Componentwise analysis of direct factorization of real symmetric and Hermitian matrices
A factorization algorithm for an \(n \times n\) symmetric matrix \(H\) is developed, \(H=GJG^{T}\), where \(G\) has full column rank and \(J=\text{diag}(\pm 1)\). This decomposition presents an extension of the Cholesky method for positive definite matrices and also a modification of the method of \textit{J. R. Bunch} and \textit{B. N. Parlett} [SIAM J. Numer. Anal. 8, 639-655 (1971; Zbl 0199.49802)]. Further in the paper, componentwise error bounds for the factorization are derived. The main result is the componentwise backward error bound: the computed \(G\) and \(J\) are the exact factors of the perturbed matrix \(H+E\), that is \[ GJG^{T} = H+E, \quad| E| \leq 91n(| H| + | G| | G| ^{T})\varepsilon, \] where \(\varepsilon\) is the machine precision. The obtained estimate reveals the error structure better and has a simpler form, which is more suitable for further applications. A componentwise forward error bound for the same symmetric factorization is also proved. This generalizes \textit{Jiguang Sun}'s studies to indefinite matrices [Linear Algebra Appl. 173, 77-97 (1992; Zbl 0808.65018), Theorem 2.2.1]. All results are extended to the case of Hermitian matrices \(H\) and to the Bunch-Parlett factorization. Finally, bounds for the scaled condition of the matrix \(G\) are found, i.e. the inequality \(\kappa(\text{scal}(G)) \leq O(n3.781^n)\) is proved irrespective of the condition and of the singularity of \(H\). All results are illustrated by a numerical example.
scientific article; zbMATH DE number 1116072
scientific article