1. Introduction to Numerical Analysis and Overview
2. Fundamentals of Vector Spaces
3. Basic Dimension and Sub-space of a Vector Space
4. Introduction to Normed Vector Spaces
5. Examples of Norms, Cauchy Sequence and Convergence, Introduction to Banach Spaces
6. Introduction to Inner Product Spaces
7. Cauchy Schwaz Inequality and Orthogonal Sets
8. Gram-Schmidt Process and Generation of Orthogonal Sets
9. Problem Discretization Using Appropriation Theory
10. Weierstrass Theorem and Polynomial Approximation
11. Taylor Series Approximation and Newton's Method
12. Solving ODE - BVPs Using Firute Difference Method
13. Solving ODE - BVPs and PDEs Using Finite Difference Method
14. Finite Difference Method (contd.) and Polynomial Interpolations
15. Polynomial and Function Interpolations, Orthogonal Collocations Method for Solving
16. Orthogonal Collocations Method for Solving ODE - BVPs and PDEs
17. Least Square Approximations, Necessary and Sufficient Conditions
18. Least Square Approximations: Necessary and Sufficient Conditions
19. Linear Least Square Estimation and Geometric Interpretation
20. Geometric Interpretation of the Least Square Solution (Contd.) and Projection
21. Projection Theorem in a Hilbert Spaces (Contd.) and Approximation
22. Discretization of ODE-BVP using Least Square Approximation
23. Discretization of ODE-BVP using Least Square Approximation and Gelarkin Method
24. Model Parameter Estimation using Gauss-Newton Method
25. Solving Linear Algebraic Equations and Methods of Sparse Linear Systems
26. Methods of Sparse Linear Systems (Contd.) and Iterative Methods for Solving
27. Iterative Methods for Solving Linear Algebraic Equations
28. Iterative Methods for Solving Linear Algebraic Equations: Convergence Analysis
29. Iterative Methods for Solving Linear Algebraic Equations:
30. Iterative Methods for Solving Linear Algebraic Equations: Convergence
31. Iterative Methods for Solving Linear Algebraic Equations: Convergence Analysis
32. Optimization Based Methods for Solving Linear Algebraic Equations: Gradient Method
33. Conjugate Gradient Method, Matrix Conditioning and Solutions
34. Matrix Conditioning and Solutions and Linear Algebraic Equations (Contd.)
35. Matrix Conditioning (Contd.) and Solving Nonlinear Algebraic Equations
36. Solving Nonlinear Algebraic Equations: Wegstein Method and Variants of Newton's Method
37. Solving Nonlinear Algebraic Equations: Optimization Based Methods
38. Solving Nonlinear Algebraic Equations: Introduction to Convergence analysis
39. Solving Nonlinear Algebraic Equations: Introduction to Convergence analysis (Contd.)
40. Solving Ordinary Differential Equations - Initial Value Problems (ODE-IVPs)
41. Solving Ordinary Differential Equations - Initial Value Problems
42. Solving ODE-IVPs : Runge Kutta Methods (contd.) and Multi-step Methods
43. Solving ODE-IVPs : Generalized Formulation of Multi-step Methods
44. Solving ODE-IVPs : Multi-step Methods (contd.) and Orthogonal Collocations Method
45. Solving ODE-IVPs: Selection of Integration Interval and Convergence Analysis
46. Solving ODE-IVPs: Convergence Analysis of Solution Schemes (contd.)
47. Solving ODE-IVPs: Convergence Analysis of Solution Schemes (contd.)
48. Methods for Solving System of Differential Algebraic Equations
49. Methods for Solving System of Differential Algebraic Equations

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