The bidiagonal singular value decomposition and Hamiltonian mechanics: LAPACK working note No. 11

Technical Report · Tue Aug 01 04:00:00 EDT 1989

OSTI ID:5841057

Deift, P; Demmel, J; Li, Luen-Chau; Tomei, C

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
• Country of Publication: United States
• Language: English

Related Subjects

657002* -- Theoretical & Mathematical Physics-- Classical & Quantum Mechanics
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
99 GENERAL AND MISCELLANEOUS
990230 -- Mathematics & Mathematical Models-- (1987-1989)
ALGORITHMS
ANALYTICAL SOLUTION
DIFFERENTIAL EQUATIONS
EIGENVALUES
EQUATIONS
HAMILTON-JACOBI EQUATIONS
HAMILTONIANS
MATHEMATICAL LOGIC
MATHEMATICAL OPERATORS
MATRICES
PARTIAL DIFFERENTIAL EQUATIONS
PERTURBATION THEORY
QUANTUM OPERATORS