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, Introduction to Matlab | Ch. 1.1,1.2,1.3, 1.4(some part) | General-Introduction, Mathematical-Preliminaries |
| 3 | Linear systems: Gaussian Elimination Method (algorithm with application), Disadvantages of GE, Consistent and Inconsistent Linear Systems with Geometrical Interpretation, Augmented Matrix, Residual and Error Vectors, Gaussian Elimination with partial and full pivoting, Scaled partial pivoting, Tri-diagonal and Banded systems, GE for tri-diagonal linear systems. | Ch. 2.1. 2.2, 2.3 (some part) | Introduction to MATLAB |
| 4 | Review of linear algebra: Vector and Matrix norms with examples, Condition number, Well-conditioning and Ill-conditioning with Hilbert matrix demonstration using MATLAB. Factorizations: LU, LDLT, Choleskey with uniqueness theorem, Doolittle and Crouts | Ch. 8.1. 8.4(some part), | myhilb.m MATLAB file |
| 5 | Numerical solution of non-linear equations: Bisection method,Regula falsi method, Fixed point/Simple iteration method, Contraction mapping theorem with application | Ch. 3.1, 3.2 | Fixed point iteration |
| 6 | Numerical solution of non-linear equations:Fixed point method, Newton Raphson method with geometrical interpretation and order of convergence, Secant method with geometrical interpretation(Order of Convergence, selfstudy), Bi-variate and multivariate taylor´s expansions, Newtons method for system of non-linear systems | Ch. 3.2, 3.3 and Ch. 1,4 (BOOK: Numerical Analysis by Endre Suli and David Mayers) | Ch1,Ch4,notes on Fixed Point |
| 7 | Application of Newtons method for system of non-linear equations with MATLAB demonstration, Simultaneous iterative scheme with existence and uniqueness theorem, Eigenvalues and Eigenvector, Power method with ratios, Aitken acceleration | Ch. 8.2, 8.3 | mynewton.m |
| 8 | Modified Power method with application, Iterative methods for solving linear systems: Rishadrson,Jacobi, Guass Seidal, SOR method, a short discussion on their convergence properties. | Ch. 8.4 | |
| 9 | Polynomial Interpolation, Lagrang interpolation, Newton´s form of interpolating polynomail | Ch. 4.1 | |
| 10 | Interpolation error, First interpolation error theorem with example, Chebyshev nodes, Neville´s algorithm, Estimation of derivatives | Ch. 4.2, 4.3 | |
| 11 | Estimation of derivatives, Forward, backward and central difference, Richardson extrapolation,Porject | Ch. 4.2, 4.3 | |
| 12 | Easter Holidays | ||
| 13 | Richardson extrapolation (general form) with theorem and example. Numerical Integration: Trapezoid rule (for both cases) with error term,Composite Trapezoid rule, Simpson rule, | Ch. 4.3, 5.1 | |
| 14 | Numerical integration: Adaptive Simpsons rule, Newton-Cotes rules, approximation errors and convergence rates, Guassian Quadrature rules with Theorem and Applications,Weighted Gaussian quadrature rules | Ch. 5.1, 5.2, 5.3. | |
| 15 | Legendre polynomials, singularities,Splines: Linear splines, natural cubic splines, Solving differential equations: Direct and Numerical methods, Initial value and boundary value problems. Euler´s method with convergence and example. | Ch. 5.4, 6.1, 6.2. (Not specific material on quadratic splines). 7.1 | |
| 16 | Improved, modified and Implicit Euler methods, Error analysis, Taylor series f(x,y), RK methods (2nd and fourth order), RK-Fehlberg method, | Ch. 7.2,7.3, 7.4 | |
| 17 | Initial value problems: Multi-step methods, implicit methods and stiff ODEs. Least squares, linear regression, and normal equations. | Ch. 7.5,9.1 |