Block Conjugate Gradient algorithms for least squares problems

In this paper, extensions for the Conjugate Gradient Least Squares (CGLS) algorithm in block forms, so-called Block Conjugate Gradient Least Squares (BCGLS), are described. Block parameter matrices are designed to explore the block Krylov subspace so that multiple right-hand sides can be treated simultaneously, while maintaining orthogonality and minimization properties along iterations. Search subspace is reduced adaptively in case of (near) rank deficiency to prevent breakdown. A deflated form of BCGLS is developed to accelerate convergence. Numerical examples demonstrate effectiveness in handling rank deficiency and efficiency in convergence accelerations in these BCGLS forms.