Condition Estimates for Matrix Functions

A sensitivity theory based on Fréchet derivatives is presented that has both theoretical and computational advantages. Theoretical results such as a generalization of Van Loan’s work on the matrix exponential are easily obtained: matrix functions are least sensitive at normal matrices. Computationally, the central problem is to estimate the norm of the Fréchet derivative, since this is equal to the function’s condition number. Two norm-estimation procedures are given; the first is based on a finite-difference approximation of the Fréchet derivative and costs only two extra function evaluations. The second method was developed specifically for the exponential and logarithmic functions; it is based on a trapezoidal approximation scheme suggested by the chain rule for the identity eX = ( eX/2ⁿ )2ⁿ. This results in an infinite sequence of coupled Sylvester equations that, when truncated, is uniquely suited to the “scaling and squaring” procedure for eX or the “inverse scaling and squaring” procedure for log X. Both the trapezoid approximation method and the more general finite-difference approach yield excellent condition estimates for a large class of problems taken from the literature. The problems in this set illustrate that condition estimates based on the Fréchet derivative have the virtue of reliability and general applicability.