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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)

Lecture 39 - Gauss-Legendre 2-point and 3-point Formula


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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