Generalized Golub–Kahan bidiagonalization and stopping criteria

The Golub–Kahan bidiagonalization algorithm has been widely used in solving least- squares problems and in the computation of the SVD of rectangular matrices. Here we propose an algorithm based on the Golub–Kahan process for the solution of augmented systems that minimizes the norm of the error and, in particular, we propose a novel estimator of the error similar to the one proposed by Hestenes and Stiefel for the conjugate gradient method and later developed by Golub, Meurant, and Strakoˇs. This estimator gives a lower bound for the error, and can be used as a stopping criterion for the whole process. We also propose an upper bound of the error based on Gauss–Radau quadrature. Finally, we show how we can transform augmented systems arising from the mixed finite-element approximation of partial differential equations in order to achieve a convergence rate independent of the finite dimensional problem size.