URV decomposition based structured methods for palindromic and even eigenvalue problems
Christian A. Schröder · 2007
In this work numerical methods for the solution of two classes of structured generalized eigenvalue problems, Ax = �Bx, are developed. Those classes are the palindromic (B = A T ) and the even (A = A T , B = −B T ) eigenvalue problems. The spectrum of these problems is not arbitrary, rather do eigenvalues occur in pairs. We will construct methods for palindromic and even eigenvalue problems that are of cubic complexity and that are guaranteed to produce eigenvalues that are paired to working precision. At the heart of both methods is a new URV-type matrix decomposition, that simultaneously transforms three matrices to skew triangular form, i.e., to a form that is triangular with respect to the Northeast-Southwest diagonal. The algorithm to compute this URV decomposition uses several other methods to reduce a single square matrix to skew triangular form: the skew QR factorization and the skew QRQ T decomposition. Moreover, a method to compute the singular value decomposition of a complex, skew symmetric matrix is presented and used.