Introduction. Matrix decompositions. Conditioning and stability. Floating point arithmetic. Error analysis.
Systems of equations. Gaussian elimination with pivoting strategies. Cholesky factorization. Stability analysis. Large systems of equations. Sparse matrix techniques. Iterative methods based on Krylov subspaces: Conjugate Gradients, GMRES, Biorthogonalization methods (BiCG and BICGstab). Convergence and spectral properties. Preconditioning.
Eigenvalues. Reduction to Hessenberg or tridiagonal forms. Rayleigh quotient and inverse iteration. QR algorithm. Lanczos iteration (symmetric case) and Arnoldi iteration (non symmetric case).