¦ ¦ v
and the associated approximate solution will be of the form
¥ ¥ ¥ ¥
5 c v)
c
¢
 ¢c
v
¢
D C
¦ Y C 6 v sC
t
¦
£
¢
¢ ¦ ¦
¥ } $
}
c c
5 X¥
¢ x 5 v
v x
D
¢
¦
¥ }
c
¢ 5 X¥
v x
D h
¢
¦
Finally, the scalars that express the approximate solution in the Krylov basis are ob
6 C
tained implicitly via inner products of vectors among the vector sequence (13.53). These
¢ Yc v
Y tY £ £
inner products are identical to those of the sequence . Therefore, these t t
VVT
TT
coef¬cients will achieve the same result as the same Krylov method applied to the reduced
Y
£
system , if the initial guess gives the residual guess .
8 £Q
D
£
A version of this proposition should allow to be preconditioned. The following result
is an immediate extension that achieves this goal.
r¤ …¨ r£
£¦ % ™ #
£¨ ¥ ¡¨
¢ £ £
Let be an approximate factorization of and
¢ #
D
de¬ne )
¥ ¥
un1 4 H¦ j
¢ 5
˜
'¢ '
D t D h
c
¢
£ ¢
¦ ¦
v
Assume that a Krylov subspace method is applied to the original system (13.1) with left
preconditioning and right preconditioning , and with an initial guess of the form
'¢ '
¥ ¥
An4 4 H¦ j
}
 5 X¥ x c v
˜
D h
¦
¦
Then this preconditioned Krylov iteration will produce iterates of the form
¥ ¥
An 4 H¦ j
}
5 X¥ x c v
¢ ¢
˜

D h
¢ ¢
¦ ¦
Moreover, the sequence is the result of the same Krylov subspace method applied to the ¢
c