A Determinant Theorem with Applications to Parallel Algorithms

We state and prove an expansion theorem for the determinant of any Hessenberg matrix. The expansion is expressed as a vector-matrix-vector product which can be efficiently evaluated on a parallel machine. We consider the computation of the first N terms of a sequence defined by a general linear recurrence. On a sequential machine this problem is $O(N^2 )$, with N processors it is $O(N)$,and with $O(N^4 )$ processors it is $O(\log ^2 N)$ using our expansion. Other applications include locating roots of analytic functions and proving doubling formulas for linear recurrences with constant co-efficients.