i e], pi1] e],singer/zr1/zr1_2008_09/as_25.pdf · 25. numerical interpolation, differentiation, and...

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 I n I S ) . .

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 ( 1 5k I 5 ) . . . . . . . . . . . . . . . . . . . . . . . . . .

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 I n 510) . .

Table 25.4. Abscissae and Weight Factors for Gaussian Integration

n=3(1)10, Exact

(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 5 n I 10). . . . . . . . . . . . . . . . . . . . . . . . . .

Table 25.7. Abscissas and Weight Factors for Gaussian Integration for Integrands with a Logarithmic Singularity (25n54) . . . . .

n=3(1)10, 8-10D

n=2(1)4, 6D

Pege 877 878 882 885 896

898

900

914

915

916

920

920

920

National Bureau of Standards. * National Bureau of Standards. (Preaently, Bell Tel. Labe., Whippeny, N.J.1

876

876 NUMERICAL ANALYSIS

Table 25.8. Abscissae and Weight Factors for Gaussian Integration of Page Momenta (1 SnS8). . . . . . . . . . . . . . . . . . . . . . 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.

Formulas

mainders.

25.1. Dif€emncea

Forwud DiiTerencea 25.1.1

bk=~$:(.--~) if n and k am of same parity.

Forward Diferencu Central Dif.tncsr

2-1 f-1

25.1.3

25.1.4

Divided Mfllerencxm

877


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

Reciprocal DHenncea

25.1.12

Remainder in Lagrmqe Interpolation Formula

25.2.3

25.2.4

25.2.5

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

n even.

n odd.

(s<€<zl)

k has the same range aa in 25.2.6.

Lyuase Two Point Interpolation Formula (Linear Interpolation)

NUMERICAL ANALYSIS 879

w a n g e Three Point Interpolation Formula

25.2.11

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 6 2 k

25.2.18 R&) .0049hyce)(€) = .0049Ae (O<p<l) .0071hef“’([) =.0071A’ (-l<p<O, l<p<2) .024hy(‘(() k.024Ae (-2<p<-l, 2<p<3)

(z-r<€<zs) Lagrange Seven Point Interpolation Formula

25.2.19 f(zo+ph)=g AJf+Re

25.2.20 i-4

.0025h7f‘f‘7)(() A..0025A7 (Ip(<l)

.0046h’f7’(€) =.0046A7 (l<Jp1<2)

.Ol9h’f7’(€) = .019A7 (2<lp1<3)

(z-a<€<zs)

Lagrange Eight Point Interpolation Formula

25.2.21

25.2.22

Rr (P) =

Let f(zlzo,zl, . . .,zk) denote the unique polynomial of k” degree which coincides in value with f(z) at a, . 25.2.23

., zr.


Taylor Erplnaion 25.2.24

25.2.25 R,,=J:f(n+l) (t) @=$ dt

(zo<€<G) (For r,, see 25.1.6.)

Newton% Forward Difference Formule

25.2.28

f(~o+Ph)=fo+P&+(G+ . . . +@*;+En

A2 xa . f a

25.2.29

(&<€<zn)

Relation Between Newton and Lagranee Coefficients

25.2.30

Everett'e Formula 25.2.31

P(P-1) (P--2) 6; 3! f (%+ PN = (1 --P)fo+ Pf1-

Relation Between Everett and Lagrange Coefficients

25.2.33

E2=A'_1 E4=A!2 Ea=AEa

$',=A: F4=A! Fa=& Everett's Formula With Throwback

(Modified Central Difference)

25.2.34

25.2.54 m=2n+l

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'

-1's Formula With Throwback 25.2.47

f(2o +ph) = (1 -p>jo+pj1+ Bz (6:. 0 + 6;. 1)

P (P -1 ) (P-)) +B36;+RJ B 2 = p v ~ B3- 6

1 f (x) "2 s+& (ak cos kx+bk s b kz) k-1

(k=OJ1,. . ,Jn)

25.2.55 m=2n

(k=OJIJ. . .,n)

b, is arbitrary.

(k=OJlJ. . .Jn-l)

Subtabulation

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 a 1 intervals respectively. Thus &=j (z,,+z) -j(;co). Assuming that the original 5* order differences are zero

25.2.56

h

At (1 - m) (1 -2m) (1 - 3m) 24m4 +

From this information we may construct the b a l tabulation by addition. For m= 10,

25.2.57 - &=.1&-.045&+.0285~-.02066~ @=.01&-.009&+.007725a g=.OOl&- .00135G

- - 2=.OOO1g

h e a r Inverse Interpolation

Find p, given jp(=j(ro+ph)). Linear

25.2.58


Quadratic Inverse Interpolation 25.2.59

(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

k=O

p=h+C2h8+C3h3+ . . ., h=(fp-%)/$

25.2.62 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 @ %+

2 3 4 - . *

G At+ 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---- +-+'+ . . .

aa=8-a - * .

%= 6-- 24 + . a .

3 6 20 30 6: ff+

-s:+s: 4+s:

a4=a+ 4 . . .

-st+%+ a,= ~ 120 . -

(Used in conjunction with 25.2.62.)

Bivariate Interpolation

Three Point Formula (Linear) 25.2.65

+ f(zo+Ph,Yo+ n 4 = (1 -P- n)fo. 0

+PfL o+ do, 1 + OW) Four Point Formula

25.2.66

i- f (%+Ph,YO+ nk) = (1 -PI (1 - n)fo.o+P(1- n v i . o

+ n(1 - Plfo, 1 +Pdl. 1 + W2) Six Point Formula

25.2.67

25.3. Dmerentiation

Lagrange's Formula

k=O 25.3.1

(See 25.2.1.)

NUMERICAL

25.3.3 rn(z) d (*+I) ) (€

t=t (%I (G<€<xn) Rb(x)= (n+ fn+l) 1) ! (€>?rb(z) +(n+l)! &j

Equally Spaced Abscissas

Three Points 25.3.4

f:=Y (zo+ph) 1

-h -- { (p- $If-* - 2p j o + (P+# f l 1 +R;

Four Points 25.3.5

f.=f’(G+ph)=k{ - 6 3P2-6P 4-2 j-l

+3p2-4p-l 3p2-2p-2 jl

3p’- 1

2 2 fo-

++.} +R;

Five Points 25.3.6

1 2p3-3p2-p+l j-2 { 12 - 4p3- 3P2---8P+4 j-l+2P3;5Pjo

j:=f’(%+Ph)=~

6

- 4p3i-3p2-8p-4 6 jl

2p3+3p2-p-1 j2) +R:

For numerical values of differentiation coeffi-

12 +

cients see Table 25.2.

Markoff’s Formulas

(Newton’s Forward Dserence Formula Differentiated)

25.3.7 p(ao+ph)-Ipo+22p-l@ 1

2

3p’-6p+2A?i+ . . . +& d P (n) &j]+R: ‘ 6

25.3.8

R‘,=hy(n+l) (6) d ( p )+hn+l ( p )” jb+l)(t) dp n+l n+l dx

(%<€<%)

1 1 1 25.3.9 hfi=&-,j@+B&-;iG+ . . .

ANALYSIS 883

11 5 12 6 25.3.10 h ~ i 2 ’ = @ - ~ + - ~ - - A ~ + . . .

25.3.11 hSjia)=&-- 3 7 15 ,&+;i&-sG+. . .

25.3.12 17 7

hy44)=&--2&+ 6g--zg+ . . .

25.3.13

h6j i5)=g- , jg+s@-s@+. 5 25 35 . .

Everett’s Formula 25.3.14

hf’(G+&) - - j o+ j ~ - 6 3p2-@+2 8; ; 3Pi-1 6:

+ - * - -[( 2n+1)]@+[(2.+1)1

5p‘-20p3+ 15p2 + 1 Op- 6 *.+5p4- 1 5p2+4 St 120 -

120 p+n--l P+n IS?

25.3.15 1 1 1 1

hji=-jo+jl-, 6:-- 6 S:+ 20 - St+- 30 St

MBFerences in Terms of Derivative8

25.3.16 h3 h4 hs

&-hji+z j ~’+~ j i a’+~ j i 4’+- j ~’’ 5!

25.3.17 7 1 & c5 h2jia) + hyia) + e h’ji4) +z hyf)

3 5 25.3.18 @=hyia’+,j h‘fi”+;i fA5)

25.3.19

25.3.20 g c5 hyi5) & c5 hyi4) + 2hyi6’

Partial Derivatives

25.3.21

+ (j1.0-f -l.o)+o(ha) -- bjo, 0 - l

bx 2h

NUMERICAL ANALYSIS 884 25.3.22 I 25.3.26

-- 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. -,,+0(h2) 1 a2.fo.0- 1 axay a 2

-- a&o-a (A 1 -f- 1.1 +A, - 1 -f- 1. - 1) + 0(h2)

25.3.23 25.3.27

1 .- ;;f&-p ~fl,l+f-l,l+fl.-l+f-l.--l

--2f1,,-2f -l,o-~fo,1-~fo,-l+~fo.o~+~~~B)

-3- c f i . o+f-1, o+ fo. 1 +fo. -1 axay 212 a2f0 O 1 ~f1,0-2f0.0+f-1,0~+0~h2~ a2f 0.0- - 1 az2 h2

-2fo.o-f1.1-f-I. -1) +O(h2)

25.3.a 25.3.28

--- -- a;+0- I& (-fa. 04- 1 6f1, o- 30f0, o a;$ ’-$ cfa, 0-4f1, O+ 6f0. 0-4f-I, o+f-2. o) + 0 (h2) + w-~. o-j-2, o) + o(h41

25.3.25 25.3.29

NUMERICAL hNALYSIS

Laplacian 25.3.33

25.3.30

acu acu V4%,0= (2 -+2 az’ay2+5do*o -

=p 1 [ ~ 0 ~ 0 . 0 - ~ ( ~ 1 , 0 + ~ 0 . I s u - l , o+% . -1)

+2(UI,l+Ul, - l+~ - l , l+~ - - I . -1)

+ (%. 2+ u2. o+u-2. O+%. -2) 1 + 0(h2)

885

Modified Trapeeoidal Rule 25.4.4

Jrf(z)dz=h [$’+fl+ . . . +fm-l+$]

h 1 lm 6 (4) ) +E [-f-1+f1+fm-1-fm+1l+720 hf (€


Simpeon'e Rule 25.4.5

Extended Simpn's Rule 25.4.6

Euler-Maclaurin Summation Formula

25.4.7

(For &k, Bernoulli numbers, see chapter 23.) If f(a+2)(z) and f("#)(z) do not change sign for

q<z<~ then lRul is less than the first neglected term. If f("+')(z) does not change sign for z,,<z<zm, ]Rut is less than twice the first neglected term.

Lapnnge Formula 25.4.8

f(z)dz=k (L:" (a) --LP (a))ft +Rn 1-0

(See 25.2.1.)

25.4.9

Equally Spaad Abecissae 25.4.11

(See Table 25.3 for A,(m).)

Newton-Cotes F O ~ U ~ M ( C l d Type)

(For Trapezoidal and Simpson's Rules see 25.4.1- 25.4.6.) 25.4.13 (Simpeon'e 3 5 rule)

25.4.14 (Bode% rule)

8 f(" (E) h7 +32js+7jJ - 945

25.4.15

275 f (*) (t)h7 12096 +75f4+19fs)-

25.4.16

25.4.17

25.4.18

25.4.19

*See page XI.

NUMERICAL

25.4.20

~of(2Mz=- 5h { 16067Cf0+f10) 299376

t 1 06300 Cf1 +fo) - 48525 (f2 +fa) +272400 C f 8 +f71

-26O55OCfi+fd +427368fsI

- 1346350 f(12) (€)h13 32691 8592

Newton-Cotes Formulas (Open 'bp) 25.4.21

3h f(*) (€)ha J;f(2)d2=F ( f I + f 2 1 + 4

25.422

28f (4) (4) h5 [f(x)dz=$ (2f,-f2+2f3)+ 90

25.4.23

95f4) (€)he ~ f ( z ) ~ z = ~ (11f1+.f2+f3+llf4)+ 144

25.4.24

c f ( x ) d z = s (1 1fi- 14j2+26fr 14f4 + 1 1fs) 6h

+41f'"(€)h7 140

25.4.25

[f(X)d%=- 7h (6llf1-453f2+562f8+562j. 1440 5257

-453f~+611f , )+~~~')(€)W 25.4.26

J;f(Z)dZ=- 945 (46Of1-954f2+2196f3-2459fr 8h

+2196fs-954f6+460f7)+~f(*)(~)hg 3956

Five Point Rule for Analytic Functions

to+ ih 25.4.27

ANALYSIS 887

J_:Df(z)dz=& {24f(z0) +4[f(z0+h) +f(zo-h)l

-[f(zo+a) +f(zo--ih)] 1 +R

1 4 7 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> + E n n 1-1

Abscissas: 2, is the itb zero of the polynomial part of

n . . .] 2s exp [------ -n n 2.322 4-52 6-72'

(See Table 25.5 for z,.)

complex. Remainder :

For n=8 and n210 some of the zeros are

+1 %=+I - (n+l) R~=J-~ (n+ 1) ! f ( ~ 2

2 2.+y(n+I) (€f 1 2 -- n(n+l>! 1-1

where (=€(d satisfies O I E I s 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 - l j ( ~ ) d ~ = ~ a= 1 wij(zt)+Rn

Related orthogonal polynomials: Legendre polynomials 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,.)

f(2n)(€) (--1<€<1) 22n+1 (n!) 4

'n=(2n+i) [(24 !13

Gauss' Formula, Arbitrary Interval

b-a n

Y"(9) Z,+(T) b+a

25.4.30 J)(Y)dY=T c 1-1 WJ(Yf) +a

*See page 11.


Related orthogonal polynomials: Pn(z), Pn(l)= 1 Abscissas: z( is the i" zero of P,(Z) Weights: wt=2/(1 -$) [P:(zt)I2

Radau's Integration Formula

25.4.31

Related polynomials:

Abscissas: xf is the im zero of

Pn-l(z)+Z'n(d z+1

1 1-zt - 1 1 Weights:

wf=2 [Pn4(zt)]2 1-z: [PL(zt)I2

Remainder:

Lobatto's Integration Formula

25.4.32

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:

zt is the i" zero of q,,(z)

Weights:

&

(See Table 25.8 for zt and wt.)

Remainder :

25.4.34


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


1 -p2n+1 (G), ~2,+1(1)=1 JCi

Abscissas: x,=l-tj where is the itb positive zero of P2,,+l(z). Weights: ~~=2[:w/*~+l) where ~ i ( " + ~ ) are the Gaussian weights of order 2n+ 1.


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=d c a 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).

Weights: w , = ~ w / ~ ~) wjZn' are the Gaussian weights of order 2n.

n 25.4.38 fo d x = x wi j(xf) +R,

-1 JiZ 1-1

Related orthogonal polynomials: Chebyshev Poly- nomials of First Kind

Abscissas: (2i- 1)r 2n Xt=COS

b+a b--a Yi=-+-x* 2 2

Related orthogonal polynomials: 1

Tn (2) t Tn (1) =F

Bbscissas: (2i-1)r 2n xf=cos

Weights: r Wa c - n

25.4.40

Related orthogonal polynomials: Chebyshev Poly- nomials of Second Kind

sin [(n+l) arccos x] * un(x)= sin (arccos x)

Abscissas: *

r i n+ 1 x*=cos -

Weights: r i *

wi=- sinZ - n+l n+l

Remainder:

(2%) !22n+1 f""'(t) (-1<t<1) 2L4.41

b+a b-a y*=-+- 2 2 Xf

un(z)= sin (arccos x)

Related orthogonal polynomials: * sin [ (n+ 1) arccos x]

Abscissas: * z U xi =cos - n+l

Weights : i * U wi=- sin2 - r n+l n+l


Abscissas: 2i-1 r xi =cos2 - - - 2n+l 2

Weights : 2r

Wf=- Xf

*See pnge 11.



Abscissas: 2i-1 r x: =cos2 - * - 2n+l 2

Weights: 2r w:=2n+l z:

Related orthogonal polynomials: polynomials orthogonal 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.)

Remainder:

f""(€> (- <€< 1 n! Jr Rn=- 2"(2n)!

Filon'a Integration Formula a 25.4.47

l+co2 e sin 28 ~(e)=2 ( 8" -- @ )

For small 8 we have

25.4.53 2e3 2e6 28' a=--- 45 315 4725 " *

2 2e2 48' 28' @-' +---+-- . . . -5 15 105 567

* For certain difficulties associated with this formula, see the article by J. W. Tukey, p. 400, "On Numerical Approximation," Ed. R. E. Langer, Madison, 1959.

*See page 11.

25.4.56 S2n-1=gf2f-1 sin (tx2f-l) I= 1

n 25-4-57 C ~ , - ~ = ~ j $ ? . ~ I= 1 COS (t22f-1)

"> m +0(hh"-2

1 1 2Tn nn - J ~(X,Y)G%=- C f(h COS mj h sin - 2uh r 2m 12-1

Circle C: i+ y2< ha. 25.4.61

(Xf, Y i) wi

(020) 112 ~=0(h4 )

(hh,O), (0,fh) 1/8

O r , Y 3 wi

(090) \12 (fh,O) 1/12 ~=0(h4)

(fz'fzJ3) h h 1/12

1 uh2 - JJ c f(x,Y)d;rdy=k i- 1 wJ(z*,yJ+R

(Xi, Y d Wt

(090) 419

( f h ,O) 1/9

( fhl fh) 1/36 R=O(h4)

/n i h \ 1 In

(xi, Yf ) Wt

(OJ 'I 1 /6

(fh,O) 1/24 R=O(h6)

Square' S: l x l l h , J Y ! S h 25.4.62

1 4h >JJf('J ?/)dsdy=& i=1 WJ(xf,Y<) +R

R

(xt, Vi) Wi

(0,O) 1/9

( J 6 - l h ,,%!,Js-Ih sin 2Tk IS+& 10 10 10 i.) 3 6 0

10 10 -) 3 6 0

('"' * ' 'J1o) -

(.,/ ! I ! $h~~~ , $+Ahs inM 10 IS-&

R=O(h'')

4 For regions, such as the square, cube, cylinder, etc., which are the Cartesian products of lower dimensional regions, one may always develop integration rules by "multiplying together'' the lower dimensional rules. Thus if

Sp1j(z)h a 5 wij(Zi) i=l

is a one dimensional rule, then

S,'S,lf(zj v)&dgs iJ-1 & wimif(zilzi) beoomea a two dimensional rule. Such rules are not neceassrily the most "economical".

893

R=o(~*) (-3tfzfi) h h 3/60

(-kt 0) 8/60

($*t 43) 8/60

PJUMERICAL ‘ANALYSIS

(A4 h , - + 4 h) 251324

<or*$ h) 10181

($ h,O) lOl8l

Equilateral Triangle T

R=O(h6)

Radius of Circumscribed Circle=h

(XC,YO 25.4.63

- JJTf(z,Y)dzdy=e Wff(2f,Yf)+R a@ 270/1200 1

5 a h 2 4 1-1

G + 1

-m+1 ((7) h,o) 155-fl

1200 R = O(h6)

G + 1 (( 14 >h’

4 7 ) 4

(Xf, YO Wf ((fly) h,*(+-) m)

( ( - F ) h , O ) I 1554 -G

0 1 1200

R=O(hs) C0,O) 314 (h, 0) 1/12 Regular Hexagon H

(+$) 25.4.64

Radius of Circumscribed Circle=h 1/12

1 - JJj(x,Y)dzdy=k 1-1 WfAXf , Yf)+R ? a h z 2 H

(Zf ,Yf ) Wf

(090) 21/36

h h 5/72 R=O(h4) ( AS’fZ d-3) (fh,O) 5/72


(xr,?/r, 23 W#

(f4 A, *& h,O)

I (O ,*&h, fdh) . - I

(*& ho, *& h) 1/15

R=O(V) ( f h, 0,O) (X#,Y#) 201

(0, *h, 0) 1/30 (0,O) 258/1008

(A+-& m) 125/1008 R=OW ((40, fh) (X#,Yf, z,) 'u)f (fh 9 9 0) 125/1008

Surface of Sphere 2: zZ+y2+z2=h* (*& h, fg h, f & h) 27/840

25.4965 1 (*.&hf&h,O)

(0, f & h, f & h)

- S s f ( x t ~ , 4 d ~ = & f r l w.f(x:,vt,Zd+R

(f .& h,O, f & h) 32/840 R=O(h*)

(fh,O, 0)

(0, f h, 0) 40/840

(090, f h)

Sphere S: $+yP+z*<h*

25.4.66

1

(21, %, 23 wi (fh,O,O) 1/6

(o,fh,o) 1/6 :*ha B ~=0(h4) - sssl(z ,Y,Z)dnlyd~=~ wd(wh , z:)+R

(O,O,fh) 1/6

(Zc,Yr,Zr) Wt

(O,O, 0) 2/5

(fh,O,o) 1/10

(0, fh,O) 1/10

(O,O, &I&) 1/10

~ = 0 ( h 4 )

Cube6 C: IzlSh

IYl S h

IzlSn.

cf,=sum of values off at the 6 points midway

cfr=sum of values o f f at the 6 centers of the

cfo=sum of values off at the 8 vertices of C. cfe=SUm of values o f f at the 12 midpoints of

edges of C. cfd=sum of values o f f at the 4 points on the

diagonals of each face at a distance of

i&h from the center of the face. 2

from the center of C to the 6 faces.

faces of c.

(zt, Yt, zo wr

(-+h,O,O) 1/6 R=o(~*)

(0, fh,O) 1/6

(O,O, *h) 1/6

25.4.68

& J$Jf(r,Y,z)ddYdz C

1 360

25.4.69

=- [-496fm+128Cf,+8cf,+5cfo]+O(h~)

1 =- 450 [91Cf1-4OCfe+ 1 6Cfd]+O(h6)

where fm=f(o, 0, 0) .

I see footnote to 26.4.62.

Tetrahedron: F 25.470

'$$Sf (w, z)dzdydz=qj 1 Cfw+z 9 CfY V

+terms of 4th order

=Gfm+% C f o + ~ C f a

+terms of 4" order

9-

32 1 4

where V: Volume of T

zfo: Sum of values of the function at the vertices

cfe: Sum of values of the function at midpoints

CfY: Sum of values of the function at the center

fm: Value of function at center of gravity of T.

of F.

of the edges of F.

of gravity of the faces of 5.

NuadERIcAL ANALYSIB 896

25.5. Ordinary DiEerential Equations'

First Order: y'=j(x, y)

Point Slope Formula

25.5.1

25.5.2

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.


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)

Runge-Kutta Method 25.5.20

1 1 Yn+l=Yn+h [y:+g (k,+kz+b> +O(h6)

Runge-Kutta Method

25.5.22 yn+,=yn+h

*See page 11.


References

Texts

(For textbooks on numerical analysis, see texts in chapter 3)

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H. Mineur, Techniques de calcul numerique, pp. 555-556 (Librairie Polytechnique Ch. BBranger, Paris, France, 1952).

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[25.9] P. C. Hammer and A. H. Stroud, Numerical evaluation of multiple integrals 11, Math. Tables Aids Comp. 12, 272-280 (1958).

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Tablee

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