2. Compute , for %% %

¥

G G

3. Replace by zero if S a f'¥ d 0S% 7` S £

}

¨

4. Compute , S

S

¦

f ¤

¥ % S

® B¬D¬B n

¬

5. Replace each by zero if a ¥ %0 `

%„ %

« §S E & SS G

§

6.

™ SS

7. If then Stop; else compute .

SS

S !

S

8. EndDo

We recognize in line 2 the same practical problem encountered in the previous section

for IC(0) for the Normal Equations. It can be handled in the same manner. Therefore, the

row structures of , , and are needed, as well as a linked list for the column structure

of .

After the -th step is performed, the following relation holds:

fd

Cw

SS ¨S£ S

S S S

f

or

q¥ ¤ v0¥ £

¦ ‘)

C w

S

£ S S

f

C

where is the row of elements that have been dropped from the row in line 5. The above

S S

equation translates into

G ¤ v0¥ £

¦ ‘)

¡

w

¢

C ¤

where is the matrix whose -th row is , and the notation for and is as before.

¤ S

¤

The case where the elements in are not dropped, i.e., the case when is the empty

E

g¤

set, is of particular interest. Indeed, in this situation, and we have the exact relation

¡

. However, is not unitary in general because elements are dropped from . If

E

SS

at a given step , then (10.81) implies that is a linear combination of the rows , £ S

f

D¬B¬

¬ £ BDB% f £

¬¬¬

, . Each of these is, inductively, a linear combination of . Therefore, £

S

d

f

DBB% f £

¬¬¬

would be a linear combination of the previous rows, which cannot be true if

d £¤%

S

f

is nonsingular. As a result, the following proposition can be stated.

‚£

¦ "0§ ¦¦ ¨£h¤¨ £¤¦ n¦

£

¢ ¥ ¤

If is nonsingular and , then the Algorithm 10.14 com-

£

z

pletes and computes an incomplete LQ factorization , in which is nonsingular

and is a lower triangular matrix with positive elements.

’ ”8 ” ’ ¤ §

$ © } ”vG

’ ˜8’ ’ ”p¥ &)0©

8 1 ¨

¤§

¡¡© &

$ ¨' %©

¥ $ &

$

¢

¡ ¡

A major problem with the decomposition (10.82) is that the matrix is not orthogonal in

general. In fact, nothing guarantees that it is even nonsingular unless is not dropped or

the dropping strategy is made tight enough.

Because the matrix of the complete LQ factorization of is identical with the

Cholesky factor of , one might wonder why the IC(0) factorization of does not always

§ §

exist while the ILQ factorization seems to always exist. In fact, the relationship between

˜ —

ILQ and ICNE, i.e., the Incomplete Cholesky for , can lead to a more rigorous §

way of choosing a good pattern for ICNE, as is explained next.

We turn our attention to Modi¬ed Gram-Schmidt. The only difference is that the row

is updated immediately after an inner product is computed. The algorithm is described

without dropping for for simplicity.

•© 0 ¨© ¤0 ¨ & ¨¥ £ q˜(

5§"0§ ¡w¤h¨ n¦ ¤ £¢

¦¢ 2 ¥£

¢ & 0 £ 9

& D!

%