Numerical Mathematics

Jeffrey S. Ovall

2024年10月 · Other Titles in Applied Mathematics 第 198 本图书 · SIAM

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This textbook introduces key numerical algorithms used for problems arising in three core areas of scientific computing: calculus, differential equations, and linear algebra. Theoretical results supporting the derivation and error analysis of algorithms are given rigorous justification in the text and exercises, and a wide variety of detailed computational examples further enhance the understanding of key concepts.

Numerical Mathematics includes topics not typically discussed in similar texts at this level, such as a Fourier-based analysis of the trapezoid rule, finite volume methods for the 2D Poisson problem, the Nyström method for approximating the solution of integral equations, and the relatively new FEAST method for targeting clusters of eigenvalues and their eigenvectors. An early emphasis is given to recognizing or deducing orders of convergence in practice, which is essential for assessing algorithm performance and debugging computational software. Numerical experiments complement many of the theorems concerning convergence, illustrating typical behavior of the associated algorithms when the assumptions of the theorems are satisfied and when they are not.

This book is intended for advanced undergraduate and beginning graduate students in mathematics seeking a solid foundation in the theory and practice of scientific computing. Students and researchers in other disciplines who want a fuller understanding of the principles underlying these algorithms will also find it useful.

The text is divided into three parts, corresponding to numerical methods for problems in calculus, differential equations, and linear algebra. Each part can be used for a one-term course (quarter or semester), making the book suitable for a two- or three-term sequence in numerical analysis or for largely independent courses on any of the three main topics.

作者简介

Jeffrey S. Ovall is a Maseeh Professor of Mathematics at Portland State University. He has held postdoctoral positions at the Max Planck Institute for Mathematics in the Sciences in Leipzig, Germany, and at the California Institute of Technology. His research primarily concerns numerical methods for partial differential equations and integral equations, with particular interest in eigenvalue problems, nonstandard discretization techniques, and effective treatment of singular solutions.