Advanced Numerical linear Algebra (45hours)

• This course provides an introductory overview of numerical Linear Algebra. It covers algorithms used to solve problems in Linear Algebra, including systems of linear equations and matrix eigenvalue problems.
• The aim of this course is to give tools (as classical algorithms) to solve numerical problems in Linear Algebra.
• The course gives classical algorithms to solve linear systems by different methods, and to find the eigenvalues of a matrix. The course details the mathematical theory behind numerical algorithms for the solution of linear systems and eigenvalue problems. The emphasis is on iterative methods suitable for large-scale problems arising, e. g., in the discretization of partial differential equations such as Stokes and Navier-Stokes equations.

Prerequisites:

Basic understanding of elementary linear algebra is necessary. Additionally, a foundational level of programming knowledge is required. While familiarity with high-level programming languages like C or Python would be beneficial, it is not mandatory.

The following topics will be covered (not necessarily in the order below)

• Introduction to Numerical Linear Algebra, Applications and Origin of Problems.
• Matrix Analysis: Linear Systems and Eigenvalue Problems.
• Direct Solution Methods.
• Iterative Methods for Linear Algebraic Systems.
• Iterative Methods for Eigenvalue Problems.
• Multigrid Methods.

Course slides and reading material will be available from the instructor.

• Optional Textbooks:
  • L. Hogben (editor). Handbook of Linear Algebra. Boca Raton, Florida: CRC Press/Taylor & Francis Group, 2014, ISBN 9781466507289.
  • B. W. Kernighan, R. Pike. The practice of programming, Reading, MA: Addison-Wesley, 1999, ISBN 020161586X.
  • Y. Saad, Iterative Methods for Sparse Linear Systems, SIAM, Philadelphia, PA.,(2003).

1. Midterm Exam: 25%
2. Project: 25%
3. Final Exam: 50%