Heath M.T. Scientific Computing: An Introductory Survey

2nd ed. — SIAM, 2018. — xx +568 p. — (Classics in Applied Mathematics, 80). — ISBN: 978-1611975574.This book presents a broad overview of numerical methods for solving all the major problems in scientific computing, including linear and nonlinear equations, least squares, eigenvalues, optimization, interpolation, integration, ordinary and partial differential equations, fast Fourier transforms, and random number generators. The treatment is comprehensive yet concise, software-oriented yet compatible with a variety of software packages and programming languages. The book features more than 160 examples, 500 review questions, 240 exercises, and 200 computer problems. Changes for the second edition include: expanded motivational discussions and examples; formal statements of all major algorithms; expanded discussions of existence, uniqueness, and conditioning for each type of problem so that students can recognize "good" and "bad" problem formulations and understand the corresponding quality of results producted; and expanded coverage of several topics, particularly eigenvalues and constrained optimization. The book contains a wealth of material and can be used in a variety of one- or two-term courses in computer science, mathematics, or engineering. Its comprehensiveness and modern perspective, as well as the software pointers provided, also make it a highly useful reference for practicing professionals who need to solve computational problems.Preface to the Classics Edition

NotationScientific ComputingApproximations in Scientific Computation
Computer Arithmetic
Mathematical Software
Historical Notes and Further ReadingSystems of Linear Equations
Linear Systems
Existence and Uniqueness
Sensitivity and Conditioning
Solving Linear Systems
Special Types of Linear Systems
Iterative Methods for Linear Systems
Software for Linear Systems
Historical Notes and Further ReadingLinear Least Squares
Linear Least Squares Problems
Existence and Uniqueness
Sensitivity and Conditioning
Problem Transformations
Orthogonalization Methods
Singular Value Decomposition
Comparison of Methods
Software for Linear Least Squares
Historical Notes and Further ReadingEigenvalue Problems
Eigenvalues and Eigenvectors
Existence and Uniqueness
Sensitivity and Conditioning
Problem Transformations
Computing Eigenvalues and Eigenvectors
Generalized Eigenvalue Problems
Computing the Singular Value Decomposition
Software for Eigenvalue Problems
Historical Notes and Further ReadingNonlinear Equations
Nonlinear Equations
Existence and Uniqueness
Sensitivity and Conditioning
Convergence Rates and Stopping Criteria
Nonlinear Equations in One Dimension
Systems of Nonlinear Equations
Software for Nonlinear Equations
Historical Notes and Further ReadingOptimization
Optimization Problems
Existence and Uniqueness
Sensitivity and Conditioning
Optimization in One Dimension
Unconstrained Optimization
Nonlinear Least Squares
Constrained Optimization
Software for Optimization
Historical Notes and Further ReadingInterpolation
Interpolation
Existence, Uniqueness, and Conditioning
Polynomial Interpolation
Piecewise Polynomial Interpolation
Software for Interpolation
Historical Notes and Further ReadingNumerical Integration and Differentiation
Integration
Existence, Uniqueness, and Conditioning
Numerical Quadrature
Other Integration Problems
Integral Equations
Numerical Differentiation
Richardson Extrapolation
Software for Integration and Differentiation
Historical Notes and Further ReadingInitial Value Problems for ODEs
Ordinary Differential Equations
Existence, Uniqueness, and Conditioning
Numerical Solution of ODEs
Software for ODE Initial Value Problems
Historical Notes and Further ReadingBoundary Value Problems for ODEs
Boundary Value Problems
Existence, Uniqueness, and Conditioning
Shooting Method
Finite Difference Method
Collocation Method
Galerkin Method
Eigenvalue Problems
Software for ODE Boundary Value Problems
Historical Notes and Further ReadingPartial Differential Equations
Partial Differential Equations
Time-Dependent Problems
Time-Independent Problems
Direct Methods for Sparse Linear Systems
Iterative Methods for Linear Systems
Comparison of Methods
Software for Partial Differential Equations
Historical Notes and Further ReadingFast Fourier Transform
Trigonometric Interpolation
FFT Algorithm
Applications of DFT
Wavelets
Software for FFT
Historical Notes and Further Reading
Random Numbers and Simulation
Stochastic Simulation
Randomness and Random Numbers
Random Number Generators
Quasi-Random Sequences
Software for Generating Random Numbers
Historical Notes and Further ReadingBibliography
IndexThis book differs from traditional numerical analysis texts in that it focuses on the motivation and ideas behind the algorithms presented rather than on detailed analyses of them. It presents a broad overview of methods and software for solving mathematical problems arising in computational modeling and data analysis, including proper problem formulation, selection of effective solution algorithms, and interpretation of results.
In the 20 years since its original publication, the modern, fundamental perspective of this book has aged well, and it continues to be used in the classroom. This Classics edition has been updated to include pointers to Python software and the Chebfun package, expansions on barycentric formulation for Lagrange polynomial interpretation and stochastic methods, and the availability of about 100 interactive educational modules that dynamically illustrate the concepts and algorithms in the book.
Scientific Computing: An Introductory Survey, Revised Second Edition is intended as both a textbook and a reference for computationally oriented disciplines that need to solve mathematical problems.