A new stable bidiagonal reduction algorithm

Abstract

A new bidiagonal reduction method is proposed for X ∈ R m×n . For m n, it decomposes X into the product X = UBV T where U ∈ R m×n has orthonormal columns, V ∈ R n×n is orthogonal, and B ∈ R n×n is upper bidiagonal. The matrix V is computed as a product of Householder transformations. The matrices U and B are constructed using a recurrence. If U is desired from the computation, the new procedure requires fewer operations than the GolubKahan procedure [SIAM J. Num. Anal. Ser. B 2 (1965) 205] and similar procedures.In floating point arithmetic, the columns of U may be far from orthonormal, but that departure from orthonormality is structured. The application of any backward stable singular value decomposition procedure to B recovers the left singular vectors associated with the leading (largest) singular values of X to near orthogonality.The singular values of B are those of X perturbed by no more than f (m, n)ε M X F where f (m, n) is a modestly growing function and ε M is the machine unit. Under certain assumptions, relative error bounds on the singular values are possible.

An augmented analysis of the perturbed two-sided Lanczos tridiagonalization process

“…The zero matrix is denoted by O, the zero column vector is denoted by o. We use Matlab-like operators triu and tril, see (5), and the overloaded operator diag which has to be understood in context. The operator diag extends to block matrices, e.g.,…”

Parallel Two-Sided Matrix Reduction to Band Bidiagonal Form on Multicore Architectures

“…Barlow et al [4] and later, Bosner et al [6], further improved the stability of the one-sided bidiagonalization technique by merging the two distinct steps to compute the bidiagonal matrix B. The computation process of the left and right orthogonal transformations is now interlaced.…”

The regularizing effect of the Golub-Kahan iterative bidiagonalization and revealing the noise level in the data

“…From the more recent literature we mention [16], where hybrid methods based on bidiagonalization are described as least-squares projection methods, and [51], where bidiagonalization is used to compute low-rank approximations of large sparse matrices. Numerical stability of the bidiagonalization algorithm was studied and new stable variants have been proposed, e.g., in [3,51], with a simplified analysis of [3] presented in [40].…”