Exponential Fitting

Abstract

Originally published by Kluwer Academic Publishers in 2004Softcover reprint ofthe hardcover lst edition 2004 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Additional material to this book can be downloaded from http://extras.springer.com Contents Preface XI Acknowledgments xiii Subroutine REGSOLV2 71 4. NUMERICAL DIFFERENTIATION, QUADRATURE AND INTERPOLATION 77 1 Numerical differentiation 77 V Vl EXPONENT/AL FITTING PrefaceExponential fitting is a procedure for an efficient numerical approach offunctions consisting of weighted sums of exponential, trigonometric or hyperbolic functions with slowly varying weight functions. Operations on such functions Iike numerical differentiation, quadrature, interpolation or solving ordinary differential equations whose solution is of this type are of a real interest nowadays in many phenomena as oscillations, vibrations, rotations or wave propagation. The behaviour of quantum particles is also described by this type of functions.We witnessed the field for many years and contributed in it. Since the total number of papers accumulated so far in this field is over 200, and these papers are spread over journals with various profiles, to mention only those of applied mathematics, of computer science or of computational physics and chemistry, we thought that the time has come for a compact and systematic presentation of this vast material. lt is hoped that in this way many persons who are faced with such problems in their own activity would be better helped than searching for the needed information in such an abundant literature. This is what we do in this book which covers a series of aspects, ranging from the theory of the procedure up to direct applications and sometimes including ready-to-use programs.The writing of this book was decided two years ago. Since our working places are not close to each other, we agreed to share the effort: the first author has taken upon him Chapters 1 to 5, while the second author was responsible for Chapter 6. However, we both share equal responsibility for the whole text.We are indebted to a number of colleagues for discussions and the collaboration in several periods of our research:

Optimizing a class of linear multi-step methods for the approximate solution of the radial Schrödinger equation and related problems with respect to phase-lag

“…Application of the method (8) to the scalar test equation leads to the difference equation (5) with = 2 and A = 0 (1) 2 given by (9). The characteristic equation associated with the difference equation is given by (10). We require equation (10) to vanish for λ = 1 (1) 4 given by (11).…”

“…The characteristic equation associated with the difference equation is given by (10). We require equation (10) to vanish for λ = 1 (1) 4 given by (11). This requirement leads to the system of equations (12).…”

“…In Figures 7, 8, 9 and 10, we present the maximum absolute error 10 (E ), where each method. Consequently, the comparison is based on the maximum absolute error which is obtained with the specific NF E, i.e.…”

Exponentially Fitted Finite Difference Schemes for Reaction-Diffusion Equations

“…In the construction of EF schemes, we will need the following EF based finite difference derivative formulas [15] …”

Multi-Parameter Exponentially Fitted, P-stable Obrechkoff Methods

“…Extending this result in [2], the technique of exponential fitting [3] is brought into the picture. Whereas a classical Obrechkoff method of order 2p has a polynomial fitting space {1,t,t 2 , .…”