Fast Parallel Band Matrix Arithmetic

AbstractFast parallel algorithms are presented for computation of the determinant, adjoint, characteristic polynomial, and rank of band matrices, and for the solution of systems of linear equations with band matrices as coefficient matrices. The algorithms can be implemented using arithmetic-boolean circuits of polynomial size and depth O(log n log m), or depth O(log n log m + log n log log n) for computations over small finite fields, where n is the order and m the band width of the matrix given as input. They can be implemented for computations over number fields and finite fields using log space uniform boolean circuits of depth O(log n log m + log n log log n) and polynomial size, for input size n and band width m. While they are not processor efficient, these algorithms can be implemented using circuits of asymptotically smaller depth than previous algorithms for these problems