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Abstract. The most widely used approach for solving the polynomial eigenvalue problem P (λ)x = m i=0 λ i A i x = 0 in n × n matrices A i is to linearize to produce a larger order pencil L(λ) = λX + Y , whose eigensystem is then found by any method for generalized eigenproblems. For a given polynomial P , infinitely many linearizations L exist and approximate eigenpairs of P computed via linearization can have widely varying backward errors. We show that if a certain one-sided factorization relating L to P can be found then a simple formula permits recovery of right eigenvectors of P from those of L, and the backward error of an approximate eigenpair of P can be bounded in terms of the backward error for the corresponding approximate eigenpair of L. A similar factorization has the same implications for left eigenvectors. We use this technique to derive backward error bounds depending only on the norms of the A i for the companion pencils and for the vector space DL(P ) of pencils recently identified by Mackey, Mackey, Mehl, and Mehrmann. In all cases, sufficient conditions are identified for an optimal backward error for P . These results are shown to be entirely consistent with those of Higham, Mackey, and Tisseur on the conditioning of linearizations of P . Other contributions of this work are a block scaling of the companion pencils that yields improved backward error bounds; a demonstration that the bounds are applicable to certain structured linearizations of structured polynomials; and backward error bounds specialized to the quadratic case, including analysis of the benefits of a scaling recently proposed by Fan, Lin, and Van Dooren. The results herein make no assumptions on the stability of the method applied to L or whether the method is direct or iterative.

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Abstract. Many models of physical and chemical processes give rise to ordinary differential equations with special structural properties that go unexploited by general-purpose software designed to solve numerically a wide range of differential equations. If those properties are to be exploited fully for the sake of better numerical stability, accuracy and/or speed, the differential equations may have to be solved by unconventional methods. This short paper is to publish composition constants obtained by the authors to increase efficiency of a family of mostly unconventional methods, called reflexive.

A new doubling algorithm-Alternating-Directional Doubling Algorithm (ADDA)-is developed for computing the unique minimal nonnegative solution of an M-Matrix Algebraic Riccati Equation (MARE). It is argued by both theoretical analysis and numerical experiments that ADDA is always faster than two existing doubling algorithms