Bidiagonal singular value decomposition and Hamiltonian mechanics

Percy Deift, James Demmel, Luen Chau Li, Carlos Tomei

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)1463-1516
Number of pages54
JournalSIAM Journal on Numerical Analysis
Volume28
Issue number5
StatePublished - Oct 1991

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Hamiltonian Mechanics
Hamiltonians
Singular value decomposition
Singular Vectors
Mechanics
Singular Values
Eigenproblem
Tridiagonal matrix
Positive definite
Perturbation Theory
High Accuracy
Differential equations
Differential equation
Eigenvalue
Predict
Computing

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics
  • Computational Mathematics

Cite this

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

In: SIAM Journal on Numerical Analysis, Vol. 28, No. 5, 10.1991, p. 1463-1516.

Research output: Contribution to journalArticle

Deift, P, Demmel, J, Li, LC & Tomei, C 1991, 'Bidiagonal singular value decomposition and Hamiltonian mechanics', SIAM Journal on Numerical Analysis, vol. 28, no. 5, pp. 1463-1516.
Deift, Percy ; Demmel, James ; Li, Luen Chau ; Tomei, Carlos. / Bidiagonal singular value decomposition and Hamiltonian mechanics. In: SIAM Journal on Numerical Analysis. 1991 ; Vol. 28, No. 5. pp. 1463-1516.
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