NUMERICAL COMPUTATION IN SCIENCE AND ENGINEERING

Transcription

1 NUMERICAL COMPUTATION IN SCIENCE AND ENGINEERING C. Pozrikidis University of California, San Diego New York Oxford OXFORD UNIVERSITY PRESS 1998

2 CONTENTS Preface ix Pseudocode Language Commands xi 1 Numerical Computation Analytical and numerical computation Hardware and software Computer arithmetic and round-off error Algorithms Computer simulation Systematic error and its reduction Iterations, numerical sequences, and their convergence 47 References 50 2 Numerical Matrix Algebra and Matrix Calculus Matrix algebra Square matrices Inverse of a matrix, cofactors, and the determinant Computation of the determinant Computation of the inverse Xt/and.LDI/decompositions QR decomposition Systems of linear algebraic equations Eigenvalues and eigenvectors Wielandt's deflation Successive linear mappings Functions of matrices 118 References Linear Algebraic Equations Significance and applications 122

3 3.2 Diagonal and triangular systems General procedures and overview Tridiagonal, pentadiagonal, and sparse systems Gauss elimination and related methods Iterative methods Jacobi, Gauss-Seidel, and the SOR method Acceleration of iterative methods by deflation Minimization and conjugate-gradients methods Overdetermined systems 178 References Nonlinear Algebraic Equations Mathematical and physical context Bracketing methods One-point iterations Newton's and higher-order methods for one equation Newton's method for a system of equations Modified Newton methods Zeros of polynomials Estimating the location of a root Difficult problems, continuation, and embedding 221 References Eigenvalues of Matrices Mathematical and physical context Complex matrices, generalized, and nonlinear eigenvalue problems Analytical results for diagonal, triangular, and tridiagonal matrices; circulants Computing the roots of the characteristic polynomial The power method Methods for computing all eigenvalues by similarity transformations Transforming a symmetric matrix to a simpler one Transforming an arbitrary matrix to a tridiagonal or Hessenberg matrix 258 References Function Interpolation and Differentiation Interpolation and extrapolation The interpolating polynomial and its computation 264

4 vii 6.3 Error and convergence of polynomial interpolation Optimal positioning of data points Selection of the interpolating variable Piecewise polynomial interpolation and splines Hermite interpolation Parametric description of lines Interpolation of a function of two variables Parametric description of surfaces Numerical differentiation of a function of one variable Numerical differentiation of a function of two variables 321 References Numerical Integration Computation of one-dimensional integrals Integration by local polynomial interpolation: Newton Cotes rules Optimal distribution of base points, and the Gauss Legendre quadrature Singular integrands Integrals over infinite domains Rapidly varying and oscillatory integrands Area, surface, and multidimensional integrals Numerical solution of Fredholm integral equations 378 References Approximation of Functions, Lines, and Surfaces Problem statement and significance Polynomial function approximation Bezier representation of lines and surfaces S-spline approximation and representation of lines and surfaces Pade approximation Trigonometric approximation and interpolation Fast Fourier transform Trigonometric approximation of a function of two variables 432 References Ordinary Differential Equations; Initial-Value Problems Problem formulation and physical context 436

5 9.2 Lnear autonomous systems Explicit and implicit methods for nonlinear systems Predictor-corrector methods Error estimate and adaptive step-size control Stiff problems 479 References Ordinary Differential Equations; Boundary-Value, Eigenvalue, and Free-Boundary Problems Two-point boundary-value problems The shooting method Finite-difference and finite-volume methods Finite-element methods Weighted-residual methods Eigenvalue problems Free-boundary problems 522 References Finite-Difference Methods for Partial-Differential Equations Introduction and procedures One-dimensional unsteady diffusion Unsteady diffusion in two and three dimensions Poisson and Laplace equations One-dimensional convection Convection in two and three dimensions Convection diffusion in one dimension Convection-diffusion in two and three dimensions 577 References 579 Appendix A: Calculus Refresher 581 Appendix B: Orthogonal Polynomials 586 Appendix C: UNIX Primer 604 Appendix D: FORTRAN Primer 611 Appendix E: FORTRAN Programs 615 Index 619