Table Of ContentMODERN
COMPUTING
METHODS
SECOND EDITION
PHILOSOPHICAL LIBRARY
NEW YORK
Engin. Library
Published 1961
by Philosophical Library, Inc.
15 East 40th Street, New York 16, X.Y.
All rights reserved
Printed in England for Philosophical Library
by John Wright <bSons Ltd., at the Stonebridge Press, Bristol
n-&5?505
.......
CONTENTS
Introduction ...... page v
1. Linear Equations and Matrices: Direct Methods 1
2. Linear Equations and Matrices: Direct Methods on Automatic
Computers 13
3. Latent Roots and Vectors of Matrices . 22
.....
4. Linear Equations and Matrices : Iterative Methods 34
....
5. Linear Equations and Matrices: Error Analysis 41
.....
6. Zeros of Polynomials 53
7. Finite-difference Methods 62
8. Chebyshev Series 71
9. Ordinary Differential Equations: Initial-value Problems . 80
10. Ordinary Differential Equations: Boundary-value Problems 93
11. Hyperbolic Partial Differential Equations 101
......
12. Parabolic and Elliptic Partial Differential Equations 112
13. Evaluation .of Lim.its;.Use .of Re.curren.ce R.elati.ons . 122
14. Evalua.tion.of In.tegra.ls ...... 130
15. Tabulation of Mathematical Functions . . . 137
Bibliography 145
Index 167
1n
INTRODUCTION
The first edition of Modern Computing Methods was based on lectures
delivered by various members of the staff of Mathematics Division,
N.P.L., as part ofa vacation course on 'Computers for Electrical Engineer
ing Problems,, organized by the Electrical Engineering Department of
the Imperial College of Science and Technology, and attended by repre
sentatives of industrial firms. The course was designed to teach the basic
principles of the use of analogue machines, high-speed digital computers,
and the techniques of numerical mathematics involved in the solution of
problems in electrical engineering.
Numerical methods are required in all branches of science, and the
techniques are generally independent of the source of the problem. For
example, the same type of differential equation may represent a problem
in physiology as well as a problem in electrical engineering. The oppor
tunity was therefore taken to present, as one of the N.P.L. series of Notes
on Applied Science, suitably edited versions of those lectures contributed
to the course by members of Mathematics Division.
The success of this first edition has encouraged the authors to under
take a complete revision, in the course of which the booklet has been
very largely rewritten. The object has been to bring the material up to
date, particularly with regard to methods suitable for automatic compu
tation, and the principal changes are as follows. The chapter on Relaxation
Methods has been replaced by one on Linear Equations and Matrices:
Iterative Methods, while the chapter in the original edition headed
Computation of Mathematical Functions has been expanded into two
chapters entitled Evaluation of Limits; Use of Recurrence Relations and
Evaluation of Integrals. Most of the other chapters have had new material
added, some being completely rewritten, and the order of the chapters
has been changed. In addition, the chapters on Linear Equations and
Matrices: Error Analysis and on Chebyshev Series are new.
The authors make no apology for the varying level of treatment of the
different topics; this is inevitable if the account is to be kept within a
comparatively small compass. Some of the new material is given in
greater detail than classical material already well catered for in available
text-books. It is hoped that the resulting booklet will prove useful both
as a working manual for those engaged in computational work and as a
basis for courses innumerical analysis inuniversities and technicalcolleges.
The first edition was written by L. Fox, E. T. Goodwin, J. G, L. Michel,
F. W. J. Olver and J. H. Wilkinson. The present edition has been pre
pared by C. W. Clenshaw, E. T. Goodwin, D. W. Martin, G. F. Miller,
F. W. J. Olver and J. H. Wilkinson, all members of the staff of Mathe
matics Division, N.P.L. Valuable criticisms and suggestions have also
been made by L. Fox, now Director of the Oxford University Computing
Laboratory, and many other members of Mathematics Division, particu
larly E. L. Albasiny, J. G. Hayes and T. Vickers. Mrs. I. Goode has
collated the material, prepared the printer,s copy and helped to see the
work through the press.
E. T. Goodwin
Superintendent
Mathematics Division
National Physical Laboratory
May 1960
VI
|
LINEAR EQUATIONS AND MATRICES:
DIRECT METHODS
DEFINITIONS AND PROPERTIES
1. A general set of n linear simultaneous algebraic equations in n un
knowns a1,a2, ...,a, can be written in the form
a112.1 +a1222 +...+a1n2n = bi,
ası”. Fazza's #... +agnºn = bar -
(1)
an121-Hanaº-H... +annºn = bn.
The coefficients are form a square matria oforder n,
011 012 ... 01n
A | “0.m “02.2 ... 0.* , (2)
0n1 0m2 ... 0nn
which is to be considered simply as an array of numbers, and the column
of constants b, similarly forms acolumn matria or vector b.The unknownsa,
form a vector x. Equations (1) can then be written in the shortened form
Ax = b. (3)
The equality sign means that each element of the product vector Ax is
equal to the corresponding element of the vector b, and the left of (1)
gives the rule for premultiplication of a vector by a matrix.
2. The solution of (1) can be written in general terms as
*1 = 0.11b1 + x1aba +...+ xinba,
*2 = 0.21b1 + x2ab, +...+ xanbn,
(4)
wn = anib, + xnaba-H...+ xnnba,
or in matrix notation as
x = A-1b, (5)
l
where A-1 has the same form as (2) with a replaced by oz.The matrix A-1
scalled the inverse orreciprocal ofthe matrix A.
i
The elements or, ofA-4 depend only on the elements of A.Itisclear
from (4) that aknowledge ofthe or, would enable the solutions of(1) to
be obtained with relative ease for any set ofconstants by,but the deter
mination ofthe or, s not trivial. One method of obtaining them s
i i
suggested by equations (4): ifinthese equations we write
bi = 1, b2 = ba =...= bn = 0,
we obtain the elements of the first column of the inverse. It follows that
the various columns ofA-" can befound insuccession by solving equa
tions (1) with the right-hand sides replaced by successive columns of
the matrix
1 0 0 ... . 0
0 1 0 ... . 0
n = O 0 1 --- - O - (6)
I
0 0 0 ... 0 1
This isthe unit matria oforder n oridentity matrix, socalled invirtue of
the relation
nx=x (7)
I
for any vector x.The suffix nisusually omitted when there is nopossible
ambiguity.
3. The main properties ofmatrices required inpractice are those of
addition, multiplication, and transposition.
Matrices can beadded only when ofthe same order, and ifBisthe
matrix (2) inwhichaisreplaced by b,then
au +bu, ala +big, ..., ain-H bin
A+B = ael +b×1, a22 +bag, “..., a2n +ban (8)
0n1+bni, 0,12+bn2, •**> 0nn +ban
eleIfmeantsmaktarrisx, tAhatisis,muevlteiprylieedlembyenat nisummbueltriplk,iedthebyrek.sulting matrix has
Square matrices ofthe same order can bemultiplied, togive
aubu-Falabel +a1aba1 +..., aubig-Falabad-Falabaº-H...,
AB =|asibil-Fassbau-Fassbal 4:..., asibia-Fassbºa-Fassbas-F ...,
------------------------------------------------------------------------------
Ifrthwecoulusmentheofnomtaattiroixn r,A,(A)a,ncd,(Ar,)c,totodendoetneotreestpheectivreeslyulttheofrmthulrtoipwlyainngd
2
corresponding elements ofrr and cs and adding the results {scalar product),
we can write (9) in the simpler form
r^A)c^B) r1(A)c2(B) ... ^(A)c^B)
AB = r2(A)Cl(B) r2(A)c2(B) ... r2(A)cn(B) (10)
rn(A)Cl(B) rn(A)c2(B) %(A)cn(B)
From (10) itis obvious that in general AB ^BA, so that the order ofmulti
plication is important. In (10) we refer to AB as B premultiplied by A, or
multiplied on the left by A, or as A postmultiplied by B, or multiplied on
the right by B.
The transposed matrix of A, called A' or Ar, is derived from A by
interchanging rows and columns. If a matrix is symmetric, so that
ars = a^, then A' = A, and
A'A = AA\
(11)
Other important cases in which the order of multiplication is immaterial
are contained in the equations
AI = IA,
(12)
AA-1 = A~1A = I. (13)
The transpose of a product is given by
(AB)' = B'A', (14)
and its inverse by ^^ ^
= A-1;
(15)
note that in both operations the order of multiplication is reversed.
4. Associated with a matrix A is its determinant, denoted by det A or
|A|. Whereas the matrix is an array of numbers and can be regarded in
manyways as an operator, the determinantis apure number. Forexample,
«i a* a*
h
bn 6o = a1 b9 -o2 b. b* +a, &i b?
c2 c3 Cl c3 Cl «2
= a1(b2c3-b3c2)-a2(b1c3-b3c1) +a3(b1c2-b2c1).
(16)
-
The sign associated with ar is ( )r+1, and the general rule for evaluation
is obvious. The determinant
obtained by omitting the row and column containing av is called a minor
of order 2 of the original determinant.
It can be shown that the inverse A-1 of A is given by
in .A
131 41
1 -A -A,
IA] 12 M2 (17)
where the minors Ars which are obtained by omitting from the original
determinant the row and column containing ar8, occur with alternate signs
and are transposed in comparison with the corresponding elements ars ofA.
5. When |A| = 0, it is clear from (17) that the matrix A has no inverse.
Such a matrix is called singular, and the corresponding linear equations
have in general no solution. If |A| = 0, the rows of A are not linearly
independent; at least one can be obtained by linear combination of the
others. For example, in the equations
-
3/1+-X%+ £3 O1;
=
Xy x^-\- 2x3 o2, (18)
Zx1 +x2+ ±x3 = 63,
it can be verified that the determinant
1 1
-1 2 = 0,
1 4
and the third of (18) is obtained by adding twice the first to the second.
In this case the equations are incompatible unless 2b1 +b2 = b3, and ifthis
holds we have effectively only two equations in three unknowns and
there is an infinity of solutions.
If
the equations are homogeneous, that is, the constants br are all zero,
the equations have no solution other than x1 = x2 = ...= xn = 0 unless the
determinant vanishes, in which case we can omit one eq-uation and solve
the rest to find the ratios of the xr, provided that the n 1 equations are
not themselves linearly dependent. For example, in the homogeneous set
ofequations corresponding to (18), we can omit the last equation and solve
xjx^+xjx^+l = 0,
(19)
-
finding x1\x3 = 1-5, xjx3 = 0-5, which also satisfy the remaining equa
tion 3x1/x3 +x2\x3 +4 = 0.
6. In solving a set of equations, it is a great advantage to be able to
use the same number of decimal places throughout any one stage of the
computation. It is a further advantage if the same number can be used
at every stage. These advantages can be gained very simply by multiply
ing the various rows and columns of coefficients by powers of ten so as
to make the largest coefficient in each row and column, including the
column of constants, Me between 0-1 and 1-0. Multiplying the rows does
not affect the solution, and multiplying the columns involves only a
trivial change in the unknowns.
The same procedure should be carried out on a matrix of which the
inverse is required. In this case, however, the multiplier of each row
must subsequently be multiplied into the corresponding column of the
inverse obtained, and the multiplier of each column into the corre
sponding row of the inverse, in order to recover the inverse of the
original matrix.
