## TANA15 Numerical Linear Algebra

#### Course Aims

The course is intended to provide basic knowledge about important matrix decompositions; such as the LU or SVD decompositions, and show how matrix decompositions can be used for analyzing and solving both practical and theoretical problems. The course also covers various important techniques from Linear Algebra, such as the Shur complement, convolutions, polynomial manipulation, or orthogonal basis generation. Both linear, and non-linear, least squares problems are also discussed in the course.

After the course students should be able to:

• Discuss the most common matrix factorizations, and explain their properties.
• Understand how the most common matrix factorizations are computed; and implement numerical algorithms for computing the most important factorizations.
• Use matrix factorizations for solving both theoretical problems and practical problems from applications.
• Discuss the usage of Linear Algebra techniques when solving important application problems, such as pattern recognition, data compression, signal processing, search engines, or model fitting.

#### Organisation

The Lectures give an overview of the theory. During the Problem seminars concrete examples, intended to illustrate the theory. are given. The Computer exercises give practical experience with implementing and using the methods.

#### Course Content

Basic operations of linear Algebra (BLAS). The LU decomposition: Solution of linear systems. Condition Number. Error estimate. The QR Decomposition: Reflection- and Rotation matrices. Least squares problems. Row Updating. The eigenvalue decomposition. Hessenberg factorization. The Power method. The QR Algoritm. Special matrices. The FFT. Functions of Matrices. The Singular value decomposition and applications. Bidiagonalization. Systems of non-linear equations: Gradient based methods. The Newton and Gauss-Newton method. Non-linear least squares problems.

#### Examination

TEN1 Written Exam (U,3,4,5) 4 hp
LAB1 Computer Exercices (U,G) 2 hp

Page responsible: Fredrik Berntsson
Last updated: 2019-11-29