The Proper Computation of the Matrix Pseudoinverse and its Impact in MVRO Filtering

There have been two new algorithms of fairly recent origin offered for the calculation of the matrix pseudoinverse. Unfortunately, nonpathological counterexamples can be constructed, as offered herein, that demonstrate the questionable nature of these two algorithms; however, a resolution is offered here to help prevent possible uncritical propagation of the questionable algorithms. As a rigorous alternative, a well-established technique (endorsed by numerical analysts) is reviewed for calculating the correct matrix pseudoinverse using a computer. Additionally, this technique possesses existent independently verified/validated and assessible software code for a convenient implementation. However, historical loose ends in calculating the associated condition number are singled out here as cause for concern and as a topic for future resolution and refinement. Finally, as the primary motivation for considering these issues, an application example is offered from estimation theory in the implementation and analysis of a minimum variance reduced-order (MVRO) filter having proper performance that critically hinges on the correct computation of the matrix pseudoinverse. While examples of applying MVRO to navigation applications were provided almost a decade ago, a clear indication of the somewhat restrictive conditions of applicability were wanting and so are elucidated here since there appears to be a resurgence of interest in this analytic technique. Another contribution is in providing a tally of the drawbacks to be incurred in using MVRO as well as its previously publicized benefits.