The Algebraic Engineering of Computer Arithmetic project established that using algebraic methods which eliminate the Jacobson radical of a finite ring used to do computer arithmetic that the multiplication and addition operations can be performed as one time-step operation. This yields a flat tree structure for the cascade transforms of operations. With suitable choices of the ring and suitable encoding further refinement is acheived.
A serious difficulty for this approach is that division has not been possible at the same speed. Therefore it is useful to consider division-free algorithms for special problems in order to obtain vast performance using our approach. An important case of this is parallel algorithms for solving systems of linear equations (for which division-free algorithms exist). Combined with the algebraic approach, this yields huge speed-ups over conventional algorithms on non-algebraically engineered hardware. Such a speed-up is also be possible in other division-free algorithms.