**Program**

*Oral exam, no written exam
will be required*

__Suggested
readings__

L.N. Trefethen, D. Bau II: “Numerical Linear Algebra”, SIAM, 1997; in the following denoted by [TB]

P.G. Ciarlet: “Introduction to Numerical Linear Algebra and Optimisation”, Cambridge University Press, 1989: in the following denoted by [C]

T. Lyche: “Lecture Notes”, downloadable here; in the following denoted by [L]

__Below,
a quite detailed list of the studied subjects, with reference to the
suggested readings__

__However,
please consider that, of course, other textbooks or references may be
used__

Matrix-vector multiplications [TB, Lecture 1]

Orthogonal vectors and unitary matrices [TB, Lecture 2]

Norms of vectors and matrices [TB, Lecture 3]

The Singular Value Decomposition of a matrix [TB, Lecture 4]

Low rank approximation [TB, Lecture 5, pagg. 35-36]

Projectors and their fundamental properties [TB, Lecture 6]

QR decomposition of a matrix [TB, Lecture 7, until pag. 52, and pag. 54]

Gram-Schmidt and (modified Gram-Schmidt) orthogonalization procedure [TB, Lecture 8, until pag. 58]

Householder triangularization procedure [TB, Lecture 10, until pag. 74]

Least square problems and the normal equations [TB, Lecture 11, until pag. 82]

Conditioning and condition numbers of a problem [TB, Lecture 12]

Brief introduction to floating point arithmetic [TB, Lecture 13, until pagg. 98-99]

Stability and backward stability of numerical algorithms [TB, Lectures 14 and 15]

Stability of the Householder triangularization [TB, Lecture 16]

Iterative methods to solve linear systems: splitting methods [L, Chapter 14: Sections 14.1, 14.2, 14.3 and 14.4 (but not the SOR method)]

General error analysis of splitting methods [L, Chapter 14: Sections 14.1, 14.2, 14.3 and 14.4 (but not the SOR method)]

The Jacobi method [L, Chapter 14: Sections 14.1, 14.2, 14.3 and 14.4 (but not the SOR method)]

The Gauss-Seidel method [L, Chapter 14: Sections 14.1, 14.2, 14.3 and 14.4 (but not the SOR method)]

Sufficient conditions for the convergence of Jacobi and Gauss-Seidel methods [L, Chapter 14: Sections 14.1, 14.2, 14.3 and 14.4 (but not the SOR method)], see also [C, Chapter 5: Section 5.3]

The gradient methods as optimization procedures for unconstrained problems. The particular case of a quadratic functional and the connection with the solution of a linear system [C, Chapter 8: Section 8.4 (but not the relaxation method)]

The conjugate gradient method and its convergence properties [L, Chapter 15]

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