Lectures on Numerical Methods
The course of lectures on numerical methods (part I) given by the author to students in the numerical third of the course of the mathematics mechanics department of Leningrad State University is set down in this volume. Only the topics which, in the opin
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I. P. MYSOVSKIH Leningrad State University
Lectures on Numerical Methods Translated by L. B. Rail Mathematics Research Center, U. S. Army University of Wisconsin
WOLTERS-NOORDHOFF PUBLISHING THE NETHERLANDS
GRONINGEN 1969
ISBN-13: 978-94-011-7485-5 e-ISBN-13: 978-94-011-7483-1 DOl: 10.1007/978-94-011-7483-1
© Copyright 1969 by Wolters-Noordhoff Publishing Groningen, The Netherlands.
Softcover reprint of the hardcover 1st edition 1969 No part of this book may be reproduced in any form, by print, fotoprint, microfilm or any other means, without written permission from the publishers.
CONTENTS
Chapter I. Numerical solution of equations 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
Finding an initial approximation The secant method. . . . . . . The method of iterations. . . . The method of iterations for systems of equations Numerical evaluation of polynomials and their derivatives Newton's method . . . . . . . . . . . . . . . . . Theorems on the convergence of Newton's method . . . Remarks on the practical application of Newton's method Lobacevskii's method. . Factorization methods . Exercises for Chapter I.
Chapter II. Algebraic interpolation 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
Introduction. . . . Finite differences. . Divided differences. The general problem of interpolation Interpolation of function values . . . The remainder term in interpolation . Interpolation at equidistant points. Newton's formulas for interpolation at the beginning and end of tables . . . Interpolation at equidistant points. The formulas of Gauss, Stirling, and Bessel . . . . . . . . . . . . . Inverse interpolation. Interpolation without differences Hermite interpolation. . Numerical differentiation Exercises for Chapter II
1 1 8 11 24 29 39 44 59 65 76 87
92 92 93 99 106 108 113 117 123 131 135 145 152
CONTENTS
Chapter III. Approximate calculation of integrals
159
1. Interpolation quadrature formulas. . . . 2. The simplest interpolation quadrature formulas . 3. Numerical integration of periodic functions and the rectangular quadrature formula. . . . . . . . . 4. Gaussian type quadrature formulas . . . . . . . . . 5. Legendre polynomials and the Gauss formula. . . . . 6. Other special cases of quadrature formula of the Gaussian type . . . . . . . . . . . . . . . 7. A. A. Markov's quadrature formulas 8. Cebysev's quadrature formula. . . . 9. Bernoulli numbers and polynomials . 10. Representation of functions by means of Bernoulli polynomials . . . . . . . . 11. The Euler-Maclaurin formula 12. Concluding remarks . . . Exercises for Chapter III. .
226 230 237 241
Chapter IV. The numerical solution of the Cauchy problem for ordinary differential equations.
247
1. Introduction. . . . . . . . . 2. The Runge-Kutta method. . . 3. On difference methods for the solution of the Cauchy problem . . . . . . . . . . . . . . . . . . . 4. The Adams extrapolation method . . . . . . . 5. The construction of the beginning of the table. 6. The Adams interpolation method . . . . . . . 7. Methods of Cowell type . . . . . ... . . . . 8. Numerical integratio
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