# InfoCoBuild

## Numerical Methods and Computation

Numerical Methods and Computation. Instructor: Prof. S. R. K. Iyengar, Department of Mathematics, IIT Delhi. This course derives and analyzes numerical methods for the solution of various problems. The course discusses the numerical solution of nonlinear system of algebraic equations. It will construct methods for finding the roots or zeros of a transcendental or a polynomial equation in one variable. Then it will extend the methods for the solution of a system of nonlinear equations. The course will consider a data or a table of values and construct the polynomial that fits this data exactly. This polynomial can be used for interpolating or predicting the value of the function, represent the data at any intermediate point. This polynomial may also be used for various other operations like differentiation and integration. In approximation the course will deal with approximation to a continuous function or to a function which represents the given data. Finally, the course will use the interpolating polynomial of a given data to find the derivative of a function, and construct methods to numerically integrate a given function or to integrate a function that represents a given data. (from nptel.ac.in)

 Errors in Computation and Numerical Instability

 Lecture 01 - Errors in Computation and Numerical Instability Solution of Nonlinear Algebraic Equations Lecture 02 - Methods for Finding the Root: Bisection Method Lecture 03 - Regula Falsi Method, Newton-Raphson Method, Chebyshev Method Lecture 04 - Muller's Method, Multipoint Iteration Methods, Convergence of the Secant Method Lecture 05 - Rate of Convergence of the Secant Method, General Iteration Method Lecture 06 - Finding the Multiple Roots of Nonlinear Equations: Newton-Raphson Method Lecture 07 - Polynomial Equations: Sturm Sequences, Sturm's Theorem Lecture 08 - Polynomial Equations: Birge Vieta Method Lecture 09 - Polynomial Equations: Bairstow Method Lecture 10 - Polynomial Equations: Graeffe's Root Squaring Method Solution of a System of Linear Algebraic Equations Lecture 11 - Introduction to Solution of a System of Linear Algebraic Equations Lecture 12 - Direct Methods: Gauss Elimination Method Lecture 13 - Gauss Elimination Method with Partial Pivoting, Gauss-Jordan Method, Triangularisation Method Lecture 14 - LU Decomposition Lecture 15 - Cholesky Method (Square Root Method), Partition Method Lecture 16 - Examples of Finding the Inverse Matrix using Partition Method, Iteration Methods Lecture 17 - Iteration Methods: Jacobi Method, Gauss-Seidel Method Lecture 18 - Generalization of the Gauss-Seidel Method, Norm of a Matrix Lecture 19 - Matrix Norms, Convergence of Iterative Methods, Optimal Relaxation Factor Lecture 20 - Determining Optimal Relaxation Factor, Eigenvalue Problems, Gershgorin Circles Lecture 21 - Finding the Eigenvalues and the Corresponding Eigenvectors: Jacobi Method Lecture 22 - Finding the Eigenvalues and the Corresponding Eigenvectors: Givens Method Lecture 23 - Finding the Eigenvalues of an Arbitrary Matrix: Rutishauser Method Lecture 24 - Eigenvalue Problems: Power Method, Convergence of the Power Method Interpolation and Approximation Lecture 25 - Introduction to Interpolation and Approximation Lecture 26 - Lagrange Interpolating Polynomial, Error of Interpolation Lecture 27 - Error of Interpolation (cont.), Divided Differences Lecture 28 - Newton's Divided Difference Interpolating Polynomial, Forward/Backward Differences Lecture 29 - Newton's Forward/Backward Difference Formula Lecture 30 - Hermite Interpolating Polynomial, Definition of Approximation, Error Norms Lecture 31 - Least Squares Approximation Lecture 32 - Quadratic Approximation, Uniform Approximation Lecture 33 - Chebyshev Polynomial Approximation Numerical Differentiation Lecture 34 - Introduction to Numerical Differentiation Lecture 35 - Uniform Mesh Spacing, Difference Operators, Method of Undetermined Coefficients Lecture 36 - Effect of Round-off Errors in Numerical Differentiation Numerical Integration Lecture 37 - Numerical Integration: Newton-Cotes Formula Lecture 38 - Trapezoidal Rule, Simpson's Rule Lecture 39 - Gauss-Legendre 2-point and 3-point Formula Lecture 40 - Gauss-Chebyshev Formula Lecture 41 - Gauss-Hermite Quadrature Rules

 References Numerical Methods and Computation Instructor: Prof. S. R. K. Iyengar, Department of Mathematics, IIT Delhi. This course derives and analyzes numerical methods for the solution of various problems.