Advanced Numerical Analysis


Lecture 1 - Introduction and Overview


Lecture 2 - Fundamentals of Vector Spaces


Lecture 3 - Basic Dimension and Sub-space of a Vector Space


Lecture 4 - Introduction to Normed Vector Spaces


Lecture 5 - Examples of Norms,Cauchy Sequence and Convergence, Introduction to Banach Spaces


Lecture 6 - Introduction to Inner Product Spaces


Lecture 7 - Cauchy Schwaz Inequality and Orthogonal Sets


Lecture 8 - Gram-Schmidt Process and Generation of Orthogonal Sets


Lecture 9 - Problem Discretization Using Appropriation Theory


Lecture 10 - Weierstrass Theorem and Polynomial Approximation


Lecture 11 - Taylor Series Approximation and Newton's Method


Lecture 12 - Solving ODE - BVPs Using Firute Difference Method


Lecture 13 - Solving ODE - BVPs and PDEs Using Finite Difference Method


Lecture 14 - Finite Difference Method (Continued...) and Polynomial Interpolations


Lecture 15 - Polynomial and Function Interpolations,Orthogonal Collocations Method for Solving ODE -BVPs


Lecture 16 - Orthogonal Collocations Method for Solving ODE - BVPs and PDEs


Lecture 17 - Least Square Approximations, Necessary and Sufficient Conditions for Unconstrained Optimization


Lecture 18 - Least Square Approximations -Necessary and Sufficient Conditions for Unconstrained Optimization Least Square Approximations ( Continued....)


Lecture 19 - Linear Least Square Estimation and Geometric Interpretation of the Least Square Solution


Lecture 20 - Geometric Interpretation of the Least Square Solution (Continued...) and Projection Theorem in a Hilbert Spaces


Lecture 21 - Projection Theorem in a Hilbert Spaces (Continued...) and Approximation Using Orthogonal Basis


Lecture 22 - Discretization of ODE-BVP using Least Square Approximation


Lecture 23 - Discretization of ODE-BVP using Least Square Approximation and Gelarkin Method


Lecture 24 - Model Parameter Estimation using Gauss-Newton Method


Lecture 25 - Solving Linear Algebraic Equations and Methods of Sparse Linear Systems


Lecture 26 - Methods of Sparse Linear Systems (Continued...) and Iterative Methods for Solving Linear Algebraic Equations


Lecture 27 - Iterative Methods for Solving Linear Algebraic Equations


Lecture 28 - Iterative Methods for Solving Linear Algebraic Equations: Convergence Analysis using Eigenvalues


Lecture 29 - Iterative Methods for Solving Linear Algebraic Equations: Convergence Analysis using Matrix Norms


Lecture 30 - Iterative Methods for Solving Linear Algebraic Equations: Convergence Analysis using Matrix Norms (Continued...)


Lecture 31 - Iterative Methods for Solving Linear Algebraic Equations: Convergence Analysis (Continued...)


Lecture 32 - Optimization Based Methods for Solving Linear Algebraic Equations: Gradient Method


Lecture 33 - Conjugate Gradient Method, Matrix Conditioning and Solutions of Linear Algebraic Equations


Lecture 34 - Matrix Conditioning and Solutions and Linear Algebraic Equations (Continued...)


Lecture 35 - Matrix Conditioning (Continued...) and Solving Nonlinear Algebraic Equations


Lecture 36 - Solving Nonlinear Algebraic Equations: Wegstein Method and Variants of Newton's Method


Lecture 37 - Solving Nonlinear Algebraic Equations: Optimization Based Methods


Lecture 38 - Solving Nonlinear Algebraic Equations: Introduction to Convergence analysis of Iterative Solution Techniques


Lecture 39 - Solving Nonlinear Algebraic Equations: Introduction to Convergence analysis (Continued...) and Solving ODE-IVPs


Lecture 40 - Solving Ordinary Differential Equations - Initial Value Problems (ODE-IVPs) : Basic Concepts


Lecture 41 - Solving Ordinary Differential Equations - Initial Value Problems (ODE-IVPs) : Runge Kutta Methods


Lecture 42 - Solving ODE-IVPs : Runge Kutta Methods (Continued...) and Multi-step Methods


Lecture 43 - Solving ODE-IVPs : Generalized Formulation of Multi-step Methods


Lecture 44 - Solving ODE-IVPs : Multi-step Methods (Continued...) and Orthogonal Collocations Method


Lecture 45 - Solving ODE-IVPs: Selection of Integration Interval and Convergence Analysis of Solution Schemes


Lecture 46 - Solving ODE-IVPs: Convergence Analysis of Solution Schemes (Continued...)


Lecture 47 - Solving ODE-IVPs: Convergence Analysis of Solution Schemes (Continued...) and Solving ODE-BVP using Single Shooting Method


Lecture 48 - Methods for Solving System of Differential Algebraic Equations


Lecture 49 - Methods for Solving System of Differential Algebraic Equations (Continued...) and Concluding Remarks