Lanczos tridiagonalization, Golub‐Kahan bidiagonalization and coreproblem

Consider an orthogonally invariant linear approximation problem Ax ≈ b. In [8] it is proved that the partial upper bidiagonalization of the extended matrix [b, A] determines a core approximation problem A11x1 ≈ b1, with all necessary and sufficient information for solving the original problem given by b1 andA11. It is shown how the core problem can be used in a simple and efficient way for solving different formulations of the original approximation problems. In [3] the core problem formulation is derived from the relationship between the Golub-Kahan bidiagonalization [2] and the Lanczos tridiagonalization [5], and from the known properties of Jacobi matrices. Here we briefly recall the approach from [3], and outline a possible direction for further research.