}

c

| 5

¥x ˜

v

D h

¦ ¦

Then this preconditioned Krylov iteration will produce iterates of the form

¥ ¥

un 4 ¦ j

}

c

¥x

£|

¢ ¢

5

˜

v

D

¢ ¢

¦ ¦

in which the sequence is the result of the same Krylov subspace method applied without¢

c

£

preconditioning to the reduced linear system with starting with ¥

£

¡

8 ¥

8

9 h

D D v

the vector .

£ %

¤¢¡

£

The proof is a consequence of the factorization ) )

¥ ¥ ¥ ¥

) )5 sn 4 ¦ j

¢ ¢

a5 ˜

˜

D h

c £

¢ ¢

£ £

¦ ¦ ¦ ¦

v

Applying an iterative method (e.g., GMRES) on the original system, preconditioned from

the left by and from the right by , is equivalent to applying this iterative method to )

'¢ '

¥

un 4 ¦ j

¢ ¤

˜ ¢

˜ ˜

c c

8

¢ v' v' D h

£

¢ ¦

The initial residual for the preconditioned system is

¥ |¥

¥ ˜

c c

' } c v'

¢ ¢x

v' v'

)

¦ ¦¥¥

¥ ¥

)¢

¥ ¥a }

D

c

¥x c ¦ ¦

£ £ 5

¦ ¦

v v t

# —t

7 u pU¢ 7 p z7 yw— y w u

{ {˜p p { p

˜

“£§

¢

! hC §¤ !

¤ …S !

§

¥

¤ ¤

¥ ¥

¢ ¢

¤

D h

¦ ¦ Y

£

48

As a result, the Krylov vectors obtained from the preconditioned linear system associated

˜

with the matrix have the form 8

¥ ¥ ¥

An¦ 4 H¦ j

¢ ¢ ¢

˜

t t ¢

Y ¦Y Yc

V§T

TT

£ £