KUK - LMS

Syllabus

MODULE NO / UNIT	COURSE SYLLABUS CONTENTS OF MODULE	NOTES
Course Code: MMATH21-308 Course Name: Elective - II ) Advanced Numerical Analysis
1	Error Analysis: Errors, Absolute, relative and percentage errors; Significant digits and numerical instability, Propagation of errors in arithmetic operations, Significant errors, Representation of numbers in computer, Normalized floating point representation and its effects. Solution of Polynomial and Transcendental Equations: Iteration methods; First order, second order and higher order methods, Accelaration of the convergence, Efficiency of a method, Newton-Raphson method for multiple roots, Modified Newton-Raphson method, Muller method and Chebyshev method, Birge-Vieta method, Bairstow method, Graeffe’s root squaring method, Solutions of systems of non-linear equations.
2	Systems of Linear Equations: Matrix inverse methods, Triangularization method, Cholesky Method, Matrix partition method, Operation count, Ill-conditioned linear systems, Moore-Penrose inverse method, Least square solutions for inconsistent systems. Iteration methods Successive over relaxation (SOR) method, Convergence analysis. Eigen values and eigen vectors, bounds on eigen values, Given’s method, Rutishauser method, Householder’s method for symmetric matrices, Power method. Numerical Differentiation based on difference formulae, Richardson’s extrapolation method, Cubic spline method, Method of undetermined coefficients.
3	Numerical Integration: Weddle’s rule, Newton-Cotes method, Gauss-Legendre, Gauss-Chebyshev, Gauss-Laguerre, and Gauss-Hermite integration methods. Composite integration method, Euler-Maclaurin’s formula, Romberg Integration, Double integration. Numerical Solution of Ordinary Differential Equations: Estimation of local truncation error of Euler and single step methods. Bounds of local truncation error and convergence analysis of multistep methods, Predictor-Corrector methods; Adams-Bashforth methods, Adams-Moulton formula, Milne-Simpson method, System of Differential Equations. Finite difference method for solving second order IVPs and BVPs, Shooting method for boundary value problems.
4	Solving Partial Differential Equations: Finite difference approximations to partial derivatives, solving parabolic equations using implicit and explicit formulae, C-N scheme and ADI methods; solving elliptic equations using Gauss-elimination, Gauss-Seidel method, SOR method, and ADI method , solving hyperbolic equations using method of characteristics, explicit and implicit methods, Lax-Wendroff’s method.