Roots of Polynomials Expressed in Terms of Orthogonal Polynomials

A technique is presented for determining the roots of a polynomial $p(x)$ that is expressed in terms of an expansion in orthogonal polynomials. The roots are expressed as the eigenvalues of a nonstandard companion matrix $B_n$ whose coefficients depend on the recurrence formula for the orthogonal polynomials, and on the coefficients of the orthogonal expansion. Some questions on the numerical stability of the eigenvalue problem to which they give rise are discussed. The problem of finding the roots of a transcendental function $f(x)$ can be reduced to the problem considered by approximating $f(x)$ by a Chebyshev polynomial. We illustrate the effectiveness of this convert-to-Chebyshev strategy by solving several transcendental equations using this plus our new algorithm. We analyze the numerical stability through both linear algebra theory and numerical experiments and find that this method is very well conditioned.