Approximating the logarithm of a matrix to specified accuracy

The standard inverse scaling and squaring algorithm for computing the matri x logarithm begins by transforming the matrix to Schur triangular form in o rder to facilitate subsequent matrix square root and Pade approximation com putations. A transformation-free form of this method that exploits incomple te Denman-Beavers square root iterations and aims for a specified accuracy (ignoring roundoff) is presented. The error introduced by using approximate square roots is accounted for by a novel splitting lemma for logarithms of matrix products. The number of square root stages and the degree of the fi nal Pade approximation are chosen to minimize the computational work. This new method is attractive for high-performance computation since it uses onl y the basic building blocks of matrix multiplication, LU factorization and matrix inversion.