The bidiagonal singular value decomposition and Hamiltonian mechanics: LAPACK working note No. 11
We consider computing the singular value decomposition of a bidiagonal matrix B. This problem arises in the singular value decomposition of a general matrix, and in the eigenproblem for a symmetric positive definite tridiagonal matrix. We show 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. We also show that the algorithm computes the singular vectors as well as the singular values to this accuracy. We also give a Hamiltonian interpretation of the algorithm and use differential equation methods 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. 36 refs., 2 figs., 2 tabs.
- Research Organization:
- Argonne National Lab., IL (USA). Mathematics and Computer Science Div.
- DOE Contract Number:
- W-31109-ENG-38
- OSTI ID:
- 5841057
- Report Number(s):
- ANL/MCS-TM-133; ON: DE89016545
- Resource Relation:
- Other Information: Portions of this document are illegible in microfiche products
- Country of Publication:
- United States
- Language:
- English
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99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE
HAMILTON-JACOBI EQUATIONS
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EIGENVALUES
HAMILTONIANS
PERTURBATION THEORY
DIFFERENTIAL EQUATIONS
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990230 - Mathematics & Mathematical Models- (1987-1989)