Ray Numerical Analysis with Algorithms and Programming

Erscheinungsjahr 2016
ISBN: 978-1-4987-4183-5
Verlag: CRC Press
Format: EPUB
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 728 Seiten

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.

Autoren/Hrsg.

Ray, Santanu Saha

Fachgebiete

• Mathematik | Informatik Mathematik Mathematik Allgemein Zahlensysteme
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle

Inhaltsverzeichnis

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.

Über Autor(innen)

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.