Band generalization of the Golub-Kahan bidiagonalization, generalized Jacobi matrices, and the core problem
| Type of publication: | Article |
| Citation: | |
| Publication status: | Published |
| Journal: | SIAM Journal on Matrix Analysis and Applications |
| Volume: | 36 |
| Number: | 2 |
| Year: | 2015 |
| Pages: | 417-434 |
| URL: | http://epubs.siam.org/doi/abs/... |
| DOI: | 10.1137/140968914 |
| Abstract: | The concept of the core problem in total least squares (TLS) problems with single right-hand side introduced in [C. C. Paige and Z. Strakoˇs, SIAM J. Matrix Anal. Appl., 27, 2006, pp. 861–875] separates necessary and sufficient information for solving the problem from redundancies and irrelevant information contained in the data. It is based on orthogonal transformations such that the resulting problem decomposes into two independent parts. One of the parts has nonzero right-hand side and minimal dimensions and it always has the unique TLS solution. The other part has trivial (zero) right-hand side and maximal dimensions. Assuming exact arithmetic, the core problem can be obtained by the Golub–Kahan bidiagonalization. Extension of the core concept to the multiple right-hand sides case AX ≈ B in [I. Hnˇetynkova´, M. Pleˇsinger, and Z. Strakoˇs, SIAM J. Matrix Anal. Appl., 34, 2013, pp. 917–931], which is highly nontrivial, is based on application of the singular value decomposition. In this paper we prove that the band generalization of the Golub–Kahan bidiagonalization proposed in this context by ˚A. Bjo¨rck also yields the core problem. We introduce generalized Jacobi matrices and investigate their properties. They prove useful in further analysis of the core problem concept. This paper assumes exact arithmetic. |
| Preprint project: | NCMM |
| Preprint year: | 2015 |
| Preprint number: | 02 |
| Preprint ID: | NCMM/2015/02 |
| Keywords: | core problem, generalized Jacobi matrices., Golub–Kahan bidiagonalization, multiple right-hand sides, total least squares problem |
| Authors | |
| Added by: | [JP] |
| Total mark: | 0 |
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