Very Fast Parallel Matrix and Polynomial Arithmetic

We present very efficient arithmetic circuits for the computation of the determinants and inverse of band matrices and for interpolation and polynomial arithmetic over arbitrary ground fields. For input size n the circuits for band matrix inversion (for constant band width) over infinite fields, and the circuits for polynomial arithmetic over arbitrary fields, have optimal depth 0(log n) and polynomial size. The algorithm for band matrix inversion is based on an extension of the method of modification (from numerical analysis) which includes a formula for the determinant of a modified matrix.