# Numerical approximation and interpolation

2022/2023
Programme:
Financial Mathematics, Second cycle
Year:
1 ali 2 year
Semester:
first or second
Kind:
core mandatory
Group:
M4
ECTS:
6
Language:
slovenian, english
Course director:
Hours per week – 1. or 2. semester:
Lectures
3
Seminar
0
Tutorial
2
Lab
0
Prerequisites

There are no prerequisites.

Content (Syllabus outline)

Approximation of functions: Spaces of approximation functions. Polynomials. Trigonometric polynomials. Piecewise polynomial functions. Stability of bases.
Weierstrass' Theorem. Positive operators.
Optimal approximation. Existence and uniqueness of the best approximation. Uniform convexity and strong normed spaces.
Uniform approximation by polynomials:
Uniqueness in the discrete and continuous case. Iteration of residuals. Construction. The first and the second Remes algorithm. Convergence. Chebyshev polynomials. Generalizations: Chebysev systems, generalized polynomials.
Continuous and discrete least squares:
Orthogonal polynomials. Three-term recurrence. Gram-Schmidt orthogonalization, basic and stable version. Reorthogonalization.
Connection between discrete and continuous case. Uniform convergence of L2-approximants.
Interpolation: Polynomial interpolation. Lagrange form. Barycentric Lagrange interpolation. Divided differences. Newton form and generalized Horner scheme. Divergence of interpolating polynomials.
Piecewise polynomial functions, splines: Euler polygons, interpolation and approximation in the second norm. Cubic splines. B-spline bases of piecewise polynomial functions. Bézier curves. Splines in two dimensions.

J. Kozak: Numerična analiza, DMFA-založništvo, Ljubljana, 2008.
R. L. Burden, J. D. Faires: Numerical Analysis, 8th edition, Brooks/Cole, Pacific Grove, 2005.
E. K. Blum: Numerical Analysis and Computation : Theory and Practice, Addison-Wesley, Reading, 1998.
Z. Bohte: Numerične metode, DMFA-založništvo, Ljubljana, 1991.
S. D. Conte, C. de Boor: Elementary Numerical Analysis : An Algorithmic Approach, 3rd edition, McGraw-Hill, Auckland, 1986.
C. de Boor: A Practical Guide to Splines, Springer, New York, 2001.
E. Isaacson, H. B. Keller: Analysis of Numerical Methods, John Wiley & Sons, New York-London-Sydney, 1994.
D. R. Kincaid, E. W. Cheney: Numerical Analysis : Mathematics of Scientific Computing, 3rd edition, Brooks/Cole, Pacific Grove, 2002.

Objectives and competences

Student supplements knowledge of analytical methods in approximation and interpolation by numerical aspects. By solving homeworks the obtained theoretical knowledge is consolidated.

Intended learning outcomes

Knowledge and understanding: Understanding of interpolation and approximation. Ability of numerical algorithms for construction of interpolating or approximating functions.
Application: Numerical construction of interpolating and approximating functions using a computer and error estimation based on theory. Interpolation and approximation are used in several fields, in particular in computer aided graphical modelling.
Reflection: Understanding of theory based through applications.
Transferable skills: Skill of using computer for solving numerical problems. Understanding differences between exact and numerical computing.

Learning and teaching methods

Lectures, exercises, homeworks, consultations.

Assessment

Homeworks or project
Written exam
Oral exam
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

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]