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.

- 2.1 Chebyshev Semi-Iterative Method
- 2.2 Modified Moments and Orthogonal Polynomials
- 2.3 Relation to the Eigenvalue Problem
- 2.4 Modified Moments from the Chebyshev Semi-iterative Method

Michael W. Berry (berry@cs.utk.edu)

Sun May 19 11:34:27 EDT 1996