### 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) |
---|---|

Pages (from-to) | 1463-1516 |

Number of pages | 54 |

Journal | SIAM Journal on Numerical Analysis |

Volume | 28 |

Issue number | 5 |

State | Published - Oct 1991 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics
- Computational Mathematics

### Cite this

*SIAM Journal on Numerical Analysis*,

*28*(5), 1463-1516.

**Bidiagonal singular value decomposition and Hamiltonian mechanics.** / Deift, Percy; Demmel, James; Li, Luen Chau; Tomei, Carlos.

Research output: Contribution to journal › Article

*SIAM Journal on Numerical Analysis*, vol. 28, no. 5, pp. 1463-1516.

}

TY - JOUR

T1 - Bidiagonal singular value decomposition and Hamiltonian mechanics

AU - Deift, Percy

AU - Demmel, James

AU - Li, Luen Chau

AU - Tomei, Carlos

PY - 1991/10

Y1 - 1991/10

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0026238244&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0026238244&partnerID=8YFLogxK

M3 - Article

VL - 28

SP - 1463

EP - 1516

JO - SIAM Journal on Numerical Analysis

JF - SIAM Journal on Numerical Analysis

SN - 0036-1429

IS - 5

ER -