Mathematics

Syllabus 2023-2024

• Basics on Scientific Computing
• Vector spaces, vector norms, matrices, and matrix norms
• Basic linear algebra: direct solution of linear systems
• Not so basic linear algebra: iterative solution of linear systems
• Polynomial interpolation
• Interpolatory Quadrature rules
• L2 projection / Least square approximation
• Introduction to Finite Difference Methods
• Introduction to Finite Element Methods

Python laboratories

• Iterations, absolute and relative errors
• Numpy, Scipy, Vectors, Matrices, and their norms
• Implementation of Gauss elimination, comparison with scipy
• Implementation of Richardson, gradient, and conjugate gradient, comparison with scipy
• Using numpy for polynomial approximation
• Using numpy for numerical integration
• Putting things together: mass matrices, least square matrices, L2 projection
• Solving ODEs and a simple PDE in 1d and 2d with finite differences
• Solving simple PDEs finite elements

Refrences and books

Scientific Computing & Numerical Analysis 

• G. Dahlquist & A. Bjorck, Numerical Methods in Scientific Computing. SIAM, 2008. 
• A. Quarteroni, R. Sacco, and F. Saleri. Numerical mathematics, Springer-Verlag, 2000. 

Numerical Linear Algebra 

• G.H. Golub & C.F. Van Loan, Matrix Computations. The Johns Hopkins University Press, third edition, 1996. 
• Lloyd N. Trefethen and David Bau III. Numerical Linear Algebra. SIAM, 1997.