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&'% What does this become in the general situation when ?

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&1 Is Cimmino™s method still equivalent to a Richardson iteration?

&32 Show convergence results similar to those of the scaled case.

3 In Section 8.2.2, Cimmino™s algorithm was derived based on the Normal Residual formulation,

i.e., on (8.1). Derive an “NE” formulation, i.e., an algorithm based on Jacobi™s method for (8.3).

4 What are the eigenvalues of the matrix (8.5)? Derive a system whose coef¬cient matrix has the

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and which is also equivalent to the original system . What are the eigenvalues of ?

Plot the spectral norm of as a function of .

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5 It was argued in Section 8.4 that when the system (8.32) is nothing but the normal

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equations for minimizing the -norm of the residual .

&'% Write the associated CGNR approach for solving this problem. Find a variant that requires

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only one linear system solution with the matrix at each CG step [Hint: Write the CG

algorithm for the associated normal equations and see how the resulting procedure can be

reorganized to save operations]. Find also a variant that is suitable for the case where the

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Cholesky factorization of is available.

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Derive a method for solving the equivalent system (8.30) for the case when and then G

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for the general case wjen . How does this technique compare with Uzawa™s method?

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6 Consider the linear system (8.30) in which and is of full rank. De¬ne the matrix

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&'% Show that is a projector. Is it an orthogonal projector? What are the range and null spaces

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of ?

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&1 Show that the unknown can be found by solving the linear system

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in which the coef¬cient matrix is singular but the system is consistent, i.e., there is a nontriv-

ial solution because the right-hand side is in the range of the matrix (see Chapter 1).

&32 What must be done toadapt the Conjugate Gradient Algorithm for solving the above linear

system (which is symmetric, but not positive de¬nite)? In which subspace are the iterates

generated from the CG algorithm applied to (8.35)?

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¢ & Assume that the QR factorization of the matrix is computed. Write an algorithm based on

the approach of the previous questions for solving the linear system (8.30).

7 Show that Uzawa™s iteration can be formulated as a ¬xed-point iteration associated with the

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Derive the convergence result of Corollary 8.1 .

8 Show that each new vector iterate in Cimmino™s method is such that

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where is de¬ned by (8.24).

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9 In Uzawa™s method a linear system with the matrix must be solved at each step. Assume that

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these systems are solved inaccurately by an iterative process. For each linear system the iterative

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process is applied until the norm of the residual is less than a R R

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certain threshold . R

&% Assume that is chosen so that (8.33) is satis¬ed and that converges to zero as tends to

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in¬nity. Show that the resulting algorithm converges to the solution. V

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Give an explicit upper bound of the error on in the case when is chosen of the form

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, where .

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