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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.

##### MSC:
 65F15 Numerical computation of eigenvalues and eigenvectors of matrices
##### Software:
Chebfun; Chebfun2; Matlab; MultiParEig; PHClab; PHCpack; rootsb
Full Text:
##### References:
  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  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  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  Muhič, A.; Plestenjak, B., On the quadratic two-parameter eigenvalue problem and its linearization, Linear Algebra Appl., 432, 2529-2542, (2010) · Zbl 1189.65070  Atkinson, F. V., Multiparameter eigenvalue problems, (1972), Academic Press New York · Zbl 0555.47001  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  Hochstenbach, M. E.; Plestenjak, B., Harmonic Rayleigh-Ritz for the multiparameter eigenvalue problem, Electron. Trans. Numer. Anal., 29, 81-96, (2008) · Zbl 1171.65378  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  Muhič, A.; Plestenjak, B., On the singular two-parameter eigenvalue problem, Electron. J. Linear Algebra, 18, 420-437, (2009) · Zbl 1190.15011  van Dooren, P., The computation of kronecker’s canonical form of a singular pencil, Linear Algebra Appl., 27, 103-141, (1979) · Zbl 0416.65026  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  MATLAB, The MathWorks, Inc.., Natick, Massachusetts, United States.  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.  Stetter, H. J., Numerical polynomial algebra, (2004), SIAM Philadelphia · Zbl 1058.65054  Stewart, G. W., Matrix algorithms volume 2: eigensystems, (2001), SIAM · Zbl 0984.65031  Jarlebring, E., The spectrum of delay-differential equations: numerical methods, stability and perturbation, (2008), TU Braunschweig, (Ph.D. thesis)  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  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  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  Townsend, A.; Trefethen, L. N., An extension of chebfun to two dimensions, SIAM J. Sci. Comput., 35, 495-518, (2013) · Zbl 1300.65010
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