Abstract
In this paper we study both direct and inverse eigenvalue problems for diagonal-plus-semiseparable (dpss) matrices. In particular, we show that the computation of the eigenvalues of a symmetric dpss matrix can be reduced by a congruence transformation to solving a generalized symmetric definite tridiagonal eigenproblem. Using this reduction, we devise a set of recurrence relations for evaluating the characteristic polynomial of a dpss matrix in a stable way at a linear time. This in turn allows us to apply divide-and-conquer eigenvalue solvers based on functional iterations directly to dpss matrices without performing any preliminary reduction into a tridiagonal form. In the second part of the paper, we exploit the structural properties of dpss matrices to solve the inverse eigenvalue problem of reconstructing a symmetric dpss matrix from its spectrum and some other informations. Finally, applications of our results to the computation of a QR factorization of a Cauchy matrix with real nodes are provided.
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Fasino, D., Gemignani, L. Direct and Inverse Eigenvalue Problems for Diagonal-Plus-Semiseparable Matrices. Numerical Algorithms 34, 313–324 (2003). https://doi.org/10.1023/B:NUMA.0000005402.66868.af
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DOI: https://doi.org/10.1023/B:NUMA.0000005402.66868.af