Topics in numerical mathematics

2022/2023
Programme:
Computer Science and Mathematics, Second Cycle
Year:
1 ali 2 year
Semester:
first or second
Kind:
optional
Group:
B
ECTS:
6
Language:
slovenian, english
Lecturers:
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.:
Numerical approximation and interpolation
Numerical integration and ordinary differential equations.
Numerical solving of partial differential equations.
Subdivision schemes.
Pythagorean-hodograph curves.
Matrix methods in data mining and pattern recognition.
Finite elements method.
Numerical methods for linear control systems.
Iterative numerical methods in linear algebra.

Readings

J. Kozak: Numerična analiza, DMFA-založništvo, Ljubljana, 2008.
R. L. Burden in J. D. Faires: Numerical Analysis, 8th edition, Brooks/Cole, Pacific Grove, 2005.
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.
B. N. Datta: Numerical Methods for Linear Control Systems, Academic Press, San Diego, 2004.
R. Barrett, M. W. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, H. van der Vorst: Templates for the Solution of Linear Systems : Building Blocks for Iterative Methods, SIAM, Philadelphia, 1994.
Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, H. van der Vorst: Templates for the Solution of Algebraic Eigenvalue Problems : A Practical Guide, SIAM, Philadelphia, 2000.

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, tutorials, homeworks, seminar, projects and consultation.

Assessment

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 rules of University of Ljubljana)

Lecturer's references

Gašper Jaklič:
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, ŽAGAR, Emil. Planar cubic G [sup] 1 interpolatory splines with small strain energy. Journal of Computational and Applied Mathematics, ISSN 0377-0427. [Print ed.], 2011, vol. 235, iss. 8, str. 2758-2765. [COBISS-SI-ID 15770969]
JAKLIČ, Gašper. On the dimension of bivariate spline space S [sub] 3 [sup] 1 ([triangle]). International journal of computer mathematics, ISSN 0020-7160, 2005, vol. 82, no. 11, str. 1355-1369. [COBISS-SI-ID 13801305]
Marjetka Krajnc:
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]
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]
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]
Bor Plestenjak:
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]
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]
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]
Emil Žagar:
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]
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]
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]