Numerical Analysis with Algorithms and Programming

Numerical Analysis with Algorithms and Programming is the first comprehensive textbook to provide detailed coverage of numerical methods, their algorithms, and corresponding computer programs. It presents many techniques for the efficient numerical solution of problems in science and engineering. Along with numerous worked-out examples, end-of-chapter exercises, and Mathematica® programs, the book includes the standard algorithms for numerical computation: Root finding for nonlinear equations Interpolation and approximation of functions by simpler computational building blocks, such as polynomials and splines The solution of systems of linear equations and triangularization Approximation of functions and least square approximation Numerical differentiation and divided differences Numerical quadrature and integration Numerical solutions of ordinary differential equations (ODEs) and boundary value problems Numerical solution of partial differential equations (PDEs) The text develops students’ understanding of the construction of numerical algorithms and the applicability of the methods. By thoroughly studying the algorithms, students will discover how various methods provide accuracy, efficiency, scalability, and stability for large-scale systems.

Errors in Numerical Computations Introduction Preliminary Mathematical Theorems Approximate Numbers and Significant Figures Rounding Off Numbers Truncation Errors Floating Point Representation of Numbers Propagation of Errors General Formula for Errors Loss of Significance Errors Numerical Stability, Condition Number, and Convergence Brief Idea of Convergence Numerical Solutions of Algebraic and Transcendental EquationsIntroduction Basic Concepts and Definitions Initial ApproximationIterative Methods Generalized Newton’s Method Graeffe’s Root Squaring Method for Algebraic Equations Interpolation Introduction Polynomial Interpolation Numerical Differentiation Introduction Errors in Computation of Derivatives Numerical Differentiation for Equispaced Nodes Numerical Differentiation for Unequally Spaced Nodes Richardson Extrapolation Numerical Integration Introduction Numerical Integration from Lagrange’s Interpolation Newton–Cotes Formula for Numerical Integration (Closed Type) Newton–Cotes Quadrature Formula (Open Type) Numerical Integration Formula from Newton’s Forward Interpolation Formula Richardson Extrapolation Romberg Integration Gauss Quadrature Formula Gaussian Quadrature: Determination of Nodes and Weights through Orthogonal Polynomials Lobatto Quadrature Method Double Integration Bernoulli Polynomials and Bernoulli NumbersEuler–Maclaurin Formula Numerical Solution of System of Linear Algebraic Equations Introduction Vector and Matrix Norm Direct MethodsIterative Method Convergent Iteration Matrices Convergence of Iterative Methods Inversion of a Matrix by the Gaussian Method Ill-Conditioned Systems Thomas Algorithm Numerical Solutions of Ordinary Differential Equations Introduction Single-Step Methods Multistep Methods System of Ordinary Differential Equations of First Order Differential Equations of Higher Order Boundary Value Problems Stability of an Initial Value Problem Stiff Differential Equations A-Stability and L-Stability Matrix Eigenvalue Problem Introduction Inclusion of Eigenvalues Householder’s Method The QR Method Power Method Inverse Power Method Jacobi’s Method Givens Method Approximation of Functions Introduction Least Square Curve Fitting Least Squares Approximation Orthogonal Polynomials The Minimax Polynomial Approximation B-Splines Padé Approximation Numerical Solutions of Partial Differential Equations Introduction Classification of PDEs of Second Order Types of Boundary Conditions and Problems Finite-Difference Approximations to Partial Derivatives Parabolic PDEs Hyperbolic PDEs Elliptic PDEs Alternating Direction Implicit Method Stability Analysis of the Numerical Schemes An Introduction to the Finite Element Method Introduction Piecewise Linear Basis Functions The Rayleigh–Ritz Method The Galerkin Method Bibliography Answers Index Exercises appear at the end of each chapter.

Dr. Santanu Saha Ray is an associate professor in the Department of Mathematics at the National Institute of Technology in Rourkela, India. He is a member of the Society for Industrial and Applied Mathematics and the American Mathematical Society. He is also the editor-in-chief of the International Journal of Applied and Computational Mathematics and the author of numerous journal articles and two books: Graph Theory with Algorithms and Its Applications: In Applied Science and Technology and Fractional Calculus with Applications for Nuclear Reactor Dynamics. His research interests include fractional calculus, mathematical modeling, mathematical physics, stochastic modeling, integral equations, and wavelet transforms. Dr. Saha Ray earned his PhD from Jadavpur University.