A new bidiagonal reduction method is proposed for X ∈ R m×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.
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics