### Lecture 1: Introduction and Overview

##### Lecture Details :

Advanced Numerical Analysis by Prof. Sachin C. Patwardhan,Department of Chemical Engineering,IIT Bombay.For more details on NPTEL visit http://nptel.ac.in

##### Course Description :

Introduction and Overview - Fundamentals of Vector Spaces - Basic Dimension and Sub-space of a Vector Space - Introduction to Normed Vector Spaces - Examples of Norms,Cauchy Sequence and Convergence, Introduction to Banach Spaces - Introduction to Inner Product Spaces - Cauchy Schwaz Inequality and Orthogonal Sets - Gram-Schmidt Process and Generation of Orthogonal Sets - Problem Discretization Using Appropriation Theory - Weierstrass Theorem and Polynomial Approximation - Taylor Series Approximation and Newton's Method - Solving ODE - BVPs Using Firute Difference Method - Solving ODE - BVPs and PDEs Using Finite Difference Method - Finite Difference Method (contd.) and Polynomial Interpolations - Polynomial and Function Interpolations,Orthogonal Collocations Method for Solving - Orthogonal Collocations Method for Solving ODE - BVPs and PDEs - Least Square Approximations :Necessary and Sufficient ConditionsLinear Least Square Estimation and Geometric Interpretation - Geometric Interpretation of the Least Square Solution (Contd.) and Projection - Projection Theorem in a Hilbert Spaces (Contd.) and Approximation - Discretization of ODE-BVP using Least Square Approximation - Discretization of ODE-BVP using Least Square Approximation and Gelarkin Method - Model Parameter Estimation using Gauss-Newton Method - Solving Linear Algebraic Equations and Methods of Sparse Linear Systems

Methods of Sparse Linear Systems (Contd.) and Iterative Methods for Solving - Iterative Methods for Solving Linear Algebraic Equations: Convergence - Iterative Methods for Solving Linear Algebraic Equations: Convergence Analysis - Optimization Based Methods for Solving Linear Algebraic Equations: Gradient Method - Conjugate Gradient Method, Matrix Conditioning and Solutions - Matrix Conditioning and Solutions and Linear Algebraic Equations and Solving Nonlinear Algebraic Equations - Solving Nonlinear Algebraic Equations: Wegstein Method and Variants of Newton's Method - Solving Nonlinear Algebraic Equations: Optimization Based Methods - Solving Nonlinear Algebraic Equations: Introduction to Convergence analysis - Solving Ordinary Differential Equations - Initial Value Problems (ODE-IVPs) - Solving Ordinary Differential Equations - Initial Value Problems - Solving ODE-IVPs : Runge Kutta Methods (contd.) and Multi-step Methods - Solving ODE-IVPs : Generalized Formulation of Multi-step Methods - Solving ODE-IVPs : Multi-step Methods (contd.) and Orthogonal Collocations Method - Solving ODE-IVPs: Selection of Integration Interval and Convergence Analysis - Solving ODE-IVPs: Convergence Analysis of Solution Schemes - Methods for Solving System of Differential Algebraic Equations

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