The Derivative of |X′X| and Its Uses

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Abstract
The determinant of matrices of the form X′X is used as the D-optimal design criterion for parameter estimation in nonlinear regression and as the estimation criterion for multiresponse parameter estimation. It is helpful to be able to calculate the gradient of this determinant so that sophisticated optimization methods can be used. A method is given to efficiently compute the determinant using modern computational linear algebra and, at the same time, to provide the necessary information to compute the gradient. The method is extended to cases in which a part of the X matrix is fixed. Examples from multiresponse parameter estimation and from experimental design are given.