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Features

Emphasizes the multidisciplinary aspect of numerical analysis involving science, computer science, engineering, and mathematics
Describes the computational implementation of algorithms, illustrating the major issues of accuracy, computational work effort, and stability
Includes the Mathematica® programming codes for each numerical method, enabling students to gain practical experience applying the methods
Gives a brief introduction to the finite element method
Contains numerous exercises that encourage students to develop their knowledge and skills in solving mathematical problems
Covers algorithms of computation; approximation; interpolation; numerical differentiation and integration; numerical solutions of nonlinear equations, linear algebraic equation systems, ODEs, and PDEs; eigenvalue problems in matrices; and approximations of functions
Summary

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.



Table of Contents

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.