Implementation of Jacobi rotations for accurate singular value computation in floating point arithmetic

Journal Article · Tue Jul 01 04:00:00 EDT 1997 · SIAM Journal on Scientific Computing

DOI:https://doi.org/10.1137/S1064827594265095· OSTI ID:522428

Drmac, Z ^[1]

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1. Univ. of Colorado, Boulder, CO (United States). Dept. of Computer Science

In this paper the author considers how to compute the singular value decomposition (SVD) A = U{Sigma}V{sup {tau}} of A = [a{sub 1}, a{sub 2}] {element_of} R{sup mx2} accurately in floating point arithmetic. It is shown how to compute the Jacobi rotation V (the right singular vector matrix) and how to compute AV = U{Sigma} even if the floating point representation of V is the identity matrix. In the case (norm of (a{sub 1})){sub 2} {much_gt} (norm of (a{sub 2})){sub 2}, underflow can produce the identity matrix as the floating point value of V, even for a{sub 1}, a{sub 2} that are far from being mutually orthogonal. This can cause loss of accuracy and failure of convergence of the floating point implementation of the Jacobi method for computing the SVD. The modified Jacobi method recommended in this paper can be implemented as a reliable and highly accurate procedure for computing the SVD of general real matrices whenever the exact singular values do not exceed the underflow or overflow limits.

• Sponsoring Organization: National Science Foundation, Washington, DC (United States); USDOE, Washington, DC (United States)
• DOE Contract Number: FG03-94ER25215
• OSTI ID: 522428
• Journal Information: SIAM Journal on Scientific Computing, Journal Name: SIAM Journal on Scientific Computing Journal Issue: 4 Vol. 18; ISSN SJOCE3; ISSN 1064-8275
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
ACCURACY
CALCULATION METHODS
CONVERGENCE
MATRICES
MODIFICATIONS