Naslov
A new stable bidiagonal reduction algorithm

Autori
Barlow, Jesse ; Bosner, Nela ; Drmač, Zlatko

Izvornik
Linear algebra and its applications (0024-3795) 397 (2005); 35-84

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
singular value decomposition, bidiagonal matrix, error analysis, orthogonality, left orthogonal matrix

Sažetak
A new bidiagonal reduction method is proposed for X ∈ Rm×n. For m ⩾ n, it decomposes X into the product X = UBVT where U ∈ Rm×n has orthonormal columns, V ∈ Rn×n is orthogonal, and B ∈ Rn×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 Golub–Kahan 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)εM∥X∥F where f(m, n) is a modestly growing function and εM is the machine unit. Under certain assumptions, relative error bounds on the singular values are possible.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

POVEZANOST RADA