observe that by (9.24) and the fact that is unitary,

f

¬ « § h ¦§ ¨ F v' £

¦‘

#”a

§

`h

f

´ h

Since the algorithm minimizes this norm over all vectors in to yield , it is clear ¡ h

w w

'W

&

& h

that the approximate solution has the smallest residual norm in h h

'

. Thus, the following result is proved. (h

#

%

‚£

¦ ¦¦ E¤¨ ¤¦ n¦

¥¨ £ £ t¶·

&W

±

The approximate solution obtained at step of FGMRES

h

w

B§ h ”D§

« ¨

minimizes the residual norm over . (h

%#

Next, consider the possibility of breakdown in FGMRES. A breakdown occurs when

E

£

¥

the vector cannot be computed in line 9 of Algorithm 9.6 because . For

C

f f

the standard GMRES algorithm, this is not a problem because when this happens then the

approximate solution is exact. The situation for FGMRES is slightly different.

‚£

¦ ¦¦ E¤¨ ¤¦ n¦

¥¨ £ £ ·

£« § C 13 E

¢ § ¨v¥

Assume that and that steps of FGMRES

E G

¢ S C f ²

have been successfully performed, i.e., that for . In addition, assume that ¥

S ¥

§ E

the matrix is nonsingular. Then is exact, if and only if . ¥

Cf

§¡

£ E

§ §¦

¥

C f ¥

If , then , and as a result

§¡ ¨ §¨

¨ f £ §#”a ` £

#Uu§

§ « §&U«u§

¬Ru§

«

f

§ f

§

If is nonsingular, then the above function is minimized for and the

a f ` fd

corresponding minimum norm reached is zero, i.e., is exact.

Conversely, if is exact, then from (9.22) and (9.23),

¤ § ¨ f ¢ ¡ ”3 ¬ ˜ f I v' £

¦‘

E ¦ £ w

¨

E E

¦ £ ¦ £ ¦ £ £

¢

We must show, by contraction, that . Assume that . Since ,,

f

§¨ Ef

f f

B˜¬ B¬ « £ h £

¬

,, , form an orthogonal system, then it follows from (9.26) that

E f

§

and . The last component of is equal to zero. A simple back-substitution for

§

the system , starting from the last equation, will show that all components of

E

hf

are zero. Because is nonsingular, this would imply that and contradict the

assumption.

The only difference between this result and that of Proposition 6.10 for the GMRES

§

algorithm is that the additional assumption must be made that is nonsingular since it is

§

no longer implied by the nonsingularity of . However, is guaranteed to be nonsingu- h

lar when all the ™s are linearly independent and is nonsingular. This is a consequence P

of a modi¬cation of the ¬rst part of Proposition 6.9. That same proof shows that the rank of

E

h¥

§

is equal to the rank of the matrix therein. If is nonsingular and ,

h h¤ h¤ h

Cf

then is also nonsingular. h

¡¶ ’ ”p£¡¨8 ³¡© &’ ”8 ” ’ ¤ §

8˜ 8¥© $ $ ©