1st Edition

Numerical Analysis with Algorithms and Programming

By Santanu Saha Ray Copyright 2016
    708 Pages 2 Color & 43 B/W Illustrations
    by Chapman & Hall

    728 Pages 2 Color & 43 B/W Illustrations
    by Chapman & Hall

    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 Equations
    Introduction
    Basic Concepts and Definitions
    Initial Approximation
    Iterative 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 Numbers
    Euler–Maclaurin Formula

    Numerical Solution of System of Linear Algebraic Equations
    Introduction
    Vector and Matrix Norm
    Direct Methods
    Iterative 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.

    Biography

    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.