Numerical methods 1

There are no prerequisites.

Introduction to numerical computing:
floating point arithmetic, errors in numerical computations, conditioning and stability, forward and backward stability of the algorithms,
Solving nonlinear equations and systems of nonlinear equations:
Bisection method, Fixed point iteration, tangent method, secant method, order of convergence, zeros of polynomials, Newton method, minimization methods,
System of linear equations:
vector and matrix norms, LU decomposition (with and without the pivoting), condition number and backward stability, a posteriory error, symmetric positive definite matrix, Cholesky decomposition,
Overdetermined systems:
Least square solution, normal equations, QR decomposition, Gram Schmidt ortogonalization, Householder reflections, Givens rotations, SVD decomposition, nonlinear least square problem,
Approximation and interpolation:
Least square approximation in vector spaces with a scalar product, interpolation problem, Lagrange and Newton form of the interpolating polynomial, interpolation error,
Computation of eigenvalues:
Schur decomposition, power iteration method, inverse power iteration method, QR iteration

1. Z. Bohte: Numerične metode, Ljubljana : Društvo matematikov, fizikov in astronomov SRS : Zveza organizacij za tehnično kulturo Slovenije, 1985, 1987.
2. Z. Bohte: Numerično reševanje sistemov linearnih enačb, Ljubljana : Društvo matematikov, fizikov in astronomov Slovenije, 1994.
3. J. W. Demmel (prevod in priredba E. Zakrajšek): Uporabna numerična linearna algebra, Ljubljana : DMFA - založništvo, 2000.
4. L. Fox, D. F. Mayers: Computing methods for scientists and engineers , Oxford : Clarendon Press, 1968.
5. E. Isaacson, H. B. Keller: Analysis of numerical methods, New York : J. Wiley, cop. 1966.
6. J. Kozak: Numerična analiza, Ljubljana : DMFA - založništvo, 2008.
7. B. Plestenjak: Razširjen uvod v numerične metode, Ljubljana : DMFA - založništvo, 2015.
8. E. Zakrajšek: Uvod v numerične metode, 2. popravljena izd. - Ljubljana : Društvo matematikov, fizikov in astronomov Slovenije, 2000.

Students acuire knowledge about different algorithms for solving problems from linear algebra like solving systems of linear and overdetermined systems of equations, computing eigenvalues, etc. Students learn to be aware of the errors that occur in numerical computations. They get familiar with the procedures for finding zeros of nonlinear equations and systems of nonlinear equations.
Students supplement a basic knowledge of the theory of approximation and interpolation.
The algorithms are tested in Matlab on problems from a real life. This provides a practical knowledge too.

Knowledge and understanding:
Knowledge and understanding of the basic concepts and stable algorithms for solving problems in linear algebra. Knowledge of methods for finding solutions of nonlinear equations. Understanding the basics of approximation and interpolation theory.
Application:
Applied numerical linear algebra finds applications in most of natural, technical and social science fields. Approximation and interpolation are used in computer graphics, geometric design and robotics.
Reflection:
Integrating theoretical and practical procedures for solving practical problems.
Transferable skills:
Selection of a stable algorithm to solve the particular problem, which arises in practice. Knowledge is transmitted to virtually all sciences: natural sciences, computer science, statistics, etc.

Lectures, exercises, homeworks, consultations

Three midterm exams instead of written exam, written exam
Oral exam. Three homeworks (required when applying for an oral exam)
Students receive two grades: one from the written exam and the other from the oral exam and homeworks.
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

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]