Leverrier's algorithm for orthogonal polynomial bases
For a given \(n \times n\) matrix \(A\), the well-known Leverrier-Fadeev algorithm can be used for simultaneous computation of the adjoint \(B(\lambda)\) and determinant \(a(\lambda)\) of the characteristic matrix \(\lambda I - A\). This paper deals with the case when both \(B(\lambda)\) and \(a(\lambda)\) are expressed relative to a basis of orthogonal polynomials. By using the standard three-term recurrence relation of orthogonal polynomials, the Leverrier-Fadeev algorithm is modified and specific formulae are derived for the cases of Chebyshev, Hermite, Legendre, and Laguerre polynomials. These results may be very useful in linear control theory and elsewhere. Obviously, further numerical analysis should be investigated.
scientific article; zbMATH DE number 871269
scientific article