f(x21)+(%)A17‘(x;1/) 1+3.) [71.-.) A: {(1,7)
+ (7?) A1160“) + (fit?) A: ‘31 1‘09?) + (%)({- -)({-) A: A1, f(x.- 1)

*(--1-—‘;)(-1)Af(11)+g—:-,-}(1,;l(:-.)a A f(x.»

 

° Steffensen, p. 205.

8.

 

3.;1 (1)111;—1>1—2—-1)(e-)(—2~1<1..—-1114131;)

We first sum (13) from s = O to s = h - l and from

‘b ==() 1H3 ti== kis- 1e

 

(16) ‘" “" '“ "' "73:11"; ,-
+ )1: E: R .

When s or t equals zero, the last term of (16) con-
tains divided differences with repeated arguments which may be shown to
be finite in value. Since the factorial coefficients vanish for s or
t equal to zero, the last term of (16) is not affected by the equality
of two of the arguments.

In the first right hand member of (16) it is permissible
to interchange the first two sumations with the last two since all the

summations are finite. Then let

(1?) L" b" (1'), (1')
°<°1 = 2Z7???) (“71”) .
5:0 t:0

From (17), ago: h k. In (16) write the term containing 0(00 apart,

obtaining

9.

h-I k-'
(13) fox+{_;1+%) = hk f(x”)
f" f" . _ 11-1 h-I
‘ 3
+ Z Z “(3&1 A1}("57) * Z Z R1
(=. fee S:0 It=°

where both i and 3 may not be zero simultaneously (1 = j 1: 0).
Nowsum (18) fromx=0 to x=m-1 andfromy=0to
y = n - 1. Since

41-! ‘14 k"

:2: 1: ;)‘(*1{-111{-)= Z Z 71%;),

x20 Y2. 8:0

we have
I'm-1 tin-I 111—1

g 2.111 ... 1171:1111

4h~l

*2

O

11-1 P-l f—l . .
Z— Zea}; A; «Adv f(ll, V)
Uzo (:9 "-0

(c. :11 fa)

 

 

111—1 11~1 111-1 11—1
+> Z Z Z R
xgo 7:0 5:0 ('20

In the second term of the right hand member of (19) we

interchange the first two summations with the last two and make use of
the fact that

111-1

(20) 11-1 ‘. _ f 5-. H 1.1
2 An A11) fiaJ V) 2' AA A311 f(u”)
V;o

&:o

 

 

 

O O

10.

Then (19) because

hh-I hn-I

(21) ;Z:f%%) z Lk::f(”;")

I "‘ "-
o

 

 

 

+-j;fl-j;——1¥ 13:15 fflulo

(.:,¢o)o

’M-I

277:71

The remainder term in (21) needs special consideration.
In (16) we find that the partial derivatives involved in the three
terms of R are each assumed to be continuous functions. Furthermore

5 (r) 15(11) ‘

the quantities (T) , (7;) , and their product do not change sign for
the values of s and t employed in the summations. It follows that
the Theorun of Mean Value may be applied to each part of the remainder
tom. Using the abbreviated notation of (17) we have

 

i-I h~l
(22) 77'“ ‘1 f},.<7)+1< (171)
I21. {:0

'— Xrt f,6(§;l

where min. ffsgé-f-é mam“); , min.{1tt3é12 s max.(1(,) and similarly

for f and 17’ . Also xéx'£x+(h-1)/h,y:y':y+(k-l)/k.
When the summationsused in (19) are applied to (22) it

is evident that there is a different i , f , and x' for each x and

a different I: , 1'1, , and y' for each y. The Theorem of Mean

 

 

 

‘

11.

Value may be applied once more since 4 , 0(°8' , and 09,31 are con-

stant during summation over x and y. Thus

(23) 1"" 4'" 11-1 t"
Z— Z. Z: Z R z 1” 09’" (MUM)
X=O 7:0 tea fzo

+111“ (Oif’t (11)“) s) ‘1'1‘ 0<Pffp)t(i)5u’) ’

where Oéfu)’, 5: m+p-2, Oéfiufl,£n+q-2, 01- {asm-l/h,
osx,en-l/k.

In formulas (21) and (23) set f(x”) : Fa mica) . Then

 

 

 

            

 

Len-I he- I 111-: 11-1
(24)
Z E— M v) — LR Z: F(11u;hv)
P- -l
+0111;

+ ,1th d)" lie ((11 K1) + Eh" K03F5(3ZJ1‘1)

Ft .
-Lkaaozwfii(§;;h;)’

where osfij, $h(m+p-2),O’:1tzl1t,é k(n+q-2),
Oéfaé hm-l, Oé’k‘< kn-l.

, -
Formula (24) is the first of the summation formulas that

we have been seeking. It may be regarded as an extension into two var-

iables of Lubbock's formula of the first type with a remainder term.o

 

0 Steffensen, p. 139.

12.

By means of (24) a double sum may be approximated by taking hk times
the sum of every hth term of one independent variable corresponding to
every kth array of the other independent variable. This amounts to us-
ing the corner points only of h by k rectangles in the xy-plane.
Corrections to this sum are determined by other quantities which in-
valve finite descending differences and a remainder term which depends
on partial derivatives of order p and q in x and y respectively.

In practical work the size of the remainder term should
be estimated first for different values of p and q, m and n, and h and
k in order that one may be aware from the start of the error in esti-
mation. If the remainder term indicates an error in the sixth decimal
place, it would be wise to carry all calculations to at least eight

decimal places.

But first the coefficients, 0‘ , in (24) must be deter-

.3.
mined. From their definition in (17)
A-I k-'

«I Z (1)
s I S l L - .

5:0 f:0

Thus 041- may be readily determined from tables for A .o

t

For h=k=5 mfind

 

4w : o<°l : ‘0 IA = °<aa ; ~ 8'

°<zozo<oz = ‘1 °<,, : L"

0(3. 2 4., Z 1 «3“ 2' '16

0(33 :' .014. .
«N.- {no
2: I
Suppose we wish to calculate X . Then
‘1

Lane rude

 

° Steffeneen, p. 140.

l3.

191: Fwy) = («+aoa)l(v+loo) and sum E— 7— POI/W) '

[(30 V:

 

Let h=k= 5, m=n=3, p=q==3o The remainder term is givenby

5-3'3 “(35 ('5) + r§f31c><u(-s) ~ 5". 314” (35) .
(i. + loo)‘ ('14, + loo) (Y1+|oo)(fll+lee) “i (33+IOO)~ (1‘34 I00)“

 

 

 

To obtain the upper limit for the error we choose {. = {a = {3
:1“: ’11,. 2 u, : o The negative sign of the error indicates that
the calculated sum will overestimate the real sum by an amount which
may be calculated to be not more than .00000155*.
The values of mu. skv ) 10 9: ”4,» are given in
Table 1. Values marked in red are used directly in the calculations;

the others are needed to calculate some of the descending differences.

 

 

 

 

 

 

 

 

 

 

 

 

v = 0 v = 1 2 2 v = v = 4

0 100000‘ 95288" 90909 “ 86957 “ 83333

1 95238 x 90703 “ 86580 “ 82816 “ 79865

a, 2 90909 ‘ 86580 " 82645 x 79051 “ 75758

5 86957 X 82816 " 79051 " 75614 .4 72464

4 83353 79865 75758 72464 69449:
Table I

The first term in the right member of (24) becomes
1 1
(25) 25: E F(9«;°'v) = 25(.000818802) .02047005.
$4-0 V 0

Only the nine values in the upper left hand corner of Table I enter into
this sum.

For i=0wehave

l4.

 

 

.2 3
(26) Z <03- 13-4 13*» FUMV)

O

13:!
( U0, 3 U1,3 U2,3 " U0,0 " 01,0“ U2,0)
(U U0,3 U1,3 U2.3 U0,0 U1,0 U2,o)°

The first term may be obtained from Table I while the second term may
be obtained from Table II which has been constructed from Table I.

 

 

 

 

 

 

 

 

 

 

 

ANA”, AVUM
0 - 4762*‘ - 3624‘
1 - 4535* - 3451x
AL 2 - 4329* - 3293‘
3 - 4141‘ - 3150‘
4 - 3968 - 3020
Table II
Then (26) becomes
10(-.000037323) - 2(.000003258) == -.000379746.
For i=1 wehave
3 3
(27) :04". A NW v::)H = “’15 WWW)”
3 3
4—1,.FUML kw): ~~8A9F(l‘“5k‘9 I.

 

 

 

O

The first term in (27) has the same value as the first term in (26)
since 1 /xy is symmetric in x and y together. Relation (27)
becomes

-.00037323 4- 4(Us.3 - "0,3 - ”3,0 "0.0)

'e8 AV(U3'3-U -U U )

15.

= -.00037323 +.0000068 + .0000001176 = -.0003663124 .

For i = 2, (24) gives
.1 . 3 3 3 a
(23) Z cg} Al A1: F(Lu;hv)J ’ = ---;2 AAA; F(Lu; In)! ,
7w , 3 3 3°
... 3 An Fan/kw)! I +- . Me A“ A, F(Lh;hv)‘ ’ .

The first and second terms of (28) have the same values as the second
term of (26) and the third term of (27) respectively. To evaluate the

third term of (28) we form Table 111 from Table II.

 

 

 

 

 

 

A’gA’ Ufl,0 A“ Avunfl
o 227 "‘ 173‘
"L x x
3 173 130
Table III

Then (28) becomes
- .000006516 + .0000001176 + .16(.000000011) = - .00000639664 .

Adding the results from (25), (25), (27), and (28) gives

“to

i: Z: x'y — .019717592 .

X=|ee ‘slee

l

 

In this particular problem the answer might have been

obtained by considering
.1

#f‘fi' = (:43?) 2 (.140416149)2

XzIOO Tzlee X3100

= .019716695',
which indicates that the actual error is almost 9 in the 7“11 deem

place while the remainder term gave an upper bound of slightly more
than 1 in the 6th decimal place. The negative sign of the error as

16.

calculated from the remainder term foretold that the calculated sum
overestimated the actual double sum.

 

 

 

 

 

 

 

 

 

 

 

Firm-e 1

Figure 1 illustrates the technique of stunning by formula
(24). The mnotional values in the left hand member of. (24) would be
represented by ordinates (not drawn in the figure) erected at the
points of intersection of the small network of rectangles. (In the’
figure, h =k =4). The first term in the right member of (24) is cal-
culated frm the values of the ordinates erected at the corner points
of the larger network of rectangles. The corrective terms are then
based on descending differences which are determined directly from the
values of these latter ordinates.

17.
3. EXTENSION OF LUBBOCK'S FORMULA OF THE SECOND TYPE

An interpolation formula ° in two variables involving

the mean and central differences may be written

[13" 3-, Lu] 3,;
(29) f(“fl' 7+?) 2 2:: [(13-7)! (157) D 8" + (I’VE) 8x

 

 

where the toms in the symbolic product operate on f(xsy). The meaning
of formula (29) for p = q z 2 has been indicated in the above refer-

ence. For later convenience, however, we choose to write the remainder

term as

f -
(so) a = (75):?! 'f(X+%"‘I"t'/” X!(P-I);I1+—-:—)

t [33]..

(7) flHiflr“ “‘P">27*Z-11w‘WM):

where the first term contains a divided difference of 2p arguments in
x and one in y; the second, one in x and Zq in y; and the third,
2p in x and Zq in y. ‘

We define

 

° Steffensen, pp. 145, 209.

18.

[2 t] [2 1]

£11; 1‘51.
(31) X : __'___,. E E (A 1:
292,- (aemm! is) 14,-...» A) (h) ,

where the summations refer to s and t which take on all values at
unit intervals from -(h - 1)/ 2 to (h - l)/2 and from -(k - l)/ 2

to (k - l)/ 2 respectively.

Now sum (29) over these values of s and t.

5‘2" Li- ,” t" ' 4'
(32) Z :JL(X+— 5 .7191) : Z— Lxlhafé; 813(0)?)
‘11?) - h: (g, to.

x._._ ..

 

 

 

+L,1R I
[20“
where the other terms in the product have vanished since (1;) and
[ad-I
H11) are odd functions.

Nextsum(32) from 1:0 to x=m~l andfromy=0
to y=n - l, andwrite the term containing )(m (zhlc) apart. But

 

 

 

since
-7 my mi; 12.7. (ac-(1:12 1.. Girl
2:} ) }(*+-i-v+‘:)= 5 7 {Hit
17:. 1;. dial). 12:1). dim) n):- “'3

 

.auasl .50

 

 

19.

K-I 1!.-

+;ZLZR

. 14,1 -(é -_l
% J.

In the second right hand member of (33) interchange the

first two summations with the last two and make use of the fact

“-1

7: 7: 5: s 374:» Sims, fl“)

sun—i- ‘._’i
4 L
Then (3:) becomes
(.43;

1 h," 44)
(34) ’31-! 11-!
2::fl-zv Lk§:E:ftv

1,: ~(‘11-I) (13-01:)

 

I

Z 2: an "'74 of: [f
“’49 ’ ‘
Z :2 m

where the remainder term needs special consideration.

In (32) the remainder term appears as a double sum of
three separate terms and remains unchanged when s is replaced by
- s, or t is replaced by - t, or both. For the first term replace
s by - s and average it with the original. For the second term re-
place t by - t and average it with the original. For the third term
three other equivalent forms may be obtained by replacing s by -s, t by
-t, and both 3 by -s and t by -t; the uitMetio mean of these four

20.

equivalent forms, 11' we omit writing the double summation sign of (32),
[ifl '( Wlhfl
:l:_[((::fl( f(x” x 2m xt(r(-»)- 4141,1144: «leg-d)

fl‘:::)li
[3' )‘53'
. (:)1P tf(x+_lx X4.” -.. X‘kU-J); 7-%lv’1tl’ "*7f_(b-I))

1‘]...
“e ,W.). ,ee w- WM)

lid"

5w
+ (T) (7;) f(x?" " ’ “'x’-"”"): WEE/7'71"” “("03

which becomes
T(z) EPJFE- W/(xi-h _.S_"’)xx+ll ._ X“'({’-'I) 7+‘El1/11+I/1771+(6-I))

_{_(_:__) lUkP(L :)&b]-‘f(Xi§g-,X,Xill - '- Xi(P-l)) 1";2’1/7‘31’ -' 7t(3”") I

or

S DIP] flit]
(T) (T) f(“f'xleu - - - WP"); 7r—g,1,1eI,-~1m-4) '

Since a divided difference of (2p + l) arguments in x
and (2g + 1) arguments in y may be expressed in terms of partial deriv-
atives of the (2p)th order in x and (2q)t'h order in y, this part of the
remainder term from (32) may be written

5?] fit]

a» {-5-} g

,2
~ Li. - K- P a.

  

  

 

 

 

wherex-ptlé)’; £x+p-l andy-q+1£u; iy-l—q-l.

Using similar properties of divided differences on the
averages for the first two terms of the remainder term in (32) we may
write

 

., IV. 4. I! II; Um..o..'W-rlrhh.v
..

Esq...

\.

21o

 

 

A-l k l
4 [2P3 E3] '
I
“’5’ > [(731%) 25,,.(£21*i-)+2$.({-) {a (“i-2%)
“M "QT [J E] I r
. 2r it

(a {—2) )5 my]

wherex-p+l<-{,l)’;é x+p-1,y-q+1ékt,h;sy+-q-1.

(4?).- (18)! (A;

The central factorials in (35) do not change sign in the

intervals of summation for s and t: and since the partial derivatives
are assumed continuous, the Theorem of Mean Value may be applied to

(35). giving
(36) 1117K kg?" Jigs (i1? +4“ 4‘ yoxf fag?» (X: 1:) - “‘1‘ klhla'fip at. (f: fil) ’

where x-p-i-léif é 1+p-l, y-q+lé;,1?’: y+q-1,
and:-%+l/h£x'éx+3§-l/h, y-%+l/ksy' sy+%-l/k.
When the summations introduced in (33) are applied to

(36), the Theorem of Mean Value may be used again, giving

(37) ’h‘» K up f1,» (fr/k 4")+ 4"” 92, o,at(£;u.) " 7““ yzp,atfzf,4f(f3i1(3))

where 1-péf.,3’,ém+p - 2, l - qéuhufisn +q-2,

and Van-£553; s m-?§-- 1/2h, 1/21: -%£1¢,s n --%- 1/21c.
Finally, in (34) and (37) replace f(x“) by

FQLa+EiLth+3§L) . Then

L“ I kfl~ l

 

 

3; gm M 2' 2(—
P4 -r .-. a _. ‘31 '3
+>—‘ X2"? 8: 5 PUMA! hwhz) ' i

22.

If ‘
+ LI 1" 1‘ kelp“: Ef,o(fu'1(') + ‘2 t1,“ kg"? C" ({3’ I“)

if ‘t
.——L I! 1am yap,” Emit—(fUt‘I) /

where h(l - p) + (h - 1)/2 éfuf, s h(m+ p - 2) + (h - 1)/2,
(39) k(l - q) +(k - 1)/2 s«.,u, g k(n + q - 2) + (k - 1)/2,
osgsrm-l, osmium-1.

Formula (38) is the second of the summation formulas
that we have been seeking. It may be regarded as an extension of
Lubbock's formula of the second type with a remainder term 0. Formula
(38) enables one to estimate a double sum by taking hk tines the sum of
a series of terms which would be situated near the midpoints of the
sides of h by k rectangles. This estimation is to be corrected by
terms involving central differences and by a remainder term involving
partial derivatives. In comparison with (24), formula (38) has the ad-
vantage that only central differences of odd order are employed. In
practice it is easier to apply if h and k are odd integers for then the
value of P need be obtained at integral points only. For h = Zc,

k = 2d, the values of F(2 on + c - is 2 dv + d - %) are not usually
found in the given table.
Before formula (38) may be used, the coefficients, x

:91) ’

must be determined. From their definition in (31), we find

 

1.; 1 A: r .1
‘ [ii 1 2’
You,” : at; . (5:) ° (2:): 2 (it?)
..lL:)_ - ‘2?)
= P - P,

2.; f)

 

°Steffenaen, P e 145 e

 

23.

where tables 0 for PM- have already been calculated. For h = k = 5,

 

we find
11,0 = Kan. : 1' Yam. 3 004
X”, = x“ = -.072 Y“: Km = -.00288
x". = K”, = .00928 X.“ =. .00020736
X : .000003444736.
5,5
liq. (It,
In order to calculate E E , let

 

F110;”): (“Hafiz/+100) and sum E— ;— PM»).

Let h = k z 5, m = n =3, p —_-_q = 3. The remainder term becomes

12:. 3‘

{egalbfi A, + i"3110,g 6! - 5' 3 k‘I‘ (é!)
(f.+'°°)w('k‘+loo) (faflooflfiauoo)? (f,+too)7 (1‘3“.0 '1

 

 

 

To obtain the upper limit for the error, we choose
3,: macaw 4, 8=".=°-

Hence the error in calculation is positive in sign and is slightly less
than 4 in the 10th decimal place. That is, the sum calculated by
formula (38) should underestimate the actual total slightly.

Table IV contains the values of F( SIN-2 ; 5' W1) X ’0?= Uh»)
The nine central values determine the first term in (38) which is
(40) 25(.000788390) = .01970975 .

Since the original entries are accurate to nine places
only, (40) may be in error in the 84Ch decimal place, a result which is

larger than the error indicated by the remainder term. This can only

 

° Steffensen, p. 145.

J

 

24.

be remedied by increasing the accuracy of Table IV.

We shall , however,

carry the present problem through in order to illustrate the technique

of calculating the central differences.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

v=-2 u=-1 v=0 9:1 v=2 v=8 v=4
-2 118147 112057 106564 101585 97050 92902 89095
-1 112057 106281 101071 96848 92047 88118 84502
0 106564 101071 96117x 91625‘ 87585" 88794 80860
1 101585 96848 91625*' 87844x 88445‘ 79879 76605
2 97050 92047 87585‘ 88445“ 79719" 76818 78185
8 92902 88118 88794 79879 76818 78051 70057
4 89095 84502 80860 76605 78185 70057 67186

Table IV
In Table V the following pattern has been used for
A = -2 to u : 4, except for the first column for which the values

may be found in each row of Table IV.

6|Jg.4
S’L’T-a
<S‘J4,4
<5‘JAH4
6 LIL-a
5 U532

8311,. , x
8'0 8 ”’7‘“;
5 u“ 5’0"
81 Uu,3 “’2

Am . I .... D lib

25.

In Table VI the pattern for V: .. gen, 2 4 is

8 U- x
I,V 8- 8U.i’y

6 U."

SUI,” UX

8 Ugly 88 27,11”

In Table VII the pattern for v = - :35, 2 :35 is

(S U‘Zlv I
83 UN,” ::1 Udi,” 8183 U-"v SIS’U

3 U-Lv 1 1 ~l,v
8 Uo,v 3 L. S 8 U0]! 8
63 U 88 U43,»

7 "V 8 S; Um.” a. z
6 Ualv 8 S? S 8 U2”! SJSSU

3 U41,” ‘ aiv
5 u.” , S 5' Um
83 U 88 U31.

For i = 0, j = l in formula (38) we have

 

( It at
41) X -'
§M+ ~fV+
on. 81'- (S; F( ‘1’ 1)!
7': t
= 8v [UoIa‘i + U|,a"i. + 01,3": '— Uo,-—‘5 _ U5"; ~ Uapfi] I
which, from Table V, is
- 8741 - 3566 - 8406 -(-4954) -(.4728) - (4512) = .000008476 .

For 1 == 1, J = O we obtain the same value.

26.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

fl: -2 A = -1 x“ = O a z 1
~6ooo ~5776 ~5498 ~5287
597 566 ,‘539 x x514 x
~5498 ~88 ~521o ~79 ~4954 ~77 ~4728 ~72
514 487 462 442
~4979 ~4728 ~4492 ~4281
~4585 ~4801 ~4090 ~8899
887 867 x;349 4 x 888 X
~4148 ~46 ~8984 ~44 ~8741 ~42 ~8566 ~41
841 828 807 292
~8807 -8511 -343444 ~8274
1‘: 2 A = 3 A = 4
~5008 ~4789 ~4598
x491 x 470 451
~4512 ~69 ~4819 ~66 ~4142 ~64
422 404 887
~409o ~8915 ~8755
~8726 ~8566 ~8420
820 x 804 292
~84oé' ~42 ~8262 ~86 ~8128 ~85
278 268 257
~8128 ~2994 ~2871
Table V
=1 -% = % V: -% U: 2 %
~5210 ~8984 ~88 ~46
256* 198* 4 2
~4954 ~8741 ~79 ~2 ~44 0
2x 5* 2 ~1’
~77 8 ~42 ~1
5 1
~4512 ~8406 , ~72 ~41
198x 144 8 -1
~4819 ~8268 ~69 o 4 ~42 7
~1 6* ~12
28518‘71 ~66 ~1 ~86 -5
2 1
~64 ~85

 

 

 

Table VII

 

 

27.

For i=0, j==2, wehave

(42) Xe“. 6: S: F'( 574+: (9+2)

’4
rl‘
\—

I

a
fi

 

 

!
v.”
3.

mic
a

=(' 072)Sy[U01i +Uy2’1, +U2,2’i ‘ Uor‘é - U'I'f - U378] 1

which, from Table V, is equal to -.000000006696. Similarly for i = 2,
1‘— 00
For i =3 =1, formula (38) gives

(43) 1% , i

)1, 15 5 H5“: we)

\

 

 

i
which, by use of Table V1, is equal to
(.o4)(144 ~ 198 ~ 198 + 256) : .00000000056 .
For i=1, 3:: 2 wehave

J. I
13. 3‘5»

(44) X2, M8 8: F(5'1¢+1o5'u+2)

 

which, by use of Table VII, is equal to
(~.00288)(6 ~ 8 ~ 2 + 2) = ~.00000000000864 .

N

Similarly for i 2, j =1.
For i :j = 2, formula (38) gives

Ii

(45)

km 5 8: HM "+)l:l

L
a.

: (':00010736)8 5, [Lin/2‘: -UL "U 1‘. +U' U],

- ‘—' o-
1 a’ J' ‘

to“

which, by use of Table VII, is equal to .9000000000010368 .
The sum to nine decimal places of terms (40) - (45) is

.019716689, while the actual sum is .019716695' so that the underesti-

34H?)J.1{.‘ mg:

28.

th

mation is 6 in the 9 decimal place. In view of the accuracy of the

original set of values this error may not be considered extreme.

The applications of formulas (24) and (38) to

“it ((9

Z x, show that the latter gives, in this particular case,

 

Xena ’:/00

the greater accuracy when each is written out to the same number of
terms. The coefficients in (38) decrease more rapidly than those of
(24), but a comparison of the two remainder terms, for the same p and
q, involves consideration of partial derivatives of different order.
The upper limits for the two errors will depend on the particular

function used and the ranges of summation.

29.

4. EXTENSION OF LUBBOCK'S FORMULA OF THE THIRD TYPE

Another interpolation formula 0 in two variables involv-

ing the mean and central differences may be written

 

 

(46) P-’ [2‘+]-I ' [it-+a (+7
f(x+i‘*f37+—'f+%)z i [5.3" (“) US“ +- (1‘.:’)!(%) 5: ]
V-l E113" a . [a ‘1’3 .
I 1‘1. 1 a 23+:
217-518) at + (454—5) 5,
J'xo

+ R 1
where the terms in the symbolic product operate on f(x ti ; 74-33) .

meaning of this formula for p = q: 2 has been indicated in the above

reference. The remainder term may be written

 

(47) R 5 EN}. {—Ll‘] 5“"
7* -—'—-—- —— - +1.11) I f , ‘
(1p)§(‘\) Limo/{5’7 1 a) (41)’( ) 4’L()(+Ih+._.1()
EP*3-l [af4]-' I r
I
(alfjl (av)l( (Z) ir'a'(3’,;14t) I
where x-P+ 135, ‘4 X+P9 Y’q-t 1211‘s y +q, and similarlyfor
LI and 11;
We define 4-2. ‘1—
< . 7T 7 MI EN"
48) S : ...—.... (s ,_
16,23' (ai)!(23')! . I.-. ~(h a. ") (h) )

where s and t take on values at unit intervals from -(h - 2)/2 to
(h - 2)/2 and from -(k - 2)/2 to (k - 2)/2 respectively.

Now sum (46) over these values of s and t.

 

° Steffensen, pp. 210, 221.

50.

 

 

L4) 4142
A a.

where the particular method of summation has caused the terms in the

)[24'+: [131-7]

h d (if) to vanish.

Add :f(x (111,459)). 4—:fl (_x+ +5. )_]£(X”)

€213.— _ -L

...—g—

product involving the odd functions (i-

to both members of (49). Keep the term containing SH =(L‘I) 0?“)
apart and make use of the fact
(50) .
l l a X;
D“ D1)((“3;7*?) “)(Mv) + AU} 7)
+_ A1;(KJV) + AXA.‘ ‘HK’I‘O
l I.

Then (49) may be written

-"— is:
‘5" E ; f(*+é+‘r>7+t+%)=

5: a} tug

31.

“25514) :44- 1) ~ w

far at. g”;

 

:5.
a

+(L-+)(k~+) [Ht-1) + ____._Ax (“51) + ___._.A~f<*+‘0 + AA Hm]
a .2 L,

14' 2."
fig 5‘97"? '38. 1:87 f(x+%;1+4i)

(l: :I :0;

0].

ZZR

~(L a.) --(h 1)

 

 

 

Nowsum(51) fromx=0tox=m-landfromy=0to

y= n — l and make a slight change in notation.

1..-: liq-r ’Mol h’k-I

(52) @thw = ZZZf/Ay-

20:0
41-” I

Z Z7182 -— ZZfl)

+(“")(k'l) :0 ;[j_(a.v) 7" AA. {5430) + Avtp'i") + éaAdg-ZRJflj

’39—! P—(

+ZZZZ5MFS :15.”pr 9)

 

\ oh III

a

32.

From Lubbook'a formula of the third type in one variabloo

 

 

 

n have
‘21-: ..-, a
(53> Zflwi) -— +Zflm) -+— 13—va
-o "’ —0 -—‘ ‘h
+ :Q": [38,: f(A/wy
1‘2: 0
+ 1t Q“ fang (Al-d),
and
[flu-I 41-! 1'9
(54) 2:]((Tlp) 1 k2 f(A2V) 4- “1’ f(Jt/V)l

 

‘
wherel-psS‘im4—p-l and l-qéq;<n+%-l. Also

~

 

 

 

a EJOG'I
_ ’ .i.
Q“ ‘ (M)! Z < h) .
5,-(4-1)
It follows from (48) that
(55) _ . _
3192‘ Z O,“ 6313 ’

 

O Steffensen, p. 1460

33.

We also make use of the following:

(56) 272.:- A‘. if!) z .31: V: flay) .,
(57) Z 7‘ W 2 71' 5—: 7L0”) at

 

 

 

 

 

film A» «a who“ ,

 

 

‘Ix

(59) 2:138 D5: {(1% w) [15:515 uh“)

Substituting the results of (53), (54), (56), (57), (58),
and (59) in (52) gives, after some simplification,

 

 

0 o

 

 

inn-I Ira-c -r a-r n-c 1"
(60> 7 7 Haw-t) = M: Z flaw) + ”—3 Zflw

+Mtlfflav

 

 

 

O

+ Z Z 0 US ink/w):

P-I t—I . 1K 1»

i [:1 S: D 8:i-‘7({A;V) ’

h_(__l«-:) :750‘19
3-1

a..-
+>: 011' US v 3%”):

I I

 

 

+
”m

 

 

+2: Ho‘tfilthtv 4— :“Q1P£h(,;)
“H 1H “:1" a?

 

The remainder term in (60), which consists of the last
three terms, may be simplified. Upon application of the Theorem of Mean
Value the first two of these terms may be written

(61) u «u 0*? 32,, (mm) + m C94. for, (315%),
where Offié m- l, l - qé‘t,<. n+q - 1, and
l-pss’,sm+p-1, Oak‘s n-l.
To evaluate the last term of the remainder term in (60)
we first apply the Theorem of Mean Value to the remainder term of (51).

Then

(62) Z : R= Sui", (‘L'H .WtfiJ

-0.» .153;
-— Swat/f. (fli’),

wherex-p+lé5-',f é x+p, y-q+ 121L172 y+q, and
x+l/héx'2x+1-l/h,y+l/kéy‘éy+l-l/ko

lhen the summations over I and y are introduced, the

 

 

 

PI‘J

Theorem of Mean Value may be applied once more, and the final form for

this part of the remainder term becomes

 

. I ... .. . .... 29‘5-‘5‘337

35.

“no: '31-

(63) > E iii—R- -— 1“» XP.£P.(£;1¢)

'150 -(‘1;3)-(h-a

 

 

 

+ a u so 13 71,33 (a u > « ,.. 5» ar 291mm I»),

where 1 -p£-3’,,3’,ém+p- l, 1 - qékhussn +q- l, and

1A2); s m- l/h, lflsa, s :11 l/k.
In (so), (61), and (63) set f(lyvhmujko). Then

Lfi-l k1!" ‘h-I 11v!

(64) Z 2%“) = M: Z Z raw)
+J’LLL:F(L‘.- ”0““), ha. ___._) :FW a»)!

4-»

+ («w-n F(““>k")/ /
‘4- O

O

 

 

 

 

 

 

+§§OIID 8:14F(Lu‘u/+:ZQ E18: Him“):
a-.. I"
+E: :75 1‘ 1’ D S: Fm kv)
((1 fiz") °

1" VP,“ x Q”, EP’°(£;KJ +k‘t1anQ‘t Ea;- (Yuk)
+ A”; x Sum Fq,,(3’,-,u,) + {tux 5%: fi‘dffim)

AP
“A k “H” Shit Hwy (ff/K").

36.

where 0 if, é h(m - l), k(l - q) éflutb‘k‘ :5 k(n +q - l),
Ml-méfifij}é-Mm+p-U,OékfisMn-U,
lea:- hm-l, lék,ékn-l.

Formula (64) is the third of the summation formulas that
we have been seeking. It may be regarded as an extension of Lubbock's
formula of the third type with a remainder term 0. The summing is done
It the corner points as in formula (24) which has the same first term
as formula (64). For identical values of p and q, more terms are
used in estimating by means of (64) than by means of (24) or (38). The
corrective terms in the present forxmlla involve the less known mean
central differences. The remainder terms of (24) and (64) are not di-
rectly comparable because they contain partial derivatives of different
orders, although it will be noticed that the remainder in (64) involves
two extra terms. The remainder term in (38) is the best of the three.

We shall illustrate the application of formula (64) to

am we '
E $ __’_._ whenh=k=5,m=n=3,andp=q=2. The
sze «lane X1

following coefficients may be found from relation (55) and tables for

00
Q1.- 0

Q; 2 -. H- 52’0 : S." 2 —"
o” .07“. SM = ..4
$9"! I: O 06-". ‘61‘ $0,? : 510,. = . a q 4 H

The value of the remainder term is

{137.0739 Us + 5"“ 5:107“) .qu
(fi+100)5(u,+loo) (fa-1- um) («not—(co)f

 

 

 

0 Steffonlen, p0 1460
°° Steffensen, p.147.

 

..enon
_

37.

5",? (.2444) 24
(if + (co) (141*4- loo)f

t a
5‘3 (2944)::
(3’31. 500)f(“,+100)

 

 

5'31(.oo5+/eu) 5'70
Ur“ '°°)‘ (“51”“);

I andwearesumming Z:F(“J°)
(A 4- (cc) (\H loo) 1 : the

To obtain the upper limit for the error we choose f, = 11 , = 0,

)

where F04”):

 

7;: f, = f;=’£,=1t,="*;= -5. and {“s(, = 1. The error is then

th

indicated as being less than 2 in the 7 decimal place.

Values of F(LA'I*V Kloq ;- U are recorded in

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4,.)
Table VIII.
V: -l V=O Vzl u=2 v:3 v=4
-1 110803 105263 100251 95694 91533 87719
0 105263 100000 95238 90909 86957 83333
1 100251 95238 90703 86580 82816 79365
A 2 95694 90909 86580 82645 79051 75758
3 91533 86957 82816 79051 75614 72464
A 4 87719 83333 79365 75758 72464 69444
Table VIII
The first term in (64) has a value of
(65) 25(.000818802) = .02047005 .
The second term becomes
(66) 10(“0,3 "' "0.0+ U1,3 " U1,0 + U2,3 " "2.0)

- .00037323,
and the third term has the same value.’

The fourth term is
(67) “113,3 - 0’3 - 113.0 ”0.0) =-— .oooooss .

The fifth term in (64) may be written

3

8 3
(5) +QIDSUF(S;5v)

3
+ Q: U 8v F003;")

 

 

 

Q). U 8y F(°’{’9

0

Z “.k D 8, [U03 + U”) fuzfi - Us”- Una ~ [13"],

which, from Table II, is equal to -.4(.000003503) :: -.0000014012 .

The sixth term has the same value.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

05,0“, 11:18.11“,
-1 o 5276 *3988
0 - 5012 ~3788
__ 05, aw... DJ. 05. vi .
1 - 4774 -3608 0 251 190
A. A
z - 4557 -3444 s 190 144;]
3 ‘- 4358 4529‘ “bl. I
4 - 4177 -3157
Table II
For i = 0, j: 1, the seventh term in (64) may be
written 3 z
(69) S [I 8“ C1 8,, Hing-Ml
0’2

 

3

: -—.9 C35,(Uo,v +2 Uw +2 Um+ 0,”)0 ,

which, from Table II, is equal to
- .8 [(-.3788) + 2(-3608) + 2(-3444) + 0-3294)
-(-5012) - 2(4774) - 2(4557) - (4358)] =- -.0000054768 .

39.

Similarly for i =1, 3: 0.

For i =j =1, formula (64) gives us
(70) .16 D $4., D Su ( UL} _. U3,o— U0,’ + U°,o) ,

which, from Table I, is equal to .0000000024 .

Combining the results of (65)-(70) gives us .0197166364
which underestimates the actual sum by slightly less than 6 in the 8th
decimal place. This checks with the fact that the error as indicated
by the remainder term is positive and has an upper limit of B in the

7th

decimal place. A larger error in estimation was made when formula
(24) was used with larger values of p and q. Because of this, the

upper bound for the error should be calculated from the remainder term
before any of the tables of differences needed in the summation formu-

las are set up.

wlj“: ii I, .

40.
5. EXTENSION 0F'WO0LHOUSE'S FORMULA OF THE FIRST TYPE

In one variable Woolhouse‘s formula of the first type is
based on the same principle as the Lubbock formulas except that the
former employs derivatives instead of differences in its corrective
terms. 0 Its derivation depends upon writing the Euler-MacLaurin sum-
mation formula in two different forms and then eliminating the integral
between them. Since the Euler-MacLaurin summation formula in two inde-
pendent variables with a remainder °° has been recently developed, it
is now possible to obtain an extension of Woolhouse's formula.

For later convenience we define a polynomial in two in-
dependent variables x and y, Bm'n(x,y), such that

11-; 4|“!

 

(71) Ax A1 8%,” (’8‘!) = 4a at X 7 ,
and
" 3' l I
2 :: a“ a. [3 . _ K, 1
(7 ) Dx D1 819., (K, ‘1) (1nd)! (“'f)'. ““1”“ ( 7)

where the first subscript of B denotes the degree of the polynomial in

z, and the second the degree in y. D, represents the 1th derivative
with respect to 1:.
Such polynomials are called Bernoulli product polynomials

from the fact that

Bm'n(x,y) = Bm(x)- Bn(y),

where Bm(x) is the Bernoulli polynomial of nth degree in one variable.°°°

 

° Steffensen, p. 148.

°° W.D. Baton, A Remainder for the Euler-Maclmurin Smnmation Formula in
Two Independent Variables, American Journal of Mathematics, Vol.LIV,
April, 1932, p. 265. Hereafter referred to as Baten.

°°°Steffensen, p. 120.

1‘

..If

41.

The values of the product polynomial for x = y = O are the values of

the Bernoulli product numbers which are designated by the symbol Bm n.
9

._ , th
It follows that Bm,n — Bm Bn where Bm is the m Bernoulli number

in one variable.
We also define a function, g'n(x,y), of period 1 in
x and y which equals Bm,n(x'y) for 05 x4 l and 0 5 y <1.
m! n!
For all x and y Bm'n(x + l,y + l) : Bm,n(x,y). The properties
of this function have been discussed elsewhere.° It has immediate use

in the Euler-MacLaurin summation formula in two variables which may be

 

written
«a-I “I." P 3' “‘9‘
7 ..
(3) Z 2 264) = Z Z gitéat-(Xd)! I
Xzo 3° (:0 3"" o o
p a 4..
~ — 011' rd!
5 ”'~ ’K

n ' tr 1‘ n+4.
JIBWLMMWLA ,

where 6131.027) ': 3t'~'/f"(x’7) . The function g(x,y) is continuous in
x and y and possesses continuous derivatives of all orders in x

and y which are integrable. The last three terms of (73) constitute

 

° Baton, p.268.

If!
4
t

V” 'fiv

42.

the remainder tam.
In (73) let g(x,y) = f(hx,ky). Then we have

 

664(51):!1" Effie/:21) where 5114,95): i._"’_q(Lx,k7) .
mar multiplying all members by hk we obtain

1». Lu kt
(74) “22M: ,) ZZ” a, 50,)“

P k“ (ma
6 ‘0'- f
-Z[Lk B‘II(O’—)Egh(x”)d’l
11.1.. [in
/‘k (Mn'i(tvd—l
f+n f" ‘k 1:
-1, h f/BM (13.)le "lawn”,

Another variation of (73) may be obtained by replacing
g(x,y) by f(x,y) and 613(x,y) by Fij (x,y) and also by replacing
m by hm and n by 1:11. Subtracting (74) from this latter variation,

we obtain
L1. -1 ‘11- -:

(75) L;f(*1)= bk : Z {We
Z 2w r 0,147"

 

43.

[‘10-—

P he.
a 1' —
- 21““) 35,401?) EMWW/

In

0

 

O

8’ In»
P . -— f—
- Z f(k (it-l) BP.1}(TIO) 5w,3'(t’7)d
1:0 °

+ geek... [/ 31%th 5+.,,+.(“5")”“’

~ I / s,,,(w5,,,t,.mde.

Formula (75) may be compared to Woolhouse's formula of
the first type in one variable. It will be noticed that the first term
in the right member of (75) is identical with the corresponding term 1m
formulas (24) and (64). The second term is the corrective term which
involves integrals and derivatives of f(x,y). Since the term in this
double sum corresponding to i =3 = O is 0, the evaluation of a
double integral is avoided. The remainder term consists of the last
four terms and offers the same difficulties as the remainder term of
the Euler-MacLaurin summation formula in two variables. For actual use
formula (75) is not as convenient as desired. It seems that further
work might be done in modifying (75) so that the remainder term would
be more adaptable to numerical evaluation. We shall, however, illus-

trate the use of the formula for a simple example where the remainder

term is zero.

9 9
.12.
TosumE 2X1 let h=k=5,m=n=2,
X30 =0

p: (12 3. Since f(x,y) :: xy, wehave (f(x/f) =_"_‘1_ and

.qu. a

44.

 

F = X113 El :: 3:33:
to 3

3
F20 = 2 X351 E; =- ”3"
F2: 2 2X72 Fla 7' 2x1“!
E, I X111 E3. :- AX,

It may be verified that all terms or (75) containing third order per-

tial derivatives vanish. We also need

 

Blo:Bo::~—:1—) BiczBozz-IT-J B.R=Ba¢:-7L5:J BH:"1~.
The first term in (75) is 65(625) = 15625.‘ The
second term becomes, from symmetry,
a 3 to [o I ,o ,0
«when + amen!)

° to l0 to

I ~2q-(’?) X174! I - 6340+”) #va ‘ )

o
O o

 

+’2(-I2v)(-3';).1x,1

 

which equals 65600. 361100 the double sum is 81225 which checks.

45.
6. EXTENSION OF HARDY'S FORMULA 0F MECHANICAL QUADRATURE

In one variable Hardy's formula of mechanical quadrature
may be obtained as a special case of the Cotes' formula which depends
upon integration of the Lagrange interpolation identity between fixed
limits.° A set of formulas for mechanical cubature may be obtained in
a similar manner by integrating the Lagrange interpolation formula for
two variables between fixed limits for x and y. We assume the
Lagrange interpolation identity °°

(76) . _ f 5 P(X)P,(w) ,v
f(x/1) — Z, 2 P (“)P “03‘ )+ R,

 

 

 

where éorfl'l _ Erna—J"
x _
30):???“ ' PW)" ‘17—’17"
and

[n+1-
R=X ] f(X,o+'-~t+;7)
71354-134

f(x/71°»: ‘)

[5'49" [Eu-’3'”

.. A ‘1 f(x/01 il/ " 1"} 7'01t0“.ts)-

We now integrate (76) from x = -m to x = m and from

y = on to y -.= n and make the following substitutions.

 

 

Then

 

° Steffensen, pp. 154, 167-168.
°° Steffensen, p. 224.

46.

(77) jfl/EMMM, = Z Z %Qf(w) + R

If we choose unity as the intervals of integration for
x and. y by letting f(x;” = F(3XT.. -, 31L; , then (77) may be

written

where

 

 

e F({i,f;) + FE???) + F(‘:"=.:I:":.)+ Witt).

tav
The remainder term in (78) may be shown to have the form
Eff-2] E15+g
R I 011-» and” (i’ 1") + 0'1“ 5154-1. (in) 1‘.)

sea heat

—-0 O

2".» 31‘ 29+1’as.*;(

is} I"),

where each { and a lies within the finite limits of integration, and

 

5". Cu... mud .II 11.

 

”all,” quw

47.

[4th] 1” E k]

= :L X 14C .
31“ (1h)! («11»)

 

1k+l

_ Edd

Values for the coefficients V“ , Vv , and 0:,“ may
be obtained from tables.° If, in formula (78), we let r = m = 3,

s 2n =3, we obtain

‘ i’ t
79
( ) [/F(X;1)d447 :_ “705100 [7397* E.

+ 73*‘f(€le+Ft-OI) + {3752(F;1°+ 60:)

+quz(fi.,o+fi:,o,) + 737 F

tn

1- 44.556 F2” 4— (as: (in

+ 553:: (Em + in)

+ (107(fi3‘+5'3) + {556(a31+ E23)]

“. 016%- [Eo (you!) + E3 (251") + .016” E"(i’;k’fl ’

where .0964 means .oooooooooe4 .

To the right hand member of (79) we add and subtract the

 

o Steffensen, p. 158.

int. :fl‘}; .7

“a ..u.‘ 5".“ as .'.

13“: z A:

48.

 

quantity ' *7 " Ax A1 E34 . The subtracted part we put

705'609

with the remainder term and make the substitution

 

Ax A E,-, = 54’1”“)-
1 e . a

The added part is combined with the functional values
after the following substitution has been made

6 6
(8°) Ax A13, = a” F: _goo(r_o+r)

+ [20(F;le+ toi)-1°(Eco+ te3)
+ 22.51;, + 36F;
—vo<ea.+e.,)+w(F +F )

to!
— 6({3 +1315)

to give

(81) /Ji/t;F(x1)¢ax1=[éeHH-Fw
, 7o!‘oo

-’

 

p.‘

- $35-$56 (Filo + Etc!) +23§871(Elo+ an)

—/iacz(fi,o+a.,) + 331m E

.II

+ 99m in + a,

M...‘ 1 'n' .—

‘- Av ...

49.

- l17009(th+ Eh”) ‘I- 232¥7(fi-37+F:t13)]

-..“ [Wm + am.) 59M] -' W7 6,10.» a) -

Formula (81) is to be compared with Hardy's formula in
one variable. Cubature formula (79) makes use of 49 ordinates. Formu-
la (81) has the advantage that 8 ordinates, En + Ft“ , have
been dropped out, although the remainder term.does contain one extra
term. Table 11 indicates which 41 of the 49 ordinates are used. The
method of this section.might be used to eliminate any set of l, 4, or
8 ordinates used in (79), but the substitution (80) leaves the coeffi-
cients in the most convenient fomm.

we shall illustrate the use of formula (81) in evaluating

the double integral ’ ’ x
[f are.

x’+ ‘
z a 7

 

For the region of integration.under consideration it may
be shown ° that the upper bound for the error is determined from the

fact

4 (IV‘l-ZA)!
- 3v+1a+c '

{I

 

'F (7,4)

343v

 

Hence the upper bound for the error is

 

 

’ . (I "1 ’
.06# 2A ' -+_#£._ +~-09‘I if" 4..Ooooooooé.
.2 J: .13 H, .2 {3’

 

0 Steffensen, p. 229.

 

‘1".

50.

The functional values needed to carry through the cube-

ture are presented in Table II.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I I _, 5'
x y=2 yzzt y=21i y=23 y=2§ y=2‘ y-3
2 1 18 8 i _2_
85 41 50 13
24‘, 3 78 39 78 39
13 365 197 425 229
2.5. 21 84 3 84 21 84 21
85 365 14 421 113 485 130
2L 10 45 90 1 90 45 10 '
2. .. __ ...—... .- _— ...—.... ......
41 197 421 5 481 257 61
23; 6 96 24 96 3 96 24
25 425 113 481 16 545 145
2% 51 102 51 102 3
229 485 257 545 17
3 .22 __.27 i2. .....27 .1.
13 130 61 145 6
Table.XI.
The required value, apart from the remainder term, is
I e L. - 35' ft JUL-r12_ +JEL4.jB_
WI: é $331+(5) 1+ 6‘ (#1: 4.1: 48! #91)

 

+ 235372 (33’— +

45‘ + 5/ + 45')
m7

1‘7? as"? 515’?

— Led—ii- ”- +__’3__
’736?(LH Lu 6! ‘1

-t 3 (
3171a “3 ‘% ['3 I“

51.

 

 

 

3 +__:L 1. fl 3

+ 79771<229 I3 .14 + ,1)
a. __I_ + 3 I

+ 3’57(/3 ‘* /5 + e)

— 1.77002 {1“ 77 f- 7’ +L+L¢ -l-—/”2 + I02 ‘79_
:5 425 345 365 a: +7; {“5 5g,—

 

 

£20c>0.1/ 34a5’

’l

Since the value of the integral is .200021343, the
th
actual error is only 2 in the 9' decimal place while the upper

bound for the error is 6 in the 9th decimal place.

 

“"

 

‘s ‘1‘ w«

52.

BIBLIOGRAPHY

Baton, W.D., A Remainder for the Euler-MacLaurin Smmnation Formula in
Two lnde endent' Variables, American USurnal of “Ethematfcs,

001. g 0. , Ipl‘II, 1932.

 

Camp, 0.0., Note on Numerical Evaluation of Double Series, The Annals
Mematical Statistics, 731. UTII, I§31e

Irwin, J .0. , 0n Quadrature and Cubature, Tracts for Computers, No. X,
ran Lon, Caflaridge Ufiversity Press, 1923.

 

Milne-Thomson, L.M., The Calculus of Finite Differences, London,
MacMillan and Company-,Tix—nitef, 1933.

 

 

Narumi, 8., Some Formulas in the Theo of Interpolation 2g Man
"Ifi'd'ependent VariaBIes, Tol¥ofi Mathemaficics Journfirfiol. 18,
1920.

Nader L. , Interpolationsformeln fur Funktionen mehrerer Arggente,
' Skandinavish AktuariEtidskriiT—T'g, 23'“.

Norlund, N.E., Lecons sur les Series d'Interpolation, Paris, Gauthier-
ViIIars at Companie, 1926.

 

Norlund, N.E., Vorlesungen uber Differenzenrechnung, Berlin, Verlag von
Julius gpringer,1924.

Scarborough, J.B., Numerical Mathematical Analysis, Baltimore, The
Johns HOpmiPress,193O.

 

Steffensen, J.F., Inter olation, Baltimore, The Williams and Wilkins
Company, 1927.

Whittaker, E.T. and Robinson, G. , The Calculus 2}: Observations, London,
Blackie and Son,Limited, 1926.

 

. i. uni-r: “vi-..nmo :» _ -

 

my“; 1|.I.I'l.

I?

 

0
1.
've
r
J I
{1

Q»
'
s. ..
..n
n - .. 1,511. .
w
.v. .
<0. ...»
.0 o
* A
a 1 .1. -«(1.11.. -

 

 

”'11'1111’1111111911111(111(1111‘11'1111611111111'ES