Numerical Methods and Numerical Analysis - Math-611

Direct solution of linear equations: Elimination and Factorization method, Ill-conditioning,, Iterative refinement. Orthogonal Factorizations: Jacobi method, Gauss-seidel method, SOR method, Conjugate Gradients method, Pre-conditioning, Chebyshev semi-iteration method. Matrix Eigenvalue Problems: Power method,  Inverse Power method, and Shifted Inverse Power method. Jacobi iterative method, Given's method, and Householder method. Sturm Sequence and QR method, Singular value decomposition.
Numerical Methods for Solving Nonlinear Equations: Bisection method, fixed point method, Newton’s method, secant method, multiple roots, modified Newton’s method, rate of convergence (error analysis), Newton's method for solving nonlinear systems. Solving Systems of Linear Equations: Gaussian elimination, Gaussian elimination with partial pivoting, LU-decomposition, Jacobi method, Gauss-Seidel method. Interpolation and Polynomial Approximations Lagrange interpolation formula, Newton's interpolation formula, interpolation using linear splines. Numerical Differentiation and Integration: Trapezoidal, and Simpson's rules. Numerical solutions of Ordinary Differential equations; Taylor methods and Runge-Kutta method of order two.