Lanczos tridiagonalization and core problems

The Lanczos tridiagonalization orthogonally transforms a real symmetric rnatrix A to symmetric tridiagonal form. The Golub-Kahan bidiagonalization orthogonally reduces a nonsyminetric rectangular matrix to upper or lower bidiagonal form. Both algorithms are very closely related. The paper [C.C. Paige, Z. Strakog, Core problems in linear algebraic systems, SIAM J. Matrix Anal. Appl, 27 (2006) 861-875] presents a new formulation of orthogonally invariant linear approximation problems A(x) approximate to b. It is proved that the partial upper bidiagonalization of the extended matrix [b, A] determines a core approximation problem A (11) x (1) approximate to b(1), with all necessary and Sufficient information for solving the original problem given by b(1) and A(11). It is further shown how the core problem call be used in a simple and efficient way for solving different formulations of the original approximation problem. Our contribution relates the core problem formulation to the Lanczos tridiagonalization and derives its characteristics from the relationship between the Golub-Kahan bidiagonalization, the Lanczos tridiagonalization and the well-known properties of Jacobi matrices. (c) 2006 Elsevier Inc. All rights reserved.