Abstract :
The pseudo-inverse (also called the MooreÐPenrose inverse or the generalized inverse) has many uses in
engineering in Þelds such as control design, structural dynamics and identiÞcation. E¦cient 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 Þnite di¤erences, or by taking the
gradient of the Singular Value Decomposition (SVD) and eigen decomposition of the appropriate matrices.
However, this is a very di¦cult 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
