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Description
This book is intended to help advanced undergraduate, graduate, and postdoctoral students in their daily work by oï¬ering them a compendium of numerical methods. The choice of methods pays signiï¬cant attention to error estimates, stability and convergence issues, as well as optimization of program execution speeds. Numerous examples are given throughout the chapters, followed by comprehensive end-of-chapter problems with a more pronounced physics background, while less stress is given to the explanation of individual algorithms. The readers are encouraged to develop a certain amount of skepticism and scrutiny instead of blindly following readily available commercial tools.
The second edition has been enriched by a chapter on inverse problems dealing with the solution of integral equations, inverse Sturm-Liouville problems, as well as retrospective and recovery problems for partial diï¬erential equations. The revised text now includes an introduction to sparse matrix methods, the solution of matrix equations, and pseudospectra of matrices; it discusses the sparse Fourier, non-uniform Fourier and discrete wavelet transformations, the basics of non-linear regression and the Kolmogorov-Smirnov test; it demonstrates the key concepts in solving stiï¬ diï¬erential equations and the asymptotics of Sturm-Liouville eigenvalues and eigenfunctions. Among other updates, it also presents the techniques of state-space reconstruction, methods to calculate the matrix exponential, generate random permutations and compute stable derivatives.
Table of Contents
Basics of numerical analysis.- Solution of nonlinear equations.- Matrix methods.- Transformations of functions and signals.- Statistical description and modeling of data.- Modeling and analysis of time series.- Initial-value problems for ordinary differential equations.- Boundary-value problems for ordinary differential equations.- Difference methods for one-dimensional partial differential equations.- Difference methods for partial differential equations in more than one dim.- Spectral methods for partial differential equations.- Inverse methods.
