A framework for structured linearizations of matrix polynomials in various bases

We present a framework for the construction of linearizations for scalar and matrix polynomials based on dual bases which, in the case of orthogonal polynomials, can be described by the associated recurrence relations. The framework provides an extension of the classical linearization theory for polynomials expressed in non-monomial bases and allows to represent polynomials expressed in product families, that is as a linear combination of elements of the form $\phi_i(\lambda) \psi_j(\lambda)$, where $\{ \phi_i(\lambda) \}$ and $\{ \psi_j(\lambda) \}$ can either be polynomial bases or polynomial families which satisfy some mild assumptions. We show that this general construction can be used for many different purposes. Among them, we show how to linearize sums of polynomials and rational functions expressed in different bases. As an example, this allows to look for intersections of functions interpolated on different nodes without converting them to the same basis. We then provide some constructions for structured linearizations for $\star$-even and $\star$-palindromic matrix polynomials. The extensions of these constructions to $\star$-odd and $\star$-antipalindromic of odd degree is discussed and follows immediately from the previous results

Robustness and perturbations of minimal bases II: The case with given row degrees

This paper studies generic and perturbation properties inside the linear space of $m\times (m+n)$ polynomial matrices whose rows have degrees bounded by a given list $d_1, \ldots, d_m$ of natural numbers, which in the particular case $d_1 = \cdots = d_m = d$ is just the set of $m\times (m+n)$ polynomial matrices with degree at most $d$. Thus, the results in this paper extend to a much more general setting the results recently obtained in [Van Dooren & Dopico, Linear Algebra Appl. (2017), http://dx.doi.org/10.1016/j.laa.2017.05.011] only for polynomial matrices with degree at most $d$. Surprisingly, most of the properties proved in [Van Dooren & Dopico, Linear Algebra Appl. (2017)], as well as their proofs, remain to a large extent unchanged in this general setting of row degrees bounded by a list that can be arbitrarily inhomogeneous provided the well-known Sylvester matrices of polynomial matrices are replaced by the new trimmed Sylvester matrices introduced in this paper. The following results are presented, among many others, in this work: (1) generically the polynomial matrices in the considered set are minimal bases with their row degrees exactly equal to $d_1, \ldots , d_m$, and with right minimal indices differing at most by one and having a sum equal to $\sum_{i=1}^{m} d_i$, and (2), under perturbations, these generic minimal bases are robust and their dual minimal bases can be chosen to vary smoothly.Comment: arXiv admin note: text overlap with arXiv:1612.0379

A class of quasi-sparse companion pencils

In this paper, we introduce a general class of quasi-sparse potential companion pencils for arbitrary square matrix polynomials over an arbitrary field, which extends the class introduced in [B. Eastman, I.-J. Kim, B. L. Shader, K.N. Vander Meulen, Companion matrix patterns. Linear Algebra Appl. 436 (2014) 255-272] for monic scalar polynomials. We provide a canonical form, up to permutation, for companion pencils in this class. We also relate these companion pencils with other relevant families of companion linearizations known so far. Finally, we determine the number of different sparse companion pencils in the class, up to permutation.This work has been partially supported by theMinisterio de Economía y Competitividad of Spain through grants MTM2015-68805-REDT and MTM2015-65798-P

Memory-efficient Arnoldi algorithms for linearizations of matrix polynomials in Chebyshev basis

Novel memory-efficient Arnoldi algorithms for solving matrix polynomial eigenvalue problems are presented. More specifically, we consider the case of matrix polynomials expressed in the Chebyshev basis, which is often numerically more appropriate than the standard monomial basis for a larger degree $d$. The standard way of solving polynomial eigenvalue problems proceeds by linearization, which increases the problem size by a factor $d$. Consequently, the memory requirements of Krylov subspace methods applied to the linearization grow by this factor. In this paper, we develop two variants of the Arnoldi method that build the Krylov subspace basis implicitly, in a way that only vectors of length equal to the size of the original problem need to be stored. The proposed variants are generalizations of the so called Q-Arnoldi and TOAR methods, which have been developed for the monomial case. We also show how the typical ingredients of a full implementation of the Arnoldi method, including shift-and-invert and restarting, can be incorporated. Numerical experiments are presented for matrix polynomials up to degree $30$ arising from the interpolation of nonlinear eigenvalue problems which stem from boundary element discretizations of PDE eigenvalue problems.Daniel Kressner; Román Moltó, JE. (2014). Memory-efficient Arnoldi algorithms for linearizations of matrix polynomials in Chebyshev basis. Numerical Linear Algebra with Applications. 21(4):569-588. doi:10.1002/nla.1913S569588214Mackey, D. S., Mackey, N., Mehl, C., & Mehrmann, V. (2006). Vector Spaces of Linearizations for Matrix Polynomials. SIAM Journal on Matrix Analysis and Applications, 28(4), 971-1004. doi:10.1137/050628350Mackey, D. S., Mackey, N., Mehl, C., & Mehrmann, V. (2006). Structured Polynomial Eigenvalue Problems: Good Vibrations from Good Linearizations. 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BIT Numerical Mathematics, 52(4), 933-951. doi:10.1007/s10543-012-0381-5Van Beeumen, R., Meerbergen, K., & Michiels, W. (2013). A Rational Krylov Method Based on Hermite Interpolation for Nonlinear Eigenvalue Problems. SIAM Journal on Scientific Computing, 35(1), A327-A350. doi:10.1137/120877556Sitton, G. A., Burrus, C. S., Fox, J. W., & Treitel, S. (2003). Factoring very-high-degree polynomials. IEEE Signal Processing Magazine, 20(6), 27-42. doi:10.1109/msp.2003.1253552Amiraslani, A., Corless, R. M., & Lancaster, P. (2008). Linearization of matrix polynomials expressed in polynomial bases. IMA Journal of Numerical Analysis, 29(1), 141-157. doi:10.1093/imanum/drm051Betcke, T., & Kressner, D. (2011). Perturbation, extraction and refinement of invariant pairs for matrix polynomials. Linear Algebra and its Applications, 435(3), 514-536. doi:10.1016/j.laa.2010.06.029Beyn, W. J., & Thümmler, V. (2010). Continuation of Invariant Subspaces for Parameterized Quadratic Eigenvalue Problems. SIAM Journal on Matrix Analysis and Applications, 31(3), 1361-1381. doi:10.1137/080723107Kressner, D. (2009). A block Newton method for nonlinear eigenvalue problems. Numerische Mathematik, 114(2), 355-372. doi:10.1007/s00211-009-0259-xLehoucq, R. B., Sorensen, D. C., & Yang, C. (1998). ARPACK Users’ Guide. doi:10.1137/1.9780898719628Hernandez, V., Roman, J. E., & Vidal, V. (2005). SLEPc. ACM Transactions on Mathematical Software, 31(3), 351-362. doi:10.1145/1089014.1089019Clenshaw, C. W. (1955). A note on the summation of Chebyshev series. Mathematics of Computation, 9(51), 118-118. doi:10.1090/s0025-5718-1955-0071856-0Stewart, G. W. (2002). A Krylov--Schur Algorithm for Large Eigenproblems. SIAM Journal on Matrix Analysis and Applications, 23(3), 601-614. doi:10.1137/s0895479800371529Su Y A compact Arnoldi algorithm for polynomial eigenvalue problems 2008 http://math.cts.nthu.edu.tw/Mathematics/RANMEP%20Slides/Yangfeng%20Su.pdfSteinbach, O., & Unger, G. (2009). A boundary element method for the Dirichlet eigenvalue problem of the Laplace operator. Numerische Mathematik, 113(2), 281-298. doi:10.1007/s00211-009-0239-1Effenberger, C., Kressner, D., Steinbach, O., & Unger, G. (2012). Interpolation-based solution of a nonlinear eigenvalue problem in fluid-structure interaction. PAMM, 12(1), 633-634. doi:10.1002/pamm.201210305Betcke, T., Higham, N. J., Mehrmann, V., Schröder, C., & Tisseur, F. (2013). NLEVP. ACM Transactions on Mathematical Software, 39(2), 1-28. doi:10.1145/2427023.2427024Grammont, L., Higham, N. J., & Tisseur, F. (2011). A framework for analyzing nonlinear eigenproblems and parametrized linear systems. Linear Algebra and its Applications, 435(3), 623-640. doi:10.1016/j.laa.2009.12.03