From the theory of summability of sequences [23], consider the more general iterative procedure

The requirement that if , then must be **x**,
results in the constraint

The iterative method resulting from the sequence will be
referred to as a * semi-iterative method* [23] and
the error vector corresponding to is given by the expression

where is an degree polynomial with and .

In order to accelerate the convergence of the semi-iterative method, it is necessary to minimize the average rate of convergence, or, equivalently, obtain the solution of the minimization problem

The solution of this problem requires * a priori* determination of the
eigenvalues. In its place, consider the new minimization problem

where . The solution of the new minimization problem is given in terms of the Chebyshev polynomials, , defined by

for

Using the 3-term recurrence for Chebyshev polynomials

and the fact that , one can generate an iteration of the form (see [23])

where and . This iterative procedure
specifies the * Chebyshev semi-iterative method* with
respect to the original iteration defined in
Equation (3).
Discussions of the convergence of this method are
presented in [13] and [22].

As shown in [10], the iteration defined by
Equation (10) can be used in combination with
the theory of modified moments
to produce approximations to the largest eigenvalues
of the matrix **M**. Specifically,
Equation (10) may be used to generate
iterates through the
iteration

Similar to the Lanczos algorithm (see [5] and [11])
the next section will show how modified moments derived from the
iterates
may be used to generate a sequence of bidiagonal matrices whose largest
singular values approximate those of the sparse matrix **A**.

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

Sun May 19 11:34:27 EDT 1996