Numerical Methods. Scientists. Engineers

Transcription

1 Third Edition Numerical Methods for Scientists and Engineers K. Sankara Rao

2 Numerical Methods for Scientists and Engineers

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4 Numerical Methods for Scientists and Engineers Third Edition K. SANKARA RAO Formerly, Professor of Mathematics Anna University, Chennai New Delhi

5 NUMERICAL METHODS FOR SCIENTISTS AND ENGINEERS, 3rd ed. K. Sankara Rao 2007 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. Tenth Printing (Third Edition) January, 2012 Published by Asoke K. Ghosh, PHI Learning Private Limited, M-97, Connaught Circus, New Delhi and Printed by Meenakshi Art Printers, Delhi

6 To my wife Leela

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8 Chapter 11 Contents Preface Preface to the Second Edition xi xiii 1. Basics in Computing Introduction Representation of Numbers Floating-point Representation Errors in Computations Inherent Errors Local Round-off Errors Local Truncation Error Problem-solving using Computers 6 2. Solution of Algebraic and Transcendental Equations Introduction Bisection Method Regula Falsi Method Method of Iteration Newton Raphson Method Muller s Method Graeffe s Root Squaring Method Bairstow Method System of Non-Linear Equations 31 Exercises Solution of Linear System of Equations and Matrix Inversion Introduction Gaussian Elimination Method Gauss Jordan Elimination Method Crout s Reduction Method Jacobi s Method Gauss Seidel Iteration Method The Relaxation Method 52 vii

9 viii Contents 3.8 Matrix Inversion Gaussian Elimination Method Gauss Jordan Method 57 Exercises Eigenvalue Problems Introduction Power Method Jacobi s Method Gerschgorin s Theorem 72 Exercises Curve Fitting Introduction Method of Group Averages The Least Squares Method Fitting a Straight Line Fitting a Parabola Fitting a Curve of the Form y = ax b Fitting an Exponential Curve Method of Moments 89 Exercises Interpolation Introduction Finite Difference Operators Forward Differences Backward Differences Central Differences Newton s Forward Difference Interpolation Formula Newton s Backward Difference Interpolation Formula Lagrange s Interpolation Formula Divided Differences Newton s Divided Difference Interpolation Formula Newton s Divided Difference Formula with Error Term Error Term in Interpolation Formulae Interpolation in Two Dimensions Cubic Spline Interpolation Construction of Cubic Spline End Conditions Maxima and Minima of a Tabulated Function Hermite Interpolation 132 Exercises 134

10 Contents ix 7. Numerical Differentiation and Integration Introduction Differentiation using Difference Operators Differentiation using Interpolation Richardson s Extrapolation Method Numerical Integration Newton Cotes Integration Formulae The Trapezoidal Rule (Composite Form) Simpson s Rules (Composite Forms) Romberg s Integration Double Integration Gaussian Quadrature Formulae Multiple Integers 169 Exercises Ordinary Differential Equations Introduction Taylor s Series Method Euler Method Modified Euler s Method Runge Kutta Methods Predictor Corrector Methods Milne s Method Adam Moulton Method Numerical Stability Stability of Modified Euler s Method 203 Exercises Parabolic Partial Differential Equations Introduction Basic Concepts in Finite Difference Methods Explicit Methods Schmidt Method Durfort Frankel Method (1953) Implicit Methods Classical Implicit Method Crank Nicolson Method (1947) Weighted Average Implicit Method The Concept of Stability Methods for Two-dimensional Equations Explicit Methods Implicit Methods Alternate Direction Implicit Method 234 Exercises 237

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