We survey the quadratic eigenvalue problem, treating its many applications, its mathematical properties, and a variety of numerical solution techniques. Emphasis is given to exploiting both the structure of the matrices in the problem (dense, sparse, real, complex, Hermitian, skew-Hermitian) and the spectral properties of the problem. We classify numerical methods and catalogue available software.

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Nonlinear eigenvalue problems arise in a variety of science and engineering applications and in the past ten years there have been numerous breakthroughs in the development of numerical methods. This article surveys nonlinear eigenvalue problems associated with matrix-valued functions which depend nonlinearly on a single scalar parameter, with a particular emphasis on their mathematical properties and available numerical solution techniques. Solvers based on Newton's method, contour integration, and sampling via rational interpolation are reviewed. Problems of selecting the appropriate parameters for each of the solver classes are discussed and illustrated with numerical examples. This survey also contains numerous MATLAB code snippets that can be used for interactive exploration of the discussed methods.

Abstract.A standard way of treating the polynomial eigenvalue problem P (λ)x = 0 is to convert it into an equivalent matrix pencil-a process known as linearization. Two vector spaces of pencils L 1 (P ) and L 2 (P ), and their intersection DL(P ), have recently been defined and studied by Mackey, Mackey, Mehl, and Mehrmann. The aim of our work is to gain new insight into these spaces and the extent to which their constituent pencils inherit structure from P . For arbitrary polynomials we show that every pencil in DL(P ) is block symmetric and we obtain a convenient basis for DL(P ) built from block Hankel matrices. This basis is then exploited to prove that the first deg(P ) pencils in a sequence constructed by Lancaster in the 1960s generate DL(P ). When P is symmetric, we show that the symmetric pencils in L 1 (P ) comprise DL(P ), while for Hermitian P the Hermitian pencils in L 1 (P ) form a proper subset of DL(P ) that we explicitly characterize. Almost all pencils in each of these subsets are shown to be linearizations. In addition to obtaining new results, this work provides a self-contained treatment of some of the key properties of DL(P ) together with some new, more concise proofs.

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.