On Accurate Quotient Singular Value Computation in Floating-Point Arithmetic

Abstract

This paper presents a new algorithm for floating-point computation of the quotient singular value decomposition of an arbitrary matrix pair $(A,B) \in \mathbf{R}^{m \times n} \times \mathbf{R}^{p \times n}$. In the case of full column rank $A$, the new algorithm computes all finite quotient singular values with high relative accuracy if $\min\{\kappa_2(AD), D \mbox{diagonal}\}$ is moderate and if an accurate rank revealing LU factorization of B is possible. Numerical experiments show that in such a case the new algorithm computes the quotient singular values of all pairs (AD,D1 BD2) with nearly the same accuracy, where D, D1, D2 are arbitrary diagonal nonsingular matrices.