Error Inequalities in Polynomial Interpolation and Their Applications

375 Pages·1993·9.42 MB·English

by  Ravi P. Agarwal

Error Inequalities in Polynomial Interpolation and Their Applications
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Volume 262
Error Inequalities in
Polynomial Interpolation
and Their Applications
by
Ravi P. Agarwal
Department ofM athematics,
National University of Singapore,
Kent Ridge, Singapore
and
Patricia 1. Y. Wong
Division ofM athematics,
Nanyang Technological Ulliversity,
Singapore
SPRINGER SCIENCE+BUSINESS MEDIA, B.V.
Library of Congress Cataloging-in-Publication Data
Agarwal, Ravi P.
Error inequalities in polynomial interpolation and their
applications I by Ravi P. Agarwal and Patricia J.Y. Wong.
p. cm. -- (Mathematics and its applications ; v. 262)
Includes bibliographical references and index.
ISBN 978-94-010-4896-5 ISBN 978-94-011-2026-5 (eBook)
DOI 10.1007/978-94-011-2026-5
1. Interpolation. 2. Approximation theory. 3. Polynomials.
1. Wong, Patricia J. Y. II. Title. III. Series: Mathematics and
its applications (Kluwer Academic Publishers) ; v. 262.
QA281 . A33 1993
511' .4--dc20 93-15913
ISBN 978-94-010-4896-5
Printed an acid-free paper
AII Rights Reserved
© 1993 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1993
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
CONTENTS
Preface ix
Chapter 1 LIDSTONE INTERPOLATION
1.1 Introduction 1
1.2 Lidstone Polynomials 2
1.3 Interpolating Polynomial Representations 17
1.4 Error Representations 17
Peano's Representation 18
Cauchy's Representation 18
1.5 Error Estimates 19
1.6 Lidstone Boundary Value Problems 21
Existence and Uniqueness 21
Picard's and Approximate Picard's Iteration 29
Quasilinearization and Approximate Quasilinearization 43
References 59
Chapter 2 HERMITE INTERPOLATION
2.1 Introduction 62
2.2 Interpolating Polynomial Representations 63
Method of Lagrange 63
Method of Newton 67
2.3 Error Representations 71
Cauchy's Representation 71
Newton's Representation 72
Peano's Representation 73
A New Representation 77
vi
2.4 Error Estimates 91
Error Estimates in Interpolation 91
Error Estimates for Derivatives 105
2.5 Some Applications 149
Generalized Maximum Principle 149
Hermite Boundary Value Problems 151
Generalized Liapunoff's Inequality 161
Generalized Hartman's Inequality 162
A Lower Bound for the Zeros of the Solutions 164
A Test for Disconjugacy 166
References 168
Chapter 3 ABEL - GONTSCHAROFF INTERPOLATION
3.1 Introduction 172
3.2 Interpolating Polynomial Representations 173
3.3 Error Representations 175
Repeated Integrals Representation 175
Peano's Representation 175
Cauchy's Representation 177
3.4 Error Estimates 178
3.5 Some Applications 186
References 189
Chapter 4 MISCELLANEOUS INTERPOLATION
4.1 Introduction 192
4.2 (n,p) and (p,n) Interpolation 192
4.3 (0,0; m, n - m) Interpolation 198
4.4 (0; m, n - m) Interpolation 203
4.5 (0,2,0; m, n - m) Interpolation 206
+
4.6 (0: 1- 1,1: 1 j - 1; m, n - m) Interpolation 208
4.7 (0; Lidstone) Interpolation 209
vii
4.8 (0,2,0; Lidstone) Interpolation 210
4.9 (1,3,0,1; Lidstone) Interpolation 211
4.10 (0: 1-1,1: 1+ j - 1; Lidstone) Interpolation 213
4.11 (0,2,1; Lidstone) Interpolation 214
References 216
Chapter 5 PIECEWISE - POLYNOMIAL INTERPOLATION
5.1 Introduction 217
5.2 Preliminaries 218
5.3 Piecewise Hermite Interpolation 228
5.4 Piecewise Lidstone Interpolation 257
5.5 Two Variable Piecewise Hermite Interpolation 266
5.6 Two Variable Piecewise Lidstone Interpolation 274
References 278
Chapter 6 SPLINE INTERPOLATION
6.1 Introduction 281
6.2 Preliminaries 282
6.3 Cubic Spline Interpolation 285
6.4 Quintic Spline Interpolation : = 4 290
T
6.5 Approximated Quintic Splines : = 4 301
T
6.6 Quintic Spline Interpolation : T = 3 309
6.7 Approximated Quintic Splines : = 3 315
T
6.8 Cubic Lidstone - Spline Interpolation 321
6.9 Quintic Lidstone - Spline Interpolation 323
6.10 L2 - Error Bounds for Spline Interpolation 327
6.11 Two Variable Spline Interpolation 334
6.12 Two Variable Lidstone - Spline Interpolation 346
6.13 Some Applications 350
Linear Integral Equations 351
References 360
Name Index 363
Preface
Given a function x(t) E c{n) [a, bj, points a = al < a2 < ... < ar =
b and subsets aj of {0,1,"',n -1} with L:j=lcard(aj) = n, the classical
interpolation problem is to find a polynomial Pn- l (t) of degree at most (n - 1)
such that
P~~l(aj) = x{i)(aj) for i E aj, j = 1,2,"" r.
In the first four chapters of this monograph we shall consider respectively
the cases: the Lidstone interpolation (a = 0, b = 1, n = 2m, r = 2, al =
a2 = {a, 2"", 2m - 2}), the Hermite interpolation (aj = {a, 1,' ", kj - I}),
the Abel - Gontscharoff interpolation (r = n, ai ~ ai+l, aj = {j - I}), and
the several particular cases of the Birkhoff interpolation. For each of these
problems we shall offer:
(1) explicit representations of the interpolating polynomial;
(2) explicit representations of the associated error function e(t) = x(t) -
Pn-l(t); and
(3) explicit optimal/sharp constants Cn,k so that the inequalities
I e{k)(t) I < C k(b - at-k max I x{n)(t) I, 0< k < n- 1
n
-, a$t$b - -
are satisfied.
In addition, for the Hermite interpolation we shall provide explicit opti
mal/sharp constants C(n,p, v) so that the inequality
II e(t) lip:::; C(n,p, v) II x{n)(t) 1111, p, v ~ 1
holds.
Although these results are of fundamental importance in every aspect of
numerical mathematics, we shall demonstrate their significance in the theory
of ordinary differential equations such as maximum principles, boundary value
problems, oscillation theory, disconjugacy and disfocality.
Polynomial interpolation often produces approximations that are wildly
oscillatory. To overcome this difficulty we divide the interval [a, bj into small
intervals and in each subinterval consider polynomials of relatively low degree
and finally 'piece together' these polynomials. This subject has steadily devel
oped over the past fifty years, and at present there are thousands of research
papers on piecewise - polynomial interpolation and their applications.
In the fifth chapter of this monograph we shall consider the piecewise Her
mite and Lidstone interpolating problems and for each of these provide:
IX
x
(i) explicit representations of the piecewise interpolating polynomial;
(ii) explicit error bounds for the derivatives of cubic and quintic piecewise
interpolation in Loo- norm; and
(iii) explicit error bounds for the derivatives of arbitrary order piecewise
interpolation in L2- norm.
In addition, these results are extended to two variable piecewise - polyno
mial interpolation.
Spline interpolation is an improvement over piecewise - polynomial inter
polation. It uses less information of the given function, yet furnishes smoother
interpolates.
In the final chapter of this monograph we shall consider the spline and
Lidstone - spline interpolating problems and for each of these give:
(a) explicit representations of the interpolating spline; and
(b) explicit error bounds for the derivatives of cubic and quintic interpo
lating splines in Loo- norm.
In addition, for the spline interpolation we shall obtain:
(Q') explicit representations of the approximated splines for the quintic case,
and precise error bounds for their derivatives in Loo-norm; and
(;3) explicit error bounds for the derivatives of arbitrary order interpolating
spline in L norm.
2-
Finally, these results are generalized to two variable spline interpolation.
Throughout final chapter, we also give numerical illustrations of the im
portance as well as the sharpness of the results which are presented.
This work is a cumulation of the authors' research in this field, extending
over a period of ten years. We hope reading this monograph will be a pleasure
and rewarding as well.
Ravi P.Agarwal
Patricia J.Y.Wong
CHAPTER 1
LIDSTONE INTERPOLATION
1.1 INTRODUCTION
In the year 1929 Lidstone [15] introduced a generalization of Taylor's series,
it approximates a given function in the neighborhood of two points instead of
one. From the practical point of view such a development is very useful; and
in terms of completely continuous functions it has been characterized in the
work of Boas [9], Poritsky [19], Schoenberg [20], Whittaker [28, 29], Widder
[30, 31], and others. In the field of approximation theory [12,27] the Lidstone
interpolating polynomial p(1.1.1) ( t) of degree (2m - 1) satisfies the Lidstone
conditions
(1.1.1) P((12.i1) .1) ( 0 ) -_ Q.. , p((21.i1). 1) ( 1 ) -- "(3 .. 0 ::; z. ::; m - 1.
The plan of this chapter is as follows : In Section 1.2 we shall introduce
+
Lidstone polynomials An(t) of degree (2n 1), provide their explicit repre
sentations and give their relations with Bernoulli and Euler's polynomials.
Here, we shall also establish several equalities and inequalities involving the
1