Managing several tools to solve systems of linear equations and eigenvalue problems.
This course aims to familiarize students with the numerical solution of three important mathematical problems in linear algebra: systems of linear equations, linear least squares 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 eigenvalues, focusing on the QR iteration.
Systems of linear equations :
Chapitre I : matrices multiplication
A. Basic Algorithms and notations
B. Taking avantage of strucutures
C. Bloc Matrices and associated algorithms
D. Vectorization
Chapitre II : Matrix analysis
A Basic concepts of linear algebra
B. Vectorial Norms
C. Matricial Norms
D. Matrix calculus and finite precision
E. Orthogonality and singular values décomposition
F. Sensibility of linear systems
Chapitre III : Linear system in general
A. Triangular systems
B. LU factorization
C. Gaussian elimination and its error
D. Pivot positions
Chapitre IV : Special cases of linear systems
A. LDM^T and LDL^T factorizations
B. Positive definite systems
C. Band systems
D. Symetrical undefinite systems
Chapitre V : Orthogonalization and least squarres
A. Householder and Givens matrices
B. QR factorzsation
C. Least squarres for full rank matrices
Chapitre VI : Iterative methods for linear systems
A. Standard methods (Jacobi -- Gauss-Seidel -- SOR)
B. Conjugate gradients
C. Preconditionners for CG
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 et Van Loan)