Handbook of Numerical Methods for the Solution of Algebraic and Transcendental Equations

202 Pages·1961·9.723 MB·English

by  V. L. Zaguskin (Auth.)

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Publisher at Pergamon Press
Handbook of Numerical
Methods for the Solution
of Algebraic and
Transcendental Equations
by
V. L. ZAGUSKIN
Translated from the Russian by
G. O. HARDING, D.Phil.
PERGAMON PRESS
OXFORD · LONDON · NEW YORK · PARIS
1961
PERGAMON PRESS LTD.
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Copyright
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1961
PERGAMON PRESS LTD.
A translation of the original volume
Spravochnik po chislennym metodam resheniya
algebraicheskikh i transtsendentnykh uravnenii
(Moscow, Fizmatgiz, 1960)
Library of Congress Card Number : 61-9787
Printed in Great Britain by
Pergamon Printing and Art Services Limited, London
FOREWORD
The solution of algebraic and transcendental equations is one of the
essential problems of applied analysis which arises in many and differ­
ent departments of physics, mechanics, engineering, and science in
general.
Much attention in existing textbooks is devoted to the explanation
of practical methods for the solution of equations, but in Russian
educational literature no special textbook exists which gives not
only a summary of the classical methods (developed already, as a
rule, in the nineteenth century), but including also specially selected
material, developed by many research workers in recent decades,
which as far as possible has been tried out in practice. *
This book is intended to fill the above mentioned gap, and at the
same time to provide a textbook, of reference type, for a wide range
of readers.
While working on the book, the author aimed in the first instance
to write a textbook which could prove to be of direct practical aid to
engineers (research workers of institutes and industrial enterprises) who
are using mathematical methods in the solution of technical problems.
With this object, the author limited himself to a selection of the
material, in order not to trouble the reader with the necessity of
choosing from the extraordinarily rich supply of methods which applied
mathematical analysis has to offer at the present time. For this
reason the author has devoted special attention to explanation, i.e. to
careful wording of the contents of the given methods, and has left
not a single method of any importance without a detailed calcula­
tion! example,
* We are not thinking here of the particular, and rather important,
question of the solution of simultaneous linear algebraic equations:
the results of numerous and fruitful researches, carried out in the
present era in this field, have been treated in special handbooks
published both abroad and in this country; see for example,
V.N.Faddeeva, Calculational methods of linear algebra, Moscow-
Leningrad, 1950, (Vychisliternye metody lineinoi algebry).
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HANDBOOK OF NUMERICAL METHODS
At the same time the author was aware of the possibility that this book
will also be used by students who are assimilating applications of
higher mathematics together with a course in pure mathematics. For
this reason the explanation of the material is accompanied as far as
possible by the theoretical basis of the methods given, or at least by
references to the relevant literature. But here also reasonable limits
were set: no problems of higher mathematics were touched on, which
lie far from the programme of Russian technical colleges (thus, for
instance, the author succeeded in finding ways of presentation
without using the theory of matrices (see section 6 chapter III,
section 4, chapter IV) and without using the theory of finite
differences (see section 4 chapter III) ).
In order to depict fully the aim of the book, it should be added
that the book will be useful not only to engineers and to students c f
technical colleges, but also to those engaged in research in the field
of applied analysis. This aim is met by the inclusion of up to date
material, and also by the original presentation of certain separate
problems developed by the author.
The subject matter of the book is shown in detail in the "table of
contents", and also in the alphabetical index, which is intended to
facilitate the use of the handbook, and at the same time suggest to
the reader a method he might use. Nevertheless it will not be super­
fluous to make a short survey of the basic sections of the book.
The first chapter is introductory. In it general information is given;
certain traditionally explained methods, which have little practical
application, (for example, Sturm's series) are not given. In this
chapter certain technical methods, which are used in the following
chapters (for instance, the method of a "moving strip"), and some
calculational schemes, are given. We point out in particular the
improvement in MacLaurin's method.
The second chapter is of special interest; in it the author outlines
a non-traditional theory of the solution of approximate equations and
using a precisely formulated definition of the complete error of a
solution, and of the conditional and unconditional errors, he indicates
a well-grounded plan for the solution of an approximate equation.
The author devotes attention to questions of " loss of accuracy " in
approximate calculations, precisely defining this concept (in existing
handbooks there is no defined use of this term at all). In the succeed­
ing chapters of the book the concept of " loss of accuracy" is used in
practice (see, for example, the results given in section 5,4 chapter III,
and in section 5.3 chapter V when the method of iteration is being
explained); however the author did not make a study of loss of accu­
racy for many of the other methods given in the book.
In chapter III approximate methods are given for the calculation of
roots of algebraic equations. The author succeeded in obtaining a
certain simplification in explaining the classical method of Loba-
chevskii at the cost of introducing the new concept of " a root with a
k -multiple modulus" ; th?.· enables one to avoid consideration of
numerous particular cases,
Xll
FOREWORD
We note in particular the exposition of the modification of
Bernoulli's method, (outlined in passing by Hildebrand); it enables
one to avoid an analysis of wearisome complications in the case of
multiple roots, these complications being inherent in the traditional
form of Bernoulli's method. We point out also that the author intro­
duces clear criteria enabling one to determine which of the four
possible types of convergence of a sequence is arising in the calcula­
tion of roots (traditional treatments of this subject lay emphasis, in
the classification of equations, on the type of the largest roots - as if
these roots are known beforehand! ). Such a scheme of investigation
leads the author to pointing out a new path for finding the second
largest roots, and successively smaller ones, which has indubitable
practical advantages over the well-known method of Aitken (to which
it is theoretically very close).
Lin's method, explained in chapter III, which has arisen only in
the last decade, is a novelty to Russian educational literature. The
author's original presentation of this method leads to the criterion of
convergence which was obtained by Aitken by use of the theory of
matrices.
In chapter IV methods for making roots more accurate are given,
which are important in the practical solution of equations. Here it
should be mentioned that in the explanation of Berstoi ' s method, the
author especially picks out the case of the application of this method
to the finding of complex roots whose moduli have been found already
as a preliminary by Lobachevskii ' s method, - this combination is a
rather happy one; it may be suggested that Berstoi's method is appre­
ciably more adaptable for this purpose than other known methods (in
particular, than the Brodetskii-Smil method, given in chapter III).
We point out also that in the presentation of the iteration method for
solving algebraic equations, a new method " of iteration with quad­
ratic convergence" is introduced, which belongs to the author,
A.Lopshits.
xiii
AUTHOR'S PREFACE
In compiling this handbook, I had in mind, first of all, the reader
who is as yet not well acquainted with calculational mathematics,
and in particular, with the solution of algebraic and transcendental
equations. In connexion with this, and also in view of the limited
size of the book, I tried to select only a small number of methods,
which are necessary for the effective solution of problems in various
situations. "Mutually interchangeable" methods, such as, for
example, the methods of Berstoi and Belostotskii for making divisors
of polynomials more accurate, appear in the book only as exceptions.
In resolving the question of which of the " mutually interchange­
able" methods to include in the book, I was guided chiefly by my
own personal attitude to these methods, which was not always,
probably, sufficiently objective. It would be very pleasing and use­
ful to me to learn the opinion of readers on which of the methods
given are of little interest in practice, and which other methods, on
the other hand, should have been included in the handbook.
I would be grateful also for information about other shortcomings
of the book.
I should like to thank all those who helped me in the work, in
particular L.V.Zaguskina for her valuable aid.
In particular I should like to thank my teacher, A.M.Lopshits, who
is also the editor of this book. His advice and help was an important
factor in my decision to write a book of this nature.
V. Zaguskin.
xv
INTRODUCTION
In this handbook we consider the solution of various equations,
such as, for example
JCe-f- 4.2240ΛΓ5 + 6.5071ΛΤ4-!- 7.5013*8 + 8.4691.*9-f
+ 3.3641* -f 1.6252 = 0, ((U)
j<* + (2 — 3i)x* — (1+0-^+4 — 2i = 0, (0.2)
o.72i*_2.831jc = 0, (0.3)
é?
aiid also simultaneous equations, for example,
Jt3 + 2**j/ + 4y* — ^,-[-1=0, 1
ex-\-2 y — 4xy = 0. J <°·4)
e
Equations of the kind (0.1) and (0.2) belong to the class of so called
algebraic equations, whose general form is
a^ + a^-1 +... + α _ χ -f a = 0, (0.5)
η γ n
Here ûo» ûi» · · · » an-v an &re giyen, real or complex, numbers (the
coefficients of the equation), and x is the required unknown number.
The expression on the left hand side of equation (0.5) is called a
polynomial. Polynomials have a series of special properties, which
enable one to apply to the solution of algebraic equations certain
methods which are not suitable in other cases. These properties are
briefly presented in chapter I. Non-algebraic equations are called
transcendental ((0.3) is an example). Functions, which are not poly­
nomials, are also called transcendental.
In the class of algebraic equations we should include also, strictly
speaking, equations which can be reduced to the form (0.5) by means
of algebraic transformations, for example
VGF+T— 2= /.**— 1 +0.1*. (0.6)
This should be taken into consideration, in particular, in the calcula­
tions of the possible number of roots of such an equation; however,
xvn
HANDBOOK OF NUMERICAL METHODS
reduction to an algebraic equation often requires such a large amount
of unproductive work that it is not always to be recommended. We
shall provisionally take equations such as (0.6) as transcendental.
Methods for the solution of equations are divided into two groups.
To the first group belong those methods which enable one by means
of various transformations to simplify the equation and write the solu­
tion in the form of a formula. The second group is made up of
numerical methods. The solution in this case is sought by means of
operations (if desired, only arithmetical ones) on numbers, in which
the answer is obtained immediately in the form of a number (usually
approximate). To the first group belong, for example, Cardano1 s
method for the solution of third order equations, the method of lower­
ing the order of recurrence equations, and so on. These methods may
sometimes be useful, but their practical value in general is not
large, since between the formula and the numerical answer there
lies, as a rule in practical problems, a large amount of calculation.
Moreover, these methods are applicable only to a comparatively
narrow class of problems. They are used chiefly when the equation
contains one or more parameters (for example, literal coefficients).
In this handbook numerical methods only are considered.
At the present time a large number of numerical methods for the
solution of equations exist, and are used. Such a diversity of methods
is connected with the following circumstances.
a). The solution of an equation is rarely carried out from begin­
ning to end by one method. At first the roots of the equation are
usually determined by some kind of method with comparatively small
accuracy, and are then made more accurate by other methods.
b). Methods do not exist which are equally effective for all
equations. Naturally, some methods are more convenient for some
equations, and other methods for other equations. This distinction
cannot always be determined beforehand. Thus, when beginning to
solve an algebraic equation we often do not know whether it has real,
or only complex, roots. Therefore, in the process of solution it is
often necessary to give up one method and change to another.
c). The solution of equations requires a rather large amount of
calculational work which occasions the need for mechanization of
the calculations. The mechanization may be of varied kinds -
slide rules and tables, desk calculating machines, electrical keyed
machines and tabulators, electronic machines. With a change in
level of mechanization of the calculational work, an estimation of
the comparative effectiveness of the methods changes. Generally the
higher the level of mechanization, the less importance is attached to
the quantity of calculations in an appraisal of the method, and the
more importance is attached to simplicity and uniformity of the cal­
culational scheme, to reliability of the result, and to minimum
introduction and extraction of auxiliary data. There are also a
number of other special properties. For instance, in the use of a slide
rule, the operations of multiplication and division take the same time,
XVlll
INTRODUCTION
while oa certain calculating machines they take different times: as
a consequence of this, a method, in which the operation of division
is often used, for example Newton's method which is considered in
chapter IV, may lead to the result quicker than the method of itera­
tion for calculations on a slide rule, but give worse results when the
calculations are by machine·
In the handbook a comparatively small number of methods are
given for the numerical solution of equations. In selecting them the
author strived to achieve the possibility of sufficiently expedient
solution of various types of problems by means of combinations of the
methods given for various levels of mechanization of the process of
calculation. Of methods similar in their calculational scheme and
purpose, only one, as a rule, is given, although others may be known
to have certain advantages over it for the solution of particular
equations.
xix