Lecture plan
Will be updated throughout the semester.
| Week | Themes | Curriculum | Relevant material |
|---|---|---|---|
| 2 | General introduction, Mathematical preliminaries, Taylor series and theorem with application, O(h) notation, Significant digits of precision, Computer arithmetic, Floating point numbers, Chopping and rounding off errors, Absolute and relative errors, Nested multiplication algorithm, Mean Value theorem,Loss of significance. | Ch. 1.1,1.2,1.3, 1.4(some part) | |
| 3 | Linear systems: Gaussian Elimination Method (algorithm with application), Disadvantages of GE, Consistent and Inconsistent Linear Systems with Geometrical Interpretation, Residual and Error Vectors. Introduction to Matlab. | Ch. 2.1. | Introduction to MATLAB |
| 4 | Gaussian Elimination with partial and full pivoting, Scaled partial pivoting, Tri-diagonal and Banded systems, GE for tri-diagonal linear systems. Vector and Matrix norms with examples, Condition number. | Ch. 2.2 and 2.3 | |
| 5 | Factorizations: LU, LDLT, Choleskey with uniqueness theorem. Eigenvalues and Gerschorin Theorem | Ch. 8.1. and 8.2 | |
| 6 | Numerical solution of non-linear equations: Bisection method, Regula falsi method, Secant method. Introduction to Fixed point iteration. | Ch3.1 and 3.3. | |
| 7 | Numerical solution of non-linear equations: Fixed point iteration method. Contraction mapping theorem with application | Ch3.3. and Notes on Fixed Point | |
| 8 | Numerical solution of non-linear equations: Newtons method | Ch3.2. | |
| 9 | Power method for numerical computations of eigenvalues and eigenvectors | Ch8.3. | |
| 10 | Polynomial Interpolation: Lagrange interpolation, Newton´s form of interpolating polynomials | Ch4.1. | |
| 11 | Errors in Polynomial Interpolation. Estimating derivatives and Richardson Extrapolation | Ch4.2. and 4.3. | |
| 12 | Numerical Integration: Trapezoid Method. Romberg Method. Simpsons Rule and Newton Cotes Rule | Ch5.1., 5.2. and 5.3. | |
| 13 | Numerical Integration: Gaussian quadrature rules. Linear splines | Ch5.4. and 6.1. | |
| 14 | No lecture - Project work | ||
| 15 | No lecture - Easter | ||
| 16 | Cubic splines. Introduction to Ordinary Differential Equations (ODEs) | 6.2. and 7.1 | |
| 17 | Initial Value Problems: Eulers method. Error analysis. Implicit Eulers Method | 7.2, 7.3 and 7.4 | |
| 18 | Stiff ODEs, Least squares, linear regression, and normal equations. | 7.5 and 9.1 |