Izbrana poglavja iz numerične matematike

Pogojev za vključitev v delo ni.

Izbrane bodo nekatere standardne teme iz podiplomske numerične matematike. Možna poglavja so: geometrijska interpolacija in aproksimacija s parametričnimi polinomskimi krivuljami in ploskvami, polinomska interpolacija v več spremenljivkah, parametrične krivulje in ploskve, ki se uporabljajo v CAGD (računalniško podprtem geometrijskem načrtovanju), valčki in uporaba v teoriji signalov in analizi slik, subdivizijske sheme za krivulje in ploskve, numerične metode za računanje matričnih funkcij, iterativne metode podprostorov in predpogojevanje, nelinearni problemi lastnih vrednosti, večparametrični problemi lastnih vrednosti, inverzni problemi lastnih vrednosti, metode zveznega nadaljevanja, večmrežne metode, teorija zlepkov, redukcija modela, slabo pogojeni problemi in regularizacija. Izbira je odvisna od interesov in raziskovalne usmeritve študentov.

1. Z. Bai ... [et al.], eds.: Templates for the solution of algebraic eigenvalue problems: a practical guide, Philadelphia : Society for Industrial and Applied Mathematics, cop. 2000.
2. R. Barrett … [et al.]: Templates for the solution of linear systems : building blocks for iterative methods, Philadelphia : Society for Industrial and Applied Mathematics, cop. 1994.
3. W. L. Biggs, V. E. Henson, S. F. McCormick: A multigrid tutorial, 2nd ed., Philadelphia, (PA) : Society for industrial and applied mathematics, cop. 2000.
4. J. P. Boyd: Chebyshev and Fourier spectral methods, 2nd ed. (revised), Mineola : Dover, 2001.
5. M. T. Chu, G. H. Golub: Inverse eigenvalue problems: theory, algorithms and applications, Numerical mathematics and Scientific Computation, Oxford : Oxford University Press, 2005.
6. N. Dyn in D. Levin: Subdivision schemes in geometric modelling, Acta Numer. 11 (2002) 73-144.
7. R. T. Farouki: Pythagorean-hodograph curves : algebra and geometry inseparable, Berlin : Springer, cop. 2008.
8. P. C. Hansen: Rank-deficient and discrete ill-posed problems : numerical aspects of linear inversion, Philadelphia : SIAM, cop. 1998.
9. N. J. Higham: Accuracy and stability of numerical algorithms, 2nd ed. - Philadelphia : Society for Industrial and Applied Mathematics, cop. 2002.
10. N. J. Higham: Functions of matrices : theory and computation, Philadelphia : SIAM, cop. 2008.

Namen predmeta je seznaniti študente z nekaterimi pomembnimi temami numerične matematike.

Znanje in razumevanje predstavljenih konceptov.
Sposobnost uporabe pridobljenega znanja in spretnosti.

Predavanja, konzultacije, reševanje problemov

Pisni izpit (domače naloge), ustni izpit
(ocene: 5 (negativno), 6-10 (pozitivno), ob upoštevanju Statuta UL)

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]
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., 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]
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]
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]