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Rational Approximation and Interpolation
Last Updated: 2026-02-05 16:22:16
Abstract
The seminar covers theory and algorithms for rational interpolation based on classical and modern literature. The various topics have to be presented by groups of students.
Objective
Participants of the seminar should acquire familiarity with the theoretical properties of approximation by means of rational functions as well as knowledge about algorithms used for computing approximating or interpolating rational functions.
Content
The simplest and most widely used function system for approximation in computational mathematics are polynomials. They are ideally suited for smooth (analytic) functions. However, in many application we encounter functions with kinks and other kinds of singularities. In this case approximation by rational functions, that is, quotients of polynomials, may be vastly superior. This is why rational approximation and interpolation is receiving increased attention for the construction of surrogate models in model order reduction. This seminar will study a number of research papers dealing with both theoretical and algorithmic aspects of rational approximation and interpolation. Topics: 1. Best approximation by rational functions 2. Best rational approximation of x 7→ |x| 3. Meinardus conjecture 4. Approximation by composite rational functions 5. Rational interpolation and linearized least-squares 6. Padé approximationj 7. Vector fitting 8. The AAA algorithm for rational approximation 9. The RKFIT algorithm for non-linear rational approximation 10. Rational minimax approximation 11. Multivariate Padé approximation 12. Fast least-squares Padé approximation Student groups will be decided and topics will be assigned during the preparatory meeting on March 1, 2023 Implementation and numerical experiments: Quite a few of the topics are algorithmic in nature. Many of the related papers mention open source implementations of the methods, mainly in MATLAB, often relying on the Chebfun library. It is desirable that groups presenting an algorithmic topic also conduct numerical experiments, those covered in the articles or others, and report their observations. More information: https://people.math.ethz.ch/~ralfh/Seminars/RAP_23/SeminarRAP_FS23.pdf
Resources
Literature
See https://people.math.ethz.ch/~ralfh/Seminars/RAP_23/SeminarRAP_FS23.pdf
Learning Materials (Links)
General Information
- Language
- English
- Levels
- BSC , MSC
Examination
- Type
- ungraded semester performance
Registration & Places
- Signup Start
- 02.01.2023
- Signup End
- 17.02.2023
Course Components
| Type | Title | Time & Place | Hours |
|---|---|---|---|
| seminar |
Rational Approximation and Interpolation
Presentation topics will be assigned during a preparatory meeting in the first weak of the teaching period.
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|
2 h weekly |
Offered In
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Seminars (NOTICE: The number of seminar places is limited, and the special selection procedure should help to allocate the places not primarily according to the registration time. For some seminars everybody is waitlisted first when he/she tries to register for a seminar in myStudies. Moreover: at most 2 mathematics seminars can be chosen per semester.)
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Seminars (NOTICE: The number of seminar places is limited, and the special selection procedure should help to allocate the places not primarily according to the registration time. For the seminars with pecial selection procedure everybody is waitlisted first when he/she tries to register for a seminar in myStudies. Moreover: At most 2 mathematics seminars can be chosen per semester. In case you need to attend 3 seminars in this semester, please take contact with the Study Administration (email: ).)
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