Topics in numerical mathematics

2022/2023
Programme:
Financial Mathematics, Second cycle
Year:
1 ali 2 year
Semester:
first or second
Kind:
optional
Group:
M4
ECTS:
6
Language:
slovenian, english
Hours per week – 1. or 2. semester:
Lectures
2
Seminar
1
Tutorial
2
Lab
0
Content (Syllabus outline)

The lecturer chooses some important topics from numerical mathematics, e.g.:

Subdivision schemes.
Pythagorean-hodograph curves.
Finite elements method.
Matrix methods in data mining and pattern recognition.
Regularization.
Tensor computation, presentation of tensors and tensor approximation.
Numerical method for multiparameter eigenvalue problems.
Nonlinear eigenvalue problems.
Numerical computation of matrix functions.

Readings

N. Dyn: Subdivision Schemes in Computer-Aided Geometric Design, Advances in Numerical Analysis II Wavelets, Subdivision Algorithms and Radial Basis Functions, W. Light (ed.), Clarendon Press, Oxford, 36-104 (1992).

R. T. Farouki: Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable, Geometry and Computing, vol. 1, Springer, Berlin, 2008.

J. N. Reddy: An introduction to finite elements method, McGraw-Hill, 1993.

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

N. J. Higham: Functions of matrices, Theory and Computation, SIAM, Philadelphia, 2008.

P. C. Hansen: Rank-Deficient and Discrete Ill-Posed Problems, SIAM, Philadelphia, 1998.

Objectives and competences

The student sees the details of one or more important areas of numerical mathematics, and learns about some recent results in the subjects.

Intended learning outcomes

Knowledge and understanding:
A deeper knowledge of one or several topics in numerical mathematics and capability of solving practical problems somehow connected with selected subjects. Knowledge of programming and usage of computer software for solving such problems.
Application: Numerical computation of mathematical problems.
Reflection: Understanding of the theory from the applications.
Transferable skills: The ability to solve mathematical problems using a computer.

Learning and teaching methods

Lectures, exercises, homeworks, consultations, projects

Assessment

homeworks or project
written exam
oral exam

Lecturer's references

Bor Plestenjak:
– HOCHSTENBACH, Michiel E., KOŠIR, Tomaž, PLESTENJAK, Bor. A Jacobi-Davidson type method for the two-parameter eigenvalue problem. SIAM journal on matrix analysis and applications, ISSN 0895-4798, 2005, vol. 26, no. 2, str. 477-497 [COBISS-SI-ID 13613401]
– MUHIČ, Andrej, PLESTENJAK, Bor. On the quadratic two-parameter eigenvalue problem and its linearization. Linear Algebra and its Applications, ISSN 0024-3795. [Print ed.], 2010, vol. 432, iss. 10, str. 2529-2542 [COBISS-SI-ID 15469913]
– HOCHSTENBACH, Michiel E., MUHIČ, Andrej, PLESTENJAK, Bor. On linearizations of the quadratic two-parameter eigenvalue problem. Linear Algebra and its Applications, ISSN 0024-3795. [Print ed.], 2012, vol. 436, iss. 8, str. 2725-2743 [COBISS-SI-ID 16095065]
Marjetka Knez:
– KRAJNC, Marjetka. Geometric Hermite interpolation by cubic G[sup]1 splines. Nonlinear Analysis, Theory, Methods and Applications, ISSN 0362-546X. [Print ed.], 2009, vol. 70, iss. 7, str. 2614-2626 [COBISS-SI-ID 15508569]
– KRAJNC, Marjetka. Interpolation scheme for planar cubic G [sup] 2 spline curves. Acta applicandae mathematicae, ISSN 0167-8019, 2011, vol. 113, no. 2, str. 129-143 [COBISS-SI-ID 16215385]
– JAKLIČ, Gašper, KOZAK, Jernej, KRAJNC, Marjetka, VITRIH, Vito, ŽAGAR, Emil. High order parametric polynomial approximation of conic sections. Constructive approximation, ISSN 0176-4276, 2013, vol. 38, iss. 1, str. 1-18 [COBISS-SI-ID 16716121]
Emil Žagar:
– ŽAGAR, Emil. On G [sup] 2 continuous spline interpolation of curves in R [sup] d. BIT, ISSN 0006-3835, 2002, vol. 42, no. 3, str. 670-688 [COBISS-SI-ID 12027993]
– 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]
– JAKLIČ, Gašper, KOZAK, Jernej, VITRIH, Vito, ŽAGAR, Emil. Lagrange geometric interpolation by rational spatial cubic Bézier curves. Computer Aided Geometric Design, ISSN 0167-8396, 2012, vol. 29, iss. 3-4, str. 175-188 [COBISS-SI-ID 16207449]