A STRUCTURED METHOD FOR THE REAL QUADRATIC EIGENVALUE PROBLEM FOR SPECIFIC GYROSCOPIC SYSTEMS

Abstract

This study examines a specific numerical approach that computes the eigenvalues (normal modes) of a Quadratic Eigenvalue Problem (QEP) of the form (&lambda^2 & middotI + &lambda · B + C)· x = 0 where B is constrained to a real skew-symmetric matrix and C is constrained to a real symmetric positive definite matrix. A widely used linearization of this QEP is the companion matrix A which is an 2n-by-2n matrix such that (1,1) block is a n-by-n skew symmetric matrix, the (1,2) block is an n-by-n symmetric positive definite matrix, (2,1) block is the Identity matrix and finally the (2,2) zero block.. The goal is to find an algorithm method which diagonalizes matrix A without contaminating the (2,2) zero block. Once this algorithm is developed, the study measures the eigenvalue error bounds and compare its efficiency to the standard symmetric QR workhorse. Also, this approach preserves the structure of the error matrix in the same form as the QEP. In ensuring that the error matrix structure is a QEP, this algorithm provides fertile ground for future analysis in sensitivity and perturbation errors in the algorithm's eigenvalues. This study concludes that the algorithm appears to have a reasonable error bound; and it is more cost efficient in finding the eigenvalues then the symmetric QR algorithm.