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2.4 Modified Moments from the Chebyshev Semi-iterative Method

Since the iteration matrix M defined defined by the canonical eigenvalue problems from Section 1.2 is symmetric, M has a complete set of orthogonal eigenvectors which can be be denoted by

Then,

where, for i>0, is the Chebyshev iterate generated by Equation (11), is the initial iterate, and is an eigenvalue of M corresponding to the eigenvector . Consider the inner product of the and iterates, i.e.,

 

Equation (18) is equivalent to the continuous integral

 

when is defined to be the following discrete non-negative distribution ([10])

By choosing l=0 in Equation (19) and , it follows that

Note that the final orthogonal polynomial has a zero at each eigenvalue i.e., . Hence, at each step of the Chebyshev semi-iterative method, we can extract the modified moment

The extraction of moments from iterates can be accelerated by using the recurrence relations for the Chebyshev polynomials defined in Equation (9). Specifically, one can show (see [22]) that

 

 

Note that the polynomial in Equation (15) associated with the modified moments and satisfies

where the coefficients of the polynomials in Equation (16) are given by

 



Michael W. Berry (berry@cs.utk.edu)
Sun May 19 11:34:27 EDT 1996