Numerical Analysis
Numerical Analysis. Instructor: Prof. R. Usha, Department of Mathematics, IIT Madras. This course on NUMERICAL ANALYSIS introduces the theory and application of numerical methods or techniques to approximate mathematical procedures (such as reconstruction of a function, evaluation of an integral) or solutions of problems that arise in science and engineering. Such approximations are needed because the analytical methods are either intractable or the problem under consideration can not be solved analytically. Explanations for why and how these approximation techniques work are provided with emphasis on accuracy and efficiency of the developed methods. The course also provides a firm foundation for further study on Numerical Analysis.
(from nptel.ac.in)
Lecture 01  Introduction 
Mathematical Preliminaries, Polynomial Interpolation 
Lecture 02  Mathematical Preliminaries, Polynomial Interpolation 
Lecture 03  Polynomial Interpolation (cont.) 
Lecture 04  Polynomial Interpolation (cont.) 
Lecture 05  Lagrange Interpolation Polynomial, Error in Interpolation 
Lecture 06  Error in Interpolation 
Lecture 07  Divided Difference Interpolation Polynomial 
Lecture 08  Properties of Divided Differences, Introduction to Inverse Interpolation 
Lecture 09  Inverse Interpolation, Remarks on Polynomial Interpolation 
Numerical Differentiation 
Lecture 10  Taylor Series Method, Method of Undetermined Coefficients 
Lecture 11  Polynomial Interpolation Method 
Lecture 12  Operator Method, Numerical Integration 
Numerical Integration 
Lecture 13  Numerical Integration: Error in Trapezoidal Rule, Simpson's Rule 
Lecture 14  Error in Simpson's Rule, Composite Trapezoidal Rule Error 
Lecture 15  Composite Simpson's Rule, Error Method of Undetermined Coefficient 
Lecture 16  Gaussian Quadrature (Two Point Method) 
Lecture 17  Gaussian Quadrature (Three Point Method), Adaptive Quadrature 
Numerical Solution of Ordinary Differential Equations 
Lecture 18  Numerical Solution of Ordinary Differential Equations 
Lecture 19  Stability, Single Step Methods, Taylor Series Method 
Lecture 20  Examples for Taylor Series Method, Euler's Method 
Lecture 21  RungeKutta Methods 
Lecture 22  Example for RKmethod of Order 2, Modified Euler's Method 
Lecture 23  PredictorCorrector Methods (AdamMoulton) 
Lecture 24  PredictorCorrector Methods (Milne) 
Lecture 25  Linear Boundary Value Problems 
Lecture 26  Boundary Value Problems: FiniteDifference Methods 
Lecture 27  Boundary Value Problems: Shooting Methods 
Lecture 28  Boundary Value Problems: Shooting Methods (cont.) 
Root Finding Methods 
Lecture 29  Root Finding Methods: The Bisection Method 
Lecture 30  The Bisection Method (cont.) 
Lecture 31  NewtonRaphson Method 
Lecture 32  NewtonRaphson Method (cont.) 
Lecture 33  Secant Method, Method of False Position 
Lecture 34  Fixed Point Methods 
Lecture 35  Fixed Point Methods (cont.) 
Lecture 36  Fixed Point Iteration Methods 
Lecture 37  Practice Problems 
Solution of Linear Systems of Equations 
Lecture 38  Solution of Linear Systems of Equations: Decomposition Methods 
Lecture 39  Decomposition Methods (cont.) 
Lecture 40  Gauss Elimination Method 
Lecture 41  Gauss Elimination Method with Partial Pivoting 
Lecture 42  GaussJordan Method 
Lecture 43  Solution of Linear Systems of Equations: Error Analysis 
Lecture 44  Error Analysis (cont.) 
Lecture 45  Iterative Improvement Method, Iterative Methods 
Lecture 46  Iterative Methods, Matrix Eigenvalue Problems, Power Method 
Lecture 47  Power Method, Gerschgorin's Theorem, Brauer's Theorem 
Lecture 48  Practical Problems 
References 
Numerical Analysis
Instructor: Prof. R. Usha, Department of Mathematics, IIT Madras. This course on NUMERICAL ANALYSIS introduces the theory and application of numerical methods or techniques to approximate mathematical procedures or solutions of problems that arise in science and engineering.
