CONTENTS

NUMERICAL METHODS AND ALGORITHMS

PREFACE

1. INTRODUCTION TO NUMERICAL METHODS

Basics of Numerical Methods, 7. Errors of Calculations, 8. Calculation Algorithm, 11. Horner's Scheme, 14. Chain Fractions, 17. Expansions of Functions in Chain Fractions, 22. Additions of Power Series, 30. Computation of Implicit Function Values, 36. Exercises, 42.

2. SYSTEMS OF LINEAR EQUATIONS

Matrix Algebra, 45. Survey of Methods and Problem Conditioning, 54. Gauss Elimination, 60. LU Decomposition, 69. Tridiagonal and Other Banded Systems, 79. Basic Iterative Methods, 89. Extremal Eigenvalues, 103. Exercises, 121.

3. NONLINEAR EQUATIONS

Introduction, 125. One-Point Iteration Methods, 128. Two-Point Iteration Methods, 137. Newton's Methods for Systems of Equations, 145. Method of Deepest Descent, 150. Exercises, 157.

4. INTERPOLATION

Preliminaries, 159. Lagrange's Interpolating Polynomial, 161. Newton's Divided-Difference Interpolating Polynomials, 168. Convergence of Polynomial Interpolation, 178. Czebyshev's Interpolation, 181. Trigonometric Interpolation, 188. Fundamentals of Spline Function Theory, 194. Cublic Spline Functions, 200. B-Splines, 219. Tension Splines, 225. Two-Dimensional Interpolation, 234. Exercises, 255.

5. DIFFERENTIATION, INTEGRATION AND APPROXIMATION

Numerical Differentiation and Approximation of Derivatives by Finite Differences, 257. Numerical Integration, 264. Types of Approximation, 285. The Least-Squares Polynomial Approximation, 288. The Least-Square Trigonometric Approximation, 307. Approximation by Spline Functions, 314. The Least-Square Approximation of Two-Dimensional Functions, 325. Exercises, 334.

6. ORDINARY DIFFERENTIAL AND INTEGRAL EQUATIONS

Initial Value Problems for ODEs: Introductory Remarks, 337; Analytical and Analytical-Numerical Methods, 340; Runge-Kutta Methods, 347; Predictor-Corrector and Multistep Methods, 366. Boundary Value Problems for ODEs: Introduction, 383; Finite Difference Approximations, 385; Cubic Spline Collocation Methods, 392. Integral Equations: Introduction, 404; Volterra Integral Equations of the Second Kind, 408; Fredholm Integral Equations of the Second Kind, 415. Exercises, 420.

7. PARTIAL DIFFERENTIAL EQUATIONS

General Remarks, 423. Requirements Imposed on Difference Schemes, 429. Parabolic Equations, 433. Hyperbolic Equations, 450. Elliptic Equations, 468. The Resolution of Discretized Elliptic Equations, 483. Exercises, 517.

APPENDIX

REFERENCES

WEB SITES

530 Contents

SPIS TREŚCI 527

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