Determination of Eigenvalues of Matrices Having Polynomial Elements

Summary. A method is described for obtaining the eigenvalues of matrices whose elements are polynomials in the eigenvalue parameter. The method is based on an iterative root-finding technique coupled with the use of estimates. The process converges with virtually arbitrary estimates while significant reduction of computing time is achieved with the use of good estimates. Only those eigenvalues of interest need be obtained if sufficiently good estimates are available. Accuracy is maintained through direct use of the original data. A subroutine employing this procedure has been applied to a large number of problems, the most elaborate involving the determination of the 32 eigenvalues of an 18th order matrix with quadratic elements.