Numerical Methods: Course Outline (MAT 408)
Number Systems and Errors; Errors, Error Estimation, Floating point arithmetic, Loss of significance and error propagation, Solution of non-linear Equations; Bisection method, Fixed point iteration, Convergence criterion for a fixed point iteration, Newton-Raphson method, Iterative methods, Secant and Regula Falsi methods, Order of convergence of Newton-Raphson and secant methods. Interpolation by Polynomials; Interpolation with equally spaced data, Newton’s forward and backward difference formulas, Hermite Polynomials, Splines, Cubic Splines. Error of the interpolating polynomial, Bessel's interpolation formula. Existence and uniqueness of the interpolating polynomial. Lagrangian interpolation, the Newton divided difference formula, System of Linear Equations; Direct Methods: Gauss elimination methods, Triangular factorization, Crout method. Iterative methods: Jacobi method, Gauss-Seidel method, SOR method, convergence of iterative methods, Numerical Differentiation and Integration; Numerical differentiation formulae based on interpolation polynomials, error estimates, Newton-Cotes formulae; Trapezoidal rule, Simpson’s formulas, Composite rules, Romberg improvement, Richardson extrapolation. Error estimation of integration formulas, Gaussian quadrature. Differentiation and integration in multidimesion, Numerical Solution of ODE’S; The Taylor series method, Euler Method, Modified Euler Method, RK-Methods upto order 4, Predictor –Corrector Methods, Eigenvalue problem; Single Dominant Eigenvalue, Power Method, Rayleigh Quotient Method.