Numerical methods 2

Completed courses Analysis 1 and Algebra 1.

Introduction to approximation theory: construction of least square approximants and best polynomial approximants. Polynomial interpolation. Lagrange form of interpolating polynomial, Newton form and divided differences.
Numerical differentiation.
Numerical integration: Newton-Cotes rules. Composite rules. Richardson extrapolation. Gaussian quadrature. Numerical approximation of multiple integrals. Monte Carlo method.
Numerical methods for ordinary differential equations (initial value problems): One-step methods (Euler method, trapezoid method, Runge-Kutta methods), multi-step methods, local and global error.
Nonsymmetric eigenvalue problem: Schur form, power iteration, inverse iteration, QR iteration. Symmetric eigenvalue problem. Singular value decomposition computation.

1. Z. Bohte: Numerične metode, Ljubljana : Društvo matematikov, fizikov in astronomov SRS : Zveza organizacij za tehnično kulturo Slovenije, 1985, 1987.
2. R. L. Burden, J. D. Faires: Numerical analysis, 6th ed., Pacific Grove (Canada) : Brooks/Cole Publ. : ITP An International Thompson Publishing Company, cop. 1997.
3. B. N. Datta: Numerical linear algebra and applications, Pacific Grove : Brooks/Cole : International Thomson Publ., cop. 1994.
4. J. W. Demmel (prevod in priredba E. Zakrajšek): Uporabna numerična linearna algebra, Ljubljana : DMFA - založništvo, 2000.
5. D. Kincaid, W. Cheney: Numerical analysis : mathematics of scientific computing, 2nd ed., Pacific Grove (California) : Brooks/Cole Publishing Company, 1996.
6. B. Plestenjak: Razširjen uvod v numerične metode, DMFA-založništvo, Ljubljana, 2015.
7. L. N. Trefethen, D. Bau: Numerical linear algebra, Philadelphia : SIAM, cop. 1997.
8. E. Zakrajšek: Uvod v numerične metode, 2. popravljena izd. - Ljubljana : Društvo matematikov, fizikov in astronomov Slovenije, 2000.

Students learn basic numerical methods for eigenvalue computation, polynomial approximation and interpolation, numerical quadrature, and methods for the ordinary differential equations. The acquired knowledge is consolidated by exercises and homework assignements.

Knowledge and understanding: Understanding of basic numerical methods for eigenvalue computation, interpolation, quadrature, and methods for the ordinary differential equations. Knowledge of computer programming and Matlab or other similar software for solving such problems.
Application: Economical and accurate numerical solution of various mathematical problems. In addition to mathematics, numerical methods are used in many other fields when the problem can be described by a mathematical model and a result in a numerical form is required. Many problems can not be solved analytically but only numerically. Also, in some cases, the numerical solution is much more economical than the analytical one.
Reflection: Understanding of the theory from the applications.
Transferable skills: The ability to select an appropriate method, solve a problem, and analize the obtained results. 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 algebra and analysis.

Lectures, lab exercises, homework, consultations

Continuing (homework and written exam): 50 %
Final (oral exam): 50 %
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Marjetka Krajnc:
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, VITRIH, Vito. Motion design with Euler-Rodrigues frames of quintic Pythagorean-hodograph curves. Mathematics and computers in simulation, ISSN 0378-4754. [Print ed.], 2012, vol. 82, iss. 9, str. 1696-1711. [COBISS-SI-ID 1024447572]
KOZAK, Jernej, KRAJNC, Marjetka. Geometric interpolation by planar cubic polynomial curves. Computer Aided Geometric Design, ISSN 0167-8396, 2007, vol. 24, no. 2, str. 67-78. [COBISS-SI-ID 14227545]
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]
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]
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]