Estimating the Condition Number of the Fréchet Derivative of a Matrix Function

ESTIMATING THE CONDITION NUMBER OF THE FRÉCHET DERIVATIVE OF A MATRIX FUNCTION∗ NICHOLAS J. HIGHAM† AND SAMUEL D. RELTON† Abstract. The Fréchet derivative Lf of a matrix function f : C n×n → Cn×n is used in a variety of applications and several algorithms are available for computing it. We define a condition number for the Fréchet derivative and derive upper and lower bounds for it that differ by at most a factor 2. For a wide class of functions we derive an algorithm for estimating the 1-norm condition number that requires O(n3) flops given O(n3) flops algorithms for evaluating f and Lf ; in practice it produces estimates correct to within a factor 6n. Numerical experiments show the new algorithm to be much more reliable than a previous heuristic estimate of conditioning.