Advanced Numerical Analysis

SYLLABUS- ADVANCED NUMERICAL ANALYSIS,
Unit I
Errors in Computation- Floating Point Representation of Numbers, Significant Digits, Rounding and Chopping a Number and Error due to these, Absolute and Relative Errors, Computation of Errors using Differentials, Errors in Evaluation of Some Standard Functions, Truncation Error.
Linear Equations-Gauss Elimination Method, LU Decomposition Method, Gauss-Jordan Method, Tridiagonal System, Inversion of Matrix, Gauss-Jacobi Method, Gauss-Seidal iterative methods and their convergence Method.
Unit II
Non-linear Equations-Iterative method, Secant method, Rate of convergence of Regula- Falsi method, Newton-Raphson method, Convergence of Newton-Raphson method for simple and multiple roots, Birge-Vieta method, Bairstow’s method and Graffe’s root squaring method for polynomial equations.
Unit III
Numerical differentiation- Differentiation methods based on Newton’s forward and backward formulae, Differentiation by central difference formula.
Numerical integration: Methodology of numerical integration, Rectangular rule, Trapezoidal rule, Simpson’s 1/3rd and 3/8th rules, Romberg Integration, Gauss-Legendre quadrature formula.
Unit IV
Algebraic Eigen Values and Eigen Vectors: Power method, jacobi’s method, Given’s method, Householder’s method Approximation: Least square polynomial approximation, polynomial approximation using orthogonal polynomials, Approximation with algebraic and trigonometric functions.
Unit V
Ordinary Differential Equations- Initial and boundary value problems, Solutions of Initial Value Problems, Single and multistep methods, Picard’s method, Taylor series method, Euler’s and Modified Euler’s methods, Runge-Kutta second order and fourth.