Algorithms for modifying recurrence relations of orthogonal polynomial and rational functions when changing the discrete inner product

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F24%3A10473005" target="_blank" >RIV/00216208:11320/24:10473005 - isvavai.cz</a>

Result on the web

<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=JOIDyYaMGH" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=JOIDyYaMGH</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.apnum.2023.07.009" target="_blank" >10.1016/j.apnum.2023.07.009</a>

Result language

angličtina

Original language name

Algorithms for modifying recurrence relations of orthogonal polynomial and rational functions when changing the discrete inner product

Original language description

Often, polynomials or rational functions, orthogonal for a particular inner product are desired. In practical numerical algorithms these polynomials are not constructed, but instead the associated recurrence relations are computed. Moreover, also typically the inner product is changed to a discrete inner product, which is the finite sum of weighted functions evaluated in specific nodes. For particular applications it is beneficial to have an efficient procedure to update the recurrence relations when adding or removing nodes from the inner product. The construction of the recurrence relations is equivalent to computing a structured matrix (polynomial) or pencil (rational) having prescribed spectral properties. Hence the solution of this problem is often referred to as solving an Inverse Eigenvalue Problem. In [34] we proposed updating techniques to add nodes to the inner product while efficiently updating the recurrences. To complete this study we present in this article manners to efficiently downdate the recurrences when removing nodes from the inner product. The link between removing nodes and the QR algorithm to deflate eigenvalues is exploited to develop efficient algorithms. We will base ourselves on the perfect shift strategy and develop algorithms, both for the polynomial case and the rational function setting. Numerical experiments validate our approach.

Czech name

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Czech description

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Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

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OECD FORD branch

10102 - Applied mathematics

Project

—

Continuities

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year

2024

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Name of the periodical

Applied Numerical Mathematics

ISSN

0168-9274

e-ISSN

1873-5460

Volume of the periodical

200

Issue of the periodical within the volume

June 2024

Country of publishing house

NL - THE KINGDOM OF THE NETHERLANDS

Number of pages

24

Pages from-to

429-452

UT code for WoS article

001238573400001

EID of the result in the Scopus database

2-s2.0-85166060392