Numerical Linear Algebra

Aim

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.

Content

• Linear algebra: LU-decomposition, SVD, psuedoinvers, orthogonal transformations, Householder transformations, projections, QR-factorisation and least squares problems.
• Eigenvalues: Normal forms, perturbation theory, Rayleigh quotient, the power method, invers iteration, transformation to Hessenberg and tridiagonal form, QR-iteration.
• Non-linear system of equations and least squares problems: Newton's and Gauss-Newton's methods. Teaching and working methods Lectures, exercises and computer laborations. Examination LAB1Laboratory work2 creditsU, GTEN1Written examination4 creditsU, 3, 4, 5The first three course aims are examined with TEN1. The fourth and fifth are examined with LAB1. Grades for examination modules are decided in accordance with the assessment criteria presented at the start of the course. Grades Four-grade scale, LiU, U, 3, 4, 5 Other information About teaching and examination languageThe teaching language is presented in the Overview tab for each course. The examination language relates to the teaching language as follows: If teaching language is “Swedish”, the course as a whole could be given in Swedish, or partly in English. Examination language is Swedish, but parts of the examination can be in English.If teaching language is “English”, the course as a whole is taught in English. Examination language is English.If teaching language is “Swedish/English”, the course as a whole will be taught in English if students without prior knowledge of the Swedish language participate. Examination language is Swedish or English depending on teaching language.OtherThe course is conducted in such a way that there are equal opportunities with regard to sex, transgender identity or expression, ethnicity, religion or other belief, disability, sexual orientation and age.The planning and implementation of a course should correspond to the course syllabus. The course evaluation should therefore be conducted with the course syllabus as a starting point. The course is campus-based at the location specified for the course, unless otherwise stated under “Teaching and working methods”. Please note, in a campus-based course occasional remote sessions could be included. Department Matematiska institutionen Course literature BooksWendland, Holger, Verfasser, (2018) Numerical linear algebra : an introduction ISBN: 9781107147133, 9781316601174, 1107147131, 131660117X