EFFICIENT DIRECT COMPUTATION OF THE PSEUDO-INVERSE AND ITS GRADIENT

The pseudo-inverse (also called the Moore-Penrose inverse or the gener alized 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 assoc iated with these techniques. In addition, the gradient of the pseudo-i nverse may be needed for sensitivity analysis or optimization. Typical methods for computing the pseudo-inverse require the singular value o r eigenvalue decomposition of the appropriate matrices. Moreover, if t he 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 direc t method of computing the gradient of the pseudo-inverse of well-condi tioned systems with respect to a scalar. The paper begins by revisitin g a direct method for computing the pseudo-inverse developed by Grevil le for matrices with independent columns. When applied to a square, fu lly populated, non-symmetric case, with independent columns, it was fo und that the approach can be up to 8 times faster than the conventiona l approach of using the SVD. Rectangular cases are shown to yield simi lar levels of speed increase. A method is then presented which is a di rect approach for computing the gradient of the pseudo-inverse that pr eviously did not exist. To help illustrate the algorithms, simple MATL AB code is provided. (C) 1997 John Wiley & Sons, Ltd.