The connection between the moments and eigenvalues is well-known (see [14,15,16]). The associated matrix equation for the set of polynomials orthogonal to the modified moments defined by Equation (15) has the form

The tridiagonal matrix of coefficients
defined in the above equation is called the ** Jacobi matrix**.
From Equation ({17) it can be inferred that the
zeros of the polynomial may be found by solving the
standard eigenvalue problem . Thus, the roots of
may be obtained as the eigenvalues of the Jacobi matrix .

As pointed out in [10], this procedure is analogous to the
Lanczos algorithm and may be used to approximate the eigenvalues of the
iteration matrix **M** in the Chebyshev semi-iterative method. A scheme
to extract modified moments based on the theory described in [10]
from the Chebyshev iterates will be
described below. The modified moments will then be used to define a Jacobi
matrix whose eigenvalues approximate those of the
iteration matrix.

Michael W. Berry (berry@cs.utk.edu)

Sun May 19 11:34:27 EDT 1996