JP-Amboage / Matrix-Numerical-Analysis

Matrix-Numerical-Analysis

BSc Mathematics 2nd year course at Universidade de Santiago de Compostela (USC)

Course objectives:

The study and application of numerical methods to solve linear and nonlinear systems of equations and to compute eigenvalues and eigenvectors associated to a matrix. Furthermore, in the computer lab sessions, students will analyze some of the studied algorithms by means of their implementation in FORTRAN 90 or MATLAB.

Contents:

Matrix overview: norms, spectral radius and Rayleigh quotient.

The need of the numerical methods for solving linear systems of equations: direct and iterative methods; matrix condition number.

Direct methods for solving a linear system: Gauss method, LU decomposition, partial pivoting strategy; Cholesky decomposition; Householder method and QR decomposition. Applications: computation of matrix determinants and inverses.

Numerical approximation of matrix eigenvalues and eigenvectors; eigenvalue estimates: Gerschgorin theorem; power and inverse iteration methods.

Iterative methods for solving a system of equations: fixed point methods; application to the linear case: Jacobi, Gauss-Seidel and relaxation methods; Newton method and variants for the nonlinear case.

Basic and complementary bibliography

Basic bibliography:

• CIARLET, P. G. [1999]: Introducción á análise numérica matricial e á optimización. Servicio de Publicacións da USC.
• GOLUB, G. H. - VAN LOAN, C. [2013]: Matrix computations. 4th ed. The Johns Hopkins University Press.
• HORN, R. A. - JOHNSON, C. R. [2013]: Matrix analysis. 2nd ed. Cambridge University Press.
• METCALF, M. - REID, J. - COHEN M. [2004]: Fortran 95/2003 explained. Oxford University Press.
• ORTEGA, J. M. [1990]: Numerical análisis: a second course. SIAM.
• STOER, J. - BULIRSCH, R. [1993]: Introduction to numerical analysis. 2nd ed. Springer-Verlag.

Complementary bibliography:

• ATKINSON, K. E. - HAN, W. [2004]: Elementary numerical analysis. John Wiley and sons.
• AUBANELL, A. - BENSENY, A. - DELSHAMS, A. [1991]: Eines bàsiques de càlcul numeric: amb 87 problemes results. Manuals de la Universitat Autònoma de Barcelona.
• GANDER, W. – GANDER M. J. – KWOK, F. [2014]: Scientific computing – An introduction using MAPLE and MATLAB. Springer.
• HEATH, M. T. [2005]: Scientific computing: an introductory survey. 2nd ed. McGraw Hill.
• KINCAID, D. - CHENEY, W. [1994]: Análisis numérico: las matemáticas del cálculo científico. Addison-Wesley Iberoamericana.
• QUARTERONI, A. [2003]: Scientific computing with MATLAB. Springer.
• QUARTERONI, A. - SACCO, R. - SALERI, F. [2000]: Numerical mathematics. Springer.
• TREFETHEN, Ll. N. - BAU, D. [1997]: Numerical linear algebra. SIAM.
• WATKINS, D. S. [2010]: Fundamentals of matrix computations. 3rd ed. Wiley.