Vatti V.B.K. Numerical Methods for Graduates

Wiley, 2021. — 343 p.Foreword
Preface
Numerical Solution of Algebraic and Transcendental Equations
Introduction
Graphical Method
Bisection Method
Iteration Method
Acceleration of Convergence
Wegstein’s Method
Aitken’s D2 Method
Extrapolated Iterative Method
Method of False-Position (or) Regula-Falsi Method
Secant Method
Newton-Raphson (N-R) Method
Some Variants of Newton-Raphson Method
Newton-Raphson Method for Multiple Roots (or) Generalized Newton-Raphson (GN-R) Method
Generalized Extrapolated Newton-Raphson (Gen-R) Method
Two-Step Iterative Methods for Solving Non-linear Equations
Exercise
Finite Differences and Interpolation
Introduction
Finite Differences
Some Other Difference Operators
Propagation of Errors on a Difference Table
Newton’s Interpolation Formulae
Central Difference Interpolation Formulae
Interpolation with Unequally Spaced Intervals
Inverse Interpolation
Exercise
Numerical Differentiation
Introduction
Finding Derivatives by Newton's Forward Formula
Finding Derivatives by Newton's Backward Formula
Finding Derivatives by Stirling's Formula
Finding Derivatives by Bessel's Formula
Exercise
Numerical Integration
Numerical Integration
Newton’s-Cote’s General Quadrature Formula
Trapezoidal Rule
Simpson’s (13) Rule
Simpson’s ( 38) Rule
Weddle’s Rule
Gaussian Quadrature Formula
Exercise
Correlation and Regression Analysis
Curve Fitting
Fitting a Straight Line
Rectification
Principle of Least Squares
Method of Least Squares
Fitting of Other Curves
Correlation
Regression
Exercise
Solution of Linear Systems by Indirect Methods
Introduction
Jacobi Iterative Method
Gauss-Seidel Iterative Method
Successive Over Relaxation (SOR) Method
Refinement of Jacobi Method
Refinement of Gauss-Seidel Method
Refinement of SOR Method
Spectral Radii Comparison of Iterative Methods
Accelerated Over-relaxation (AOR) Method
Accelerated Gauss-Seidel (AGS) Method
Extrapolated Accelerated Gauss-Seidel (EAGS) Method
EAGS Method of Second Degree
Alternating Direction Implicit Iteration Method
Spectral Radius of the Variant of ADI Matrix
Numerical Finding of Largest and Smallest Eigenvalues: Power Method
Exercise
Solution Non-linear Systems by Indirect Methods
Introduction
Successive Approximation (or) Iteration Method
Multi-variable Newton’s Method
Extrapolated Successive Approximation (ESA) Method
Accelerated Multi-variable Newton’s (AMVN) Method
Fixed Point Accelerated Multi-variable Newton’s (AMVN) Method
Numerical Finding of Complex Roots
Exercise
Numerical Solution of Ordinary Differential Equations
Introduction
Taylor Series Method
Picard’s Method
Euler’s Method
Modified Euler’s Method
Runge-Kutta Method of Order Two
Runge-Kutta Method of Order Four
Runge-Kutta Method for Simultaneous Differential Equations
Predictor-Corrector Methods
Exercise
Iterative Solution of Partial Differential Equations
Introduction
Finite-Difference Approximations to Partial Derivatives
Condition for the Negativeness of the Eigenvalues of the Jacobian Matrix
An Upper Bound for the Spectral Radius of the Jacobi Matrix
Convergence of the Modified Jacobi Method
Numerical Solution of Laplace Equation
Refinement of Jacobi Method for the Solution of Laplace Equation
Refinement of Gauss-Seidel Method for the Solution of Laplace Equation
Refinement of SOR Method for the Solution of Laplace Equation
Alternating Direction Implicit Iteration
Bounds for the Eigenvalues of H, V, D–1 H, D–1V
Parabolic Equations
Crank-Nicolson Method
Hyperbolic Equations
Exercise
Index