Linear Algebra. Calculus I. It is advisable to have some basic knowledge of MATLAB.

The student will get familiar with the basic algorithms for solving the main four problems of numerical linear algebra (NLA), namely: (1) the solution of linear systems, (2) the solution of least squares problems, (3) the computation of eigenvalues and eigenvectors, and (4) the computation of the singular value decomposition (SVD). Also, he/she will acquire techniques and tools from NLA that can be useful either in his/her professional performance, in areas like data analysis and pattern recognition, or in a scientific career in the field of applied and computational mathematics. In particular, the student will learn and will be able to manage: - The basic MATLAB commands in the context of the four main problems in NLA mentioned above. - The fundamentals of numerical analysis (conditioning, stability, and computational complexity). - The error analysis in numerical methods, in particular those appearing in NLA. - The basic facts of the floating point arithmetic. - The basic notions of matrix norms, together with their relevance in the numerical computations that involve the use of matrices. - The tools and the theory underlying the algorithms that are currently employed for the solution of linear systems, both for matrices with small to moderate size (direct methods) and for large scale matrices (iterative methods). - The tools and the theory underlying the algorithms that are currently employed for the computation of eigenvalues and eigenvectors, both for matrices with small to moderate size (direct methods) and for large scale matrices (iterative methods). - The theory and tools in the computation of the SVD, as well as an approach to the basic algorithms for computing such decomposition. - The basic theory and tools in the solution of least squares problems. - Some of the applications of the SVD in both theoretical and applied frameworks, like the distance to the set of matrices with smaller rank or the image compression and the principal component analysis. - Some of the standard applications of NLA in applied contexts, like data analysis or image recognition.

1. Fundamentals on numerical analysis. 1.1 Introduction to floating point arithmetic. 1.2 Conditioning and stability. 1.3 Computational cost. 2. Vector and matrix norms. 2.1 Vector norms. 2.2 Induced matrix norms. 2.3 General matrix norms. 2.4 The spectral radius. 3. The Schur normal form. 3.1 Unitary matrices. 3.2 The Schur triangular form. 3.3 Some applications. 4. Direct methods for solving linear systems. 4.1 Gaussian elimination and LU factorization. 4.1.1 Gaussian elimination and LU without pivoting. 4.1.2 LU with pivoting: partial and total pivoting. 4.1.3 Stability analysis of the LU factorization. The growth factor. 4.2 Symmetric matrices: The Cholesky factorization. 4.3 QR factorization. 4.3.1 QR factorization and Gram-Schmidt. 4.3.2 QR factorization using Householder reflections. 5. Direct methods for computing eigenvalues and eigenvectors. 5.1 Reduction to Hessenberg and tridiagonal form. 5.2 The QR algorithm. 5.2.1 The power and inverse power method with shifts. 5.2.2 Subspace iteration. 5.2.3 The QR algorithm. 5.2.4 The QR algorithm in practice. 5.2.5 Algorithms for symmetric matrices. 5.3 Sensitivity of the eigenvalue problem. 6. The SVD: computation and some applications. 6.1 The SVD: existence and basic features. 6.2 Algorithms for computing the SVD. 6.3 Applications of the SVD in low-rank approximation problems. 6.3.1 Distance to singularity. 6.3.2 Image compression. 6.3.3 Principal Component Analysis. 6.3.4 Other applications. 7. Least squares problems. 7.1. Least squares problems: basic properties. 7.2 Solution of least squares problems. The pseudoinverse. 7.3 Normal equations vs QR for solving least squares problems. 7.4 Least squares problems using the SVD. 8. Iterative methods for solving linear systems and computing eigenvalues. 8.1 Krylov-type methods. 8.1.1 The Krylov methods. 8.1.2 The conjugate gradient method. 8.1.3 Other methods. 8.2 Lanczos and Arnoldi methods for computing the eigenvalues. 9. Further applications of Numerical Linear Algebra. 9.1 PageRank. 9.2 Matrix completion. 9.3 Data Mining.

The course is conceived as a practical course from the computational point of view. There will be 14 theory lectures and 14 practical sessions that will be carried out, presumably, in the computer laboratory and will address practices with the MATLAB program. The participation in these practices will be fundamental in order to acquire the objectives of the course. The practices will be developed in small groups and the solutions will be provided to the professor before the end of the quarter. The course includes a basic training in the MATLAB program, and the basic tools, codes, and basic programming skills will be provided to the students. The notes of the course will be available for the students from the very beginning, with all the contents that will be addressed during the course. These notes will include a series of exercises that will allow the students to go deeper in the contents of the course, and/or will give them the opportunity to solve by themselves some of the open questions in the notes. There will be some teaching ours, according to the official rules of the university.