Padé Approximants Method and Its Applications to Mechanics

265 Pages·1976·2.93 MB·English

by  George A. Baker Jr. (auth.)

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Edited by .J Ehlers, MLinchen, .K Hepp, Zerich,
.H .A Weidenm~ller, Heidelberg, and .J Zittartz, n16K
Managing Editor: W. BeiglbSck, Heidelberg
74
Pade Approximants Method
dna stI Applications to Mechanics
Edited yb .H Cabannes
Springer-Verlag
Berlin. Heidelberg. New York 91 7 6
Editor
Prof. Henri Cabannes
Universit@ Pierre et Marie Curie
Mecanique Theorique
Tour 66, 4, Place Jussieu
75005 Paris/France
Library of Congress Cataloging in Publication Data
Main entry under title:
Fade approximants method and its applications ot.
mechanics.
(Lecture notes in physics ; 47)
Bibliography: p.
Includes index.
I. Pade approximant. .2 Fluid mechanics.
I. Cabannes, Henri. II. Series.
QC20.7.P3P35 515 .01' 532' 75-46504
ISBN 3-540-07614-X Springer-Verlag Berlin - Heidelberg • New York
ISBN 0-387-07614-X Springer-Verlag New York • Heidelberg • Berlin
This ,,york is subject to copyright, All rights are reserved, whether the whole
or part of the material is concerned, specifically those of translation, re*
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Under § 54 of the German Copyright Law where copies are made for other
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be determined by agreement with the publisher.
© by Springer-Verlag Berlin • Heidelberg 1976
Printed in Germany
Printing and binding: Offsetdruckerei Beltz, Hemsbach/Bergstr.
Henri Pad~ (1863 - 1953)
Henri Euggne Pad~ was born in Abbeville (France) on Dec. 17th 1863.
Admitted in 1883 to the Ecole Normale Supgrieure, he left it in 1886 with the
highest teacher's degree (Agr~gation) in Mathematics. After teaching at the classi-
cal secondary school in Limoges, Carcassonne and Montpellier, he was granted a leave
in 1889 in order to study in Germany, first in Leipzig, then in G~ttingen. On June
21st 1892, before the University of Paris, he defended his doctorate thesis on the
approximate representation of a function by rational fractions.
Henri Pad~ was appointed lecturer at the Faculty of Sciences of Lille
in 1897, Professor of Rational and Applied Mechanics at the Faculty of Sciences of
Poitiers in 1902 and Professor of Mechanics at the Faculty of Bordeaux in 1903. In
1906, he was elected Dean of the Faculty of Sciences of Bordeaux and became Laureate
with the major prize for mathematical sciences awarted by the Academy of Sciences
on the report of Emile Picard. In 1908 he was named Rector of the Academy of
Besan~on, he was then the youngest rector in France. In 1917 he became rector of the
Academy of Dijon and in 1923 rector of the Academy of Aix-Marseilles, an office he
kept until he retired in 1934.
Henri Pad~ died in 1953 at the age of 89.
Henri Pad~ (1863 - 1953)
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Editor's Preface
In 1892, in the Scientific Transactions of the Ecole Normale Sup~rieure in
Paris, the french mathematician Henri Pad~ published an article concerning the ap-
proximate representation of a function by rational fractions. Three-quarters of a
century later, the advent of arithmetical computers led scientists to consider va-
rious methods of representing functions, especially rapidly converging functions.
In a paper published in 1955, D. Shanks showed the advantages of pad~'smethod,
which makes it possible to deduce from any converging or diverging series of powers
a table of rational approximations of the functions represented by these series.
Since then many physicists and applied mathematicians have studied the representa-
tion of functions by means of rational fractions, the main object being to obtain
ever faster computing algorithms. The most celebrated and now classic example is
for calculating the sum of the following series:
4 + .
4 4 4 + (_)n 2 n + I ""
(I) S = 4 - 3-- - + 5 7 "'"
The sum S is equal to 7. But to obtain from series (I) -- which is a very poor al-
gorithm -- the value of~correct to eight digits, the first term which is neglected
-8
must be less than I0 in absolute value, which means thatwe must consider 20 mil-
lion terms. If however one applies Padg's D,I~ transformation repeatedly to the
sum S of the first n terms, that is if one associates with the sequence S the new
n n
sequence
- S 2
(2) v Sn-I Sn+l n
Ln = + - 2 S '
Sn- 1 Sn+ I n
then to obtain the same accuracy it will suffice to consider only the first nine
terms of series (1) and to apply the transformation (2) five times.
The underlying principle of Pad~'s method is particularly simple. Given a
power series
oc
(3) ~ c k x k ,
k=0
Pad~ proposed finding the closest approximation to the sum, by defining a rational
fraction Pm(X) / Qn(X) , with
m n
(4) Pm(X) = ~ a k x k , Qn(X) = I + ~ b k x k ,
k=0 k=l
in which the numerator and the denominator are polynomials of degrees m and n res-
pectively. These polynomials Pm(X) and Qn(X) are determinated from the identity
oo oc
(5) Qn (x) k= 0~ Ck xk - Pm(X) = xm+n+I k= O~ Yk xk
This identity leads to m+n+l linear equations from which the unknown m+n+l values
a k and b k (b = 1) can be determined. Pad@ obtained the following result:
0
em-n+ 1 Cm_n+ 2 -.. Cm+ l
(6) A Pm(X) = c m Cm+ 1 ..- Cm+ n
m
xj m xj ~ c. x j
j!O ej-n Cj-n+ 1 ...
j=O j=0 j
Cm-n+ 1 Cm-n+2 " " " Cm+ 1
(7) A Qn(X) c m Cm+ 1 Cm+n
n n-I
x x
In these two formulae, it is necessary to take c. = 0 when j is negative, and A is
J
the minor obtained by eliminating the last line and the last column.
Pad@'s rational approximations are widely used in computer calculations,because
they are generally more efficient than polynomial approximations. They almost halve
the number of operations required• These approximations are particularly convenient
when one takes m = n ( defining the diagonal of the Pad@ table .) For in this case
the coefficients Yk of the right-hand side of (5) usually decrease so quickly that
the first term Y0 x 2n+I , divided by Qn(X), constitues an excellent approximate
value for the absolute error introduced if one uses Pm(X) / Qn(X) instead of
XI
co
c k x k. For lxI < 1 , the value of Qn(X) differs very little from unity because
k=0
the b k coefficients usually decrease very rapidly. It will therefore suffice
to calculate y , whence
0
&
n
(8) T
0 n
c 1 c 2 e n+ 1
c 2 c 3 Cn+ 2
A
n
Cn+ 1 Cn+ 2 C2n+l
6n is the minor obtained by eliminating the last line and the last column.
In an excellent article written for the collective work entitled "Mathematical
Methods for Digital Computers" (published by John Wiley), Kogbetliantz shows that,
irrespective of accuracy required, the Pad6 method is the best method for construc-
ting programs for calculating sin x and cos x:
(_)k
(9) cos x = ~ (2k) ! z k with z = x 2
k=0
Taking m = n, we have
PI 12 - 5 x 2
3 1
Q1 12 + x 2 ' Y0 2 6!
P2 15120 - 6900 x 2 + 313 x 4
59 1
, Y
Q2 15120 + 660 x 2 + 13 x 4 0 42 I0!
As we can see on the figure, the approximation P2 / Q2 which is constructed with the
IIX
five first terms of the series (I) is better than the sum S 5 of those terms.
Sum S s
/
P2
/ / Q2
Exact value
! ! I I IX
5
-I
Approximations of cos x
For m = n = 3, we obtain
P3 I + a I x + a 2 x 2 + a 3 x 3
Q3 + bl x + b 2 x 2 + b 3 x 3
3665 229
with al
7788 bl 7788
71| I
a2 25960 b2 2360
2 923 127
a 3 = -
7 850 304 b3 39 251 520