Numerical Methods

Solution of Equations and Eigenvalue problems Solution of equation - Fixed point iteration : x = g(x) method - Newton's method - Solution of linear system by Gaussian elimination and Gauss Jordon methods - Iterative methods - Gauss Seidel methods - Inverse of matrix by Gauss Jordon method - Eigen value of a matrix by power method and by Jacobi method for symmetric matrix. Iterpolation and Approximation Lagrangian polynomials - Divided differences - Interpolating with a cubic spline - Newton's forward and backward difference formulas. Numerical Differentiation and Integration Differentiation using interpolation formulae - Numerical integration by trapezoidal and Simpson's 1/3 and 3/8 rules - Romberg's method - Two and Three point Gaussian quadrature formulas - Double integrals using trapezoidal and Simpson's rules. Initial Value Problems for Ordinary Differential Equations Single step methods : Taylor series method - Euler's methods for first order, Runge - Kutta method for solving first and second order equations - Multistep methods : Milne's and Adam's predictor and corrector methods. Boundary Value Problems in Ordinary and Partial Differential Equations Finite difference solution of second order ordinary differential equation - Finite difference solution of one dimensional heat equation by explicit and implicit methods - One dimensional wave equation and two dimensional Laplace and Poisson equations.