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 $e^X = ( e^{X/2^n } )^{2^n } $. This results in an infinite sequence of coupled Sylvester equations that, when truncated, is uniquely suited to the “scaling and squaring” procedure for $e^X $ or the “inverse scaling and squaring” p...