The effect of the number of processors on the convergence of the parallel block Jacobi method

Elsner L, Neumann M, Vemmer B. The effect of the number of processors on the convergence of the parallel block Jacobi method. Linear Algebra and its Applications. 1991;154-156:311-330.We investigate the effect, on the rate of convergence of a model of the asynchronized parallel iteration method, of allowing the number of processors to differ from the number of splittings. Under certain regularization assumptions we prove that decreasing the number of processors increases the convergence rate. Our interpretation of this result for the model is as follows: increasing the number of processors means that each processor updates the global approximation in the host node with a local iteration which is computed from older global data. Hence the convergence rate is reduced. To prove our results we develop theorems for comparison of the spectral radii for certain nonnegative matrices which are of interest in their own right. We provide numerical examples to illustrate our results