Nearly Optimal Block-Jacobi Preconditioning

Abstract

The goal of Jacobi preconditioning of a symmetric positive definite matrix by a diagonal matrix is to choose to minimize the condition number . In 1969, van der Sluis proved that choosing so that the diagonal entries of are all ones reduces to within a factor of the minimum possible, where the factor depends on both the dimension and the norms used to define the condition number. We extend this result in two ways to block-Jacobi preconditioning, where is a block-diagonal matrix with blocks of given sizes, and we consider instead of to maintain the symmetric positive definite (spd) property. First, we extend van der Sluis’s original bound to include block-Jacobi. Second, we define a new norm in which choosing so that the corresponding diagonal blocks of are identity matrices minimizes the condition number. We use this to show that the condition number in the 2-norm of this optimally scaled is at least as large as the condition number in the new norm, and at most a factor larger, where is the number of diagonal blocks. We give an example where the optimal 2-norm condition number nearly attains this new upper bound, which for this example is tighter than van der Sluis’s bound by a factor equal to the matrix dimension. Finally, all these results generalize to the case of one-sided scaling of a full row-rank matrix .