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New numerical algorithm for deflation of infinite and zero eigenvalues and full solution of quadratic eigenvalue problems

In this talk we will present a new method for computing all eigenvalues and eigenvectors of the quadratic eigenvalue problem. It is an upgrade of the quadeig algorithm by Hammarling, Munro and Tisseur, which attempts to reveal and remove by deflation certain number of zero and infinite eigenvalues before QZ iterations. Proposed modifications of the quadeig framework are designed to enhance backward stability and to make the process of deflating infinite and zero eigenvalues more numerically robust. Using an upper triangular version of the Kronecker canonical form proposed algorithm deflates additional infinite and zero eigenvalues, in addition to those conducted from the rank of the corresponding leading coefficient matrix and constant coefficient matrix. Finally, we present examples which confirms superior numerical performances of the proposed method.

eigenvalues ; quadratic eigenvalue problem ; QR factorization ; QZ method ; backward error ; infinite eigenvalues

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