Theory and Applications of Numerical Analysis

Theory and Applications of Numerical Analysis

G M Phillips and P J Taylor



This book has remained in print since it was first published in 1973. For the publisher's details about the Second Edition, click on Phillips and Taylor on the Web .


From Reviews of the Second Edition:

"The first edition was an outstanding work, and the additions that have been put in the Second Edition are very appropriate and have been written up in exemplary fashion."
Philip J. Davis, Brown University



CONTENTS:

Introduction: What is Numerical Analysis? Numerical Algorithms. Properly Posed and Well-Conditioned Problems.

Basic Analysis: Functions. Limits and Derivatives. Sequences and Series. Integration. Logarithmic and Exponential Functions.

Taylor's Polynomial and Series: Function Approximation. Taylor's Theorem. Convergence of Taylor Series. Taylor Series in Two Variables. Power Series.

The Interpolating Polyomial: Linear Interpolation. Polynomial Interpolation. Accuracy of Interpolation. The Neville-Aitken Algorithm. Inverse Interpolation. Divided Differences. Equally Spaced Points. Derivatives and Differences. Effect of Rounding Error. Choice of Interpolation Points. Examples of Bernstein and Runge.

"Best" Approximation: Norms of Functions. Best Approximations. Least Squares Approximations. Orthogonal Functions. Orthogonal Polynomials. Minimax Approximation. Chebyshev Series. Economization of Power Series. The Remez Algorithms. Further Results on Minimax Approximation.

Splines and Other Approximations: Introduction. B-splines. Equally-Spaced Knots. Hermite Interpolation. Padé and Rational Approximation.

Numerical Integration and Differentiation: Numerical Integration. Romberg Integration. Gaussian Integration. Indefinite Integrals. Improper Integrals. Multiple Integrals. Numerical Differentiation. Effect of Errors.

Solution of Algebraic Equations of One Variable: Introduction. The Bisection Method. Interpolation Methods. One-Point Iterative Methods. Faster Convergence. Higher Order Processes. The Contraction Mapping Theorem.

Linear Equations: Introduction. Matrices. Linear Equations. Pivoting. Analysis of Elimination Method. Matrix Factorization. Compact Elimination Methods. Symmetric Matrices. Tridiagonal Matrices. Rounding Errors in Solving Linear Equations.

Matrix Norms and Applications: Determinants, Eigenvalues, and Eigenvectors. Vector Norms. Matrix Norms. Conditioning. Iterative Correction From Residual Vectors. Iterative Methods.

Matrix Eigenvalues and Eigenvectors: Relations Between Matrix Norms and Eigenvalues; Gerschgorin Theorems. Simple and Inverse Iterative Method. Sturm Sequence Method. The QR Algorithm. Reduction to Tridiagonal Form - Householder's Method.

Systems of Non-Linear Equations: Contraction Mapping Theorem. Newton's Method.

Ordinary Differential Equations: Introduction. Difference Equations and Inequalities. One-Step Methods. Truncation Errors of One-Step Methods. Methods Based on Numerical Integration; Explicit Methods. Convergence of One-Step Methods. Effect of Rounding Errors on One-Step Methods. Methods Based on Numerical Integration; Implicit Methods. Iterating with the Corrector. Milne's Method of Estimating Truncation Errors. Numerical Stability. Systems and Higher Order Equations. Notes Comparing Step-by-Step Methods.

Boundary Value and Other Methods for Ordinary Differential Equations: Shooting Method for Boundary Value Problems. Boundary Value Problem. Extrapolation to the Limit. Deferred Correction. Chebyshev Series Method.

Appendices

References

Solutions to Selected Problems

Subject Index