E], 1OD - Pmf

25. Numerical Interpolation, Differentiation,
and Integration
PHILIP J. DAVIS AND IVAN POLONSKY *
Contente
Formulas
25.1. Differences . . . . . . . . . . . . . . . . . . . . . .
25.2. Interpolation . . . . . . . . . . . . . . . . . . . . .
25.3. Differentiation . . . . . . . . . . . . . . . . . . . .
25.4. In tepa tion . . . . . . . . . . . . . . . . . . . . . .
25.5. Ordinary Differential Equations . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . .
Table 25.1. +Point Lagrangian Interpolation Coefficients (3 In IS) . .
n=3,4,p=- [“;‘I - (.01) E], Exact
n=5, 6, p=-p$] (.01) E], 10D
n=7,8, p=- pi1] - (.1) E], 1OD
Table 25.2. +Point coefficients for k-th Order Daerentiation
(15kI5) . . . . . . . . . . . . . . . . . . . . . . . . . .
k=1, n=3(1)6, Exact
k=2(1)5, n=k+1(1)6, Exact
Table 25.3. n-Point Lagrangian Integration Coe5cients (3 In 510) . .
n=3(1)10, Exact
Table 25.4. Abscissae and Weight Factors for Gaussian Integration
(2In596). . . . . . . . . . . . . . . . . . . . . . . . . .
n=2(1)10, 12, 15D
n= 16(4)24(8)48(16)96, 21D
Table 25.5. Abscissas for Equal Weight Chebyshev Integration
(21n59) . . . . . . . . . . . . . . . . . . . . . . . . . .
n=2(1)7, 9, 10D
Table25.6. Abscissae and Weight Factors for Lobatto Integration
(3 5nI 10). . . . . . . . . . . . . . . . . . . . . . . . . .
n=3(1)10, 8-10D
Table 25.7. Abscissas and Weight Factors for Gaussian Integration
for Integrands with a Logarithmic Singularity (25n54) . . . . .
n=2(1)4, 6D
National Bureau of Standards.
* National Bureau of Standards. (Preaently, Bell Tel. Labe., Whippeny, N.J.1
Pege
877
878
882
885
896
898
900
914
915
916
920
920
920
876

876
NUMERICAL ANALYSIS
Table 25.8. Abscissae and Weight Factors for Gaussian Integration of
Momenta (1 SnS8). . . . . . . . . . . . . . . . . . . . . .
Page
921
k=0(1)5, n=1(1)8, 10D
Table 25.9. Abscissae and Weight Factors for Laguerre Integration
(2SnS15). . . . . . . . . . . . . . . . . . . . . . . . . . 923
n=2(1)10, 12, 15, 12D or S
Table 25.10. Abscissae and Weight Factors for Hermite Integration
(2Sn120). . . . . . . . . . . . . . . . . . . . . . . . . . 924
n=2(1)10, 12, 16, 20, 13-15D or S
Table 25.11. Coefficienta for Filon’a Quadrature Formula (0 5e-g 1) . , 924
e=o(.oi).i (.1) 1 , 8D

25. Numerical Interpolation, Differentiation, and Integration
Numerical analyste have a tendency to ac-
cumulate a multiplicity of tools each designed for
highly specialized operations and each requiring
special knowledge to use properly. From the
vast stock of formulas available we have culled
the present selection. We hope that it will be
useful. As with all such compendia, the reader
may miss his favorites and find others whose
utility he thinks is marginal.
We would have liked to give examplea to
illuminate the formulas, but this haa not been
feasible. Numerical analysis is partially a ecience
and partially an art, and short of writing a text-
book on the subject it has been imposeible to
indicate where and under what circumstances the
various formulas are useful or accurate, or to
elucidate the numerical difficulties to which one
might be led by uncritical use. The formulas are
therefore issued together with a caveat against
their blind application.
mainders.
25.1.1
Formulas
25.1. Dif€emncea
Forwud DiiTerencea
bk=~$:(.--~) if n and k am of same parity.
Forward Diferencu Central Dif.tncsr
25.1.3
25.1.4
2-1 f-1
Divided Mfllerencxm
877

878 NUMERICAL ANALYSIS
25.1.6 where *n(Z)=(Z-%) (Z-ZI) . . . (Z-zn)
and r:(z) is ita derivative:
25.1.7
*:(z&)= (zt-5) . * * (z&-zt-l)(Zt-zt+l)
. . . (zt-z.)
Let D be a simply connected domain with a
piecewise smooth boundary C and contain the
points a, . . ., zn in its interior. Let f(z) be
qalytic in D and continuous in D+C. Then,
25.1.10
25.1.11
25.1.12
Reciprocal DHenncea
25.2.3
25.2.4
25.2.5
Remainder in Lagrmqe Interpolation Formula
The conditions of 25.1.8 are assumed here.
For n odd, (-; (n-l)Skl 3 1 (n-1)).
25.2.7
25.2.8
k has the same range aa in 25.2.6.
n even.
n odd.
(s

25.2.11
wange Three Point Interpolation Formula
f(zo+~h)=A-if-i+Aofo+Alfi+R2
25.2.12
Rib) .065hy(€) = .065Aa (IpI 5 1)
Lagrange Four Point Interpolation Formula
25.2.13
f (&+ph)=A-i f-i+Aojo+Ai fi+Ao fa+Ra
-P(P--l) (P--2)f-l+(p’--1)(P--2)fo
k
6 2
NUMERICAL ANALYSIS 879
25.2.18 R&)
.0049hyce)(€) = .0049Ae (O

880 NUMERICAL ANALYSIS
25.2.24
Taylor Erplnaion
25.2.25 R,,=J:f(n+l) (t) @=$ dt
(For r,, see 25.1.6.)
25.2.28
(zo

f ( + ~ ph) = (1 - p )jo +pji + E&. o +F&, 1
+ E4%, O+F4$. 1 +
25.2.44 8:s 6'- .0131266+ .004368-.001610
25.2.45 6&= 9- .278276'+ .06856'- .01661°
25.2.% R = .00000083lp$,l+ .00000946'
25.2.47
-1's Formula With Throwback
f(2o +ph) = (1 -p>jo+pj1+ Bz (6:. 0 + 6;. 1)
P(P-1) (P-))
+B36;+RJ B2=pv~ B3- 6
1
f (x) "2 s+& (ak cos kx+bk sb kz)
k-1
25.2.54 m=2n+l
25.2.55 m=2n
(k=OJIJ. . .,n)
b, is arbitrary.
(k=OJlJ. . .Jn-l)
Subtabulation
(k=OJ1,. . ,Jn)
Let j(x) be tabulated initially in intervals of
width h. It is desired to subtabulate j(z) in
intervals of width h/m. Let A and designate
differences with respect to the original and the
h
ha1 intervals respectively. Thus &=j (z,,+z)
-j(;co). Assuming that the original 5* order
differences are zero
25.2.56
(1 - m) (1 -2m) (1 - 3m)
+ 24m4
From this information we may construct the b al
tabulation by addition. For m= 10,
25.2.57
-
-
&=.1&-.045&+.0285~-.02066~
@=.01&-.009&+.007725a - g=.OOl&- .00135G
2=.OOO1g
hear Inverse Interpolation
Find p, given jp(=j(ro+ph)).
25.2.58
Linear
At

882 NUMERICAL ANALYSIS
25.2.59
Quadratic Inverse Interpolation
(f1- 2fo+f- l)P8 + (fl -f- l)P + 2 (fo-fp) = 0
Inverse Interpolation by Revemion of !hries
OD
25.2.60 Given f(zo+ph) =fp=C akpk
25.2.61
25.2.62
k=O
p=h+C2h8+C3h3+ . . ., h=(fp-%)/$
ca= -a~lai
-a, 3a; 2l& 14at
c5=-+>+---
al ai a; ai +- at
Inversion of Newton's Forward Difference Formula
25.2.63
%=fo
ale&--+--- Ai @ %+
G At+
2 3 4 - . *
cca=s-4 ..*
a4=a+. % . .
(Used in conjunction with 25.2.62.)
25.2.64
Inversion of Everett'r Formula
%=jo
s: s: s: 6'
aI=s4---- +-+'+
3 6 20 30
. . .
6: ff+
aa=8-a - * .
-s:+s: 4+s:
%=6--
24 +.a.
Lagrange's Formula
a4=a+ 4 . . .
a,= ~
-st+%+
120 .-
(Used in conjunction with 25.2.62.)
25.2.65
Bivariate Interpolation
+
Three Point Formula (Linear)
f(zo+Ph,Yo+ n4 = (1 -P- n)fo. 0
25.2.66
Four Point Formula
i-
+PfL o+ do, 1 + OW)
f (%+Ph,YO+ nk) = (1 -PI (1 - n)fo.o+P(1- nvi. o
+ n(1 - Plfo, 1 +Pdl. 1 + W2)
25.2.67
25.3.1
(See 25.2.1.)
Six Point Formula
25.3. Dmerentiation
k=O

{ {
25.3.3
rn(z) d (*+I) )
Rb(x)= (n+ fn+l) 1) ! (€>?rb(z) +(n+l)! &j (€
t=t (%I (G

884
NUMERICAL ANALYSIS
25.3.22 I 25.3.26
-- a&o-a (A
1 a2.fo.0- 1
1 -f- 1.1 +A, - 1 -f- 1. - 1) + 0(h2)
25.3.23 25.3.27
25.3.a
a2f0 -3- O 1 ~f1,0-2f0.0+f-1,0~+0~h2~ a2f 0.0- - 1
az2 h2
-- a;?-& ~f1.1-~f0.I+f-l,l+f1.0-2f0.0+f~~.0
+f1.-1--2fo, -I+f-l.-l,+O(h”>
Cfl.l-~l,-l-f-l.l+f-l. axay -,,+0(h2)
a 2
axay 212
25.3.28
--- a;+0- I& (-fa. 04- 1 6f1, o- 30f0, o -- a;$ ’-$
+ w-~. o-j-2, o) + o(h41
25.3.25 25.3.29
cfi. o+f-1, o+ fo. 1 +fo. -1
-2fo.o-f1.1-f-I. -1) +O(h2)
cfa, 0-4f1, O+ 6f0. 0-4f-I, o+f-2. o) + 0 (h2)
1 .-
;f&-p ~fl,l+f-l,l+fl.-l+f-l.--l
--2f1,,-2f -l,o-~fo,1-~fo,-l+~fo.o~+~~~B)

25.3.30
acu acu
V4%,0= (2 -+2 az’ay2+5do*o -
NUMERICAL hNALYSIS
Laplacian 25.3.33
=p 1 [ ~0~0.0-~(~1,0+~0. Isu-l,o+%. -1)
+2(UI,l+Ul, -l+~-l,l+~--I. -1)
+ (%. 2+ u2. o+u-2. O+%. -2) 1 + 0(h2)
25.4.4
Modified Trapeeoidal Rule
Jrf(z)dz=h [$’+fl+ . . . +fm-l+$]
885
h 1lm 6 (4) )
+E [-f-1+f1+fm-1-fm+1l+720 hf (€

886 NUMERICAL ANALYSIS
25.4.5
25.4.6
25.4.7
Simpeon'e Rule
Extended Simpn's Rule
Euler-Maclaurin Summation Formula
(For &k, Bernoulli numbers, see chapter 23.)
If f(a+2)(z) and f("#)(z) do not change sign for
q

25.4.20
{ ~of(2Mz=- 5h
299376
16067Cf0+f10)
t 1 06300 Cf1 +fo) - 48525 (f2 +fa) +272400 Cf8 +f71
-26O55OCfi+fd +427368fsI
- 1346350 f(12) (€)h13
32691 8592
25.4.21
25.422
25.4.23
Newton-Cotes Formulas (Open 'bp)
3h f(*) (€)ha
J;f(2)d2=F (fI+f21+4
28f (4) (4) h5
[f(x)dz=$ (2f,-f2+2f3)+ 90
~ f ( z ) ~ z = ~
25.4.24
NUMERICAL ANALYSIS 887
95f4) (€)he
(11f1+.f2+f3+llf4)+ 144
6h
cf(x)dz=s (1 1fi- 14j2+26fr 14f4 + 1 1fs)
25.4.25
+41f'"(€)h7
140
[f(X)d%=- 7h (6llf1-453f2+562f8+562j.
1440
5257
-453f~+611f,)+~~~')(€)W
25.4.26
8h
J;f(Z)dZ=- 945 (46Of1-954f2+2196f3-2459fr
+2196fs-954f6+460f7)+~f(*)(~)hg 3956
Five Point Rule for Analytic Functions
25.4.27
to+ ih
J_:Df(z)dz=& {24f(z0) +4[f(z0+h) +f(zo-h)l
-[f(zo+a) +f(zo--ih)] 1 +R
147
IRI S a 9 If'"(z)l, S designates the square
withvertices zo+i'h(k=O, 1,2,3); hcan becomplex.
Chebyahev's Equal Weight Integration Formula
2 n
25.4.28 J",j(x)dx=- c j(st> +En
n 1-1
Abscissas: 2, is the itb zero of the polynomial part
of
2s exp [------
-n n
n . . .]
2.322 4-52 6-72'
(See Table 25.5 for z,.)
For n=8 and n210 some of the zeros are
complex.
Remainder :
+1 %=+I
- (n+l)
( ~ 2
R~=J-~ (n+ 1) ! f
--
2
n(n+l>! 1-1
2 2.+y(n+I) (€f 1
where (=€(d satisfies OIEIs and O l E t l s
(i=l, . . . ,n)
Integration Formulas of Gaussian Type
(For Orthogonal Polynomials see chapter 22)
Gaud Formula
1
2504.29 J-lj(~)d~=~
a= 1 wij(zt)+Rn
Related orthogonal polynomials: Legendre poly-
nomials Pn(z) 9 Pn(l)=l
Abscissas: x, is the ith zero of Pn(x)
Weights : w, = 2/( 1 - 8) [Pn ( 2, ,I2
(See Table 25.4 for z1 and w,.)
22n+1 (n!) 4
'n=(2n+i) [(24!13
f(2n)(€) (-1

888 NUMERICAL ANALYSIS
Related orthogonal polynomials: Pn(z), Pn(l)= 1
Abscissas: z( is the i" zero of P,(Z)
Weights: wt=2/(1 -$) [P:(zt)I2
25.4.31
Related polynomials:
Radau's Integration Formula
Abscissas: xf is the im zero of
Weights:
Remainder:
25.4.32
Pn-l(z)+Z'n(d
z+1
1 1-zt - 1 1
[Pn4(zt)]2 1-z: [PL(zt)I2
wf=2
Lobatto's Integration Formula
Related polynomials: P:-l(z)
Abscissas: zr is th6 (i-lPzero of P:,,(z)
Weights:
2
W{= (X'P f 1)
n(n- 1) [Pn- 1 ( 4 I'
(See Table 25.6 for z: and wi.9
Remainder:
Related orthogonal polynomials:
(21 =&~ZGRP~~O) (1-22)
(For the Jacobi polynomials P!". O' see chapter 22.)
Abscissas:
Weights:
zt is the i" zero of q,,(z)
(See Table 25.8 for zt and wt.)
Remainder :
25.4.34
Related orthogonal polynomials:
Abscissas: zt=l-[j where is the im positive
zero of P2n+l(z).
Weights: ~,=2[jw:~"+~) where w12"+') are the
Gaussian weights of order 2n+l.
Remainder :
25.4.35
yt=a+ (b-ulzt
Related orthogonal polynomials:
1
JCi
-p2n+1 (G), ~2,+1(1)=1
Abscissas: x,=l-tj where is the itb positive
zero of P2,,+l(z).
Weights: ~~=2[:w/*~+l) where ~i("+~)
Gaussian weights of order 2n+ 1.
are the
&

Related orthogonal polynomials:
-
~2~ (Jl-x), P2n(1)=1
Abscissas: xi=l-t: where ti is the i" positive
zero of PZn (x) .
Weights: wf=2wjZn) wiZn) are the Gaussian weights
of order 2n.
Remrticder :
25.4.37 ey
dy=dca 2 i-1 wJ(yt) +Rn
yf=a+ (b-ah
Related orthogonal polynomials:
-
Pzn (41 - z) ,P2n (1) = 1
Abscissas:
xi=l-t: where tf is the it" positive zero of PZn(x).
2L4.41
Weights: w,=~w/~~) wjZn' are the Gaussian weights
of order 2n.
25.4.38 fo dx=x wi j(xf) +R,
n
-1 JiZ 1-1
Related orthogonal polynomials: Chebyshev Poly-
nomials of First Kind
Abscissas:
Xt=COS
(2i- 1)r
2n
b+a b--a
Yi=-+-x* 2 2
Related orthogonal polynomials:
Tn (2) t Tn (1) =F
1
NUMERICAL ANALYSIS 889
Bbscissas:
Weights:
25.4.40
(2i-1)r
xf=cos
2n
r
Wac-
n
Related orthogonal polynomials: Chebyshev Polynomials
of Second Kind
sin [(n+l) arccos x] *
un(x)= sin (arccos x)
Abscissas:
i
x*=cos - r
n+ 1
*
Weights:
r
wi=- sinZ
i
-
n+l n+l
*
Remainder:
(2%) !22n+1 f""'(t) (-1

890
Related orthogonal polynomials:
Abscissas:
Weights:
2i-1 r
x: =cos2 - * -
2n+l 2
2r
w:=2n+l z:
Related orthogonal polynomials: polynomials or-
thogonal with respect the weight function --In z
Abscissas: See Table 25.7
Weights: See Table 25.7
25.4.45
Related orthogonal polynomials: Laguerre polynomials
Ln(z).
Abscissas: z: is the i* zero of Ln(Z)
Weights:
*
'Zr
wi= (~+~)TL+~(s:)I*
(See Table 25.9 for xf and wt.)
Remainder:
25.4.46
Related orthogonal polynomials: Hermite polynomials
Hn(z).
Abscissas: z: is the it" zero of Hn(z)
Weights:
2"-Ln! 6
n2[Hn - 1(zd l2
(See Table 25.10 for zf and toc.)
*See page 11.
NUMERICAL ANALYSIS
Remainder:
25.4.47
n! Jr
Rn=-
2"(2n)! f""(€> (-

25.4.56 S2n-1=gf2f-1
sin (tx2f-l)
I= 1
n
25-4-57 C~,-~=~j$?.~ I= 1 COS (t22f-1)
1 1 2Tn nn
"> m
- J ~(X,Y)G%=- C f(h COS mj h sin -
2uh r 2m 12-1
Circle C: i+ y2< ha.
25.4.61
1
uh2 - JJ c f(x,Y)d;rdy=k i- 1 wJ(z*,yJ+R
+0(hh"-2
(Xf, Y i) wi
(020) 112 ~=0(h4)
(hh,O), (0,fh) 1/8
Or, Y 3 wi
(090) \12
(fh,O) 1/12 ~=0(h4)
(fz'fzJ3) h h 1/12

(xi, Yf) Wt
(OJ 'I 1 /6
(fh,O) 1/24 R=O(h6)
(xt, Vi) Wi
(0,O) 1/9
(J6-lh ,,%!,Js-Ih sin 2Tk IS+&
10 10 10 i.) 360
-
('"' * ' 'J1o)
(.,/!I!$h~~~,$+AhsinM 10 IS-&
10 10 -) 360
R=O(h'')
Square' S: lxllh,JY!Sh
25.4.62
1
4h >JJf('J ?/)dsdy=& i=1 WJ(xf,Y

(A4 h,-+4 h) 251324
h’
G+1
1200
R = O(h6)
47)
4
(Xf, YO
I
Wf
((-F)h,O)
1554-G
0 1 1200 ((fly)
h,*(+-) m)
C0,O)
(h, 0)
314
1/12
R=O(hs)
Regular Hexagon H
Radius of Circumscribed Circle=h
(+$)
1/12
25.4.64
1
- JJj(x,Y)dzdy=k 1-1 WfAXf, Yf)+R
?ahz 2 H
(Zf ,Yf ) Wf
(090) 21/36
( AS’fZ h h d-3) 5/72 R=O(h4)
(fh,O) 5/72

894 NUMERICAL ANALYSIS
(xr,?/r, 23
(f4 A, *& h,O)
W#
(*& ho, *& h) 1/15
I (O,*&h,fdh) .- I
(X#,Y#) 201
( f h, 0,O)
R=O(V)
(0,O)
(A+-&
258/1008
m) 125/1008 R=OW
(0, *h, 0)
((40, fh)
1/30
(fh
-
9 9 0) 125/1008
Surface of Sphere 2: zZ+y2+z2=h*
25.4965
1
Ssf(xt~,4d~=& frl w.f(x:,vt,Zd+R
(X#,Yf, z,)
(*& fg h, h, f & h)
(*.&hf&h,O)
(f .& h,O, f & h)
'u)f
27/840
32/840 R=O(h*)
(21, %, 23 wi
(0, f & h, f & h)
(fh,O, 0)
(0, f h, 0) 40/840
(090, f h)
Sphere S: $+yP+z*

(Zc,Yr,Zr) Wt
(O,O, 0) 2/5
(fh,O,o) 1/10
(0, fh,O) 1/10
(O,O, &I&) 1/10
Cube6 C: IzlSh
(zt, Yt, zo wr
(-+h,O,O) 1/6
(0, fh,O) 1/6
(O,O, *h) 1/6
25.4.68
& J$Jf(r,Y,z)ddYdz
C
IYl Sh
IzlSn.
~=0(h4)
R=o(~*)
1
=- 360 [-496fm+128Cf,+8cf,+5cfo]+O(h~)
25.4.69
1
=- 450 [91Cf1-4OCfe+ 1 6Cfd]+O(h6)
where fm=f(o, 0, 0) .
I see footnote to 26.4.62.
cf,=sum of values off at the 6 points midway
from the center of C to the 6 faces.
cfr=sum of values off at the 6 centers of the
faces of c.
cfo=sum of values off at the 8 vertices of C.
cfe=SUm of values off at the 12 midpoints of
edges of C.
cfd=sum of values off at the 4 points on the
diagonals of each face at a distance of
i&h from the center of the face.
2
25.470
'$$Sf (w,
V
9-
where
zfo:
cfe:
Tetrahedron: F
z)dzdydz=qj
1 Cfw+z 9 CfY
+terms of 4th order
32 1 4
=Gfm+% C f o + C~ f a
V: Volume of T
+terms of 4" order
Sum of values of the function at the vertices
of F.
Sum of values of the function at midpoints
of the edges of F.
Sum of values of the function at the center
of gravity of the faces of 5.
fm: Value of function at center of gravity of T.
CfY:

896
25.5.1
25.5.2
25.5. Ordinary DiEerential Equations'
First Order: y'=j(x, y)
Point Slope Formula
Y .+l 'Y .+ hY: + 0(h2)
Y n+1 =yn--l+ 2hY:+ o(ha)
Trapeddal Formula
h
25.5.3 Y,+l=y.+z (Y:+,+Y:) +o(m
6 The reader is cautioned against possible instabilities
especially in formulas 25.5.2 and 25.5.13. See, e.g.
(25.1 1 I, [ 25.121.
NuadERIcAL ANALYSIB

Systems of Differential Equations
First Order: y’=f(r,y,z), z’=g(z,y,z).
Second Order Runge-Kutta
25.5.17
1
yn+l=yn+2 (k1+kz)+O(h3),
1
Zn+l=zn+S (h+&) +O(h3)
NUMERICAL ANALYSIS 897
Runge-Kutta Method
25.5.20
1
Yn+l=Yn+h [y:+g (k,+kz+b> +O(h6)
25.5.22 yn+,=yn+h
*See page 11.
Runge-Kutta Method
1

898
Texts
(For textbooks on numerical analysis, see texts in
chapter 3)
[25.1] J. Balbrecht and L. Collats, Zur numerischen
Auswertung mehrdimensionaler Integrale, Z.
Angew. Math. Mech. 38, 1-15 (1958).
[25.2] Berthod-Zaborowski, Le calcul des integrales de
L1
la forme: f(x) log x dx.
NUMERICAL ANALYSIS
References
H. Mineur, Techniques de calcul numerique, pp.
555-556 (Librairie Polytechnique Ch. BBranger,
Paris, France, 1952).
[25.3] W. G. Bickley, Formulae for numerical integration,
Math. Gas. 23, 352 (1939).
[25.4] W. G. Bickley, Formulae for numerical differentiation,
Math. Gas. 25, 19-27 (1941).
[25.5] W. G. Bickley, Finite difference formulae for the
square lattice, Quart. J. Mech. Appl. Math., 1,
35-42 (1948).
[25.6] G. Birkhoff and D. Young, Numerical quadrature
of analytic and harmonic functions, J. Math.
Phys. 29,217-221 (1950).
[25.7] L. Fox, The use and construction of mathematical
tables, Mathematical Tables vol. I, National
Physical Laboratory (Her Majesty’s Stationery
Office, London, England, 1956).
[25.8] S. Gill, Process for the step-by-step integration of
differential equations in an automatic digital
computing machine, Proc. Cambridge Philos.
SOC. 47, 96-108 (1951).
[25.9] P. C. Hammer and A. H. Stroud, Numerical
evaluation of multiple integrals 11, Math.
Tables Aids Comp. 12, 272-280 (1958).
[25.10] P. C. Hammer and A. W. Wymore, Numerical
evaluation of multiple integrals I, Math. Tables
Aids Comp. 11.59-67 (1957).
[25.11] P. Henrici, Discrete variable methods in ordinary
differential equations (John Wiley & Sons, Inc.,
New York, N. Y., 1961).
[25.12] F. B. Hildebrand, Introduction to numerical
analysis (McGraw-Hill Book Co., Inc., New
York, N.Y., 1956).
[25.13] 2. Kopal, Numerical analysis (John Wiley & Sons,
Inc., New York, N.Y., 1955).
[25.14] A. A. Markoff, DifTerensenrechnung (B. G.
Teubner, Leipsig, Germany, 1896).
[25.15] S. E. Mikeladze, Quadrature formulas for a regular
function, SoobFrE. Akad. Nauk Grusin. SSR. 17,
289-296 (1956).
[25.16] W. E. Milne, A note on the numerical integration
of differential equations, J. Research NBS 43,
537-542 (1949) RP2046.
[25.17] D. J. Panov, Formelsammlung zur numerischen
Behandlung partieller Differentialgleichungen
nach dem Differenzenverfahren (Akad. Verlag,
Berlin, Germany, 1955).
r25.181 E. Radau, &tudes sur les formules d’approximation
qui servent B calculer la valeur d’une intBgrale
dBfinie, J. Math. Pures Appl. (3) 6, 283-336
(1880).
[25.19] R. D. Richtmeyer, Difference methods for initial-
value problems (Interscience Publishers, New
York, N.Y., 1957).
[25.20] M. Sadowsky, A formula for approximate com-
putation of a triple integral, Amer. Math.
Monthly 47, 539-543 (1940).
[25.21] H. E. Salser, A new formula for inverse interpola-
tion, Bull. Amer. Math. SOC. SO, 513-516
(1944).
[25.22] H. E. Salser, Formulas for complex Cartesian
interpolation of higher degree, J. Math. Phys.
28,200-203 (1 949).
[25.23] H. E. Salser, Formula for numerical integration of
first and second order differential equations in the
complex plane, J. Math. Phys. 29, 207-216
(1950).
[25.24] H. E. Salser, Formulas for numerical differentia-
tion in the complex plane, J. Math. Phys. 31,
155-169 (1952).
[25.25] A. Sard, Integral representations of remainders,
Duke Math. J. 15,333-345 (1948).
[25.26] A. Sard, Remainders: functions of several variables,
Acta Math. 84,319-346 (1951).
[25.27] G. Schulz, Formelsammlung sur praktischen
Mathematik (DeGruyter and Co., Berlin,
Germany, 1945).
[25.28] A. H. Stroud, A bibliography on approximate
integration, Math. Comp. 15, 52-80 (1961).
[25.29] G. J. Tranter, Integral transforms in mathematical
physics (John Wiley & Sons, Inc., New York,
N.Y., 1951).
[25.30] G. W. Tyler, Numerical integration with several
variables, Canad. J. Math. 5, 393-412 (1953).
Tablee
[25.31] L. J. Comrie, Chambers’ six-figure mathematical
tables, vol. 2 (W. R. Chambers, Ltd., London,
England, 1949).
[25.32] P. Davis and P. Rabinowits, Abscissa and weights
for Gaussian quadratures of high order, J.
Research NBS 56, 35-37 (1956) RP2645.
[25.33] P. Davis and P. Rabinowits, Additional abscissas
and weights for Gaussian quadratures of high
order: Values for n=64, 80, and 96, J. Research
NBS 60,613-614 (1958) RP2875.
[25.34] E. W. Dijkstra and A. van Wijngaarden, Table
of Everett’s interpolation coefficients (Elcelsior’s
Photo-offset, The Hague, Holland, 1955).
[25.35] H. Fishman, Numerical integration constants,
Math. Tables Aids Comp. 11, 1-9 (1957).
[25.36] H. J. Gawlik, Zeros of Legendre polynomials of
orders 2-64 and weight coefficients of Gauss
quadrature formulae, A.R.D.E. Memo (B) 77/58,
Fort Halstead, Kent, England (1958).
[25.37] Gt. Britain H.M. Nautical Almanac Office, Interpolation
and allied tables (Her Majesty’s
Stationery Office, London, England, 1956).

[25.38] I. M. Longman, Tables for the rapid and accurate
numerical evaluation of certain infinite integrals
involving Bessel functions, Math. Tables Aids
Comp. 11, 166-180 (1957).
[25.39] A. N. Lowan, N. Davids, and A. Levenson, Table
of the zeros of the Legendre polynomials of
order 1-16 and the weight coefficients for Gauss’
mechanical quadrature formula, Bull. Amer.
Math. soc.48, 739-743 (1942).
[25.40] National Bureau of Standards, Tables of La-
grangian interpolation Coefficients (Columbia
Univ. Press, New York, N.Y., 1944).
[25.41] National Bureau of Standards, Collected Short
Tables of the Computation Laboratory, Tables
of functions and of zeros of functions, Applied
Math. Series 37 (U.S. Government Printing
05ce, Washington, D.C., 1954).
[25.42] P. Rabinowitz, Abscissas and weights for Lobatto
quadrature of high order, Math. Tables Aids
Comp. 69, 47-52 (1960).
(25.431 P. Rabinowitz and G. Weiss, Tables of abscissas
and weights for numerical evaluation of integrals
of the form
Math. Tables Aids Comp. 68, 285-294 (1959).
NUMERICAL ANALYSIS 899
[25.44] H. E. Salzer, Tables for facilitating the use of
Chebyshev’s quadrature formula, J. Math.
Phys. 26, 191-194 (1947).
[25.45] H. E. Salzer and R. Zucker, Table of the zero8 and
weight factors of the first fifteen Laguerre
polynomials, Bull. Amer. Math. SOC. 55, 1004-
1012 (1949).
[25.46] H. E. Salzer, R. Zucker, and R. Capuano, Table of
the zeros and weight factors of the first twenty
Hermite polynomials, J. Research NBS 48,
111-116 (1952) RP2294.
[25.47] H. E. Salzer, Table of coefficients for obtaining the
first derivative without differences, NBS Applied
Math. Series 2 (U.S. Government Printing
Office, Wmhington, D.C., 1948).
[25.48] H. E. Salzer, Coefficients for facilitating trigonometric
interpolation, J. Math. Phys. 27,274-278
(1949).
[25.49] H. E. Salzer and P. T. Roberson, Table of coefficients
for obtaining the second derivative without
differences, Convair-Astronautics, San Diego,
Calif. (1957).
[25.50] H. E. Salzer, Tables of osculatory interpolation
coefficients, NBS Applied Math. Series 56
(U.S. Government Printing Office, Washington,
D.C., 1958).