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Rounding error and perturbation bound for the symplectic QR factorization

To compute the eigenvalues of a skew-symmetric matrix $A$, we can use a one-sided Jacobi-like algorithm to enhance accuracy. This algorithm begins by a suitable Cholesky-like factorization of $A$, $A = G^{T} J G$. In some applications, $A$ is given implicitly in that form and its natural Cholesky-like factor $G$ is immediately available, but ``tall'', i.e., not of full row rank. This factor $G$ is unsuitable for the Jacobi-like process. To avoid explicit computation of $A$, and possible loss of accuracy, the factor has to be preprocessed by a QR-like factorization. In this paper we present the symplectic QR algorithm to achieve such a factorization, together with the corresponding rounding error and perturbation bounds. These bounds fit well into the relative perturbation theory for skew-symmetric matrices given in factorized form.

Symplectic QR factorization; Rounding error bounds; perturbation bounds

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