On generating Sobolev orthogonal polynomials

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F23%3A10473004" target="_blank" >RIV/00216208:11320/23:10473004 - isvavai.cz</a>

Výsledek na webu

<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=ca7c6saHCX" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=ca7c6saHCX</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s00211-023-01379-3" target="_blank" >10.1007/s00211-023-01379-3</a>

Jazyk výsledku

angličtina

Název v původním jazyce

On generating Sobolev orthogonal polynomials

Popis výsledku v původním jazyce

Sobolev orthogonal polynomials are polynomials orthogonal with respect to a Sobolev inner product, an inner product in which derivatives of the polynomials appear. They satisfy a long recurrence relation that can be represented by a Hessenberg matrix. The problem of generating a finite sequence of Sobolev orthogonal polynomials can be reformulated as a matrix problem, that is, a Hessenberg inverse eigenvalue problem, where the Hessenberg matrix of recurrences is generated from certain known spectral information. Via the connection to Krylov subspaces we show that the required spectral information is the Jordan matrix containing the eigenvalues of the Hessenberg matrix and the normalized first entries of its eigenvectors. Using a suitable quadrature rule the Sobolev inner product is discretized and the resulting quadrature nodes form the Jordan matrix and associated quadrature weights are the first entries of the eigenvectors. We propose two new numerical procedures to compute Sobolev orthonormal polynomials based on solving the equivalent Hessenberg inverse eigenvalue problem.

Název v anglickém jazyce

On generating Sobolev orthogonal polynomials

Popis výsledku anglicky

Sobolev orthogonal polynomials are polynomials orthogonal with respect to a Sobolev inner product, an inner product in which derivatives of the polynomials appear. They satisfy a long recurrence relation that can be represented by a Hessenberg matrix. The problem of generating a finite sequence of Sobolev orthogonal polynomials can be reformulated as a matrix problem, that is, a Hessenberg inverse eigenvalue problem, where the Hessenberg matrix of recurrences is generated from certain known spectral information. Via the connection to Krylov subspaces we show that the required spectral information is the Jordan matrix containing the eigenvalues of the Hessenberg matrix and the normalized first entries of its eigenvectors. Using a suitable quadrature rule the Sobolev inner product is discretized and the resulting quadrature nodes form the Jordan matrix and associated quadrature weights are the first entries of the eigenvectors. We propose two new numerical procedures to compute Sobolev orthonormal polynomials based on solving the equivalent Hessenberg inverse eigenvalue problem.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2023

Kód důvěrnosti údajů

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

Název periodika

Numerische Mathematik

ISSN

0029-599X

e-ISSN

0945-3245

Svazek periodika

2023

Číslo periodika v rámci svazku

3-4

Stát vydavatele periodika

DE - Spolková republika Německo

Počet stran výsledku

29

Strana od-do

415-443

Kód UT WoS článku

001288845800001

EID výsledku v databázi Scopus

2-s2.0-85175347798