For the large generalized singular value decomposition (GSVD) computation, given three left and right searching subspaces, we propose a class of general projection methods that works on (A, B) directly, and computes approximations to the desired GSVD components. Based on it, we propose a CJ-FEAST GSVDsolver to compute a partial generalized singular value decomposition (GSVD) of a large matrix pair (A, B) with the generalized singular values in any given interval. The solver itself is a highly nontrivial extension of the FEAST eigensolver for the standard or generalized eigenvalue problem and the CJ-FEAST SVDsolvers for the singular value decomposition (SVD) problem. We exploit the Chebyshev–Jackson (CJ) series to construct an approximate spectral projector of the matrix pair <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="211_2025_1466_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="92" /> </InlineMediaObject> <EquationSource Format="TEX">$(A^TA,B^TB)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msup> <mi>A</mi> <mi>T</mi> </msup> <mi>A</mi> <mo>,</mo> <msup> <mi>B</mi> <mi>T</mi> </msup> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> associated with the generalized singular values of interest, use subspace iteration on it to generate a right subspace, and premultiply it with A and B to obtain two left subspaces. The spectral projector and its approximations are unsymmetric, and the convergence problems and algorithmic implementations on the CJ-FEAST GSVDsolver are far more difficult and complicated than those on the two available CJ-FEAST SVDsolvers. We derive accuracy estimates for the approximate spectral projector and its eigenvalues, and establish a number of convergence results on the underlying subspaces and the approximate GSVD components obtained by the CJ-FEAST GSVDsolver. We propose general-purpose choice strategies for the series degree and subspace dimension. Numerical experiments illustrate that (1) the CJ-FEAST GSVDsolver is practical and it is much more robust and accurate than its contour integral-based variant with the trapezoidal rule and the Gauss–Legendre quadrature and speeds up the latter several dozen to hundred times, and (2) it is competitive with and has huge advantage over a very best Jacobi–Davidson GSVDsolver when the number of desired GSVD components is no more than dozens and is more than one hundred, respectively. The CJ-FEAST GSVDsolver is directly adaptable to the generalized eigenvalue problem of a large symmetric positive definite pair.

A CJ-FEAST GSVDsolver for computing a partial GSVD of a large matrix pair with the generalized singular values in a given interval

期刊论文
发表日期：

2025年4月22日

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Zhongxiao Jia,
Kailiang Zhang

摘要

For the large generalized singular value decomposition (GSVD) computation, given three left and right searching subspaces, we propose a class of general projection methods that works on (A, B) directly, and computes approximations to the desired GSVD components. Based on it, we propose a CJ-FEAST GSVDsolver to compute a partial generalized singular value decomposition (GSVD) of a large matrix pair (A, B) with the generalized singular values in any given interval. The solver itself is a highly nontrivial extension of the FEAST eigensolver for the standard or generalized eigenvalue problem and the CJ-FEAST SVDsolvers for the singular value decomposition (SVD) problem. We exploit the Chebyshev–Jackson (CJ) series to construct an approximate spectral projector of the matrix pair $(A^TA,B^TB)$ $(A^{T} A, B^{T} B)$ associated with the generalized singular values of interest, use subspace iteration on it to generate a right subspace, and premultiply it with A and B to obtain two left subspaces. The spectral projector and its approximations are unsymmetric, and the convergence problems and algorithmic implementations on the CJ-FEAST GSVDsolver are far more difficult and complicated than those on the two available CJ-FEAST SVDsolvers. We derive accuracy estimates for the approximate spectral projector and its eigenvalues, and establish a number of convergence results on the underlying subspaces and the approximate GSVD components obtained by the CJ-FEAST GSVDsolver. We propose general-purpose choice strategies for the series degree and subspace dimension. Numerical experiments illustrate that (1) the CJ-FEAST GSVDsolver is practical and it is much more robust and accurate than its contour integral-based variant with the trapezoidal rule and the Gauss–Legendre quadrature and speeds up the latter several dozen to hundred times, and (2) it is competitive with and has huge advantage over a very best Jacobi–Davidson GSVDsolver when the number of desired GSVD components is no more than dozens and is more than one hundred, respectively. The CJ-FEAST GSVDsolver is directly adaptable to the generalized eigenvalue problem of a large symmetric positive definite pair.