Numerical methods 2

2022/2023
Programme:
Financial mathematics, First Cycle
Year:
2 year
Semester:
second
Kind:
mandatory
ECTS:
5
Language:
slovenian
Hours per week – 2. semester:
Lectures
2
Seminar
0
Tutorial
2
Lab
0
Prerequisites

Completed courses Analysis 1 and Algebra 1.

Content (Syllabus outline)

Nonsymmetric eigenvalue problem. Schur form. Power iteration. Inverse iteration. QR iteration.
Symmetric eigenvalue problem. Condition numbers. Tridiagonal QR iteration. Rayleigh quotient. Jacobi method. Genearlized eigenvalue problem.
Singular value decomposition computation. QR iteration for bidiagonal matrices. Jacobi method.
Data approximation. Least squares problems. Approximation of periodic data. Construction of empirical formulas.
Polynomial interpolation. Lagrange interpolation. Linear interpolation. Successive linear interpolation. Divided differences. Newton interpolation. Numerical differentiation.
Numerical integration. Newton-Cotes rules. Composite rules. Romberg extrapolation. Gaussian quadrature.
Numerical methods for ordinary differential equations. Methods for initial value problems. One-step methods. Runge-Kutta methods. Multi-step methods. Boundary problems.

Readings

J. W. Demmel: Uporabna numerična linearna algebra, DMFA-založništvo, Ljubljana, 2000.
B. N. Datta: Numerical Linear Algebra and Applications, Brooks/Cole, Pacific Grove, 1995.
Z. Bohte: Numerične metode, DMFA-založništvo, Ljubljana, 1991.
L. N. Trefethen, D. Bau: Numerical Linear Algebra, SIAM, Philadelphia, 1997.
D. R. Kincaid, E. W. Cheney: Numerical Analysis : Mathematics of Scientific Computing, 3rd edition, Brooks/Cole, Pacific Grove, 2002.
R. L. Burden, J. D. Faires: Numerical Analysis, 8th edition, Brooks/Cole, Pacific Grove, 2005.
E. Zakrajšek: Uvod v numerične metode, DMFA-založništvo, Ljubljana, 2000.

Objectives and competences

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.

Intended learning outcomes

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.

Learning and teaching methods

Lectures, lab exercises, homework, consultations

Assessment

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

Lecturer's references

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]