Since is unitary, the rank of is that of , which equals the rank of

¤

¢ ¢ ¢ ¢

¢ ¢

"U

W ¬

since these two matrices differ only by a zero row (the last row of ). If then ¨ Y

¢ ¢¢

¢

² § !

²

is of rank and as a result is also of rank . Since is of full

¢ ¢ ¢

!

y y

§

rank, this means that must be singular.

b

$ $

The second part (2), was essentially proved before the proposition. For any vector , T

£ b £Xb

$¤

¡ ¡

²b W $ ¢ S

²W ¬

£ £

FS ¢ ¢ ¢

¡ F

¡

£

¬ ² £

¢& ¢

r± a „ ®

i

£ £ b ¢

&

¬ ² £

w0

¡

0 ¢ ¢ ¢

"U

W

The minimum of the left-hand side is reached when the second term in the right-hand side

¢ ¢

b ¬

of (6.37) is zero. Since is nonsingular, this is achieved when . &

¢ ¢ W “¢

To prove the third part (3), we start with the de¬nitions used for GMRES and the

b ¢ w

©¬© $

relation (6.21). For any ,

¡ $

b¢

¬ ©t§y¨

² ²W

FS ¢

¡

"UW ¡

²b W $ ¢ S b¢

¬ $

¢¤ ¢¤ ¢ F

¡

"U

W

¬ ² $

& $

¢ ¢¤

¢ ¢

"U

W $ b¢¢

²

As was seen in the proof of the second part above, the 2-norm of is minimized & ¢

b

when annihilates all components of the right-hand side except the last one, which is & ¢

equal to . As a result, ¡

T

¢

’U

W X

¬ tc¨

©§ ² ¤ "U ¡

¢ ¢ ¢ ¢¡

¢

W "U

W W"U

which is (6.35). The result (6.36) follows from the orthonormality of the column-vectors

of . ¤ W"U

¢ ¢

¢b

So far we have only described a process for computing the least-squares solution

| 7¥

¥ C

§

of (6.24). Note that this approach with plane rotations can also be used to solve the linear

system (6.15) for the FOM method. The only difference is that the last rotation must ¢

be omitted. In particular, a single program can be written to implement both algorithms

using a switch for selecting the FOM or GMRES options.

It is possible to implement the above process in a progressive manner, i.e., at each step

of the GMRES algorithm. This approach will allow one to obtain the residual norm at every

step, with virtually no additional arithmetic operations. To illustrate this, start with (6.30),

i.e., assume that the ¬rst rotations have already been applied. Now the residual norm is !

©

available for and the stopping criterion can be applied. Assume that the test dictates that

¥

$ $

§

further steps be taken. One more step of the Arnoldi algorithm must be executed to get ¡©

¦

¡ ¢

and the -th column of . This column is appended to which has been augmented by ¥

¡

¦

( £

a zero row to match the dimension. Then the previous rotations , , are applied ¥

W

to this last column. After this is done the following matrix and right-hand side are obtained:

¥ ¥ ¥ ¥ ¥ ¥

P £¥ G

P ¤¥ G P ¥¥ G

PG P ¥G P ¡¥ G £

£ £ £ £ £ ¡

rW