×

zbMATH — the first resource for mathematics

Tridiagonal test matrices for eigenvalue computations: two-parameter extensions of the Clement matrix. (English) Zbl 1354.65075
Summary: The Clement or Sylvester-Kac matrix is a tridiagonal matrix with zero diagonal and simple integer entries. Its spectrum is known explicitly and consists of integers which makes it a useful test matrix for numerical eigenvalue computations. We consider a new class of appealing two-parameter extensions of this matrix which have the same simple structure and whose eigenvalues are also given explicitly by a simple closed form expression. The aim of this paper is to present in an accessible form these new matrices and examine some numerical results regarding the use of these extensions as test matrices for numerical eigenvalue computations.

MSC:
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
15A18 Eigenvalues, singular values, and eigenvectors
65F50 Computational methods for sparse matrices
15B36 Matrices of integers
Software:
testmatrix
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Sylvester, J., Théoreme sur LES déterminants, Nouv. Ann. Math., 13, (1854), 305
[2] Kac, M., Random walk and the theory of Brownian motion, Amer. Math. Monthly, 54, 7, 369-391, (1947) · Zbl 0031.22604
[3] Clement, P. A., A class of triple-diagonal matrices for test purposes, SIAM Rev., 1, 1, 50-52, (1959) · Zbl 0117.14202
[4] Taussky, O.; Todd, J., Another look at a matrix of mark Kac, Linear Algebra Appl., 150, 341-360, (1991) · Zbl 0727.15010
[5] A. Edelman, E. Kostlan, The road from kacs matrix to kacs random polynomials, in: J. Lewis (Ed.), Proceedings of the Fifth SIAM Conference on Applied Linear Algebra, 1994, pp. 503-507. · Zbl 0817.15014
[6] Boros, T.; Rózsa, P., An explicit formula for singular values of the Sylvester-Kac matrix, Linear Algebra Appl., 421, 2, 407-416, (2007) · Zbl 1121.15012
[7] Bevilacqua, R.; Bozzo, E., The Sylvester-Kac matrix space, Linear Algebra Appl., 430, 11, 3131-3138, (2009) · Zbl 1167.15014
[8] Nomura, K.; Terwilliger, P., Krawtchouk polynomials, the Lie algebra sl(2), and leonard pairs, Linear Algebra Appl., 437, 1, 345-375, (2012) · Zbl 1261.33001
[9] Gregory, R. T.R. T.; Karney, D. L., A collection of matrices for testing computational algorithms, tech. rep., (1978)
[10] N.J. Higham, The test matrix toolbox for MATLAB (Version 3.0), University of Manchester Manchester, 1995.
[11] Abdel-Rehim, E., From the Ehrenfest model to time-fractional stochastic processes, J. Comput. Appl. Math., 233, 2, 197-207, (2009) · Zbl 1185.60048
[12] Igelnik, B.; Simon, D., The eigenvalues of a tridiagonal matrix in biogeography, Appl. Math. Comput., 218, 1, 195-201, (2011) · Zbl 1255.15009
[13] Cuminato, J. A.; McKee, S., A note on the eigenvalues of a special class of matrices, J. Comput. Appl. Math., 234, 9, 2724-2731, (2010) · Zbl 1195.15020
[14] Yueh, W.-C., Eigenvalues of several tridiagonal matrices, Appl. Math. E-Notes, 5, 66-74, 210-230, (2005)
[15] Oste, R.; Van der Jeugt, J., Doubling (dual) Hahn polynomials: classification and applications, Symmetry Integrability Geom. Methods Appl., 12, 003-027, (2016) · Zbl 1333.81141
[16] Jafarov, E.; Stoilova, N.; Van der Jeugt, J., Finite oscillator models: the Hahn oscillator, J. Phys. A, 44, 26, (2011) · Zbl 1220.81091
[17] Oste, R.; Van der Jeugt, J., A finite oscillator model with equidistant position spectrum based on an extension of su(2), J. Phys. A, 49, 17, (2016) · Zbl 1344.81077
[18] Koekoek, R.; Lesky, P. A.; Swarttouw, R. F., Hypergeometric orthogonal polynomials and their q-analogues, (2010), Springer Science & Business Media · Zbl 1200.33012
[19] Galántai, A.; Hegedűs, C., Hymans method revisited, J. Comput. Appl. Math., 226, 2, 246-258, (2009) · Zbl 1181.65053
[20] Maeda, K.; Tsujimoto, S., A generalized eigenvalue algorithm for tridiagonal matrix pencils based on a nonautonomous discrete integrable system, J. Comput. Appl. Math., 300, 134-154, (2016) · Zbl 1342.65115
[21] Brockman, P.; Carson, T.; Cheng, Y.; Elgindi, T.; Jensen, K.; Zhoun, X.; Elgindi, M., Homotopy method for the eigenvalues of symmetric tridiagonal matrices, J. Comput. Appl. Math., 237, 1, 644-653, (2013) · Zbl 1256.65028
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.