Managing several tools to solve systems of linear equations and eigenvalue problems.
This course aims to familiarize students with the numerical solution of systems of linear equations and eigenvalue problems. It also aims to make the students aware of the relevant issues in selecting appropriate methods and to teach them to be critical with respect to these methods (in terms of error analysis, quality versus cost of the numerical solution, etc.).
After an introduction to matrix algebra, the first part of the course deals with the solution of general and special linear systems of equations, using direct and iterative methods. It also covers the solution of overdetermined linear systems. The second part of the course presents methods for the solution of eigenvalue problems, focusing on the QR iteration.
Systems of linear equations :
Chapitre I : Matrix multiplication
A. Basic algorithms and notation
B. Taking advantage of structures
C. Bloc Matrices and associated algorithms
D. Vectorization
Chapitre II : Matrix analysis
A. Basic concepts of linear algebra
B. Vector norms
C. Matrix norms
D. Matrix calculus and finite precision
E. Orthogonality and the singular value decomposition
F. Sensitivity of linear systems
Chapitre III : General linear systems
A. Triangular systems
B. LU factorization
C. Gaussian elimination and roundoff error
D. Pivoting
Chapitre IV : Special linear systems
A. LDM^T and LDL^T factorizations
B. Positive definite systems
C. Banded systems
D. Symmetric indefinite systems
Chapitre V : Orthogonalization and least squares
A. Householder and Givens transformations
B. QR factorization
C. Full-rank least squares problems
Chapitre VI : Iterative methods for linear systems
A. Standard methods (Jacobi -- Gauss-Seidel -- SOR)
B. Conjugate gradient method
C. Preconditioning
The final score is the mean of two examinations:
An oral examination for the theoretical part of the course, and the oral defense of an assignment for the practical part.
Matrix computations (Golub and Van Loan)