Generalized Scans and Tri-Diagonal Systems

Motivated by the analysis of known parallel techniques for the solution of linear tridiagonal system, weintroduce generalized scans, a class of recursively de#ned lengthpreserving, sequence-to-sequence transformations that generalize the well-known pre#x computations #scans#. Generalized scan functions are described in terms of three algorithmic phases, the reduction phase that saves data for the third or expansion phase and prepares data for the second phase which is a recursiveinvocation of the same function on one fewer variable. Both the reduction and expansion phases operate on bounded numberofvariables, a key feature for their parallelization. Generalized scans enjoya property, called here protoassociativity, that gives rise to ordinary associativity when generalized scans are specialized to ordinary scans. We show that the solution of positive de#nite block tridiagonal linear systems can be cast as a generalized scan, thereby shedding light on the underlying structure enabling k..