A new stable bidiagonal reduction algorithm

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 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)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.