The Conditioning of Linearizations of Matrix Polynomials

Abstract

Abstract. The standard way of solving the polynomial eigenvalue problem of degree m in n × n matrices is to "linearize" to a pencil in mn × mn matrices and solve the generalized eigenvalue problem. For a given polynomial, P , infinitely many linearizations exist and they can have widely varying eigenvalue condition numbers. We investigate the conditioning of linearizations from a vector space DL(P ) of pencils recently identified and studied by Mackey, Mackey, Mehl, and Mehrmann. We look for the best conditioned linearization and compare the conditioning with that of the original polynomial. Two particular pencils are shown always to be almost optimal over linearizations in DL(P ) for eigenvalues of modulus greater than or less than 1, respectively, provided that the problem is not too badly scaled and that the pencils are linearizations. Moreover, under this scaling assumption, these pencils are shown to be about as well conditioned as the original polynomial. For quadratic eigenvalue problems that are not too heavily damped, a simple scaling is shown to convert the problem to one that is well scaled. We also analyze the eigenvalue conditioning of the widely used first and second companion linearizations. The conditioning of the first companion linearization relative to that of P is shown to depend on the coefficient matrix norms, the eigenvalue, and the left eigenvectors of the linearization and of P . The companion form is found to be potentially much more ill conditioned than P , but if the 2-norms of the coefficient matrices are all approximately 1 then the companion form and P are guaranteed to have similar condition numbers. Analogous results hold for the second companion form. Our results are phrased in terms of both the standard relative condition number and the condition number of Dedieu and Tisseur for the problem in homogeneous form, this latter condition number having the advantage of applying to zero and infinite eigenvalues.

Polynomial Eigenvalue Problems: Theory, Computation, and Structure

“…Unless the block structure of the linearization is respected (and it is not by standard algorithms), the conditioning of the larger linear problem can be worse than that of the original matrix polynomial, since the class of admissible perturbations is larger. For example, eigenvalues that are wellconditioned for P(λ ) may be ill-conditioned for L(λ ) [39,41,78]. Ideally, when solving (20) via (21) we would like to have κ P (λ ) ≈ κ L (λ ).…”

“…This is done for example in [39] for pencils L ∈ DL(P), where minimization of the ratio over L is considered. Backward errors characterize the stability of a numerical method for solving a problem by measuring how far the problem has to be perturbed for an approximate solution to be an exact solution of the perturbed problem.…”

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

“…This will be a consequence of the following result. (12), with U and u partitioned as in (14), the relation…”

“…This yields an equivalent dn×dn linear eigenvalue problem L 0 −λL 1 , to which standard linear eigenvalue solvers like the QZ algorithm or the Arnoldi method [11] can be applied. Many different linearizations are possible, and much progress has been made in the last few years in developing a unified framework [21], which allows, e.g., for structure preservation [20] and a unified sensitivity analysis [14,1].…”

Pseudospectra of Matrix Polynomials that Are Expressed in Alternative Bases

“…When working with matrix polynomials, it is standard to use a strong linearization to convert the polynomial eigenvalue problem to a generalized eigenvalue problem [3,14,15]. A strong linearization, as opposed to weaker linearizations, preserves eigenstructure at infinity as well as the finite eigenstructure.…”