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