Preface. 2 Linear Equations and Eigenvalue Problem 22

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1 Contents Preface xv 1 Errors in Computation Introduction Floating Point Representation of Number Binary Numbers Binary number representation in computer Significant Digits Rounding and Chopping a Number Errors due to Rounding/Chopping Measures of Error in Approximate Numbers Errors in Arithmetic Operations Computation of Errors Using Differentials Errors in Evaluation of Some Standard Functions Truncation Error and Taylor s Theorem 15 Exercise 1 20 References and Some Useful Related Books/Papers 21 2 Linear Equations and Eigenvalue Problem Introduction Ill-conditioned Equations Inconsistency of Equations Linear Dependence Rank of a Matrix Augmented Matrix Methodology for Computing A 1 by Solving Ax = b Cramer s Rule Inverse of Matrix by Cofactors Definitions of Some Matrices Properties of Matrices Elementary Transformations Methods for Solving Equations (Direct Methods) Gaussian elimination method (Basic) Gaussian elimination (with row interchanges) 35

2 vi Contents 2.14 LU Decomposition/Factorisation By Gaussian elimination method Crout s method Cholesky s method Reduction to PA = LU Gauss Jordan (or Jordan s) Method Tridiagonal System Inversion of Matrix Number of Arithmetic Operations in Gaussian Elimination Eigenvalues and Eigenvectors Power Method to Find Dominant Eigenvalue/Latent Root To find smallest eigenvalue by power method Determination of subdominant eigenvalues Iterative Methods Gauss Jacobi method Gauss Seidel method Condition for Convergence of Iterative Methods Successive Over-Relaxation (S.O.R.) Method Norms of Vectors and Matrices Vector norm Matrix norm Forms of matrix norm Compatibility of matrix and vector norms Spectral norm Sensitivity of Solution of Linear Equations 97 Exercise References and Some Useful Related Books/Papers Nonlinear Equations Introduction Order of Convergence of Iterative Method Method of Successive Substitution Bisection Method (Method of Halving) Regula Falsi Method (or Method of False Position) Secant Method Convergence of Secant/Regula Falsi Methods Newton Raphson (N R) Method Evaluation of some arithmetical functions 118

3 Contents vii Convergence of Newton Raphson method Convergence when roots are repeated Simultaneous Equations Method of successive substitution Newton Raphson method Complex Roots Bairstow s Method 129 Exercise References and Some Useful Related Books/Papers Interpolation Introduction Some Operators and their Properties Linearity and commutativity of operators Repeated application and exponentiation of operators Interrelations between operators Application of operators on some functions Finite Difference Table Propagation of error in a difference table Error in Approximating a Function by Polynomial Justification for approximation by polynomial Newton s (Newton Gregory) Forward Difference (FD) Formula Error in Newton s FD formula Newton s (Newton Gregory) Backward Difference (BD) Formula Central Difference (CD) Formulae Gauss s Backward (GB) formula Gauss s Forward (GF) formula Stirling s formula Bessel s formula Everett s formula Steffensen s formula Comments on central difference formulae General Comments on Interpolation Lagrange s Method Divided Differences (DD) Divided differences are independent of order of arguments Newton s Divided Difference (DD) formula Lagrange s Formula Versus Newton s DD Formula 188

4 viii Contents 4.12 Hermite s Interpolation 191 Exercise References and Some Useful Related Books/Papers Numerical Differentiation Introduction Methodology for Numerical Differentiation Differentiation by Newton s FD Formula Error in differentiation Differentiation by Newton s BD Formula Differentiation by Central Difference Formulae At tabular points At non-tabular points Method of Undetermined Coefficients Comments on Differentiation Derivatives with Unequal Intervals Forward Difference formulae Backward Difference formulae Central Difference formulae 221 Exercise References and Some Useful Related Books/Papers Numerical Integration Introduction Methodology for Numerical Integration Rectangular Rule Trapezoidal Rule Simpson s 1/3 rd Rule Comments on Simpson s 1/3 rd rule Simpson s 3/8 th Rule Weddle s Rule Open-Type Formulae Newton Cotes (or Cotes) Formulae Method of Undetermined Coefficients Euler Maclaurin Formula Richardson s Extrapolation Romberg Integration 256

5 Contents ix 6.14 Comments on Numerical Integration Gaussian Quadrature Gauss Legendre quadrature formula Gauss Chebyshev quadrature formulae Gauss Laguerre formula Gauss Hermite formula 271 Exercise References and Some Useful Related Books/Papers Ordinary Differential Equations Introduction Initial Value and Boundary Value Problems (IVP and BVP): Solution of IVP Reduction of Higher-Order IVP to System of First Order Equations Picard s Method (Method of Successive Approximations) Taylor s Series Method Numerical Method, its Order and Stability Euler s Method Modified (Improved) Euler s Method Runge Kutta (R K) Methods Application to first order simultaneous equations Predictor Corrector (P C) Methods Milne s method Adams Bashforth method Boundary Value Problem (BVP) BVP as an Eigenvalue Problem 308 Exercise References and Some Useful Related Books/Papers Splines and their Applications Introduction A Piece-Wise Polynomial Spline Approximation Uniqueness of Cubic Spline Construction of Cubic Spline (Second Derivative Form) Construction of Cubic Spline (First Derivative Form) Minimal Property of a Cubic Spline Application to Differential Equations 331

6 x Contents 8.9 Cubic Spline: Parametric Form Introduction to B-Splines Bezier Spline Curves Convex Polygon and Convex Hull 349 Exercise References and Some Useful Related Books/Papers Method of Least Squares and Chebyshev Approximation Introduction Least Squares Method Normal Equations in Matrix Form Approximation by Standard Functions Over-Determined System of Linear Equations Approximation by Linear Combination of Functions Approximation by Orthogonal Polynomials Chebyshev Approximation 370 Exercise References and Some Useful Related Books/Papers Eigenvalues of Symmetric Matrices Introduction Compact Form of Eigenvalues and Eigenvectors Eigenvalues of Powers of a Matrix Eigenvalues of Transpose of a Matrix Theorem: Eigenvectors of A and A T are Biorthogonal Corrollary: Eigenvectors of Symmetric Matrix form Orthogonal Set Theorem: Eigenvalues of Hermitian Matrix are Real Product of Orthogonal Matrices is an Orthogonal Matrix Eigenvalues of S T AS when S is Orthogonal Eigenvectors of S T AS when S is Orthogonal Methods to find Eigenvalues of Symmetric Matrix Jacobi s Method (Classical) Convergence of Jacobi method Cyclic Jacobi method Givens Method Householder s Method Matrix S is symmetric Matrix S is orthogonal 406

7 Contents xi Similarity transformation First transformation General procedure Sturm Sequence and its Properties Sturm sequence Theorem Eigenvalues of Symmetric Tridiagonal Matrix Upper and Lower Bounds of Eigenvalues Gerschgorin s theorem Corollary Brauer s theorem Determination of Eigenvectors LR Method QR Method 426 Exercise References and Some Useful Related Books/Papers Partial Differential Equations Introduction Some Standard Forms Boundary Conditions Finite Difference Approximations for Derivatives Methods for Solving Parabolic Equation Explicit method/scheme/formula Fully Implicit scheme/method Crank Nicolson s (C N) scheme Comparison of three schemes Compatibility, stability and convergence Compatibility of explicit scheme Stability of explicit scheme Stability of C N scheme Further comparison of schemes Derivative boundary conditions Zero-time discontinuity at endpoints Parabolic equation in two dimensions Alternating Direction Implicit (ADI) method Non-rectangular space domains Methods for Solving Elliptic Equations 478

8 xii Contents Solution by Gauss Seidel and Gaussian elimination Solution by SOR method Solution of elliptic equation by ADI method Methods for Solving Hyperbolic Equations Finite difference methods Explicit method Implicit method Stability analysis Characteristics of a partial differential equation Significance of characteristics Method of characteristics for solving hyperbolic equations Hyperbolic Equation of First Order Finite difference methods Lax Wendroff s method Wendroff s method Other explicit/implicit methods Solving second order equation by simultaneous equations of first order Solution of first order hyperbolic equation by method of characteristics 521 Exercise References and Some Useful Related Books/Papers Finite Element Method Introduction Weighted Residual Methods Galerkin s method Least squares method Subdomain method Collocation method Non-homogeneous Boundary Conditions Variational Methods Functional and its variation Rayleigh Ritz (or Ritz) method Equivalence of Rayleigh Ritz and Galerkin Methods (1 D) Construction of Functional Preliminaries from vector calculus Minimum Functional Theorem (MFT) 549

9 Contents xiii Application of MFT to one-dimension problem Equivalence of Rayleigh Ritz and Galerkin Methods (2 D) Pre-requisites for Finite Element Method Shape functions Normalised/natural coordinates Finite Element Method Ordinary differential equation Elliptic equation Node-wise (point-wise) assembly Higher order elements Element of rectangular shape Parabolic equation (one dimension) Parabolic equation (two dimensions) Hyperbolic equation 616 Exercise References and Some Useful Related Books/Papers Integral Equations Introduction Fredholm Integral Equations Volterra Integral Equations Green s Function Solution of Differential Equation Represented by Integral and Vice-Versa Reduction of Differential Equation to Integral Equation Reduction of a BVP to Fredholm equation Reduction of IVP to Volterra equation Methods for Solving Fredholm Equations Analytical method Classical iterative method Numerical method Methods for Solving Volterra Equation Numerical method Taylor s series method Iterative method 650 Exercise References and Some Useful Related Books/Papers 658

10 xiv Contents 14 Difference Equations Introduction Method of Solution To find y H To find y P Simultaneous Difference Equations and Exponentiation of Matrix Property of constant Row-sum (Column-sum) 673 Exercise References and Some Useful Related Books/Papers Fourier Series, Discrete Fourier Transform and Fast Fourier Transform Introduction Fourier Series Fourier Series with Other Intervals Half-Range Fourier Series Fourier Series for Discrete Data Fourier Transform Discrete Fourier Transform (DFT) Representation of Transforms in Matrix Form Complex Roots of Unity Fast Fourier Transform (FFT) Fast Fourier Transform via Inverse Transform (Author s Comments) 699 Exercise References and some useful related books/papers Free and Moving Boundary Problems: A Brief Introduction Introduction Moving Boundary Problems Moving Grid Method (MGM) MGM with interpolations MGM without interpolations Free Boundary Problem 720 References and Some Useful Related Books/Papers 721 Appendices 723 Appendix A: Some Theorems and Formulae 723 Appendix B: Expansions of Some Functions 726 Appendix C: Graphs of Some Functions 727 Answers to Exercises 730 Index 755