A tridiagonal matrix construction by the quotient difference recursion formula in the case of multiple eigenvalues
type: Item
instance of: scholarly article
title: A tridiagonal matrix construction by the quotient difference recursion formula in the case of multiple eigenvalues
zbMATH Open document ID: 1329.65080
author: Kanae Akaiwa
author: Masashi Iwasaki
author: Koichi Kondo
author: Yoshimasa Nakamura
published in: Pacific Journal of Mathematics for Industry
publication date: 2015-11-10T00:00:00Z
review text: The authors consider the inverse eigenvalue problem to find, from (multiple) eigenvalues, a tridiagonal matrix with these eigenvalues, including their multiplicities. For this, they use the classical quotient difference (qd) algorithm, see, e.g. [\textit{M. H. Gutknecht} and \textit{B. N. Parlett}, IMA J. Numer. Anal. 31, No. 3, 741--754 (2011; Zbl 1222.65032)]. This possibility seems to be have known to \textit{H. Rutishauser} [Lectures on numerical mathematics. Ed. by Martin Gutknecht with the assistance of Peter Henrici, Peter Läuchli and Hans-Rudolf Schwarz. Transl. by Walter Gautschi. Boston etc.: Birkhäuser (1990; Zbl 0699.65002)], but the present authors prove that it works. They need, along with the eigenvalues, their multiplicities and also the degrees of the characteristic polynomial and of the minimal polynomial as obtained if the eigenvalues are written in a Jordan form matrix. To start the qd algorithm, they also need two vectors the choice of which is said to be not problematic. The authors prove that the minimal polynomial of the tridiagonal matrix obtained is equal to its characteristic polynomial, containing the prescribed eigenvalues, and that the tridiagonal matrix is diagonalizable if and only if there are no multiple eigenvalues. The algorithm is thought to be used for symbolic computation because of rounding errors and their consequences under floating point computations. Several examples are provided for the algorithm.
reviewed by: Gisbert Stoyan
Mathematics Subject Classification ID: 65F18
Mathematics Subject Classification ID: 65F50
Mathematics Subject Classification ID: 68W30
zbMATH DE Number: 6506869
zbMATH Keywords: inverse eigenvalue problem
zbMATH Keywords: quotient difference algorithm
zbMATH Keywords: multiple eigenvalues
zbMATH Keywords: tridiagonal matrix
zbMATH Keywords: characteristic polynomial
zbMATH Keywords: minimal polynomial
zbMATH Keywords: numerical example
zbMATH Keywords: symbolic computation
Wikidata QID: Q59396835
MaRDI profile type: Publication
full work available at URL: s40736-014-0010-0
OpenAlex ID: W2015416660
cites work: Q4681781
cites work: From qd to LR, or, how were the qd and LR algorithms discovered?
cites work: Finding zeros of a polynomial by the Q-D algorithm
cites work: Q4072022
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cites work: Q4485695
cites work: Q4840770
cites work: Bestimmung der Eigenwerte und Eigenvektoren einer Matrix mit Hilfe des Quotienten-Differenzen-Algorithmus
cites work: Q3996941
DOI: 10.1186/S40736-014-0010-0
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description: scientific article; zbMATH DE number 6506869