## 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 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.

Original language | English (US) |
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Pages (from-to) | 35-84 |

Number of pages | 50 |

Journal | Linear Algebra and Its Applications |

Volume | 397 |

Issue number | 1-3 |

DOIs | |

State | Published - Mar 1 2005 |

## All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics