Introduction to numerical computations. Sources of errors in numerical computing. Stability of problems, convergence of methods and stability of computational processes. Error bounds.
Solving of nonlinear systems. Bisection. Iteration. Tangent and secant method. Solving of algebraic equations. Solving systems of nonlinear equations. Iteration. Newton method.
Solving of systems of linear equations. Vector and matrix norms. Stability. Error bounds. Gauss elimination. Pivoting. Special linear systems.
Linear least squares problem. Predetermined systems. Normal equations. QR decomposition. Givens rotations and Householder reflections. Singular value decomposition. Pseudoinverse. SVD applications. Regularization. Nonlinear least square problem.
Computation with multidimensional matrices (tensors). Introduction to different forms and methods for approximation with low rank tensors.
Numerical methods
B. Plestenjak: Razširjen uvod v numerične metode, DMFA – založništvo, Ljubljana, 2015
M. T. Heath, Scientific Computing: An Introductory Survey, McGraw-Hill, Boston, 2002.
J. W. Demmel: Uporabna numerična linearna algebra, DMFA-založništvo, Ljubljana, 2000.
G.H. Golub, C. F. Van Loan: Matrix Computations, 4th edition, Johns Hopkins Univ. Press, Baltimore, 2013
B. N. Datta: Numerical Linear Algebra and Applications, Brooks/Cole, Pacific Grove, 1995.
Student learns basic facts on numerical computation and analytical methods for solving nonlinear equations and systems of linear equations with some of well known numerical methods. In tutorial and homework the gained knowledge is increased and computer software, used by numerical computations, is applied.
Knowledge and understanding:
Understanding of computation in floating point arithmetics and causes of errors in numerical computations. Knowledge of basic algorithms for solving systems of linear equations and systems of nonlinear equations. Programming and use of Matlab and other similar tools for solving these kinds of problems.
Lectures, tutorial, homework, consultations.
Type (examination, oral, coursework, project):
written and oral exam, 2 homeworks
grades: 5 (negative), 6-10 (positive) (by The statues 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:
MEERBERGEN, Karl, PLESTENJAK, Bor. An Sylvester-Arnoldi type method for the generalized eigenvalue problem with two-by-two operator determinants. Numerical linear algebra with applications, ISSN 1070-5325, 2015, vol. 22, iss. 6, str. 1131-1146. [COBISS-SI-ID 17494105]
PLESTENJAK, Bor, GHEORGHIU, C. I., HOCHSTENBACH, Michiel E. Spectral collocation for multiparameter eigenvalue problems arising from separable boundary value problems. Journal of computational physics, ISSN 0021-9991, 2015, vol. 298, str. 585-601. [COBISS-SI-ID 17347417]
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]
Emil Žagar:
KOVAČ, Boštjan, ŽAGAR, Emil. Some new G[sup]1 quartic parametric approximants of circular arcs. Applied mathematics and computation, ISSN 0096-3003. [Print ed.], 2014, vol. 239, str. 254-264. [COBISS-SI-ID 17031769]
JAKLIČ, Gašper, KOZAK, Jernej, KRAJNC, Marjetka, VITRIH, Vito, ŽAGAR, Emil. An approach to geometric interpolation by Pythagorean-hodograph curves. Advances in computational mathematics, ISSN 1019-7168, 2012, vol. 37, no. 1, str. 123-150. [COBISS-SI-ID 16051289]
JAKLIČ, Gašper, KOZAK, Jernej, KRAJNC, Marjetka, VITRIH, Vito, ŽAGAR, Emil. Hermite geometric interpolation by rational Bézier spatial curves. SIAM journal on numerical analysis, ISSN 0036-1429, 2012, vol. 50, no. 5, str. 2695-2715. [COBISS-SI-ID 16449369]
