Lecture Description
- The CosmoLearning Team
Course Index
- Introduction to Numerical Analysis 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 Conditions
- Least Square Approximations: Necessary and Sufficient Conditions
- Linear 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
- Iterative Methods for Solving Linear Algebraic Equations: Convergence Analysis
- Iterative Methods for Solving Linear Algebraic Equations:
- 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 (Contd.)
- Matrix Conditioning (Contd.) 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 Nonlinear Algebraic Equations: Introduction to Convergence analysis (Contd.)
- 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 (contd.)
- Solving ODE-IVPs: Convergence Analysis of Solution Schemes (contd.)
- Methods for Solving System of Differential Algebraic Equations
- Methods for Solving System of Differential Algebraic Equations
Course Description
This is an advanced course on Numerical Analysis by Prof. Sachin C. Patwardhan, Department of Chemical Engineering, IIT Bombay. It has been designed with the following learning objectives in mind:
- Clearly bring out role of approximation theory in the process of developing a numerical recipe for solving an engineering problem
- Introduce geometric ideas associated with the development of numerical schemes
- Familiarize the student with ideas of convergence analysis of numerical methods and other analytical aspects associated with numerical computation
It is shown that majority of problems can be converted to computable forms (discretized) using three fundamental ideas in the approximation theory, namely Taylor series expansion, polynomial interpolation and least square approximation. In addition, the student is expected to clearly understand role of the following four fundamental tools:
- Linear Algebraic Equation
- Nonlinear Algebraic Equations
- Ordinary Differential Equations- Initial Value Problem
- Optimization