¬¬ ® BDB%

¬¬¬

be a partition of the set and, for each , let be the matrix (%

% %

G "

obtained by extracting the columns of the identity matrix whose indices belong to .

S

Going back to the projection framework, de¬ne . If an orthogonal projection S#

!

!

method is used onto to solve (8.1), then the new iterate is given by

¤

I ‘ x¤£

¦

Gs

ut w

¡ S ¢S !

S

˜ ˜S ¬ ˜S qea S ˜S x` C ˜ ˜S P ‘ x¤£

¦

C

f d

S¢ !§f'd a S $

`

! !

Each individual block-component can be obtained by solving a least-squares problem

S¢ W

C

&'% Wu§ ¢ S ©¨ §

¬«

(

An interpretation of this indicates that each individual substep attempts to reduce the resid-

ual as much as possible by taking linear combinations from speci¬c columns of . Similar

S

to the scalar iteration, we also have

s

uvs C 8 C

t

¨ ¡ S

S

f &

where now represents an orthogonal projector onto the span of .

S S

s

f% DB¬BUf% f

¬¬ %«

Note that is a partition of the column-set and this parti-

C 0)0)0)C f ¦S ( S

¤

tion can be arbitrary. Another remark is that the original Cimmino method was formulated

¶ 8˜ ’ ”p20("'&}%#”8³ª%¡"p!¡©’ ”W

8¥1 )© ¥ ’$ ©˜ ’8 © 8¥ ˜ 8©

¡¡

¡¨ ¨¦ ¤

© §¥ &

$

for rows instead of columns, i.e., it was based on (8.1) instead of (8.3). The alternative

algorithm based on columns rather than rows is easy to derive.

fE•–”—H‘—w5x “ovV‚m¢¤u”U£“ XfP

˜ …“ ’ " … •" …¢

¥¦v™

A popular combination to solve nonsymmetric linear systems applies the Conjugate Gra-

dient algorithm to solve either (8.1) or (8.3). As is shown next, the resulting algorithms can

be rearranged because of the particular nature of the coef¬cient matrices.

ecAC¥

6 3¨31

2 §2 W8

We begin with the Conjugate Gradient algorithm applied to (8.1). Applying CG directly C

to the system and denoting by the residual vector at step (instead of ) results in the

P S S

following sequence of operations:

”a % ˜ x¡` a w P % $P` ©© a % £a P % P `

f

`

f ©

‘

˜ ¨ P P

f

a P % $£a f P % f P ` ©

P`

w P5 ©

.

f

f

C

S ‘ S

¨ C

If the original residual must be available at every step, we may compute

˜

C S C

S

¨

the residual in two parts: and then which is

P P

S

f f f f

the residual for the normal equations (8.1). It is also convenient to introduce the vector

S

. With these de¬nitions, the algorithm can be cast in the following form.

S

n ! " wE¤©n¦§¥£¢

¢ ¡ ¨¦ ¤¢

˜ ¢ P ¨‚¢ C C

E

1. Compute , , .

P

¬BD¬B% ª

¬

2. For , until convergence Do:

«« §S r « f§S S P ªS S

3.

§z §

4.

w 2 2

5. S

S

S "S

S ¨ S C f S C

6. S

S C ˜ ¢ f S P

«« s§S P pf§ «« § S P #f ™2

7.

§ S