Computation of the exponential and other isotropic tensor functions and their derivatives

Abstract

In the present paper we focus on numerical aspects of the computation of isotropic tensor functions and their derivative. In the general case of non-symmetric tensor arguments only two numerical algorithms appear to be appropriate. The first one represents a recurrent procedure resulting from the Taylor power series expansion of an isotropic tensor function. The second algorithm is based on a recently proposed closed-form representation which can be obtained from the definition of an isotropic tensor function either by the tensor power series or by the Dunford–Taylor integral. To improve the accuracy in the case of nearly equal eigenvalues a series expansion of this closed formula is proposed. Both algorithms are finally illustrated by an example of the exponential tensor function where emphasis is placed on the precision issue.