Computational Methods

Transcription

1 Numerical Computational Methods Revised Edition P. B. Patil U. P. Verma Alpha Science International Ltd. Oxford, U.K.

2 CONTENTS Preface List ofprograms v vii 1. NUMER1CAL METHOD, ERROR AND ALGORITHM Nature of Solution of a Problem Numerical Method Characteristics of Numerical Computing Decimal Number System Binary Number System Limitations of Representing N umbers in Computer Absolute, Relative and Percentage Errors Chopping off Error Versus Rounding off Error Significant Digits in Approximation The Non-Commutative Arithmetic Truncation Error (h") Order of Error Propagation of Error Algorithm Golden Rules for Algorithm Design Programming Flowcharting 16 Questions NUMERICAL SOLUTIONS OF TRANSCENDENTAL EQUATIONS Introduction Definition of Root or Zero of a Function Concept of Iterative Methods Search Method for Initial Guess Bisection Method False Position Method Newton-Raphson Method Secant Method Method ofsuccessive Iteration Comparison of Different Root Methods Properties of Roots of Polynomial 52

3 X Contents 2.12 Roots of Polynomial by Newton-Raphson Method 53 Questions ELIMINATION METHODS FOR SOLVING SIMULTANEOUS EQUATIONS Introduction Matrix Notation of Set of Equations Gauss Elimination Method' Pivoting Condition Number Gauss-Jordan Method General Principle of Decomposition Methods Doolittle Method Crout Method Inverse of a Matrix Choleskey's Method Given's Rotation Method 96 Questions SOLUTIONS OF SIMULTANEOUS EQUATIONS: ITERATIVE METHODS Introduction Jacobi Method Gauss-Seidel Method Over Relaxation Method System of Nonlinear Equations Newton-Raphson Method for "Nonlinear Equations 122 Questions SOLUTIONS OF EIGEN-EQUATIONS Concept of Eigen-System Polynomial Method The Fadeev-Leverrier Method Graeffs Root Squaring Method for Finding Roots of a Polynomial Power Method to Find Eigen Value and Eigen Vector QR Iterative Method 140 Questions INTERPOLATION Introduction Polynomial Interpolation Newton-Gregory Forward Difference Interpolation Newton-Gregory Backward Difference Interpolation Central Difference Formulae Gauss Forward Interpolation Gauss Backward Interpolation Stirling Interpolation Formula Bessel Interpolation Formula Laplace-Everett Interpolation Formula Algorithm of Central Difference Interpolations Choice of the Interpolation Formula 182

4 Contents XI 6.12 Newton's Divided Difference Interpolation Lagrange's Interpolation Error Propagation in Difference Scheme Spline Interpolation Cubic Spline Interpolation Double Interpolation Applications of Interpolations 208 Questions LEAST SQUARE CURVE FITTING Concept ofbest Fit Criteria ofbest Fit and Least Square Fit Linear Regression Linearisation of Non-Linear Relation for Fitting Polynomial Regression Multiple Linear Regression Non-Linear Regression 238 Questions NUMERICAL DIFFERENTIATION Introduction First Order Derivative by a Two-Point Formula First Order Derivative by Three Point Formulae Second and Higher Order Derivatives Role of h in Decreasing Error of Derivative Calculation Richardson's Extrapolation The Cubic Spline Method 263 Questions NUMERICAL INTEGRATION Introduction Trapezoidal Rule of Integration Simpson's 1/3 Rule of Integration Newton-Cotes Formulae ofintegration Gaussian Integration Formulae Gaussian Two-Point Formula Gauss Legendre Formulae Romberg Integration The Cubic Spline Method of Integration Algorithms for Integrating Known Functions Integration of Known Function by Trapezoidal Rule Integration of Known Function by Simpson's 1/3 Rule Multiple Integrals Change of Variables for Multiple Integration and Jacobian 299 Questions 302

5 Xll Contents 10. SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS Introduction Picard Method Taylor Series Method Euler's Method Heun's Method Polygon Method or Modified Euler's Method Runge-Kutta Methods (General Concept) Second Order Runge-Kutta Methods Fourth Order R-K Method (RK-4) Single Step and Multi-Step Methods Acquiring Required Accuracy in Single Step Milne-Simpson Method Adams-Bashforth-Moulton Method Modifiers in Multi-Step Methods System of First Order Differential Equations Second Order Differential Equation Higher Order Differential Equations 346 Questions NUMERICAL SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS Nature ofpartial Difference Equation Fundamentals of Finite Difference (FD) Method Representation of Derivatives as Difference Expressions Expressing Differential Equation as Difference Equation Expressing Boundary Conditions in Finite Difference Procedure Modification in Nonrectangular Mesh Special Triangular Grid General Approach to Solve the Set offd Equations Liebmann's Iterative Method to Solve Laplace Equation Bender-Schmidt Method to Solve the Heat Equation Crank-Nicholson Method The Solution of the Wave Equation Solution of Helmholtz Equation in Two Dimensions 3 92 Questions OPTIMIZATION Introduction Optimization Problem Formulation Single Variable Optimization Algorithm Multi Variable Optimization Algorithm Steepest Descent Method Conjugate Gradient Method 411 Questions 418 APPENDIX A: MONTE CARLO METHODS 421 Introduction 421

6 Contents XIII Distinctive Features of Monte Carlo Methods 421 Random Variables 422 Pseudo Random Number Generators 424 Transformation of Random Variables 425 Major Components of the Monte Carlo Algorithm 426 Hit and Miss Integration 426 Monte Carlo Integration Using Distributions 429 Merits and Demerits of Monte Carlo Method 432 APPENDIX B: ESSENTIALS OF PROGRAMMING LANGUAGE 433 Introduction 433 Character Set 433 Data Types 434 Variables and Functions Names 434 Operators Used in Programming 434 Internal Function 435 Input/Output Statement 435 Control Statement 436 Looping 438 Nesting of Loops 439 Arrays 439 Subprogram 439 APPENDIX C: HOW TO START VISUAL BASIC PROGRAMMING 440 APPENDIX D: ABOUT C++PROGRAMMING 441 Introduction 441 Important Points Related to C++ Programming 441 APPENDIX E: PROGRAMS IN VISUAL BASIC 443 APPENDIX F: PROGRAMS IN FORTRAN 525 APPENDIX G: PROGRAMS IN C Bibliography 663 Index 664