v £¢ ¢ v ¢ # # AAA

H

¢ ¢£V v ¢ BBCB v £

, where is the -th column of the identity

¢ ¤ ¥$¤

matrix.

£C

¡© The ™s are orthonormal.

¤ #

C BB0B v C

The vectors are equal to the Arnoldi vectors produced by the Gram-Schmidt ver-

AAA

sion, except possibly for a scaling factor.

2 Rewrite the Householder implementation of the Arnoldi algorithm with more detail. In particu-

# 5

lar, de¬ne precisely the Householder vector used at step (lines 3-5). ¥

3 Consider the Householder implementation of the Arnoldi algorithm. Give a detailed operation

count of the algorithm and compare it with the Gram-Schmidt and Modi¬ed Gram-Schmidt

algorithm.

¢

4 Derive the basic version of GMRES by using the standard formula (5.7) with and

¦

¢

.

5 Derive a version of the DIOM algorithm which includes partial pivoting in the solution of the

Hessenberg system.

§ 1

6 Show how the GMRES and FOM methods will converge on the linear system when

„

„

§

‚

‚

¤

and with .

7 Give a full proof of Proposition 6.11.

8 Let a matrix have the form ¨g

§

A

g

Assume that (full) GMRES is used to solve a linear system, with the coef¬cient matrix . What

is the maximum number of steps that GMRES would require to converge?

9 Let a matrix have the form: ¨g

§

A

‚

Assume that (full) GMRES is used to solve a linear system with the coef¬cient matrix . Let

v !

©¤

¤!

!

©¤

© ¤ !

be the initial residual vector. It is assumed that the degree of the minimal polynomial of

‚

with respect to (i.e., its grade) is . What is the maximum number of steps that GMRES¦

v ¢ £ q3 £ ¥§ £ ¤ !9£

£

¤

would require to converge for this matrix? [Hint: Evaluate the sum

£ £

£ ‚

¤ ¨ £ © ¤ !

where is the minimal polynomial of with respect to .]

§

¨ ¡¡µ£ „ ¢

§|5¥ „yq¢| ¢ £ ¥§

5| j C§¦£¥

5

£ £ "–© ¡"

§

1B

10 Let ¨g

§

g §

„

..

g

.

A

§

g

„t g

v

§

‚

g

e g 3

9

Show that .

`

Assume that (full) GMRES is used to solve a linear system with the coef¬cient matrix .

What is the maximum number of steps that GMRES would require to converge?

¢ ¡ ¢

¢ ¢ ¢

11 Show that if is nonsingular, i.e., is de¬ned, and if , then , i.e., ! ¢!

both the GMRES and FOM solutions are exact. [Hint: use the relation (6.46) and Proposition

6.11 or Proposition 6.12.]

12 Derive the relation (6.49) from (6.47). [Hint: Use the fact that the vectors on the right-hand side

of (6.47) are orthogonal.]

13 In the Householder-GMRES algorithm the approximate solution can be computed by formulas

(6.25-6.27). What is the exact cost of this alternative (compare memory as well as arithmetic

£C

requirements)? How does it compare with the cost of keeping the ™s?

14 An alternative to formulas (6.25-6.27) for accumulating the approximate solution in the House-

£C ¢

holder-GMRES algorithm without keeping the ™s is to compute as

¢

¢ A0BA v E ¤ ¡

A

¡ ¡

where is a certain -dimensional vector to be determined. (1) What is the vector for the

¤

¢