zbMATH — the first resource for mathematics

Jacobi-Davidson methods for polynomial two-parameter eigenvalue problems. (English) Zbl 06448790
Summary: We propose Jacobi-Davidson type methods for polynomial two-parameter eigenvalue problems (PMEP). Such problems can be linearized as singular two-parameter eigenvalue problems, whose matrices are of dimension \(k(k+1) n / 2\), where \(k\) is the degree of the polynomial and \(n\) is the size of the matrix coefficients in the PMEP. When \(k^2 n\) is relatively small, the problem can be solved numerically by computing the common regular part of the related pair of singular pencils. For large \(k^2 n\), computing all solutions is not feasible and iterative methods are required.
When \(k\) is large, we propose to linearize the problem first and then apply Jacobi-Davidson to the obtained singular two-parameter eigenvalue problem. The resulting method may for instance be used for computing zeros of a system of scalar bivariate polynomials close to a given target. On the other hand, when \(k\) is small, we can apply a Jacobi-Davidson type approach directly to the original matrices. The original matrices are projected onto a low-dimensional subspace, and the projected polynomial two-parameter eigenvalue problems are solved by a linearization.

65F15 Numerical computation of eigenvalues and eigenvectors of matrices
Full Text: DOI
[1] Jarlebring, E.; Hochstenbach, M. E., Polynomial two-parameter eigenvalue problems and matrix pencil methods for stability of delay-differential equations, Linear Algebra Appl., 431, 369-380, (2009) · Zbl 1170.65063
[2] Meerbergen, K.; Schröder, C.; Voss, H., A Jacobi-Davidson method for two-real-parameter nonlinear eigenvalue problems arising from delay-differential equations, Numer. Linear Algebra Appl., 20, 852-868, (2013) · Zbl 1313.65083
[3] Hochstenbach, M. E.; Muhič, A.; Plestenjak, B., On linearizations of the quadratic two-parameter eigenvalue problems, Linear Algebra Appl., 436, 2725-2743, (2012) · Zbl 1245.65042
[4] Muhič, A.; Plestenjak, B., On the quadratic two-parameter eigenvalue problem and its linearization, Linear Algebra Appl., 432, 2529-2542, (2010) · Zbl 1189.65070
[5] Atkinson, F. V., Multiparameter eigenvalue problems, (1972), Academic Press New York · Zbl 0555.47001
[6] Hochstenbach, M. E.; Košir, T.; Plestenjak, B., A Jacobi-Davidson type method for the nonsingular two-parameter eigenvalue problem, SIAM J. Matrix Anal. Appl., 26, 477-497, (2005) · Zbl 1077.65036
[7] Hochstenbach, M. E.; Plestenjak, B., Harmonic Rayleigh-Ritz for the multiparameter eigenvalue problem, Electron. Trans. Numer. Anal., 29, 81-96, (2008) · Zbl 1171.65378
[8] Hochstenbach, M. E.; Plestenjak, B., A Jacobi-Davidson type method for a right definite two-parameter eigenvalue problem, SIAM J. Matrix Anal. Appl., 24, 392-410, (2002) · Zbl 1025.65023
[9] Muhič, A.; Plestenjak, B., On the singular two-parameter eigenvalue problem, Electron. J. Linear Algebra, 18, 420-437, (2009) · Zbl 1190.15011
[10] van Dooren, P., The computation of kronecker’s canonical form of a singular pencil, Linear Algebra Appl., 27, 103-141, (1979) · Zbl 0416.65026
[11] Hochstenbach, M. E.; Sleijpen, G. L.G., Harmonic and refined Rayleigh-Ritz for the polynomial eigenvalue problem, Numer. Linear Algebra Appl., 15, 35-54, (2008) · Zbl 1212.65150
[12] MATLAB, The MathWorks, Inc.., Natick, Massachusetts, United States.
[13] P. Dreesen, K. Batselier, B. De Moor, Back to the roots: Polynomial system solving, linear algebra, systems theory, in: Proceedings of the 16th IFAC Symposium on System Identification, Brussels, Belgium, 2012, pp. 1203-1208.
[14] Stetter, H. J., Numerical polynomial algebra, (2004), SIAM Philadelphia · Zbl 1058.65054
[15] Stewart, G. W., Matrix algorithms volume 2: eigensystems, (2001), SIAM · Zbl 0984.65031
[16] Jarlebring, E., The spectrum of delay-differential equations: numerical methods, stability and perturbation, (2008), TU Braunschweig, (Ph.D. thesis)
[17] Guan, Y.; Verschelde, J., Phclab: A MATLAB/octave interface to phcpack, (Stillman, M.; Verschelde, J.; Takayama, N., Software for Algebraic Geometry, The IMA Volumes in Mathematics and its Applications, vol. 148, (2008), Springer New York), 15-32 · Zbl 1148.68578
[18] Verschelde, J., Algorithm 795: phcpack: a general-purpose solver for polynomial systems by homotopy continuation, ACM Trans. Math. Software, 25, 251-276, (1999) · Zbl 0961.65047
[19] Nakatsukasa, Y.; Noferini, V.; Townsend, A., Computing the common zeros of two bivariate functions via Bézout resultants, Numer. Math., 129, 181-209, (2015) · Zbl 1308.65076
[20] Townsend, A.; Trefethen, L. N., An extension of chebfun to two dimensions, SIAM J. Sci. Comput., 35, 495-518, (2013) · Zbl 1300.65010
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.