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2. Derivation of CSI-MSVD Method

As discussed in [10], the basic iteration


can be used to solve the system of linear equations


where M is either the matrix or the matrix defined by Equation (2). Also assume that the matrix M is suitably scaled so that its spectral radius () is less than 1. If the Chebyshev semi-iterative method [13] is used to solve systems defined by Equation (4), the iteration in Equation (3) will be of the form


where is a polynomial of degree k in M, and is a column vector of dimension or depending on whether M is the two-cyclic matrix of Equation (2) or the matrix . In this section, a procedure (CSI-MSVD) for estimating the eigenvalues of M (corresponding to the largest singular values of A) using Equation (5) with the method of modified moments is discussed. A more formal review of the theory of iterative methods which addresses issues such as convergence criteria and rates of convergence to establish the optimality of the Chebyshev semi-iterative method is given in [22]. In succeeding sections, will be taken to indicate the Euclidean norm, unless stated otherwise.

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