Rescaling the GSVD with application to ill-posed problems
The generalized singular value decomposition (GSVD) decomposes a matrix pair \((A,L)\in(\mathbb{R}^{m\times n},\mathbb{R}^{p\times n})\), \(m\geq n\geq p\), as \(A=U\Sigma X^T\) and \(L=VMX^T\) with \(U\) and \(V\) orthogonal, \(X\in\mathbb{R}^{n\times n}\) nonsingular, and \(\Sigma\) and \(M\) are diagonal matrices satisfying \(\Sigma^T\Sigma+M^TM=I_n\), see [\textit{Z. Bai} and \textit{J. W. Demmel}, SIAM J. Sci. Comput. 14, No. 6, 1464--1486 (1993; Zbl 0789.65024)]. In this paper, the GSVD is rescaled such that the \(X\) matrix has columns of unit norm. This improves the condition of \(X\) which can be exploited to solve ill-conditioned systems \(Ax=b\) (with \(L\) as regularization matrix) and in a variant of the truncated GSVD of \textit{P. C. Hansen} [BIT 29, No. 3, 491--504 (1989; Zbl 0682.65021)].
generalized singular value decomposition
truncated generalized singular value decomposition
scientific article; zbMATH DE number 6418189
scientific article