¤ DA 7 §¡¤

C

A test run of FOM with no preconditioning.

The column labeled Iters shows the total actual number of matrix-vector multiplications

(matvecs) required to converge. The stopping criterion used is that the 2-norm of the resid-

ual be reduced by a factor of relative to the 2-norm of the initial residual. A maximum £Y

¢

y

of 300 matvecs are allowed. K¬‚ops is the total number of ¬‚oating point operations per-

formed, in thousands. Residual and Error represent the two-norm of the residual and error

vectors, respectively. In this test, was taken to be 10. Note that the method did not suc- !

ceed in solving the third problem.

¢

©D7 ¥©DC7 B8 C7¤

¨

§ I 'B ¤ E 'B A 3 8 ©DB©C8

E

I'©

¢

A second alternative to FOM is to truncate the Arnoldi recurrence. Speci¬cally, an integer

is selected and the following “incomplete” orthogonalization is performed.

W

“6

˜Q¤ v ¢

¡¦ ¤ 7¦ ¢

©7

DE U©5 C¢

5S B Y8

S D PD £7@©¢Q B @4PD

6

6 QS ¢ ¢ ©E Y8

5D

3 A

' 7$

%# ) 2

1

}

¬ 1. For Do:

#V¡™(

!( ( ˜

u§ 3 ¥

y §¬

2. Compute ¦ T

¬vE

²

3. For Do:

wW Y( § P H 6

Y((t©

XR y

y

¬

uR

4. ¦( ¥

£

3

(¬

² R ¦ vR £

u

5. ¥ ¥

6. EndDo

¬ ¬

£

u u u¦ u£ u

7. Compute and

¥ ¥

£

’U

W "U

W ’U

W

8. EndDo

The number of directions against which to orthogonalize may be dictated by mem- W

ory limitations. The Incomplete Orthogonalization Method (IOM) consists of performing

the above incomplete orthogonalization procedure and computing an approximate solution

using the same formulas (6.14) and (6.15).

1 h“

˜Q¤ v ¢

¡¦ ¤¦

S Q¨8 D B

3 A

' 7$

%# ) 2

1

Run a modi¬cation of Algorithm 6.4 in which the Arnoldi process in lines 3 to 11

is replaced by the Incomplete Orthogonalization process and every other compu-

tation remains unchanged.

R¦

It is now necessary to keep only the previous vectors. The others are not needed W

in the above process and may be discarded. However, the dif¬culty remains that when

E

¬

R¦

the solution is computed by formula (6.14), all the vectors for are !$V˜ (

((

y

required. One option is to recompute them at the end, but essentially this doubles the cost

of the algorithm. Fortunately, a formula can be developed whereby the current approximate

© ©

solution can be updated from the previous approximation and a small number

¢ ¢

W“

C´ C

´

yq

„ 5| mV ¡ ® ky5„" @¦£ t ©

| § |

±

¨ ¡ © B ¦£

§

!–¡ 1B

§

of vectors that are also updated at each step. This progressive formulation of the solution

leads to an algorithm termed Direct IOM (DIOM) which we now derive.

¡

The Hessenberg matrix obtained from the incomplete orthogonalization process ¢

¬ ¡!

¬

has a band structure with a bandwidth of . For example, when and , it is

¡

wW W

y

of the form

£

£ £ £