Numerical Linear Algebra

Recommended Prerequisites

Linear Algebra and Numerical Analysis.

Teaching Methods

Theoretical classes with detailed presentation of the subjects, highlighting the strong interaction between theoretical concepts and their practical application.

Learning Outcomes

This course gives an overview of numerical linear algebra and describes the main theoretical notions and algorithms used in matrix computations for solving linear systems, linear least squares problems or eigenvalue problems. It also addresses the issue of stability and accuracy in scientific computing and some challenges encountered in high-performance computing with the advent of new computer architectures.

The main skills to be developed are: capacity for analysis and synthesis; competence in oral and written communication and problem-solving; competence in working in an international context; autonomous learning; adaptability to new situations; creativity; competence to research; critical thinking.

Work Placement(s)

No

Syllabus

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).

Head Lecturer(s)

Adérito Luís Martins Araújo

Assessment Methods

Assessment
Research work: 20.0%
Synthesis work: 20.0%
Frequency: 20.0%
Exam: 20.0%
Resolution Problems: 20.0%

Bibliography

J.W. Demmel, Applied Numerical Linear Algebra, SIAM, 1997.

J.J. Dongarra, I.S. Du, D.C Sorensen, and H.A Van der Vorst, Numerical Linear Algebra for High-Performance Computers, SIAM, 1998.

G.H. Golub and C.F. Van Loan, Matrix Computations, John Hopkins University Press, 2013.

N.J. Higham, Accuracy and Stability of Numerical Algorithms, SIAM, 2002.

C.D. Meyer, Matrix Analysis and Applied Linear Algebra, SIAM, 2000.

L.N. Trefethen and D. Bau, Numerical Linear Algebra, SIAM, 1997.