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Let now so that is symmetric for all (see Section 10.5.5). Assume that, at "

a given step the matrix is positive de¬nite. Show that

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4

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(&

¢ ¡¢

7 7

§

9 c5

4 9 4c5 ¤

£

© § ¥

¡ ¡¢

4

in which and are, respectively, the smallest and largest eigenvalues of

§

¦

£

.

17 In the two-sided version of approximate inverse preconditioners, the option of minimizing

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¢ i

–

T ©! ¢

¢

¢

was mentioned, where is unit lower triangular and is upper triangular.

5 9

¢

§

&% What is the gradient of ?

&1 Formulate an algorithm based on minimizing this function globally.

C

18 Consider the two-sided version of approximate inverse preconditioners, in which a unit lower

¢ ¢ i

– ©

¨

triangular and an upper triangular are sought so that . One idea is to use an

i ¢

–

alternating procedure in which the ¬rst half-step computes a right approximate inverse to ,

which is restricted to be upper triangular, and the second half-step computes a left approximate

¢–

inverse to , which is restricted to be lower triangular.

¢

&% Consider the ¬rst half-step. Since the candidate matrix is restricted to be upper trian-

gular, special care must be exercised when writing a column-oriented approximate inverse

algorithm. What are the differences with the standard MR approach described by Algorithm

10.10?

¢ 9 ™ 5

¢ –

&1 Now consider seeking an upper triangular matrix such that the matrix is close to

the identity only in its upper triangular part. A similar approach is to be taken for the second

half-step. Formulate an algorithm based on this approach.

19 Write all six variants of the preconditioned Conjugate Gradient algorithm applied to the Normal

Equations, mentioned at the end of Section 10.7.1.

eg– –

TX¤ T £ Y¤ T

T

£

20 With the standard splitting , in which is the diagonal of and

its lower- and upper triangular parts, respectively, we associate the factored approximate inverse

85

C 85 9

C

e 9 RSP

factorization,

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&% Determine and show that it consists of second order terms, i.e., terms involving products

7

Y¤

of at least two matrices from the pair .

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£

Now use the previous approximation for ,

R

7

85

C 59 85 9

C 9

e

R $PR £

£ RSPR £ R ¤ R R

£

R

7 7

Show how the approximate inverse factorization (10.83) can be improved using this new

approximation. What is the order of the resulting approximation?

NOTES AND REFERENCES. A breakthrough paper on preconditioners is the article [149] by Mei-

¤

jerink and van der Vorst who established existence of the incomplete factorization for -matrices

and showed that preconditioning the Conjugate Gradient by using an ILU factorization can result in

an extremely ef¬cient combination. The idea of preconditioning was exploited in many earlier papers.

For example, in [11, 12] Axelsson discusses SSOR iteration, “accelerated” by either the Conjugate

pp

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Gradient or Chebyshev acceleration. Incomplete factorizations were also discussed in early papers,

for example, by Varga [212] and Buleev [45]. Thus, Meijerink and van der Vorst™s paper played an

essential role in directing the attention of researchers and practitioners to a rather important topic and

marked a turning point. Many of the early techniques were developed for regularly structured matri-

ces. The generalization, using the de¬nition of level of ¬ll for high-order Incomplete LU factoriza-

tions for unstructured matrices, was introduced by Watts [223] for petroleum engineering problems.

Recent research on iterative techniques has been devoted in great part to the development of

better iterative accelerators, while “robust” preconditioners have by and large been neglected. This

is certainly caused by the inherent lack of theory to support such methods. Yet these techniques

are vital to the success of iterative methods in real-life applications. A general approach based on

modifying a given direct solver by including a drop-off rule was one of the ¬rst in this category

[151, 157, 235, 98]. More economical alternatives, akin to ILU( ), were developed later [179, 183,

68, 67, 226, 233]. ILUT and ILUTP, are inexpensive general purpose preconditioners which are

fairly robust and ef¬cient. However, many of these preconditioners, including ILUT and ILUTP, can