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& By identifying the diagonal elements of with those of , derive an algorithm for generat-

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ing the elements of the diagonal matrix recursively.

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Establish that if for then . Is it true in general that for all ?

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Assume that for we have . Show a suf¬cient condition under ¢

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which . Are there cases in which this condition cannot be satis¬ed for any ?

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Assume now that all diagonal elements of are equal to a constant, i.e., for

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Show a condition on under which ¢!

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12 Show the second part of (10.79). [Hint: Exploit the formula where

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are the -th columns of and , respectively].

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13 Let a preconditioning matrix be related to the original matrix by , in which

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is a matrix of rank . "

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&'% Assume that both and are Symmetric Positive De¬nite. How many steps at most are

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required for the preconditioned Conjugate Gradient method to converge when is used as

a preconditioner?

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Answer the same question for the case when and are nonsymmetric and the full GM-

RES is used on the preconditioned system.

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14 Formulate the problem for ¬nding an approximate inverse to a matrix as a large

linear system. What is the Frobenius norm in the space in which you formulate this problem?

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15 The concept of mask is useful in the global iteration technique. For a sparsity pattern , i.e., a set

e

¢

5 9

of pairs and a matrix , we de¬ne the product to be the matrix whose elements

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¥ h

¥§ G §

¢

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are zero if does not belong to , and otherwise. This is called a mask operation

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since its effect is to ignore every value not in the pattern . Consider a global minimization of

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the function . ¢

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&'% What does the result of Proposition 10.3 become for this new objective function?

&1 Formulate an algorithm based on a global masked iteration, in which the mask is ¬xed and

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equal to the pattern of .

&32 Formulate an algorithm in which the mask is adapted at each outer step. What criteria would

you use to select the mask?

16 Consider the global self preconditioned MR iteration algorithm seen in Section 10.5.5. De¬ne

the acute angle between two matrices as 0

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1

0

56

2

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Following what was done for the (standard) Minimal Residual algorithm seen in Chapter 5,

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establish that the matrices and produced by global MR without 7

dropping are such that

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