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2.2 Modified Moments and Orthogonal Polynomials

  Let be a weight function represented in terms of the moments defined by

 

where for . Gautschi [8] describes a procedure for the recursive computation of the coefficients and for the three-term recurrence

 

where the set of polynomials are orthogonal on with respect to the weight function . That is,

Whereas Gautschi's procedure in [8] is numerically unstable for computing the recurrence in Equation (14), a more stable procedure may be obtained if is codified in terms of the modified moments

 

where is a set of orthogonal polynomials satisfying a recurrence relation of the form

 

It has been shown in [10] that some simplifications to this more stable procedure are possible when the polynomials are chosen to be the Chebyshev polynomials. The simplified procedure is described in Section 3 as part of the CSI-MSVD algorithm.


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