H W

X

¥

and where is the matrix,

w

¢ ¢ ! !

y

Y

Y

.

y .

²

.

$ „

y² yY

a± „ ®

i

¬ ¥

. .

.. ..

¢ y y ¦ 54

H y ( y ( y

¢ . .

. .

. .

¢

. ¡W “ W

.

‚ ²

.

y y

²

Y

y

¢

The columns of can be rescaled, for example, to make each of them have a 2-norm

¢

¢

"U

W

equal to one, by multiplying to the right by a diagonal matrix. Let this diagonal ¢

’U

W

matrix be the inverse of the matrix

¬ t& ¢ X ( X (t X $ ¦ 54

H

(

¢

"U

W W

$

Then,

R±` „ ®

°i

¢

§ ¬

¤ ¢ ’U

W “¢ ¢

¢ ¢ ¢

"U "U

W W

W

$

With this, equation (7.56) becomes

a±0 „ ®

i

$

¢

¬ ¤ § t ¬

²¨ ²W

¢

¨ ¢ ¢ ¢ ¢ ¢ ¢

¡

a±a „ ®

i

"U ¢

W

¬ ²

¡ X ¢ "U

W “¢

¢ ¢ ¢ ¢

W"U W"U

W W

$ $

By analogy with the GMRES algorithm, de¬ne

¡ $ ¢

! ¢ "U

¢ ¢

W

¡ ¡

Similarly, de¬ne to be the matrix obtained from by deleting its last row. It is easy

¢ ¢

© !

¬

to verify that the CGS iterates (now de¬ned for all integers ) satisfy the ¡™( ¡`

(˜ (Y

¢

y

same de¬nition as FOM, i.e., T

± ¯ „ ®

i

¡¢¤ X

(¬ ©

© ¡ X W “¢

w

¢

W

It is also possible to extract a GMRES-like solution from the relations (7.61) and

(7.63), similar to DQGMRES. In order to minimize the residual norm over the Krylov

subspace, the 2-norm of the right-hand side of (7.63) would have to be minimized, but $ ¢

this is not practical since the columns of are not orthonormal as in GMRES. W “¢

¢

"U

W

² W"U

However, the 2-norm of can be minimized over , as was done for the ¡X ¢ ’U

¢ ¢

W W

QMR and DQGMRES algorithms.

This de¬nes the TFQMR iterates theoretically. However, it is now necessary to ¬nd a

formula for expressing the iterates in a progressive way. There are two ways to proceed.

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