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3. The CSI-MSVD Algorithm

  Let be a weight function codified in terms of the 2n modified moments as defined in Equations (21) and (22). A procedure to compute the coefficients of polynomials orthogonal with respect to is desired. From Equation (14), the polynomials are of the form

Following [10] with the choice , the coefficients and may be determined using the recurrences below.

For ,
For ,



Here, and are defined by Equation (23), and initially,

The computation of 's and 's , effectively constructs the elements of the Jacobi matrix from Equation (17), whose eigenvalues approximate those of the iteration matrix M. Thus, by setting M to either of the two canonical matrices described in Section 1.2, one can approximate singular triplets of a general matrix by solving an equivalent symmetric eigenvalue problem. One implementation of this method (referred to as CSI-MSVD) was first considered in [2]. A more complete discussion of the method for each of the canonical eigenvalue problems from Section 1.2 is given in [22].

Michael W. Berry (
Sun May 19 11:34:27 EDT 1996