Let M be in the form of a weakly cyclic matrix of index 2,
defined in Equation (2). Partition
the Chebyshev iterate 
into
the 
vector 
and the 
vector 
, and
the vector b from Equations (4) into
the 
vector 
and the 
vector 
so that
Equation (11) can be re-written as
The elements of successive iterates generated by Equations (26)
and (27) are related by the dependency shown in
Figure (1),
so that choosing 
when 
forces 
.
If 
denotes the 
iterate, then
Thus, at each step, only one of the Equations (26) and
(27) needs to be computed, reducing the number of computations
by a factor of two. Also, this
new iteration requires only one initial vector approximation to 
as opposed to the two approximations (
and 
)
required for the general case.
The simplifications provided by Equation (28)
also reduce the number of computations involved in the calculation
of the coefficients of the Jacobi matrix in Equation (17).
From Equations (28) and (22), it follows that
, with 
, so that 
.
It can be shown by induction that
which forces 
for all k. Hence, the only unknown quantities
in the Jacobi matrix are the elements of the sub-diagonal, 
.
As demonstrated in [12],
the eigenvalues of the 
Jacobi matrix are the same
as the singular values of the matrix
The QR-iteration for bidiagonal matrices [7]
may be applied to the matrix 
to obtain its singular values, thus
approximating the singular values of the original 
matrix A.
