Numerical linear algebra

Mathematics, First Cycle
3 year
Hours per week – 2. semester:

Completed course Introduction to numerical methods.

Content (Syllabus outline)

Nonsymmetric eigenvalue problem. Perturbation theory. Implicit QR iteration with single and double shifts.

Symmetric eigenvalue problem. Rayleigh quotient. Min-max theorem. Overview of numerical methods for the symmetric eigenvalue problem.

Generalized eigenvalue problem. Polynomial eigenvalue problem.

Singular value decomposition. Pseudoinverse. Application in linear least squares problems. Approximation with low rank matrices. Regularization. Overview of numerical methods for the singular value decomposition.

Computation with multidimensional matrices (tensors). Approximation with low rank tensors and applications in data mining.

  • B. Plestenjak: Razširjen uvod v numerične metode, DMFA – založništvo, Ljubljana, 2015.

  • J. W. Demmel: Uporabna numerična linearna algebra, DMFA-založništvo, Ljubljana, 2000.

  • L. Elden: Matrix Methods in Data Mining and Pattern Recognition, SIAM, Philadelphia, 2007.

  • G. H. Golub, C. F. Van Loan: Matrix Computations, 4rd edition, Johns Hopkins Univ. Press, Baltimore, 2013.

  • B. N. Datta: Numerical Linear Algebra and Applications, Brooks/Cole, Pacific Grove, 1995.

  • L. N. Trefethen, D. Bau: Numerical Linear Algebra, SIAM, Philadelphia, 1997.

Objectives and competences

Students learn numerical methods for the computation of eigenvalues and eigenvectors. New knowledge constructively complements the content of courses Algebra 1 and Introduction to numerical methods. The acquired knowledge is consolidated by homework assignements and solving problems using programs Matlab and Mathematica.

Intended learning outcomes

Knowledge and understanding: Understanding of numerical algorithms for the computation of eigenvalues and eigenvectors. The ability to choose an appropriate algorithm. Knowledge of computer programming package Matlab or other similar software for solving such problems.
Applications: Economical and accurate numerical computation of eigenvalues and eigenvectors.
Reflection: Understanding of the theory from the applications.
Transferable skills: The ability to solve mathematical problems using a computer. Understanding the differences between the exact and the numerical computation. The subject enriches constructively the knowledge of linear algebra.

Learning and teaching methods

Lectures, exercises, homework, consultations


Type (examination, oral, coursework, project):
Continuing (homework, midterm exams, project work)Final (written and oral exam)
Grading: 6-10 pass, 5 fail (according to the Statute of UL)

Lecturer's references

Bor Plestenjak:

  • GHEORGHIU, C. I., HOCHSTENBACH, Michiel E., PLESTENJAK, Bor, ROMMES, Joost. Spectral collocation solutions to multiparameter Mathieu's system. Applied mathematics and computation, ISSN 0096-3003. [Print ed.], 2012, vol. 218, iss. 24, str. 11990-12000. [COBISS-SI-ID 16484185]
  • PLESTENJAK, Bor, BAREL, Marc van, CAMP, Ellen van. A Cholesky LR algorithm for the positive definite symmetric diagonal-plus-semiseparable eigenproblem. V: CHING, Wai-Ki (ur.). Second international conference on structured matrices : Hong Kong Baptist University, 08-11 June 2006, (Linear algebra and its applications, ISSN 0024-3795, Vol. 428, Issues 2-3, 2008). New York: North Holland, 2008, vol. 428, iss. 2-3, str. 586-599. [COBISS-SI-ID 14475097]
  • PLESTENJAK, Bor. Numerical methods for the tridiagonal hyperbolic quadratic eigenvalue problem. V: Fifth international workshop on accurate solution in eigenvalue problems : hagen, Germany from June 29 to July 1, 2004. Philadelphia: SIAM, 2006, vol. 28, no. 4, str. 1157-1172. [COBISS-SI-ID 14367833]
    Emil Žagar:

  • JAKLIČ, Gašper, ŽAGAR, Emil. Curvature variation minimizing cubic Hermite interpolants. Applied mathematics and computation, ISSN 0096-3003. [Print ed.], 2011, vol. 218, iss. 7, str. 3918-3924. [COBISS-SI-ID 16049241]

  • JAKLIČ, Gašper, KOZAK, Jernej, KRAJNC, Marjetka, VITRIH, Vito, ŽAGAR, Emil. Hermite geometric interpolation by rational Bézier spatial curves. SIAM journal on numerical analysis, ISSN 0036-1429, 2012, vol. 50, no. 5, str. 2695-2715. [COBISS-SI-ID 16449369]
  • KOZAK, Jernej, ŽAGAR, Emil. On geometric interpolation by polynomial curves. SIAM journal on numerical analysis, ISSN 0036-1429, 2004, vol. 42, no. 3, str. 953-967. [COBISS-SI-ID 13398617]