Computational Methods
ES-201
Review of Taylor Series, Rolle's Theorem and Mean Value Theorem, Approximations and Errors in numerical computations, Data representation and computer arithmetic, Loss of significance in computation Location, Location of roots of equation - Bisection method (convergence analysis and implementation), Newton Method(convergence analysis and implementation), Secant Method (convergence analysis and implementation), Unconstrained one variable function minimization by Fibonacci search, Golden Section Search and Newton’s method, Multivariate function minimization by the method of steepest descent, Nelder- Mead Algorithm
Interpolation: Assumptions for interpolation, errors in polynomial interpolation, Finite differences, Gregory-Newton’s Forward Interpolation,Gregory-Newton’s backward Interpolation, Lagrange’s Interpolation,Newton’s divided difference interpolation, Numerical Integration: Definite Integral, Newton-Cote’s Quadrature formula, Trapezoidal Rule, Simpson’s one-third rule, Simpson’s three-eight rule, Errors in quadrature formulae, Romberg’s Algorithm, Gaussian Quadrature formula
System of Linear Algebraic Equations: Existence of solution, Gauss elimination method and its computational effort, concept of Pivoting, Gauss Jordan method and its computational effort, Triangular Matrix factorization methods: Dolittle algorithm, Crout’s Algorithm, Cholesky method, Eigen value problem: Power method, Approximation by Spline Function: First-Degree and second degree Splines, Natural Cubic Splines, B Splines, Interpolation and Approximation
Numerical solution of ordinary Differential Equations: Picard’s method, Taylor series method, Euler’s and Runge-Kutta’s methods, Predictor-corrector methods: Euler’s method, Adams-Bashforth method, Milne’s method, Numerical Solution of Partial Differential equations: Parabolic, Hyperbolic, and elliptic equations