BAND GENERALIZATION OF THE GOLUB-KAHAN BIDIAGONALIZATION, GENERALIZED JACOBI MATRICES, AND THE CORE PROBLEM
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. Strakos, SIAM J.
Matrix Anal. Appl., 27 (2005), 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 approximate to B in [I. Hnetynkova, M.
Plesinger, and Z. Strakos, 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 Bjorck 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.