Name
Abstract | ||
|---|---|---|
This paper concerns accurate computation of the singular value decomposition (SVD) of an mxn matrix A . As is well known, cross-product matrix based SVD algorithms compute large singular values accurately but generally delivers poor small singular values. A new novel cross-product matrix based SVD algorithm is proposed: (a) use a backward stable algorithm to compute the eigenpairs of A TA and take the square roots of the large eigenvalues of it as the large singular values of A ; (b) form the Rayleigh quotient of A TA with respect to the matrix consisting of the computed eigenvectors associated with the computed small eigenvalues of A TA ; (c) compute the eigenvalues of the Rayleigh quotient and take the square roots of them as the small singular values of A . A detailed quantitative error analysis is conducted on the method. It is proved that if small singular values are well separated from the large ones then the method can compute the small ones accurately up to the order of the unit roundoff. An algorithm is developed that is not only cheaper than the Golub-Reinsch and Chan SVD algorithms but also can accurately update or downdate the SVD when a row is appended to or deleted from A and compute certain refined Ritz vectors for large matrix eigenproblems at low cost. Several variants of the algorithm are proposed that compute some or all parts of the SVD. Typical numerical examples confirm the high accuracy of the algorithm. |
Year | DOI | Venue |
|---|---|---|
2006 | 10.1007/s11075-006-9022-x | Numerical Algorithms |
Keywords | Field | DocType |
cross-product matrix,eigenvalue,eigenvector,finite precision arithmetic,Rayleigh quotient,singular value,singular vector,SVD,65F15,65F30,65G05,15A12,15A18 | Singular value decomposition,Rayleigh quotient,Combinatorics,Mathematical optimization,Singular value,Cross product,Mathematical analysis,Matrix (mathematics),Square root,Eigenvalues and eigenvectors,Mathematics,Computation | Journal |
Volume | Issue | ISSN |
2001 | 6 | Mathematics Preprint Archive |
Citations | PageRank | References |
2 | 0.43 | 1 |
Authors | ||
1 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Zhongxiao Jia | 1 | 121 | 18.57 |