iterate from Equation
(3), 
, can be written as
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.
