Methods in Approximation Techniques for Mathematical Modelling
Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The He
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Mathematics and Its Applications
Managing Editor: M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Editorial Board: F. CALOGERO, Universita degli Studi di Roma, Italy Yu. I. MANIN, Steklov Institute of Mathematics, Moscow, U.S.S.R. A. H. G. RINNOOY KAN, Erasmus University, Rotterdam, The Netherlands G.-C. ROTA, M.I. T., Cambridge, Mass., U.S.A.
Methods in Approximation Techniques for Mathematical Modelling
Richard E. Bellman Department of Electrical Engineering. University of Southern California. Los Angeles. U.S.A.; Center for Applied Mathematics. The University of Georgia. Athens. Georgia. U.S.A.
and
Robert S. Roth Boston. U.S.A.
D. Reidel Publishing Company A MEMBER OF THE KLUWER ACADEMIC PUBLISHERS GROUP
Dordrecht / Boston / Lancaster / Tokyo
Libnry or Congress Cataloging in Publication Data
Bcllmun , Rich J rd Ernest, 1920Met hod s in Jpproximation. and its upplication~) Includes bibliographies Jnd index. I. Approximation theory. I. Roth, Robert, 1930II. Title. Ill. Series: Mathem atic s and its upplicatio ns (D. Reidel Publ b hing Co mpany ) QA22I.B36 1986 511'.4 86- 461 (Muthcmatic~
ISBN·13 :97IJ.94'()1 0·8544·1 001: 10. 1007/978-94-009-4600-2
e-ISBN-13: 978-94-009-4600-2
Published by D. Reidel Pub lishing Company, P.O. Box 17, 3300 AA Do rdrecht, Holland Sold and distributed in the U.S.A. and Canada by Kluwe r Academic Publishers, 190 O ld IUrby Street, Hingham, MA 02043, U.S.A. In all othe r cou ntries, sold and distributed by K1uwer Academic Publishers Group, P.O. Box 322, 3300 AH Dordrechl, Holland
2- 0388- 200 ts
All Rights Rese rved () 1986 by D. Reidel Publishing Company, Dordrecht, HoUand Softcover reprint of the hardcover 1st edition 1986 No parI of the material protected by this copyright notice may be reproduced o r utilized in any fonn or by a ny means, electronic or mechanical, including photocopying, .eeo.dinj: or by any infonnation storage and retrieval system, without written pe rmission from the copy.ight owne r.
CONTENTS EDITOR'S PREFACE PREFACE
ix xi
Chapter 1.
BASIC CONCEPTS
• 1
Introduction • . . . • .•••• 1 Integral Domains,Fields and Vector Spaces • . • . • .•••• 2 Subspaces, Bases and Inner Products • 4 Spaces, Subspaces and Approximation • 7 The Continuous Function . • . • • . • 8 Polynomial Subspaces • . . • • • . . 9 Spaces Generated by Differential Equations . . • . • . • • 10 13 The Piecewise Linear Function • Discussion •• • • . . . 15 Bibliograpy and Comments 16 2.
POLYNOMIAL APPROXIMATION • • • . Introduction • . • . • . • • • • • Piecewise Linear Functions • • Curve Fitting by Straight Lines. . A One Dimensional Process in Dynamic Programming . . . • . • • . • The Functional Equation • . • • •• The Principle of Optimality. • •• A Direct Derivation . . . . • • • • Curve Fitting by Segmented Straight Lines • . • • . • • • . • A Dynamic Programming Approach A Computational Procedure . Three Dimensional polygonal Approximation • • • • • • The Orthogonal Polynomials • • • • The Approximation Technique
17 17 22 24 26
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