Efficient direct computation of the pseudo-inverse and its gradient

Abstract

The pseudo-inverse (also called the Moore–Penrose inverse or the generalized inverse) has many uses in engineering in fields such as control design, structural dynamics and identification. Efficient computation of the pseudo-inverse can greatly ease the computational burden associated with these techniques. In addition, the gradient of the pseudo-inverse may be needed for sensitivity analysis or optimization. Typical methods for computing the pseudo-inverse require the singular value or eigenvalue decomposition of the appropriate matrices. Moreover, if the gradient is required, it is either computed with finite differences, or by taking the gradient of the Singular Value Decomposition (SVD) and eigen decomposition of the appropriate matrices. However, this is a very difficult task, if possible at all. This paper develops a direct method of computing the gradient of the pseudo-inverse of well-conditioned systems with respect to a scalar. The paper begins by revisiting a direct method for computing the pseudo-inverse developed by Greville for matrices with independent columns. When applied to a square, fully populated, non-symmetric case, with independent columns, it was found that the approach can be up to 8 times faster than the conventional approach of using the SVD. Rectangular cases are shown to yield similar levels of speed increase. A method is then presented which is a direct approach for computing the gradient of the pseudo-inverse that previously did not exist. To help illustrate the algorithms, simple MATLAB code is provided. © 1997 John Wiley & Sons, Ltd.