83. P. F. Yip, a Computationally Efficient Method for Evaluating the Gradient of 2d

Matrix exponentials and their derivatives play an important role in the perturbation analysis, control and parameter estimation of linear dynamical systems. The well-known integral representation of the matrix exponential’s directional derivative, , enables us to derive a number of new properties of this derivative, along with spectral, series and exact representations. Many of these results extend to arbitrary analytic functions of a matrix argument, for which we have also derived a simple relation between the gradients of their entries and the directional derivatives in the elementary directions. Based on these results, we construct and optimize two new algorithms for computing the directional derivative. We have also developed a new algorithm for computing the matrix exponential, based on a rational representation of the exponential in terms of the hyperbolic function , which is more efficient than direct Padé approximation. Finally, these results are illustrated by an application to a biologically important parameter estimation problem which arises in nuclear magnetic resonance spectroscopy. exp t τ – ( ) A ( ) Vexp τA ( ) dτ 0 t ∫