®3

. Thus, outer iterations can be de¬ned which sweep over the matrix, as well as

©¥

3

inner iterations which compute each column. At each outer iteration, the initial guess for

each column is taken to be the previous result for that column.

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2 3¢6

2¡ He

Y e H RP W U T 9

W

¡

The ¬rst theoretical question which arises is whether or not the approximate inverses ob-

¡

tained by the approximations described earlier can be singular. It cannot be proved that

is nonsingular unless the approximation is accurate enough. This requirement may be in

con¬‚ict with the requirement of keeping the approximation sparse.

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Assume that is nonsingular and that the residual of the ap-

¡

proximate inverse satis¬es the relation

GGF v0¥ £

¦ ‘)

² § ¡ ¨ 8 § G

¡

„§¬R§

where is any consistent matrix norm. Then is nonsingular.

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§

The result follows immediately from the equality

GGF v0¥ £

¦µ ‘)

¨8 ¡ 8 8

¡ ¨

f ¨

¬8

` a

§³ §

² ¨

Since , Theorem 1.5 seen in Chapter 1 implies that is nonsingular.

G

The result is true in particular for the Frobenius norm which is consistent (see Chapter 1).

¡

It may sometimes be the case that is poorly balanced and as a result can be ¤

¡

large. Then balancing can yield a smaller norm and possibly a less restrictive condi-

¡

tion for the nonsingularity of . It is easy to extend the previous result as follows. If is

«

nonsingular and two nonsingular diagonal matrices exist such that ¨%f ¨

¦6 F ‘v0¥ £

² §v« ¨ ¡£ f ¨ ¨ 8 § )

G

¡

„§¬R§

where is any consistent matrix norm, then is nonsingular.

Each column is obtained independently by requiring a condition on the residual norm

of the form

GF F v0¥ £

¦ ‘)

§ ¨ § D 3D

± %

„§R§

¬

for some vector norm . From a practical point of view the 2-norm is preferable since it is

related to the objective function which is used, namely, the Frobenius norm of the residual

8

£f¨

¡ . However, the 1-norm is of particular interest since it leads to a number of simple

theoretical results. In the following, it is assumed that a condition of the form

GI F v0¥ £

¦ ‘)

§ U©¨ § 3f

±

is required for each column.

The above proposition does not reveal anything about the degree of sparsity of the

¡

resulting approximate inverse . It may well be the case that in order to guarantee nonsin-

¡

gularity, must be dense, or nearly dense. In fact, in the particular case where the norm in

the proposition is the 1-norm, it is known that the approximate inverse may be structurally

¡

dense, in that it is always possible to ¬nd a sparse matrix for which will be dense if

² f § ¡ ¨ 8 §

.

G ¡

Next, we examine the sparsity of and prove a simple result for the case where an

assumption of the form (10.56) is made.

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£

¥ "0§

¢ S

#§

Let and assume that a given element of satis¬es

§

fd

the inequality

GP F v0¥ £

¦ ‘)

% ¡ S ¡ s % %

¡ S ¡

Cf

S

±

then the element is nonzero.

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