Iterative methods or those methods by which approximations are improved until one receives an accurate value comprise an important learning objective in mathematics. Keeping this in mind, the main objective of this book is to incorporate important iterative methods in a single volume, at an appropriate depth. This will enable students and researchers to apply these iterative techniques to scientific and engineering problems. The text discusses in detail the methods of solving linear systems of equations, nonlinear equations, system of non-linear equations, initial value problems and partial differential equations of all the three types by the use of iterative methods. The text is substantiated with 335 problems including many solved examples and exercises with answers at the end of each chapter.
1. Numerical Solution of Linear Systems of Equations 1.1 Introduction 1.2 Jacobi Iterative Method 1.3 Gauss-Seidel Iterative Method 1.4 Successive Over Relaxation (SOR) Method 1.5 Refinement of Jacobi Method 1.6 Refinement of Gauss-Seidel Method 1.7 Refinement of SOR Method 1.8 Spectral Radii Comparison of Iterative Methods 1.9 Accelerated Over-relaxation (AOR) Method 1.10 Accelerated Gauss-Seidel (AGS) Method 1.11 Extrapolated Accelerated Gauss-Seidel (EAGS) Method 1.12 EAGS Method of Second Degree 1.13 Alternating Direction Implicit Iteration Method 1.14 Spectral Radius of the Variant of ADI Matrix 1.15 Numerical Finding of Largest and Smallest Eigenvalues: Power Method 2. Numerical Solution of Non-linear Equations 2.1 Introduction 2.2 Graphical Method 2.3 Bisection Method 2.4 Iteration Method 2.5 Acceleration of Convergence 2.6 Wegstein’s Method 2.7 Aitken’s 2 Method 2.8 Extrapolated Iterative Method 2.9 Method of False-Position (or) Regula-Falsi Method 2.10 Secant Method 2.11 Newton-Raphson (N-R) Method 2.12 Some Variants of Newton-Raphson Method 2.13 Newton-Raphson Method for Multiple Roots (or) Generalized Newton Raphson (GN-R) Method 2.14 Generalized Extrapolated Newton-Raphson (GEN-R) Method 2.15 Two-Step Iterative Methods for Solving Non-linear Equations 3. Numerical Solution of System of Non-linear Equations 3.1 Introduction 3.2 Successive Approximation (or) Iteration Method 3.3 Multi-variable Newton’s Method 3.4 Extrapolated Successive Approximation (ESA) Method 3.5 Accelerated Multi-Variable Newton’s (AMVN) Method 3.6 Fixed Point Accelerated Multi-Variable Newton’s (AMVN) Method 3.7 Numerical Finding of Complex Roots 4. Numerical Solution of Initial Value Problems4.1 Introduction 4.2 Taylor Series Method 4.3 Picard’s Method 4.4 Euler’s Method 4.5 Modified Euler’s Method 4.6 Runge-Kutta Method of Order Two 4.7 Runge-Kutta Method of Order Four 4.8 Runge-Kutta Method for Simultaneous Differential Equations 4.9 Predictor-Corrector Methods 5. Numerical Solution of Partial Differential Equations 5.1 Introduction 5.2 Finite-Difference Approximations to Partial Derivatives 5.3 Condition for the Negativeness of the Eigenvalues of the Jacobian Matrix 5.4 An Upper Bound for the Spectral Radius of the Jacobi Matrix 5.5 Convergence of the Modified Jacobi Method 5.6 Numerical Solution of Laplace Equation 5.7 Refinement of Jacobi Method for the Solution of Laplace Equation 5.8 Refinement of Gauss-Seidel Method for the Solution of Laplace Equation 5.9 Refinement of SOR Method for the Solution of Laplace Equation 5.10 Alternating Direction Implicit Iteration 5.11 Bounds for the Eigenvalues of H, V, D–1H, D–1V 5.12 Parabolic Equations5.13 Crank-Nicolson Method5.14 Hyperbolic EquationsBibliography Index
V.B.K. Vatti :- Dr. V.B.K. Vatti, Professor of Engineering Mathematics in College of Engineering, Andhra University, obtained his Ph.D. from Indian Institute of Technology Bombay, Mumbai after pursuing graduation and post-graduation at Andhra University. He has about 32 years of academic experience apart from research, teaching various mathematical subjects including computer programs and computational methods to graduate and postgraduate students of engineering. He has nearly 50 publications to his credit in various national and international journals. He has guided more than a dozen research awardees which include both Ph.Ds and M.Phils. He has also served the university as Co-ordinator of University Grant Commission, Chief of Employment and Guidance Bureau, and Head of the Department of Engineering Mathematics. He has also co-authored a few books on mathematics for the distance education students of Andhra University.