# Budapest University of Technology and Economics, Faculty of Electrical Engineering and Informatics

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Numerical Methods of Linear Algebra

A tantárgy neve magyarul / Name of the subject in Hungarian: A lineáris algebra numerikus módszerei

Last updated: 2023. szeptember 6.

 Budapest University of Technology and Economics Faculty of Electrical Engineering and Informatics Numerical Methods of Linear Algebra
 Course ID Semester Assessment Credit Tantárgyfélév VIMAD041 ősz 4/0/0/v 5 1/2
3. Course coordinator and department Dr. Pach Péter Pál,
Web page of the course http://cs.bme.hu/lanm
4. Instructors
 Dr. Friedl Katalin associate professor Department of Computer Science and Information Theory Dr. Pach Péter Pál associate professor Department of Computer Science and Information Theory
5. Required knowledge

Linear algebra, mathematical analysis.

6. Pre-requisites
Ajánlott:
None.
7. Objectives, learning outcomes and obtained knowledge We would like to give an overview of the most commonly used methods of numerical algebra and numerical analysis. The aim of the class is to make the methods used for solving numerical problems in practice familiar to the students, who will learn how to choose the procedure which fits the given problem the best.
8. Synopsis

1) Vector and matrix norms, applications to some important estimates, Rayleigh quotient.

2) Localisation of the eigenvalues, Gershgorin circles.

3) Singular values of a matrix. Singular value decomposition.

4) Moore-Penrose pseudoinverse.

5) Linear equation systems, condition number.

6) Numerical solution of linear equation systems. Direct methods: Crout version of Gaussian elimination.

7) Solving a linear equation system when the coeffient matrix is tridiagonal.

8) Conjugate gradient method.

9) Iterative methods: Gauss-Seidel method; Successive over-relaxation. Alternating direction method.

10) Application: solving the Poisson equation. Tensor product.

11) Numerical solution of the eigenvalue problem.

12) The power iteration and the inverse iteration method, Mises theorem.

13) The eigenvalue problem of real symmetric matrices.

14) The Householder transformation       .

15) The eigenvalue problem for tridiagonal matrices, Sturm's theorem.

16) The eigenvalue problem for nonsymmetric matrices.

17) Transformation to Hessenberg form.

18) QR transformation.

19) Applications of the Courant-Fischer theorem.

20) Lánczos method.

21) Different kind of applications depending on time and interest, e.g. numerical solutions of differential equations, cluster analysis, PageRank.

9. Method of instruction

4 hours of lecture per week.

10. Assessment

Signature: 1 homework.

Final: oral exam.

12. Consultations

In office hours or by appointment.

13. References, textbooks and resources

Rózsa Pál: Lineáris algebra és alkalmazásai. 3. átdolgozott kiadás. Tankönyvkiadó, Budapest, 1991.

H. Golub – C.F. Van Loan: Matrix Computations, The John Hopkins University Press, 1989.

14. Required learning hours and assignment
 Kontakt óra 56 Félévközi készülés órákra 28 Felkészülés zárthelyire 0 Házi feladat elkészítése 16 Kijelölt írásos tananyag elsajátítása 10 Vizsgafelkészülés 40 Összesen 150
15. Syllabus prepared by

Dr. Rózsa Pál, professor, Department of Computer Science and Information Theory