Investigation of Inverse Vandermonde Matrix Calculation for Linear System Applications.

Efficient inversion of a 2n x 2n Vandermonde matrix is a key requirement of a new algorithm developed for parameter identification in linear time-invariant systems. It has features very desirable for application to large systems with many unknown parameters, such as adaptive flight control systems. A generalized algorithm for inverting the Vandermonde matrix was proposed. It was chosen for study because its structure parallels that of the first algorithm. The algorithm was coded into a computer subroutine called VANINV, and 43 eigensystems were used to test its computational accuracy and efficiency. Four potential problem areas tested were large system orders, eigenvalues near each other, eigenvalues near zero, and eigenvalues with large magnitude differences. For a comparison, the Vandermonde matrix for each system was also inverted using routines from the International Mathematical and Statistical Library IMSL. Test results indicated that VANINV is not quite as fast or as accurate as the IMSL routines. However, the tests were limited to systems with real distinct eigenvalues because the IMSL routines cannot handle other types. VANINV in its present form can handle systems having any combination of complex andor repeated eigenvalues. Therefore, recommendations for further research and several possible means of improving VANINV are outlined.