On condition numbers of polynomial eigenvalue problems with nonsingular leading coefficients

In this paper, we investigate condition numbers of eigenvalue problems of matrix polynomials with nonsingular leading coefficients, generalizing classical results of matrix perturbation theory. We provide a relation between the condition numbers of eigenvalues and the pseudospectral growth rate. We obtain that if a simple eigenvalue of a matrix polynomial is ill-conditioned in some respects, then it is close to be multiple, and we construct an upper bound for this distance (measured in the euclidean norm). We also derive a new expression for the condition number of a simple eigenvalue, which does not involve eigenvectors. Moreover, an Elsner-like perturbation bound for matrix polynomials is presented.Comment: 4 figure

Conditioning and backward errors of eigenvalues of homogeneous matrix polynomials under Möbius transformations

We present the first general study on the effect of Möbius transformations on the eigenvalue condition numbers and backward errors of approximate eigenpairs of polynomial eigenvalue problems (PEPs). By usingthe homogeneous formulation of PEPs, we are able to obtain two clear andsimple results. First, we show that if the matrix inducing the Möbius transformation is well-conditioned, then such transformation approximately preservesthe eigenvalue condition numbers and backward errors when they are definedwith respect to perturbations of the matrix polynomial which are small relativeto the norm of the whole polynomial. However, if the perturbations in eachcoefficient of the matrix polynomial are small relative to the norm of that coefficient, then the corresponding eigenvalue condition numbers and backwarderrors are preserved approximately by the Möbius transformations induced bywell-conditioned matrices only if a penalty factor, depending on the norms ofthose matrix coefficients, is moderate. It is important to note that these simple results are no longer true if a non-homogeneous formulation of the PEP isused

Structured Hölder condition numbers for multiple eigenvalues

The sensitivity of a multiple eigenvalue of a matrix under perturbations can be measured by its Hölder condition number. Various extensions of this concept are considered. A meaningful notion of structured Hölder condition numbers is introduced, and it is shown that many existing results on structured condition numbers for simple eigenvalues carry over to multiple eigenvalues. The structures investigated in more detail include real, Toeplitz, Hankel, symmetric, skewsymmetric, Hamiltonian, and skew-Hamiltonian matrices. Furthermore, unstructured and structured Hölder condition numbers for multiple eigenvalues of matrix pencils are introduced. Particular attention is given to symmetric/skew-symmetric, Hermitian, and palindromic pencils. It is also shown how matrix polynomial eigenvalue problems can be covered within this framework. © by SIAM

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