Numerical Calculus 9781400875900

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Numerical Calculus

Numerical Calculus Approximations, Interpolation, Finite Differences, Numerical Integration, and Curve Fitting

By William Edmund tjYCilne

1949 PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY

COPYRIGHT, I949, BY PRINCETON UNIVERSITY PRESS LONDON: GEOFFREY CUMBERLEGE, OXFORD UNIVERSITY PRESS

PRINTED IN THE UNITED STATES OF AMERICA

PREFACE The growth of computational facilities in the period since World War I has already been phenomenal, and the possibilities in the near future are beyond imagining. The number of excellent calculating machines now avail­ able in almost every office or laboratory, to say nothing of the amazing Sequence Controlled Calculator, or the equally marvelous Electronic Numerical Integrator and Calculator, makes it possible to solve whole categories of problems that only yesterday were prohibitively diffi­ cult. In consequence the subject of numerical analysis is surely destined to make enormous strides in the decades to come. On the other hand our traditional courses in college and graduate mathemati.es too often turn out students poorly trained in the art of translating' theoretical analysis into the concrete numerical results generally required in practical applications. As a physicist friend of mine once said, 'Tou mathematicians know how to solve this problem but you can't actually do it". The aim of this book is to aid in bridging the considerable gulf between classroom mathematics and the numerical applications. It is designed to provide rudi­ mentary instruction in such topics as solution of equa­ tions, interpolation, numerical integration, numerical solution of differential equations, finite differences, approximations by Least Squares, smoothing of data, and simple equations infinite differences. The presentation is intentionally elementary so that anyone with some knowledge of calculus and differential equations can read it understandingly. Mathematical elegance and rigor have frequently been sacrificed in favor of a purely naive treatment. ν

PREFACE Prom the wealth of material available the decision to include or to exclude specific topics has posed a vexing problem, and personal interest rather than object­ ive logic has doubtless influenced the choice. It is hoped that enough has been included to meet a fairly general need and to stimulate wider study of this interesting and useful branch of mathematics. The author is indebted to Mr. James Price for the calculation of Table VI, to Professor William M. Stone for some of the material in Article 82, and to Professor Burns W. Brewer for criticisms of Chapter VIII. WilliamEdmund Milne, Ph.D., D.Sc. Professor of Mathematics Oregon State College Corvallis, Oregon March 19^8

CONTENTS Page Chapter I.

SIMULTANEOUS LINEAR EQUATIONS

1 . Determinants

3 3

2.

Solution of Linear Equations by Determinants . .

3.

Homogeneous Equations

11

4.

The Method of Elimination

15

5.

Numerical Solution of Linear Equations

17

6.

Symmetrical Equations

2b

7

7.

Check on Accuracy of Computation

25

8.

Evaluation of Determinants

26

9.

10.

Calculation of Cofactora

27

IVbgnitude of Inherent Errors

29

Chapter II. SOLUTION OP EQUATIONS BY SUCCESSIVE APPROXIMATIONS

36

11 .

One Equation in One Unknown

36

12.

Exceptional Cases

M

13.

Simultaneous Equations

1l·.

Successive Substitutions

^7

15.

Exceptional Cases

k9

16.

Complex Roots of Algebraic Equations

53

17·

Solutions of λ-determinants

57

Chapter III. INTERPOLATION

63

18.

Interpolating Functions

63

19.

Linear Interpolation

68

20.

Aitken's Process of Iteration

69

21.

22.

Neville's Method of Iteration Inverse Interpolation

73

23.

The Error in Polynomial Interpolation

78

2k.

Lagrange's Interpolation Formula

83

25.

Equally Spaced Points

89

72

Page Chapter

IV.

NUMERICAL D I F F E R E N T I A T I O N AND INTEGRATION

26.

Numerical

27.

Differentiation Points.

93

Differentiation

.

Formulas

93 for

Equally

Spaced

,

96

28.

Numerical

Integration

29.

Undetermined

30.

Investigation

31 .

C a l c u l a t i o n of

32.

The T r a p e z o i d a l

100

Coefficients of

the

the

10k

Error

108

Error

114

Rule

116

35-

Simpson's

3^.

Newton-Cotes

Rule Quadrature

Formulas,

Closed Type

.

122

120

35.

Newton-Cotes

Quadrature

Formulas,

Open Type

.

126

36.

Summary

.

127

Chapter V.

NUMERICAL S O L U T I O N OF D I F F E R E N T I A L EQUATIONS

37-

First

38.

Second Method

39-

Equations

ko.

Special

M .

Simultaneous

b2.

Use o f

of

13k Second

Order

for

138

Second-Order

Equations

.

.

Equations

Five-term

k3.

Factorial

The B i n o m i a l

b5.

Finite

h6.

Detection

1 k2

Formulas

1^5

Polynomials

11*5

Coefficients

150

Differences

Newton's Forward Newton's

of

.

Errors

Binomial

Interpolation

.

1 51+

.

158

Formula,-

Differences Binomial

1^0 1U1

F I N I T E DIFFERENCES

41)-.

U8.

132

Formulas

Chapter V I .

^7.

131

Method

1 60

Interpolation

l>ormula,

Backward D i f f e r e n c e s

161

^9.

Gauss's

I n t e r p o l a t i o n Formulas

50.

Central

Differences.

Stirling's

viii

163 Formula.

. . .

172

Page 51. 52.

53· 54. 55. 56.

57· 58.

Everett's Central Difference Formula Bessel1s Formula

178

Tabulationof Polynomials Subtabulation Derivatives in Terms of Differences Integral of Newton's InterpolationFormula ... Symmetric Integrals of Stirling's Formula. . . . Integral of Everett's Formula

1 83 185 19 1

Chapter VII. DIVIDED DIFFERENCES 5 9 . Definition of Divided Differences 6 0 . The Interpolating Polynomial in Terms of Divided Differences 61. Other Forms of the Interpolating Polynomial. . . Chapter VIII. RECIPROCAL DIFFERENCES 6 2 . Approximation by Means of Rational Fractions . . 6 3 . Reduction of the Determinant 6 k . Reciprocal Differences 6 5 . Further Properties of Reciprocal Differences . . 6 6 . Exceptional Cases

195 196 1 97 201 201

207

215 219 219 221 226 232 236

Chapter IX. 67. 68. 69. 70. 71. 72. 73.

lk. 75.

POLYNOMIAL APPROXIMATION BY LEAST SQUARES Least Sauares Further Investigation of Least Squares Least Squares for Integrals Orthogonal Polynomials .... Use of Orthogonal Polynomials Orthogonal Polynomials. Equally Spaced Points . Graduation, or Smoothing of Data Alternative Treatment of Smoothing Formulas. . . Gauss's Method of Numerical Integration

181

Chapter X. OTHER APPROXIMATIONS BY LEAST SQUARES. 7 6 . General Problem of Approximation by Least Squares

2^2 2^2 250 255 257 263 265 275 280 285 291 291

Page 77.

Fundamental

78.

Trigonometric

79-

Harmonic

80.

Calculation

81 .

The G r a m - C h a r l i e r

82.

Case o f

Chapter 83.

Approximation

of

300

Coefficients

317

S I M P L E D I F F E R E N C E EQUATIONS

8^.

Calculus

Differences

85.

The D i f f e r e n c e Exact Linear

88.

Linear

the

Appendix

309

Approximation

a Difference

86.

302

Points

S o l u t i o n of

87.

29U

Analysis

Discrete

XI.

292

Theorem

of

32k

Equation

32U •

328

Eouation

331

Equations

337

Difference

Equations

of

Order

Higher

Than

First

.

Equations,

A.

Variable

Coefficients

3^6

N O T A T I O N A N D SYMBOLS

351

Appendix B.

TEXTS,

Appendix

C.

C I A S S I F I E D G U I D E TO FORMUIAS AND METHODS.

Tables.

I. II.

T A B L E S , AND B I B L I O G R A P H I E S

Binomial

. . . .

353

Coefficients

Interpolation Newton's

for

Interpolation

Formula

366

I I I .

Everett's

IV.

Lagrange's Equally

Interpolation Coefficients

Spaced

Coefficients for

Legendre's

VI.

Orthogonal Polynomials

Polynomials

.

369

Five 371

(Adapted

to

the 373

Interval

Equally

.

Points.

V.

VII.

359 365

Coefficients

Binomial

341

Spaced

Integrals

of

for

n

+ 1

Points

Binomial

375 Coefficients, 382

VIII.Gamma INDEX

a n d Digamma F u n c t i o n s

'

383 385

x

NUMERICAL CALCULUS

Chapter I SIMJIiTANEOUS LIKEAE EQUATIONS A great variety of problems in pure mathematics and in the several branches of applied mathematics involve either directly or indirectly seta of simultaneous linear equations.

Hence, it is appropriate to devote this first

chapter to methods of solving such equations and to an estimate of the attainable accuracy of the solution. For theoretical investigations and for the case of equations with literal coefficients we commonly employ determinants.

For the case of numerical coefficients we

uae a systematic scheme of elimination adapted to machine calculations.

While the two methods are not fundamentally

distinct, they differ so much in details of operation that it is convenient to treat them separately.

However, since

a knowledge of determinants is almost essential for a full understanding of the method of elimination it is not practical to make the two treatments entirely independent. 1.

DETERMINANTS

We present here without proof a brief account of the elementary properties of determinants. (a)

The value of a two-rowed determinant is given

by the formula,

a

(b)

I

b

2

"

a

2

b

i

The value of a three-rowed determinant is given

by the formula,

h

1.

(c) n-rowed

The

determinant

determinant

column is row

SIMULTANEOUS L I N E A R

called

and t h e

by

the

j-th

of

n-1

striking

minor

column.

EQUATIONS

of

rows

out

the

Thus

obtained

the

i-th

element

in

the

from

row

in

an

and

the

j-th

i-th

determinant

(1 )

the

minor

In a columns nal

of

similar

determinant,

j-th

By t h e column is

by

elements by

by

the

by

(e)

small letter

The

of

cofactor meant It

same

striking

of

two

out

three

letters

order of

the

w i l l

two

less

minor be

of

products

in

that

from

and

the

three

i-th

element to

two

origi-

etc. row

multi-

designate

corresponding

Thus

of

and

less,

convenient

and t h e i r

rows

than the

rows

three

an element

capitalized.

sum o f

out

order

striking

we o b t a i n m i n o r s

(d)

plied

mariner b y

we o b t a i n m i n o r s

columns

and

is

cofactors

(1)

we

have

each element

of

a

row

or

1.. (or

column)

ant.

For

(f)

by

its

example,

The

column)

by the

(or

column)

is

(g) are in

two

If

of

products

to

of

zero.

columns

interchanged,

the

5

equal

to

the

determin-

(1)

corresponding

equal

two

is

(or

value

each element

of

cofactors

another

of

a

row row

Thus two

of

rows)

the

of

a

determinant

determinant

is

changed

sign. (h)

If

corresponding

elements

rows)

of

a

are

determinant

two

from

aum o f

(or

DETERMINANTS

own c o f a c t o r

is

determinant

of

equal,

two the

columns value

the

columns

(or

zero.

(i)

If

corresponding

elements

rows)

of

a determinant

are proportional,

the

determinant

(j)

is

The v a l u e

of

a determinant

Rows a n d c o l u m n s

(2)

A factor

•changing t h e

(3)

row

(or is

The

elements

row

(k) of

two

the

order

value

of

they

a row

(or

a d d i t i o n of

column).

rows

(or

if without

occur.

elements

resulting

by t h i s

corresponding

determinant.

unchanged

i n which

and t h e

of

by the

the

is

interchanged

removed f r o m a l l

column)

(or

The number the

are

multiplied

changed

order

is

ant

times

of

zero.

(1)

0

(or

of

of

determin-

factor. column) a

are

constant

element

of

another

Thus,

columns)

a

is

called

the

6

1. (1 )

Rank o f

determinant said

to

order to

SIMULTANEOUS L I N E A R

be

of

of of

be t h e

a determinant.

n-th rank

the

order n.

is

If

the

non-vanishing

rank

of

EQUATIONS

the

not

If zero

the the

determinant

minor

of

value

of

a

determinant is

highest

zero

is

the

order

is

said

row as

in

(e).

determinant.

Exercises Evaluate

by

(b)

k.

Evaluate

1,

2,

3 by

cofactors

of

first

5.

Evaluate

1,

2,

3 by

cofactors

of

second

6.

Verify

3,

elements

by

(f)

in

cofactors

1, of

2,

7.

Evaluate

by

8.

Evaluate

7 by method

9.

Show

10.

Show t h a t

is

first

row

in x

whose

row.

(e)

an algebraic

Find 1,

of

in(e).

(j,

3).

that

roots 11.

taking

second

c o l u m n as

2,

are

a,

the- rank 3,

7-

equation of

b,

c.

of

the

third

determinants

degree

in

Exercises

1. 2.

SIMULTANEOUS L I N E A R EQUATIONS

S O L U T I O N OF L I N E A R EQUATIONS B Y

Let

ua

consider

a

set

of

linear

7

DETERMINANTS

equations

in

n

unknowns.

(1 )

Let of

the the

value x's

in

multiply

the

etc.,

addition,

the

determinant

be denoted b y D,

element

,

of

this

f i r s t

let

determinant

of

the

by

equations.

the

by the

"

of

Now l e t

us

,

The

properties

the

result

of

coefficients

cofactor

be

e q u a t i o n above

and add t h e i n view

and

formed

the

the

second of

by

the

cofactors,

is

(2) It

is

the

readily

expansion

seen that in

the

elements

of

expression on the the

first

right

column of

is

the

determinant

that

is,

first

it

is

the

determinant

column replaced

determinant

by

determinant

obtained

by

the

Three

column of

cases

are

,

by

and

the

Let

i n general

let

from D by

b's.

now t o

D with

b's.

us

replacing

T h e n we h a v e

be

the

the

distinguished.

elements

of

denote

this

denote

the

the

i-th

equations,

the

column

8

1. SIMULTANEOUS LINEAR EQUATIONS I. D ^ 0. In this case the equations have one and onl? one solution, which is given by

That this is actually a solution may be proved by direct substitution back into the original equations. II. D=O, not all D^ = 0. Then we have the im­ possible equation, 0 = D^, and the equations have no solution. III. D=O, all D^ = 0. Let r be the rank of D and r^ the rank of D^. Then (a) If any r^ is greater than r, there is no solution. (b) If no r^ is greater than r, then r of the unknowns can be obtained in terms of the remaining η - r unknowns, as follows: Rearrange (if necessary) the order of the equations and the order of the unknowns in the equations so that the r-rowed determinant in the upper left hand corner of D ( as rearranged) will not be zero. In the first r equations transpose to the right all the variables beyond the first r, and solve the first r equations for the first r unknowns as in Case I. The solutions thus found will satisfy the remaining equations also and will contain η - r undetermined variables. Nbtrix. A system of ran quantities arranged in a rectangular array of m rows and η columns is called a matrix. In contrast to a determinant, which is a quantity determined from its matrix by certain rules, a matrix is not a quantity at all but an array of quantities. Por example two determinants with different elements may have the same value, but two matrices with different elements are different matrices. We shall not be concerned with the general properties of matrices, but we do find it convenient to define and use the term rank of a matrix. From any given matrix it

2.SOHJTION OF LINEAR EQUATIONS BY DETERMINANTS 9 is possible to set up determinants by striking out (if necessary) certain rows and columns from the matrix. The determinant of highest order that can be so formed has an order equal to the lesser of the two numbers m and n, if m and η are unequal, or equal to m if m and η are equal. The highest rank of the determinants of highest order is said to be the rank of the matrix. Thesquare matrix consisting of the array of co­ efficients in the left-hand members of equations (1) is called the matrix of the coefficients. The matrix a 11

aI 2

aIn

bI

a21

a22

a2n

b2

arn

an2 '

ann

bn

···

with η rows and η + 1 columns is called the augmented matrix. Using this terminology we can express the results Ij II, III above in the following'way. I. If the matrix of the coefficients is of rank η the equations have a unique solution. II. If the rank of the augmented matrix is greater than the rank of the matrix of the coefficients the equations have no solutions. Ill· If the ranks of the augmented matrix and of the matrix of the coefficients are both equal to η - k there are an infinite number of solutions expressible in terms of k arbitrary parameters. Exercises Determine whether the following equations have solutions, and obtain solutions when possible. Use determinants.

10

1 . SIMULTANEOUS LINEAR EQUATIONS

1.

2.

3.

7.

8 9.

Solve

the

set

of

equations

SSo ol vl ve e t ht he e s se et t o of f e eq qu ua at it oi on ns s

2. TO.

S O L U T I O N OP L I N E A R EQUATIONS B Y DETERMINANTS

Show t h a t

always

have

simple 11.

equations

a unique

of

the

solution,

11

type

which

c a n be f o u n d

by

substitutions.

Show t h a t

10 m a y b e

the

solution

of

the

equations

in

Exercise

written

3. A set

of

unknowns

of

i n which

the

n

the

HOMOGENEOUS EQUATIONS

simultaneous

linear

equations

with

n

form

constant

terms

are

a l l

zero

always

has

at

12

1.

least the

is

one

solution,

not

is

of the

zero rank

the r

where r

side,

and

If

the

D are

express to

zero. the

set of

row

not

are

where

k

is

actually of

remaining

Investigate

the

equations

the

of In

one row a l l

determinant

this

1 , we

1,

get

-

not

a l l

case a more

solve n

-

r

rank

of

the

to

the

cofactor

(provided

the

cofactors

since

for

r

variables.

We

the

right

s y m m e t r i c a l way

proportional D

D

cofactors

of

of

If

2.

to

solutions

the

homogeneous of

to

equations the

of

j-th

this

Thus

constant.

solutions

this

of

1,

n

to

say z,

D is

an arbitrary

Article

D is

one v a r i a b l e , obtaining

zero).

furnish

possible

x and y ,

non-zero

Applying Example

is the

each x .

element

solution.

it

for

rank

other

of

transpose

solve

no

< n,

that

may t h e r e f o r e

have

terms

in

1.

H e r e we f i n d

(f)

namely

equations

variables

Example

is

EQUATIONS

determinant

of

of

SIMULTANEOUS L I N E A R

follows

That at

in this'case

procedure

to

the

these

once D =

values

from

0.

equations

of

(e)

and

3.HOMOGENEOUS EQUATIONS

or

if

This

we

13

let

result

is

seen to

be e q u i v a l e n t

of

problem that

to

the

one

previously

obtained. A type variety

of

applications

is

arises

frequently

illustrated

in

by the

a

wide

following

example: Example the

2.

For what

non-zero

the

parameter

A

y,

we

w i l l

solutions.

Collecting; the

that

not

zero

the

the

determinant

which

the

determinant

I n order

the

Solving

the

the

determinant

D must

equation for

given equations

x,

equations

we o b t a i n t h e must

of

homogeneous

homogeneous

parameter

this

coefficients D of

z,

a

vanish.

Upon cubic

, we f i n d

a

have

following

.37228, have

and

obtain

equations

solution expanding equation

satisfy:

.62772, the

of

equations

have

as

value

set

of

three for

non-zero

values each of

which

solutions.

u

1.

SIMULTANEOUS L I N E A R

EQUATIONS

Exercises 1.

Solve

2.

Solve

3.

Determine

and f i n d

the

corresponding

non-zero

solutions:

k.

Determine

and f i n d

the

corresponding

non-zero

solutions:

5.

Show t h a t

the

equat ions

sin

have

no

real

non-zero

solutions.

6 ^ 0 .

1 . SIMULTANEOUS LINEAR EQUATIONS

15

I k THE METHOD OF E L I M I N A T I O N The use

of

determinants

for

equations

becomes

excessively

equations

exceeds

four

coefficients

are

practical

method of

elimination.

of

four

can r e a d i l y should

the

extend

Let

in

the

the

of

is

not

zero.

the

the

next

to

if

of the

digits. a

systematic

given for as

the

general steps

make c l e a r

actual

equations

is

detailed

given to

shown i n

set

the

to

so

to

unknowns,

procedure

a pattern for

the

four

linear

number

several

The e x p l a n a t i o n with

i n mind t h a t

process as

if

and e s p e c i a l l y

expressed

discussion are

short

intended

s o l u t i o n of

c a l c u l a t i o n we r e s o r t

equations

be k e p t

following

five

numbers

Hence f o r

case

or

the

cumbrous

case. in

the

Article,

the

reader It

the basis

and a r e

for

not

computation. be

solved

be w r i t t e n

as

follows:

(1 )

(2)

(3) (*)

where first

equation

equations

is

choosing

coefficient

for

of

is

The f i r s t

stage

following

steps:

a)

Divide

equation

b)

Multiply

tract

from

t h e n by of

the

- and

the

coefficient

we r e a r r a n g e the

not of

(1)

(2);

subtract

first

the

of

order

e q u a t i o n one

in of

the

the

i n which

the

zero.) the

elimination

by

resulting

equation

equations

(If

zero,

is

effected

by

the

; equation f i r s t

then by from

(4).

and The

by

and

subtract result

is

sub-

from the

(3); set

16

1 . SIMULTANEOUS L I N E A R

EQUATIONS

(5)

(6) (7) (8)

where

the

b's

srs

obtsjLn6d f r o m

the

a's

by th©

formulas

(9) (10)

To p e r f o r m t h e operate

same m a n n e r (5),

second

on equations a s we d i d

and

(K),

The

third

(6),

stage (7),

i n the

of

and

case

the (8)

of

elimination in precisely

equations

(1),

on equations

(12)

we the (2),

obtaining

(11 ) (12)

(13) where (U)

(15)

(13)

(16)

(17) where

(18)

(19)

giving

stage

is

performed

and

k. The f o u r t h by

,

THE METHOD OP E L I M I N A T I O N stage

consists

simply

in

17

dividing

(17)

giving

(20)

i n which

The v a l u e s equations

(20),

of

the

(16),

x's

are

(11),

now f o u n d

and

(5),

in

as

turn

from

follows:

(21 )

(22) (23) (21+)

Simple s u b s t i t u t i o n s In without to

case the

follow

is

necessary

aid

of

a

the

5. If

the that

(A)

the to

calculating outlined

s o l u t i o n by

solve

a

set

machine,

operations the

equations

compressed

machine

into

is

used

described

entire

elimination.

of

it

equations

may b e

best

above.

NUMERICAL S O L U T I O N OF L I N E A R

telescope

be

steps

a calculating

an extent original

complete

it

the

final

the

following

is

possible

in Article

computation,

and

EQUATIONS

it

4 to

including

values compact

of

the

form:

to

such

the

x's,

can

1 . SIMULTANEOUS LINEAR EQUATIONS 18

(B)

(C)

The m a t r i x (1) of

to

(k)

numbers

(C)

of

are

tions

is

(C), the

augmented m a t r i x

at

various

consisting

x's.

inserted

and t o

the

The m a t r i x

occurring

The m a t r i x values

(A)

Article

The

to

facilitate

Copy t h e

[2]

Divide

first

the

row

equations

the

consist

elimination.

gives in

order

the

(B) of

and

opera-

explanation.

column of

first

of

only,

the

of

seen to

i n brackets

indicate

the

is

stages

one

numbers

solely

[1]

of

(B)

row

of

(A). (A)

(excepting

by

Note the

[3]

Compute

that

the

row and

desired the

, comes

1

or

its

number

-

bar

is

of

numerator

is

number

s t i l l

is

and t h e

to

on the in

by the

the

found

put

by

then recorded

The

performed

is

according

Compute

(A) w h i l e

column of

The

complement)

[! P·

1021,

November, 19^7-

Chapter II SOLUTION OP EQUATIONS BY SUCCESSIVE APPROXIMATIONS It is usually impractical to solve either transcend­ ental equations or algebraic equations of higher than the second degree by means of direct analytical operations. Practically all of the many methods which have been devised to solve such equations are in effect methods of approximation whereby a crude guess at the root is used to obtain a closer value, the latter again used to obtain a still closer value, and so on, until the desired accuracy is secured.

Some of the most useful ways of

carrying out the method of successive approximations are described and illustrated in this chapter. 1 1 .O N E E Q U A T I O N I N O N E U N K N O W N Suppose that the equation to be solved is (1 )

y = f(x) = 0.

Suppose also that by means of a rough graph or otherwise we have ascertained that there is a root of the equation in the vicinity of

χ = xQ.

The method of successive

approximations consists in finding a sequence of numbers xQ, X1, x2, ..., converging to a limit f(a) = 0.

a

such that

The recurrence relation by which xn+1 is

calculated after Xn has been obtained may be expressed in the form (2)

xn+1 = xn - fUn)M

in which m denotes the slope of a suitably chosen line. The ideal choice of m is obviously the slope of the chord joining the point (xn, y ), where yn = f(xn), to the point

11. ONE EQUATION IN ONE UNKNOWN

37

(a, 0), for then x n + 1 = a, and the problem Is solved. Since of course the point

a

is unknown this ideal value

of m is also unknown and we are obliged to use some type of approximation for m.

There are several ways of

choosing an approximate value for m. 1)

The slope of the tangent to the curve

y = f(x)

at x = x n gives

2)

The slope of the chord joining two points

already calculated, say

3)

If

and

, gives

and x = x g are values of x for which

f(x1 ) and f(x 2 ) have unlike .signs, so that the desired root lies between x 1 and x g , it is frequently satisfactory to use

throughout the successive steps.

Since m is calculated

once for all, this choice saves considerable labor. 4)

Similarly if the curvature is not too great

near the root it may suffice to use

throughout the successive steps.

This is especially

advantageous if the computation of successive values of :

laborious. 5)

If the curvature does not change sign near the

root, it is clear from a consideration of the graph that a value of m between those given by 3) and 4) will often

38

II. SOLUTION BY SUCCESSIVE APPROXIMATIONS

be better than either one.

Hence, we may employ the

arithmetic mean

as our approximate value of m. No inflexible rule can be given for the best choice of m.

The computer need only remember that the ideal

value is the slope of the chord joining a known point (x^, y 1 )'on the curve with the point (a, 0), and then make the wisest choice available in the particular problem. The several methods mentioned above will now be illustrated by numerical examples. Example 1.

Find the positive root of

Here we have, using method l),

By substituting a few trial values of x we find that indicating a root between x = 1 and x = 2, probably nearer x = 2. x Q = 2.

Hence X q is chosen as

The computation may be arranged as shown below:

The desired root to 5 decimal places is

x = 1.79632.

11. ONE EQUATION IN ONE UNKNOWN Example 2.

Find the smaller positive root of

To five places the desired root is Example 5•

39

x = 0.44881

Find the positive root of

By method 1)

From a rough sketch we conclude that the root is somewhere near

x = 2.

The computation may be arranged as

shown:

The last number in the right-hand column is the root to nine places.

40

II. SOLUTION BY SUCCESSIVE APPROXIMATIONS Example 4.

Find the largest positive root of

By trial substitutions we locate the largest positive root between

x = 4

and x = 5 .

Since

f(4) = - 2 3 ,

f(5) = 94, approximately, the slope of the chord is 117. The approximate value of

is 50.

Using method 5)

we take a value of m about,half way between 50 and 117, say

m = 8 5 , and let

x Q = 4.

The substitutions are

performed by the usual method of synthetic substitution. This can be done in a continuous operation on the calculating machine.

(In the case of machines where the

carry-over is not effective over all the dials in certain positions it may be necessary to replace negative numbers* on the dials by their complements, in which case the change of sign must be carefully watched.)

The work may

be arranged as shown:

It is possible to prove that with suitable limitations on f(x) and on the choice of the Initial value X q the process of successive approximations will give a sequence X q , x 1 , x g , ... which converges to a root of f(x) = 0.

For the practical computer, however, such a

theorem is of somewhat academic interest, since the numerical process itself either converges with reasonable rapidity to a value which is obviously a root, or else by its behavior gives warning that x Q was poorly chosen

11 . ONE EQUATION IN ONE UNKNOWN

41

or that f(x) haa some peculiarity near the supposed root. Some peculiar cases will be considered in the next article. Exercises Find to five places all real roots of

12. EXCEPTIONAL CASES If the derivative f'(x) ?anishes at or near a root of f(x) = 0, the process of approximation encounters trouble becauae the divisor m is small.

In such a case it is

frequently best to obtain the root of

first

of all, especially if f"(x) is not near zero.

Suppose

that we have found the root x = a of the equation f'(x) = 0.

We next calculate f(a) and f"(a). the quantity

a

Then

is a double root,

and f(a) and f"(a) have like signs, there is no root of f(x) = 0 in the vicinity of x = a. For under these conditions the curve

is

concave away from the x-axis and cannot cross the axis in the neighborhood of x = a. 3)

If f(a) + 0 and f(a) and f"(a) have unlike signs,

we may expect to find two roots of f(x) = 0, one greater than

a

f(x) at

and one less than

a.

Using Taylor's series for

x = a, noting that f'(a) = 0

and neglecting

terms of third and higher degree we obtain the approximate values

b2

II. SOLUTION BY SUCCESSIVE APPROXIMATIONS

for the two roots. Each one of these may now be refined by successive approximations in the usual manner. Example 1.

Tabulating a few values of f(x) and Its derivatives we have

These values show a root of and

between

x = 1

x = 2, with the possibility of two roots of in the same interval.

tion applied to

The method of approxima-

f'(x) = 0 gives

We next find and these values in conjunction with formulas (1) and (2) above yield as a first approximation for the two roots of f(x) = 0

For the first value x^

12. EXCEPTIONAL CASES

10

and the method of approximation gives the improved value

Similarly

Example 2.

Here f'(x) = 3 - 6 cos x, which vanishes for Also l/2f"(a) = 2.598. Use of formula (1) gives as an approximate value for the root

The method of approximation applied to this value now yields The other root is found in a similar manner. When all three quantities, f(x), f'(x), f"(x) vanish close to the same point additional complications occur. As such instances are rare and the analysis is lengthy these cases will be omitted.

II. SOLUTION BY SUCCESSIVE APPROXIMATIONS Exercises Find to five places the roots of

13. SIMULTANEOUS EQUATIONS The method of approximation may also be extended to the solution of simultaneous equations.

Thus if

are two equations in two unknowns, and if a" point close to a solution has been determined by graphical methods or otherwise a closer point (x, y ) can usually be obtained as follows: Let

Expand F(x, y) and

B(x, y ) in Taylor's series to terms of the first degree, and assume that (x, y) is a solution, i. e., Then, approximately,

in which for brevity we have set etc.

The two equations are

solved for 6x and rfy and the new approximation to the solution is given by

The process is repeated until the desired accuracy is secured.

13· SIMULTANEOUS EQUATIONS The following scheme of computation is convenient for the case of two equations.

F

D

GX

Fy O

G

Px

-Sx

+Sj

X

J

The values of F, G, Fx etc. are calculated for χ= , y = Jr1 and inserted in the proper places in the scheme. Then the first column is covered and the remaining two-rowed determinant D=F G - F G χ y y x is evaluated on the machine by cross multiplication and entered in the place shown. Next the second column is covered and the remaining two-rowed determinant FGy - GFy is evaluated on the machine, and while the result is still on the dials the division by D is performed. This gives - Sx, which is entered as shown. Finally the third column is covered, the determinant FGΛv - GF-Λ.v is evaluated, the result is divided by D, and recorded for Sj· The improved values of χ and y are now given by χ = X1 + Sx, y = y1 + Sj and are recorded. The next step will be an exact repetition of the foregoing with the new values of χ and y. If it is found that the values of Fx, F , ΰχ, Gy are not much changed in successive calculations, we need not recompute them at every step but merely copy them, together with D, from the previous step.

46

II. SOHJTION BY SUCCESSIVE APPROXIMATIONS Example.

Find a solution (different from the

obvious solution

x = 0,

y = 0) of the equations

An examination shows that in the first quadrant a solution occurs in each interyal in which both and

cos x

sin x

are positive, i. e., for etc.

Accordingly, we first make the

change of v a r i a b l e w h i c h

transforms

the given equation to

We take as initial estimates for x' and y the values and carry out the computation according to the scheme described above.

1 3 • SIMULTANEOUS EQUATIONS x1 = -1.279

0.2

y

47 =

-8.879

0.8107

1 . 1 0 0

-79.49 x'

=

-0.029562 0.15277 -58.751

-8.879

-0.8107

- 0 . 1 3 6

-O.132

0.336

y

=

x

=

0.001496 0.0004830 -58.51

-7.4883

-7.4883

-1.6359 0.00049

0.35629

y

=

1

=

2.76849

1.790

"7-437

-7-^37

"1.790

-0.00002 x

2.768

1.6359

-0.02029 1

2.9

0.00020

0.35631

y

=

2.76869

O.OOOO563

1.79*

-7.44*

0.0000455

-7.44*

-1-79*

-58.5*

-0.00000

*Not recomputed.

x'

=

X

=

x =

0.00001

0.35631

7.85398

-

y =

2.76870

y =

2.7687

x'

7-^977

14. SUCCESSIVE SUBSTITUTIONS It may be possible by suitable manipulation and combination of the given equations y) = 0

F(x, y ) = 0,

to transform them to a n equivalent pair

such that for values of x and y near a common solution the following inequalities are satisfied:

46 II. SOHJTION BY SUCCESSIVE APPROXIMATIONS where k denotes some positive constant less than unity. Whenever this is possible a process of successive substitutions will lead to a solution. Example.

The pair of equations used in the example

of Article 13 can be expressed in the form

We take

x' = 0

in the first equation, compute y, put

this y in the second equation and compute x", put this x' in the first equation and compute y, etc.

This process

gives us the following sequence of values:

These values check closely with the values obtained for x 1 and y in the Example of Article 13• Exercises Solve by the methods of Articles 13 or 14.

II. SOLUTION BY SUCCESSIVE APPROXIMATIONS 15. EXCEPTIONAL CASES When the determinant

1+9

F D= G„ vanishes at or near a supposed solution of F(x, y)= 0, G(x, y) = O we may anticipate difficulty with the method described in Article 13· The vanishing of D indicates (a) multiple solutions, (b) two or more solutions close together, or (c) no solution at all. The locus of points defined by the equation D(x, y) = 0 is the curve on which the loci of F(x, y) = const., G(x, y) = const. have either common tangents or singular points. If the two determinants D y (DG) = (DF) = F„ G„ are not both zero in the vicinity of the point in question we may proceed as follows: I. Construct a graph of the curves showing the approximate point of intersection or tangency, and also showing the signs of the functions F(x, y) and G(x, y) in the several regions bounded by the curves. II. Solve the simultaneous set F=O, D=O (or G=O, D=O, whichever seems easier) by successive approximations. We suppose that the intersection of F=O, D=O is found to be χ = a, y = b. Then cal­ culate G(a,b). If G(a, b) = 0, the curves are tan­ gent at (a, b) and the point (a, b) is the desired solution. If G(a, b) + 0 its sign, compared with the signs of F and G as shown in I, will determine whether a) There is no solution, or b) Two nearby solutions.

50

II. SOLUTION BY SUCCESSIVE APPROXIMATIONS III.

G(x, y )

If case b) occurs we expand

F(x, y )

in Taylor's series at (a,b), assume

G(x, y) = 0;

recall that

F(a, b) = 0,

and

F(x.,y) = o,

and obtain

where < The solution of this pair of simultaneous quadratic equations for

(fx and

task in itself. 1.

f j normally would be a considerable

Here however we are aided by two facts.

Since

we have

so that if we multiply (1) by k and subtract from (2) we eliminate all terms of first degree and obtain

(3)

in which 2.

If we set

will differ only slightly f

we may expect that jn r

o

m

t

h

e

slope of

the tangent to Therefore setting in the form

and equation (5) in the form

we rewrite equation (1)

15• EXCEPTIONAL CASES

(If F

y

is small compared to F x it is better to set and solve for v

and

gj. ) The two equations

(k) and (5) are easily sdlved by successive substitutions. We then have approximations to the two points of intersection IV.

Each of these trial pairs is now refined to the

desired degree of accuracy by the usual method of successive approximations. Example.

Investigate the solutions of

The stages of the investigation as outlined above are as follows. I.

The graph of the curves, showing the signs of

F and G, is given in Fig. 1 and indicates points of tangency at approximately (1.9, +2.7)-

Figure 1

52

II. SOLUTION BY SUCCESSIVE APPROXIMATIONS

II. The general expression for D as a function of x and y proves to be

Accordingly, we solve the simultaneous equations

(It happens here that these equations can be solved analytically, but usually we must resort to the method of approximations.)

For these value?

The solution in the first quadrant is

G(a, b) =

-0.018586,

and the fact that

G(a, b) is negative shows, by a consideration of the signs in Fig. 1, that there are two nearby points of intersection. III.

At the point (a, b) we now find

Using these values we set up equations (4) and (5) as in the text.

They prove to be

Starting with the trial value /i= -0.35356 in (5) and

15- EXCEPTIONAL CASES

53

using successive substitutions we get (choosing the negative sign)

This gives IV.

x = 1.88371,

y = 2.71592.

Returning to the original equations P = 0

and G = 0, we solve by successive approximations, starting with the values

x = 1 .88371,

y = 2.71592, and after

three steps obtain

The other solution is found in a similar manner. Exercises Solve the sets of equations.

1 6. COMPLEX ROOTS OP ALGEBRAIC EQUATIONS Corresponding to a pair of complex roots of the algebraic equation

with, real coefficients there is a real quadratic factor Instead of determining the complex root directly by the method of Article 11, we may avoid all substitutions of complex numbers by determining the p and q of the corresponding quadratic factor.

To do this

in a systematic manner with the calculating machine we set up a procedure for synthetic division by a quadratic factor

5k

II. SOLUTION BY SUCCESSIVE APPROXIMATIONS

analogous to the well known synthetic division by a linear factor.

The coefficients

a column.

are arranged in

Then a second column of b's is computed by the

formulas

The b's are the coefficients in the quotient. P

same way we divide the quotient by taining a column of c's.

z

In the

+ pz + q,

ob-

The complete computational

setup is shown below:

The recursion formulas are

16. COMPLEX ROOTS OP ALGEBRAIC EQUATIONS

55

Also

The quantities D,

dp, and - tfq are computed on the

calculating machine just as mere the D,

-

e

s

second degree p o l y n o m i a l f o r etc.

Since i n each case t h e p o l y n o m i a l

clear that

the order of

subscripts

is

is

unique i t

immaterial,

so

is that

),etc. Of t h e v a r i o u s ways i n w h i c h t h e p o l y n o m i a l may be e x p r e s s e d , machine c a l c u l a t i o n i s

is

machine by cross m u l t i p l i c a t i o n , on the d i a l s , Example.

is

Given

e a s i l y evaluated on the and t h e r e s u l t ,

sin

Evaluating the determinant removed f r o m

for

following:

while

d i v i d e d by

From t h e t r i g o n o m e t r i c

cos

interpolating

t h e most c o n v e n i e n t

probably the

For the two-rowed determinant still

linear

f i n d cos t a b l e s we h a v e

a n d d i v i d i n g b y 2k we

obtain

N o t e t h a t a n y common f a c t o r may b e and

Hence,

1 9 . LINEAR INTERPOLATION we may i g n o r e

the decimal point

as i n t e g e r s . in

and

Note a l s o t h a t will

69

and t r e a t

common d i g i t s

also occur i n y .

those

quantities

on the

Thus i n t h e

a b o v e we c o u l d h a v e i g n o r e d t h e d i g i t s

left

example

825 a n d

written

only

so t h a t

again

cos

interpolations may e f f e c t

_.

I n t h e case o f

repeated

w i t h n u m b e r s c a r r i e d t o many f i g u r e s

a worth-while

saving of

this

labor.

Exercises The s t u d e n t

s h o u l d a p p l y t h e above method

interpolation i n tables functions,

etc.,

until

of

logarithms,

the process

is

to

trigonometric thoroughly

mastered. 2 0 . A I T K E N ' S PROCESS OF ITERATION When l i n e a r

interpolation fails

a c c u r a c y more p o i n t s

to give

may b e u s e d , w i t h a n

interpolating

p o l y n o m i a l of h i g h e r degree t h a n t h e f i r s t . the

adequate Consider

expression

Note f i r s t and formed f o r polynomial

that

this

expression i s formed f o r

i n e x a c t l y t h e same m a n n e r a s and y 2 . i n x of

Secondly,

degree 2.

it

is

Finally,

evidently if

is a

70.

III.

IOTERP0LATI0N

and i f it

Prom t h e s e

follows

that

I n o t h e r w o r d s , we may o b t a i n t h e s e c o n d d e g r e e polating polynomial

by a p p l y i n g

interpolation to could obtain the

and same r e s u l t

. from either

the interpolating polynomial f o r o b t a i n e d by l i n e a r

of

t h e two

In

points

i n t e r p o l a t i o n a p p l i e d t o two

i n t e r p o l a t i n g polynomials For

n + 1

example

each formed f o r

inter-

linear

E v i d e n t l y we

T h i s p r o c e s s n a y be e x t e n d e d i n d e f i n i t e l y .

points.

results

n of

forms

general is different the

given

20.

A I T K E N ' S PROCESS OP ITERATION

71

etc.

In Aitken'a la

process a n a r r a y of

set up a c c o r d i n g t o t h e f o l l o w i n g

The c a l c u l a t i o n s

linear scheme:

are e a s i l y performed on the

c u l a t i n g machine by c r o s s m u l t i p l i c a t i o n and since the elements of

interpolations

cal-

division,

the numerator determinant

occur

in

t h e a r r a y i n t h e same r e l a t i v e p o s i t i o n s r as i n the 1 s t degree 2 n d degree 3 d degree x^ - X x i determinant. E x0 a. 2m9p8l 5 e0 1. From t h e t a b u l a t e d v a l u e s o f t h e- 1 6 2 0.3 f0.1+ u n c t i o0.3961+6 n S i ( x ) a 0.1+57195 t i n t e r v a l s of 0.1 c a l c u l a t e S i ( 0 -. 6426 2 ) . T h0.1+9311 e c o m p l e t e c o m p u t a t i o0.1+56537 n appears below. The c o38 lumns 0.1+56134 0.5 headed and a r e t a k e n f r o m t h e t a b l e a n d t h e r e 138 0.456484 0.456557 0.1+51+900 0.6 0.58813 m a i n i n g v a l u e s a r0.1+53502 e c a l c u l a t e0d. 4 5i 6n 4 3a2c c o r d0a.n4c5e6 5w e 5 7i t h t h 238 0.68122 0.7 s t nd r d scheme a b o v e . x^ 1 degree 2 degree 3 d e g r e e x.^ - x 0.3

0.29850

-162

OA

0.3961+6

0.457195

0.5

0.1+931 1

0.1+5613^

0.1+56537

-62

0.6 0.7

0.58813 0.68122

0.1+51+900 0.1+53502

0.1+561+81+ 0.1+561+32

38 0.1+56557 0.1+56557

138 238

72

III.

INTERPOLATION

The c o m p u t a t i o n s h a v e b e e n c a r r i e d t o one decimal place errors is

due t o n e g l e c t e d d i g i t s .

n o t r e a l l y needed b u t

fact

that

and a l s o

is a partial indicates

unnecessary. places,

The l i n e f o r

x =

s e r v e s as a c o n t r o l ,

t h e two e n t r i e s

identical

additional

i n order t o reduce the accumulation of

i n the

the

degree column are

check on t h e n u m e r i c a l

that

0.7

since

computation

a h i g h e r degree p o l y n o m i a l

is

Since t h e o r i g i n a l data were g i v e n t o

five

t h e i n t e r p o l a t e d v a l u e s h o u l d a l s o be r o u n d e d

to f i v e places, value of

giving

Si(0.462)

to

Si(0.462) = 0.45656. seven places

our i n t e r p o l a t e d value

is really

is

The

0.4565566,

quite

off

true so

that

close to the

true

value. Example 2 . radians at

Prom a t a b l e o f

sin x for x given

intervals

of

0.1 f i n d

The c a l c u l a t i o n

is

shown b e l o w

at 1 s t ddeeggrreeee

X X

s il nt t ix

4 0 4 .. 0

-0.75680250

4.1

.8182771' .81 827711

-.9031 - . 9 0 3 1 11207 207

4.2

.87157577 .87157577

--.89338269 .89338269

2 nn dd ddeeggrreeee 2

the value of

3 r d ddeeggrreeee 3

sin

4 tt hh d de eg g rr e ee e 4

in

4.238.

th 5 5th d d ee gg rr ee ee

xx^ - x x _ 2 J 8 -238

i4t .. 3 3

.. 9 4 91 16 61 16 65 59 94

-.88323083 -.88323083

939401 939401

4 .4 4.4

.95160207 .95160207

-.87270824

91 12 26 63 31 1 9

4.5

•97753018 .97753018

-.86186885 -.86186885

888316 888316

The v a l u e should be

-,38 -138 -38 -38

-.88968553 -.88968553 -0.88957475 -0.88957475 7928 7928 8390 8390

62 -0.88957194

-0.88957194 11

-- 00 .. 88 8 5 77 11 99 99 89 95

162 162 262 262

-0.88957200.

Exercises 1.

From t h e t a b l e o f

values

calculate Si(0.45); 2.

Prom t h e t a b l e calculate

of

s i n ^.25;

g i v e n i n Example 1

Si(0.4-7);

Si(043).

v a l u e s g i v e n i n Example 2 s i n 4.275;

2 1 . N E V I L L E ' S METHOD OP

sin

A v a r i a t i o n of the f o r e g o i n g process Neville.

Its

basic principle

is

4.23.

ITERATION is

due

t h e same as i n

to Aitken's

method b u t t h e c o m p u t a t i o n a l s e t u p i s m o d i f i e d as

follows:

2 1 . N E V I L L E ' S METHOD OP ITERATION

Cd,re must be e x e r c i s e d t o s e l e c t the column same l i n e

of

the determinant.

c o m p u t a t i o n as t h e e n t r i e s

Example.

entries

,1«stt degree

ain x

For

for

lie

the

in

from the

left-hand

instance

The p r o b l e m g i v e n i n E x a m p l e 2 o f A r t .

i s here worked by N e v i l l e ' s

4 4.. 00

the correct

s i n c e t h e s e do n o t a l w a y s

column of

x X

73

20

Method.

n d degree 2ru3

33rr n.

If k is odd

If k is even

In either case there are k + 2 constants, one of which is arbitrary, since numerator and denominator may be divided by any non-zero coefficient. Theorem 1. If i are k + 1 points with distinct x's, there cannot exist two distinct irreducible rational fractions y = N 1 (x)/D 1 (x) and y = N (x)/D 2 (x) of order k both of which are satisfied by the given k + 1 pairs of values (x^, y^). For if both fractions are satisfied then the equation

65. PROPERTIES OF RECIPROCAL DIFFERENCES 235 is true for k + 1 distinct values of x . But each member of this equation is a polynomial of degree not exceeding k , and hence, the two members must be identically equal and must contain identical linear factors (real or imaginary). Moreover, since N ^ x ) and D ^ x ) have no common factors, all the linear factors of N ^ x ) must be in N g (x) and all the factors of D ^ x ) must be in D 2 (x). Likewise, all linear factors of -N2(x) must be in N n (x) and all linear factors of D 2 ( x ) must be in D ^ x ) . Hence, N ^ x J / D ^ x ) and N 2 (x)/D 2 (x) are identical fractions. Theorem 2. If y = R lc (x) is. an irreducible rational fraction of order k , the reciprocal- differences of order k are constant. For using we get by the reduction of Article 6b

while using

we have

th

that If where formed because constant, In for of this order the Theorem This Rexample for (x) way, of determinant kkmay nare symmetry.

P5(a)

204 68 -28 -89 -120 -126 -1 12 "83 -44 0 44 83 1 12 126 1 20 89 28 -68 -204

612 -68 -388 "453 "354 -168 42 227 352 396 352 227 42 -168 -354 -453 -388 -68 612

1 02 -68 -98 -58 3 54 79 74 44 0 -44 -74 -79 -54 -3 58 98 68 -1 02

213180

3

2 288132

891 48

V I . ORTHOGONAL POLYNOMIALS

381

n = 19 (20 points) s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 s» m

P., (a) 19 17 15 13 11 9 7 5 3 1 -1 -3 -5 -7 -9 -11 "13 -15 "17 -19 2660

Pa(s) 57 39 23 9 "3 -13 -21 -27 -31 -33 -33 -31 -27 -21 -13 -3 9 23 39 57 17556

P,(a) 969 357 -85 "377 -539 -591 -553 -445 -287 -99 99 287 445 553 591 539 377 85 -357 -969 4903140

1938 -1 02 -1 122 -1402 -1187 -687 "77 503 948 1188 1188 948 503 "77 -687 -1187 -1 402 -1 122 -1 02 1938 22881320

1938 -1 1 22 -1 802 -1 222 -187 771 1351 1441 1076 396 -396 -1076 -1441 -1351 -771 187 1 222 1802 1 122 -1938 31201800

n = 20 (21 points) s

P,(a)

Pa(8)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

10 9 8 7 6 5 4 3 2 1 0 -1 -2 "3 -4 -5 -6 -7 -8 -9 -1 0

1 90 135 82 37 -2 -35 -62 -83 -98 -1 07 -1 1 0 -107 -98 -83 -62 "35 -2 37 82 133 1 90

S

770

™ m

201894

P5(a)

P,(S) 570 228 -24 -196 -298 -340 -332 -284 -206 -1 08 0 108 206 284 332 340 298 1 96 24 -228 "570 1730520

969 0 -51 0 -6 80 -61 5 -406 -1 30 150 385 540 594 540 385 1 50 -130 -406 -61 5 -680 -510 0 969 5720330

3876 -1938 -3468 -261 8 -788 1 063 2354 2819 2444 1 404 0 -1 404 -2444 -281 9 -2354 -1 063 788 2618 3468 1938 -3876 121687020

Table VII IHTEORAIB OP BINOMIAL COEFFICIEHTS, ^

X

0

1

2

-1

-1

1 2

12

1 8

720

1

1

1 2

-1

12

1 55

2

2

2

1

0

3

3

3

2 2

i

1

It

4

8

20

8 3

5

5

sa 2

III

^

6

6

18

27

2 It

7

7

42 2

8

8

32

3

12

12

1223 24

208

96

522.

3

dt

6

7

iff

-19087 6o4eo

17280

720

tIo

60480

-1

1 90

37^0

4

3

^

90

5

-863

;29_ 2240 14 55

425

0

-8

945

-273

8

-1070017 3628800

2571 J 89600

241 92

362^00

1036800

955

8_

"112 16200

T?oo

sfs

-369 44800

35^

8 955

14175

-107

,94 14175

35^5

5257

275

T5t

2§f

1 2096

241 92

27?

20736

121

21

41 i"5o

0

1 400

10

26117

720

10 1 60

2336 45

382

30919

?257

8640

17280

18128

736 189

9h

9

-8183 518400 3953 14175

8183

1^00 8183 1036800 0

Table V I I I GAMMA AND DIGAMMA FUNCTIONS 0 .02 .01+ .06 .08

1.00000 .98884 .97844 .96874 •95973

-.57722 -.54480 -.51327 -.48263 -.45280

• 50 • 52 • 54 • 56 .58

.88623 .88704 .88818 .88964 .89142

.03649 .05502 .07323 .09114 .10874

.10 .12 .14 .16 .18

.95135 .94359 .93642 .92980 .92373

.60 .62 .64 .66 .68

•89352 .89592 .89864 .90167 .90500

.12605 .14308 .15983 .17633 .19256

.20 .22 .24 .26 .28

.91817 .91311 .90852 •90440 .90072

-.42375 -.39546 -.36787 -.34095 -.31469 -.28904 -.26398 -.23949 -.21555 -.19212

.70 • 72 • 74 .76 .78

•90864 .91258 .91683 .92137 .92623

.20855 .22429 .23980 .25509 .27015

.30 • 32 .34 • 36 • 38

.89747 .89464 .89222 .89018 .88854

-.16919 -.14674 -.12475 -.10321 -.08209

.80 .82 .84 .86 .88

.93138 .93685 .94261 .94869 •95507

.40 .42 .44 .46 .48

.88726 .88636 .88581 .88560 .88575

-.06138 -.04107 -.02114 -.00158 +.01763

• 90 • 92 • 94 • 96 • 98

.50

.88623

.03649

1 .00

.96177 .96877 .97610 .98374 .99171 1.00000

.28499 .29963 .31406 .32829 •34233 .35618 .36985 .38334 .39666 .40980 .42278

For values beyond the range of the tables use the formulas

383

INDEX Page 24

Accuracy of solution of linear equations Aitken's method of interpolation

69

Algebraic equations complex roots of -

53

linear -

3,7,11,15,17,24,29

Approximat ion - by exponential formula

248

- by Gram-Charlier series

309

- by least squares

242,255,292

- by polynomials

66,69,72,78,83,160,161 163,172,178,181,207

- by power formula

247

- by rational fractions

219

- to frequency function at discrete points

317

successive -

36,41,44,47,53,57

trigonometric -

294

Augmented matrix

. . .

Beasel's interpolation formula Binomial coefficients

9 1 81

150,365

Binomial coefficient functions Calculus of differences

150,366,369 328

Central derivatives, formulas for Central difference interpolation Central difference notation

99 172,178,181 172

Coefficients binomial -

150

effect of errors in - of linea,r equations

29

- in central differentiation formulas

99

- in closed quadrature formulas

385

122

386 Coefficients,

INDEX (continued; 97,98,99

- in differentiation formulas - in fifth degree smoothing formulas

279

- in Gauss method of numerical integration. - in open quadrature formulas

288 126

- in orthogonal polynomials

259,260^.267,375

- in third degree smoothing formulas - in trigonometric approximation Lagrangian Stirling's -

. . . .

278 296,301 83,371

•

tables of Lagrangian undetermined Cofactor £>f a determinant Cofactors, calculation of Complex roots of algebraic equations

146 90 104 4 27 53

Conditions for solution of linear equations

8,9

Continued fraction, interpolation by -

229

Continuous solution of difference equations

326

Convergents of continued fraction

230

Corrector formula

135

Definite operator

115

Degenerate set of points

240

Derivatives formulas for formulas for central - in terms of differences Detection of errors Determinants evaluation of solution of equations by Difference equations

97,98 99 191 158 3 26 7 324

Differences finite -

154

calculus of -

328

central -

172

divided -

201

INDEX Differences

38?

(continued)

reciprocal -

219,226

Differential equations

five-term formulas for -

1 42

second order -

138

simultaneous -

141

solution of -

131

special formulas for second order -

1 4o

three-term formulas for -

135

Differentiation, numerical

93,191

Digamma function

330,383

Discrete solution of difference equation

326

Divided differences definition of -

201

interpolation by -

207

Elimination

15

Elimination, rule for -

19

Equally spaced points, orthogonal polynomials for

.

265

Equations homogeneous linear linear non-linear -

11 3,7,11,15,17,24 36,41,44,47,49,53,57

symmetrical linear -

24

Error detection of -

1 58

inherent - in numerical differentiation - in polynomial interpolation - in solution of linear equations - of linear formulas in general Evaluation of determinants Everett's central difference formula

29 .

94 78 29 108 26 178

Everett's formula, integral of -

197

Exponential formula, approximation by -

248

Factorial polynomials Finite differences

145 1^5,154

*NDEX

388 Finite differences

(continued)

formulas for -

328

notation for -

154,172

Five-term formulas for differential equations

142

Formula Bessel's interpolation -

181

Everett's central difference . . . . . . . . . . . . .

178

exponential -

248

- for central derivatives

97

- for central difference interpolation . . .

172,178,181

- for differential equations

135,140,143

- for error in differentiation

94

- for error i n interpolation

78

- for error in linear formulas

111

- for interpolation with reciprocal differences. - for interpolation by divided differences - for numerical differentiation

225,229

. . . . . .

207

97,98,99

Gauss'3 interpolation -

163

Gregory' s

196

Lagrange's interpolation -

83

Newton-Cotes integration -

122,126

Newton's interpolation -

160,161

power -

.

247

predictor -

135

smoothing -

278,279

S t i r l i n g ' s interpolation-

176

Fourier coefficients Frequency function, approximation at discrete points Functions, interpolating, defined Fundamental theorem for least squares Gamma function Gauss'3 interpolation formula

296 .

317 63 292

330,383 163

Gauss's method of numerical integration

285

General Solution of difference equation

327

Graduation of data

275

Gram-Charlier approximation

309

Gregory's formula

196

INDEX

389

Harmonic analysis

300

Hermitian polynomial

310

Homogeneous equations

11

Indefinite operator

115 29

Inherent errors Integrals,

(See Numerical integration) 100

Integration, numerical (See also Numerical integration)

63

Interpolating function polynomial -

6k

rational -

6k 65

trigonometric Interpolation

63 69

Aitken's method of Bessel'a formula for -

181

- by continued fractions.

225

- by convergents

230

- by divided differences

207,215

- by reciprocal differences

225,230 78

error of Everett's formula for -

178

Gauss's formula for -

163

inverse -

73

,

Lagrange's formula for -

83

linear -

68

Neville's method of -

72

Newton's binomial formula for -

160

Stirling's formula for -

176

Inverse interpolation

.

73

Iteration Aitken'a process of -

69

Neville's method of -

72

Lagrange's interpolation formula

83

Lagrangian coefficients

83

Lambda determinant

13

390

INDEX 57

Lambda determinants, solution of Least square approximation, fundamental theorem of-.

292

Least squares

242

- for integrals

255

Legendre polynomials Linear equations

261 3>7,11,15,17,24,29

accuracy of solution of -

24

elimination in

15

numerical solution of - . . . . ,

17

solution of - by determinants

7

Linear interpolation

68

Linear operator

108 9

Matrix . . - of coefficients. Minor of a determinant Moments Neville's method of interpolation

. .

9 4 313 72

Newton-Cotes quadrature formulas - closed type

122

- open type

126

Newton's binomial interpolation formula - forward differences

160

- backward differences

1 61

Notation - for divided differences

201

- for finite differences

154

- for reciprocal differences

226

- for factorial polynomials

1 45

- for binomial coefficient functions

150

Numerical differentiation

93

Numerical differentiation, error of -

94

Numerical integration

1 00

error in -

1 08

Gauss's method of -

285

Gregory's formula for -

196

Numerical integration (continued)

Page

modified trapezoidal rule for -

118

Newton-Cotsa closed formulas for -

122

Newton-Cotes open formulas for -

126

Simpson's rule for -

120

• trapezoidal rule for using differences Weddle's rule for -

116 193,196,197 125

Numerical solution - of difference equations

325

- of differential equations

131

- of linear equations - of non linear equations

17 36-62

Operator definite -

115

indefinite -

115

linear -

1 08

Order of a rational fraction

234

Orthogonal polynomials

257

Orthogonal polynomials, equally spaced points

265

Orthogonality, roots of A-equations

58

Pentagamma function

330

Polynomial approximation by least squares

242

Polynomial interpolating function

64

Polynomials factorial -

145

Hermit ian -

31 0

orthogonal -

257

properties of -

66

summation of -

332

tabulation of -

183

Power formula, approximation by -

247

Predictor formula

135

Rank - of a determinant

6

- of a matrix

9

INDEX

392 Rational fractions approximation by -

219

order of -

23U

summation of -

332

Rational interpolating function

64,227

Reciprocal differences

219,226

Rule - for elimination

19

modified trapezoidal -

118,119

Sheppard's -

164,215

Simpson'3 -

120

trapezoidal -

116

Weddle's -

125

SensitLvity, measure of -

33

Sheppard's central difference notation Sheppard's rules

172 164,215

Short method of solving differential equations . . .

132

Simpson's Rule

120

Simultaneous differential equations Simultaneous non linear equations,

141 successive

approximations

44

Smoothing

275

Solution of a difference equation

324

discrete -

325

continuous -

326

general -

327

Solution of equations - by determinants - by elimination - numerical

17

- by successive approximations - using

7 15

determinants

36-57 57

Starter formulas

135

Stirling's central difference formula

176

integral of -

196

INDEX

393

Stirling's numbers

146

Substitutions, successive -

47

Subtabulation

185

Successive approximations

56

Successive substitutions

47

Summation

332

Symmetric integrals of Stirling's formula

196

Symmetrical equations

24

Tabulation of polynomials

183

Tetragamma function

330

Three-term formulas, for differential equations.

. .

Trapezoidal rule

135 116

Trigamma function

330

Trigonometric approximation

294

Trigonometric interpolation

. . .

65

Undetermined coefficients

104

Weddle's Rule

125

Weight function

291>