¢

¡ ¡

"U

W

¬

u

In particular, if then the residual norm must be equal to zero which means that the

¢ Y

solution is exact at step .

¨ ¡¡µ£ „ ¢

§|5¥ yq¢| ¢ £ ¥§

„ 5| j C¦£¥

5§

9 C "–© ¡"

§

1B

¡' © C8 5 2%

¢

¢ ¡563 I ' E

3

1 F

¢

If Algorithm 6.9 is examined carefully, we observe that the only possibilities of breakdown

¬ ¬

u¦ u `u

in GMRES are in the Arnoldi loop, when , i.e., when at a given step . Y Y

£

W"U W"U

In this situation, the algorithm stops because the next Arnoldi vector cannot be generated.

However, in this situation, the residual vector is zero, i.e., the algorithm will deliver the

exact solution at this step. In fact, the converse is also true: If the algorithm stops at step

¬ u ©V§ A¨

² ¬

u `u

with , then . Y Y

£

W"U

h¤ ¡ v A£

£¦ ¡CDA

' ' ©' )$ )

§

§

Let be a nonsingular matrix. Then, the GMRES algorithm

¬ ©

u u u

breaks down at step , i.e., , if and only if the approximate solution is exact.

Y`

£

W"U

£ A™¤ ¢¡6

£

¬ ¬

u `u Y` u

To show the necessary condition, observe that if , then . Indeed, ¢

Y

£

u "U

W

§ ¬

uu¨

since is nonsingular, then is nonzero by the ¬rst part of Proposition 6.9 P W “ u GB£

u

¬ ¬

Y` u Y` u ¨

and (6.31) implies . Then, the relations (6.36) and (6.40) imply that .

¢

To show the suf¬cient condition, we use (6.40) again. Since the solution is exact at step

m

² ¬ ¬

`u u $u

and not at step , then . From the formula (6.31), this implies that . ¢ Y Y

£

"U

W

y

¡A 2% F ©D'C79763

¢5

¢ £

¢ I '( EPP5

E B A8&5 5 ¡6QI ©C8

E

F53

¢

If the last row of the least-squares system in (6.38) is deleted, instead of the one in (6.39),

i.e., before the last rotation is applied, the same approximate solution as FOM would ¡

result. As a practical consequence a single subroutine can be written to handle both cases.

This observation can also be helpful in understanding the relationships between the two

algorithms.

We begin by establishing an interesting relation between the FOM and GMRES iter-

ates, which will be exploited in the next chapter. A general lemma is ¬rst shown regarding

the solutions of the triangular systems

¢

¢b ¢ ¬ $

& ¢

¡

obtained from applying successive rotations to the Hessenberg matrices . As was stated ¢

¢b

before, the only difference between the vectors obtained in GMRES and Arnoldi is

¢

T

that the last rotation is omitted in FOM. In other words, the matrix for the two ¢ ¢

X

methods differs only in its entry while the right-hand sides differ only in their last $( !

!

components. $

’ ¥w

¢ $ ¡

CDA ¢

P#

1 1

¥T!

Let

be the upper part of the matrix and, as before, ¢¥ ¤ ¢ ¢

!

¡

¢ “

&W

¥

let be the upper part of the matrix . Similarly, let be the vector of ¤ ¢¥

¢ ¢ ¢

! ! X

T

the ¬rst components of and let be the vector of the ¬rst components &

¤ FS

¢ ¢