A Golub-Kahan-type reduction method for matrix pairs

We describe a novel method for reducing a pair of large matrices {A;B} to a pair of small matrices {H;K}. The method is an extension of Golub-Kahan bidiagonalization to matrix pairs, and simplifies to the latter method when B is the identity matrix. Applications to Tikhonov regularization of large linear discrete ill-posed problems are described. In these problems the matrix A represents a discretization of a compact integral operator and B is a regularization matrix. Keywords: Generalized Golub-Kahan bidiagonalization, eneralized Lanczos bidiagonalization, generalized Krylov method, matrix pair decomposition, ill-posed problem, Tikhonov regularization, multi-parameter regularization.