Table Of ContentExponential Fitting
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Volume 568
Exponential Fitting
by
Liviu Gr. lxaru
Nationalinstitute for Research and Development for Physics and Nuclear Engineering,
"Horia Hulubei", Department ofTheoretical Physics, Bucharest, Romania
and
Guido Vanden Berghe
University of Gent,
Department of Applied Mathematics and Computer Science, Gent, Belgium
SPRINGER SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
Additional material to this book can be downloaded from http://extras.springer.com
ISBN 978-90-481-6590-2 ISBN 978-1-4020-2100-8 (eBook)
DOI 10.1007/978-1-4020-2100-8
Printed on acid-free paper
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© 2004 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 2004
Softcover reprint ofthe hardcover lst edition 2004
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Contents
Preface XI
Acknowledgments xiii
1. INTRODUCTION 1
2. MATHEMATICAL PROPERTIES 11
1 A reference differential equation 11
2 A regularization procedure 16
2.1 Subroutine REGSOLV 23
3 An outline of the exponential fitting procedure 28
4 The theory of the exponential fitting approach 38
4.1 The C operator 38
4.2 Dimensional consistency 41
4.3 Evaluation of the coefficients 42
4.4 Error formulae 49
3. CONSTRUCTION OF EF FORMULAE
FOR FUNCTIONS WITH
OSCILLATORY OR HYPERBOLIC VARIATION 53
1 The Schrödinger equation 53
2 Functions "ls(Z), s = -1, 0, 1, 2, ... 58
2.1 Subroutines GEBASE and GEBASEV 63
2.2 Subroutines CGEBASE and CGEBASEV 64
3 A six-step flow chart 66
3.1 Subroutine REGSOLV2 71
4. NUMERICAL DIFFERENTIATION,
QUADRATURE AND INTERPOLATION 77
1 Numerical differentiation 77
V
Vl EXPONENT/AL FITTING
1.1 Three-point formulae for the first derivative 77
1.2 The five-point formula for the firstderivative 83
1.3 The three-point formula for the seeond derivative 87
1.3.1 Subroutine EFDER 90
2 Quadrature 91
2.1 Simpson formula 92
2.2 Quadrature rules with predetermined abscissa points 94
2.2.1 Symmetrie abscissas 95
2.2.2 Nonsymmetrie abscissas 101
2.2.3 Subroutines EFQS and EFQNS 102
2.3 Extended quadrature rules with predetermined abscissa
points 105
2.3.1 Subroutine EFEXTQS 114
2.4 Integration of a produet of funetions 118
2.4.1 Subroutine CENCl 118
2.5 Gauss-Legendre quadrature rule 121
2.5.1 Subroutines EFGAUSS and EFGQUAD 130
3 Interpolation 134
3.1 A simple seheme 135
3.1.1 Subroutine EFINT 139
5. LINEAR MULTISTEP SOLVERS FüR
ORDINARY DIFFERENTIAL EQUATIONS 145
First order equations 146
1.1 Exponential fitting versions of the two-step
bdf algorithm 147
1.2 Theoretieal issues 150
1.2.1 Loeal prob lern: order of the method and eonsisteney 154
1.2.2 Propagation problern and zero-stability 157
1.2.3 Linear stability theory 160
1.3 Frequeney evaluation and stability properties for two-
step bdf algorithms 164
1.4 The flexible two-step bdf algorithm AEF 169
1.4.1 Numerieal evaluation of the derivatives 169
1.4.2 Choosing the appropriate version 172
1.4.3 Numerieal illustrations 173
1.5 Variable step form of the AEF 178
1.5.1 Choosing the stepsize 183
1.5.2 Numerieal illustrations 184
1.6 Function fitting form of the two-step bdf algorithm 186
Contents Vll
2 Second order equations 192
2.1 The Numerov method 192
2.1.1 Subroutine EFNUM 198
2.2 Linear stability theory and P-stability 200
2.3 Symmetrie four-step methods 210
2.3.1 Subroutine EF4STEP 212
2.4 Other methods 213
2.5 Function fitting methods 215
6. RUNGE-KUTTA SOLVERS FOR
ORDINARY DIFFERENTIAL EQUATIONS 223
1 Formalism and construction scheme for classical Runge-Kutta
methods 223
1.1 General formulation of explicit Runge-Kutta methods 223
2 Exponential-fitted explicit methods 231
2.1 Introduction 231
2.2 The explicit exponential-fitted methods 232
2.3 Error analysis 240
2.4 An approach of Sirnos 241
2.5 Some simple examples 245
2.6 Error and steplength control 246
2.6.1 Local error estimation and a good choice for J-L 246
2.6.2 Some numerical experiments 249
2.6.3 An embedded pair of exponential-fitted explicit Runge-
Kutta methods 250
Derivation of the embedded pair 250
Numerical experiments 251
2.6.4 Frequency determination and steplength control 253
The methods 254
J-L-determination and steplength control 257
Numerical experiments 260
3 Exponential-fitted implicit methods 261
3.1 Classical implicit Runge-Kutta methods 262
3.2 A detailed study of the exponential-fitted two-stage
implicit Runge-Kutta methods 271
3.2.1 Order 2 methods 271
3.2.2 Order 3 methods 273
3.2.3 Order 4 methods 277
3.2.4 Numerical examples 281
3.2.5 Fixed versus J-L-dependent knot points 284
viii EXPONENT/AL FITTING
3.2.6 Conclusion 288
3.3 A specific implementation of implicit methods 288
3.3.1 The special case of a one-stage EFRK method 289
3.3.2 A special two-stage implicit EFRK method 290
3.3.3 Technical details for implementation 293
Calculation of the derivatives 293
Starting procedure 293
Choosing between A 1 and A2 294
A flow chart of the procedure 294
Numerical illustration 295
Conclusions 297
4 Some related research: Runge-Kutta-Nyström methods 298
CD contents 305
Topic Index 307
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