Complexity of parallel matrix computations

We estimate parallel complexity of several matrix computations under both Boolean and arithmetic machine models using deterministic and probabilistic approaches. Those computations include the evaluation of the inverse, the determinant, and the characteristic polynomial of a matrix. Recently, processor efficiency of the previous parallel algorithms for numerical matrix inversion has been substantially improved in (Pan and Reif, 1987), reaching optimum estimates up to within a logarithmic factor; that work, however, applies neither to the evaluation of the determinant and the characteristic polynomial nor to exact matrix inversion nor to the numerical inversion of ill-conditioned matrices. We present four new approaches to the solution of those latter problems (having several applications to combinatorial computations) in order to extend the suboptimum time and processor bounds of (Pan and Reif, 1987) to the case of computing the inverse, determinant, and characteristic polynomial of an arbitrary integer input matrix. In addition, processor efficient algorithms using polylogarithmic parallel time are devised for some other matrix computations, such as triangular and QR-factorizations of a matrix and its reduction to Hessenberg form.