Padé error estimates for the logarithm of a matrix

The error of Padé approximations to the logarithm of a matrix and related hypergeometric functions is analysed. By obtaining an exact error expansion with positive coefficients, it is shown that the error in the matrix approximation at X is always less than the scalar approximation error at x, when ∥X∥ < x. A more detailed analysis, involving the interlacing properties of the zeros of the Padé denominator polynomials, shows that for a given order of approximation, the diagonal Padé approximants are the most accurate. Similarly, knowing that the denominator zeros must lie in the interval (1,∞) leads to a simple upper bound on the condition number of the matrix denominator polynomial, which is a crucial indicator of how accurately the matrix Padé approximants can be evaluated numerically. In this respect the Padé approximants to the logarithm are very well conditioned for ∥X∥ < 0·25. This latter condition can be ensured by using the ‘inverse scaling and squaring’ procedure for evaluating the logarithm.