Stewart, G. W.This paper describes a divide-and-conquer strategy for solving block Hessenberg systems. For dense matrices the method is a little more efficient than Gaussian elimination; however, because it works almost entirely with the original blocks, it is be much more efficient for sparse matrices or matrices whose blocks can be generated on the fly. For Toeplitz matrices, the algorithm can be combined with the fast Fourier transform to give a new superfast algorithm. (Also cross-referenced as UMIACS-TR-92-109)en-USTechnical Report