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\title{A New Application of Leverrier's Algorithm For Orthogonal Polynomial Bases 
\thanks{This project was suppported by the National Natural Science Foundation of China}}
\author{Guorong Wang and Lin Qiu \\
Department of Mathematics, Shanghai Normal University\\
100 Guilin Road, Shanghai 200234, China}
\date{}
\maketitle
An algorithm is presented which is an extension of the Leverrier's algorithm for simultaneous
computation of the adjoint $B(\lambda)$ and determinant $a(\lambda)$ of the matrix 
$\lambda E -A$, where $E$ is singular but $\det(\lambda E-A) \neq 0$. And we express the
adjoint $B(\lambda)$ and the determinant $a(\lambda)$ relative to a basis of orthogonal 
polynomials. Special formulas are derived for the cases of Chebyshev, Hermite, Legendre
and Laguerre polynomials.  We have solved the open problem by S. Barnett[2].

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