¤

©¬ © w

¢ ¢ ¢ ¢

¡

W“

where is de¬ned above. This gives the following algorithm, called the Direct Incom-

¢

¡

plete Orthogonalization Method (DIOM).

h“

˜Q¤ v ¢

¡¦

3 A

' 7$

%# ) 2

1

3 3

©

A¨ ¬g ¨

² V§

© ¬ ¬

¦£

1. Choose and compute , , .

¨ S

¨

S

W

™( ¬!

2. For , until convergence Do: (˜

²#$!( y § QI6 E R y

PH ¬

3. Compute , and as in wW ¦

#(V©

! (

¢ ¢

£

"U

W

y

4. lines 2-7 of Algorithm (6.6). ¡

5. Update the LU factorization of , i.e., obtain the last column ¢

¬¤

6. of using the previous pivots. If Stop. £

¤ Y

¢ ¢¢

W

¬¤ ¢

¬! ¢²

7. if then else S

(

¢ ¢ ¢

§ ©

W “ R W “

y £ ¢

¬ ²¤ ¢R ¢R $! R

8. ( for set )

£ £ £

W ¢“ R

¦ E Y Y

¢ ¢

¢ W “¢

¡ ¡ ¡

"QH “

WU

¦¬ ©

©

9. w

¢ ¢ ¢ ¢

¡

W“

10. EndDo

Note that the above algorithm is based implicitly on Gaussian elimination without

¡ ¢b ¢ ¬

pivoting for the solution of the Hessenberg system . This may cause a pre- FS

¡

W

mature termination in line 6. Fortunately, there is an implementation based on Gaussian

elimination with partial pivoting. The details of this variant can be found in [174]. DIOM

can also be derived by imposing the properties that are satis¬ed by the residual vector and

R

the conjugate directions, i.e., the ™s. ¡

Observe that (6.4) is still valid and as a consequence, Proposition 6.7, which is based

on it, still holds. That is because the orthogonality properties were not used to derive the

two relations therein. Since the residual vector is a scalar multiple of and since the ¦ ¢

W"U

R¦™s are no longer orthogonal, IOM and DIOM are not orthogonal projection techniques.

They can, however, be viewed as oblique projection techniques onto and orthogonal ¢

to an arti¬cially constructed subspace.

h¤ ¡ v A£

£¦ A

' ' ©' )$ )

§

IOM and DIOM are mathematically equivalent to a projection

process onto and orthogonally to

¢

W § QH ¬ © ¢ ( £ (

£¡

¢

C (

¢

where T

X

² R ¬ R vE

¬

¢¦(R

¦ ¦ ¦ ( $(V(

!

¢

"U

W "U

W y

´

| 7¥

C

§

£ 6 A

£

The proof is an immediate consequence of the fact that is a multiple of ¨ ¦

¢ ¢

"U

W

R

and by construction, is orthogonal to all ™s de¬ned in the proposition.

¦ ¢

"U

W

The following simple properties can be shown:

E

¬

RT ¨

The residual vectors , , are “locally” orthogonal,

$V(

!(

yX ¥

¬ 0E

² ¬E

R¨( u¨ for ¨

0

¡Y

( (W

§

u