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2.3 Relation to the Eigenvalue Problem

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