# Numerical Analysis with Algorithms and Programming

## Preview

## Book Description

**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.

## Author(s)

### 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.