are generated. Approximations to the left singular vector corresponding
to the largest singular value are generated by the sequence
, while the corresponding right singular
vector is approximated by the sequence 
.
An estimate of the error in these singular vector approximations is
desired. Let
and consider the error vectors defined by
These vectors measure the error in the singular vector approximations
but do not reflect the error in the
singular triplet, i.e., 
and
. The vectors 
and 
are generated as part of the solution of the system
of equations defined by Equation (4) with b=0 and
is approximating the singular value
of A (now scaled by 
) closest to 1.
The vectors 
and 
are
generated by substituting m=k+1 in Equation (26) and m=k+2 in
Equation (27) (with 
) to obtain
Hence, the quantity 
is calculated as an intermediate
result in the calculation of 
, and 
can be calculated
at step k+2.
An analogous result for 
is harder to derive since
the right multiplication
of 
is by 
which is calculated in
Equation (33)
(rather than 
). Consider the intermediate product
From Equation (31), 
and so
Since 
by definition,
it follows that 
.
Substituting this expression for 
into Equation (35) yields
where 
,
, and hence
is a perturbation of a vector in the desired direction
. If this perturbation is suitably small, i.e.
0, then 
may be approximated (see [22]) by
where 
is the normalized version of 
.
A pseudo-code for the two-pass CSI-MSVD algorithm to approximate
singular values and corresponding singular vectors with
implicit error estimation is provided
in Figures 4 and 5.
Figure 4: Pseudo-code for one PASS of the CSI-MSVD algorithm.
Figure 5: Pseudo-code for two-pass CSI-MSVD algorithm with implicit error
estimation.
