Numerical methods 2

Practical Mathematics
3 year
first or second
Hours per week – 1. or 2. semester:
Content (Syllabus outline)

Approximation of continuous functions by polynomials:
normed spaces, element of best approximation,
Bernstein polynomials, Weierstrass theorem, element of best uniform approximation
Functional and parametric splines:
functional and parametric C1 and C2 splines, local Hermite polynomial, parametric G1 and G2 polynomial splines, planar and spatial Bezier curves, some basic Bezier splines
Numerical derivation and integration:
numerical derivation, trapezoidal rule, midpoint rule, Simpson's rule, some simple quadrature formulae, numerical computation of multiple integrals
Numerical solution of (systems) ordinary differential equations:
numerical solution of ordinary differential equations of first order (Euler's method, trapezoidal rule, Runge-Kutta methods), numerical solution of systems of differential equations, numerical solution of higher order differential equations


Z. Bohte: Numerična analiza, Višja matematika III, DMFA založništvo, Ljubljana, 1976.
Z. Bohte: Numerične metode, DMFA založništvo, Ljubljana, 1987.
E. Zakrajšek: Uvod v numerične metode, DMFA založništvo, Ljubljana 1998.
Z. Bohte: Numerično reševanje sistemov linearnih enačb, DMFA založništvo, Ljubljana, 1994.
J.W. Demmel, Priredba E. Zakrajšek: Uporabna numerična linearna algebra, DMFA založništvo, Ljubljana, 2000.
L. Fox, D.F. Mayers: Computing Methods for Scientists and Engineers, Clarendon Press, Oxford, 1968.
E. Isaacson, H.B. Keller: Analysis of Numerical Methods, John Wiley & Sons, Inc., 1966.
J. Kozak: Numerična analiza, DMFA založništvo, Ljubljana 2008.

Objectives and competences

Students will acquire knowledge about algorithms for approximation of continuous functions by polynomials. They will learn about Wierstrass' theorem and about algorithms for the construction of the element of best uniform approximation.
Students will learn about functional and parametric splines, in particular special cases of splines based on Bezier curves.
They will acquire basic knowledge of numerical derivation and integration. Some particular rules of numerical integration will be presented in detail. They will learn about numerical solution of ordinary differential equations of the first and higher order.

Intended learning outcomes

Knowledge and understanding: Knowledge and understanding of the basic concepts and stable algorithms for solving problems in numerical analysis. Understanding of methods for construction of splines, algorithms for numerical derivation, integration and numerical solution of differential equations.
Application: Numerical analysis finds its application in almost all natural sciences, technical sciences, and computer sciences. Splines are among the most important structures in computer aided geometric design. Numerical integration and solution of differential equations are basic problems on several applicative fields of research.
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, technical sciences, computer sciences, economy, etc.

Learning and teaching methods

Lectures, exercises, homeworks, consultations


Type (examination, oral, coursework, project):
two obligatory homeworks required to apply for an oral exam,
two midterm exams instead of written exam, written exam
oral exam
Grading: 1-5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

Marjetka Knez:
JAKLIČ, Gašper, KOZAK, Jernej, KNEZ, 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]
KNEZ, 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]
KNEZ, 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]
Jan Grošelj:
GROŠELJ, Jan, KNEZ, Marjetka. A construction of edge B-spline functions for a C^1 polynomial spline on two triangles and its application to Argyris type splines. Computers & Mathematics with Applications. [Print ed.]. Oct. 2021, vol. 99, str. 329-344. [COBISS-SI-ID 98324739]
GROŠELJ, Jan, SPELEERS, Hendrik. Super-smooth cubic Powell–Sabin splines on three-directional triangulations: B-spline representation and subdivision. Journal of Computational and Applied Mathematics 386, 23 pages, 2021. [COBISS-SI-ID 58260483]
GROŠELJ, Jan. Argyris type quasi-interpolation of optimal approximation order. Computer Aided Geometric Design 79, 2020. [COBISS-SI-ID 21571843]