F I F T H E D I T I O N. Introductory Methods of Numerical Analysis. S.S. Sastry
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1 F I F T H E D I T I O N Introductory Methods of Numerical Analysis S.S. Sastry
2 Introductory Methods of Numerical Analysis
3
4 Introductory Methods of Numerical Analysis Fifth Edition S.S. SASTRY Formerly, Scientist/Engineer SF Vikram Sarabhai Space Centre Trivandrum New Delhi
5 INTRODUCTORY METHODS OF NUMERICAL ANALYSIS, Fifth Edition S.S. Sastry 2012 by PHI Learning Private Limited, New Delhi. All rights reserved. No part of this book may be reproduced in any form, by mimeograph or any other means, without permission in writing from the publisher. ISBN The export rights of this book are vested solely with the publisher. Forty-fifth Printing (Fifth Edition) June, 2012 Published by Asoke K. Ghosh, PHI Learning Private Limited, M-97, Connaught Circus, New Delhi and Printed by Rajkamal Electric Press, Plot No. 2, Phase IV, HSIDC, Kundli , Sonepat, Haryana.
6 To My Grandsons Venkata Bala Nagendra Venkata Badrinath
7
8 Contents Preface xiii 1. Errors in Numerical Calculations Introduction Computer and Numerical Software Computer Languages Software Packages Mathematical Preliminaries Errors and Their Computations A General Error Formula Error in a Series Approximation 14 Exercises 19 Answers to Exercises Solution of Algebraic and Transcendental Equations Introduction Bisection Method Method of False Position Iteration Method Newton Raphson Method Ramanujan s Method Secant Method Muller s Method Graeffe s Root-Squaring Method Lin Bairstow s Method Quotient Difference Method 58 vii
9 viii Contents 2.12 Solution to Systems of Nonlinear Equations Method of Iteration Newton Raphson Method 64 Exercises 68 Answers to Exercises Interpolation Introduction Errors in Polynomial Interpolation Finite Differences Forward Differences Backward Differences Central Differences Symbolic Relations and Separation of Symbols Detection of Errors by Use of Difference Tables Differences of a Polynomial Newton s Formulae for Interpolation Central Difference Interpolation Formulae Gauss Central Difference Formulae Stirling s Formula Bessel s Formula Everett s Formula Relation between Bessel s and Everett s Formulae Practical Interpolation Interpolation with Unevenly Spaced Points Lagrange s Interpolation Formula Error in Lagrange s Interpolation Formula Hermite s Interpolation Formula Divided Differences and Their Properties Newton s General Interpolation Formula Interpolation by Iteration Inverse Interpolation Double Interpolation 118 Exercises 119 Answers to Exercises Least Squares and Fourier Transforms Introduction Least Squares Curve Fitting Procedures Fitting a Straight Line Multiple Linear Least Squares Linearization of Nonlinear Laws Curve Fitting by Polynomials Curve Fitting by a Sum of Exponentials 135
10 Contents ix 4.3 Weighted Least Squares Approximation Linear Weighted Least Squares Approximation Nonlinear Weighted Least Squares Approximation Method of Least Squares for Continuous Functions Orthogonal Polynomials Gram Schmidt Orthogonalization Process Approximation of Functions Chebyshev Polynomials Economization of Power Series Fourier Approximation Fourier Transform Discrete Fourier Transform (DFT) Fast Fourier Transform (FFT) Cooley Tukey Algorithm Sande Tukey Algorithm (DIF FFT) Computation of the Inverse DFT 174 Exercises 176 Answers to Exercises Spline Functions Introduction Linear Splines Quadratic Splines Cubic Splines Minimizing Property of Cubic Splines Error in the Cubic Spline and Its Derivatives Surface Fitting by Cubic Splines Cubic B-splines Representation of B-splines Least Squares Solution Applications of B-splines 203 Exercises 204 Answers to Exercises Numerical Differentiation and Integration Introduction Numerical Diferentiation Errors in Numerical Differentiation Cubic Splines Method Differentiation Formulae with Function Values Maximum and Minimum Values of a Tabulated Function Numerical Integration Trapezoidal Rule Simpson s 1/3-Rule Simpson s 3/8-Rule 222
11 Introductory Methods Of Numerical Analysis 30% OFF Publisher : PHI Learning ISBN : Author : SASTRY, S. S. Type the URL : Get this ebook
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Romberg Integration and Gaussian Quadrature P. Sam Johnson October 17, 014 P. Sam Johnson (NITK) Romberg Integration and Gaussian Quadrature October 17, 014 1 / 19 Overview We discuss two methods for integration.