### Abstract

Computing the singular value decomposition of a bidiagonal matrix B is considered. This problem arises in the singular value decomposition of a general matrix, and in the eigenproblem for a symmetric positive-definite tridiagonal matrix. It is shown that if the entries of B are known with high relative accuracy, the singular values and singular vectors of B will be determined to much higher accuracy than the standard perturbation theory suggests. It is also shown that the algorithm in [Demmel and Kahan, SIAM J. Sci. Statist. Comput., 11 (1990), pp. 873-912] computes the singular vectors as well as the singular values to this accuracy. A Hamiltonian interpretation of the algorithm is also given, and differential equation methods are used to prove many of the basic facts. The Hamiltonian approach suggests a way to use flows to predict the accumulation of error in other eigenvalue algorithms as well.

Original language | English (US) |
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Pages (from-to) | 1463-1516 |

Number of pages | 54 |

Journal | SIAM Journal on Numerical Analysis |

Volume | 28 |

Issue number | 5 |

DOIs | |

State | Published - 1991 |

### ASJC Scopus subject areas

- Numerical Analysis
- Computational Mathematics
- Applied Mathematics

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## Cite this

*SIAM Journal on Numerical Analysis*,

*28*(5), 1463-1516. https://doi.org/10.1137/0728076