v

one projection step from and let , , and .

!

Show that

£ £ 0( U£ ! 3 49 " " q3 9 & " " q3

)' 9

A

Does this equality, as is, establish convergence of the algorithm?

` Assume now that is selected at each projection step to be the index of a component of

1& §

largest absolute value in the current residual vector . Show that !

Wv

642 % " 2

5 3 92 " 2

3

’

9eq3

8¤

7

1¨! X£ @

fq3

9

in which is the spectral condition number of . [Hint: Use the inequality

72 2 A@

uW„t ¤

v .] Does this prove that the algorithm converges?

!

§ 1

2 Consider the linear system , where is a Symmetric Positive De¬nite matrix. Consider

B ¢

£ &

D(R ¨¥¦1

C© § C

a projection step with where is some nonzero vector. Let be the new

E § „t

v & "

%FE § Tt

v

iterate after one projection step from and let , and . "

Show that

9

& " "

¡q3 ' 9 C ! 3

C A9 C

q3

q3 49 "

"

Does this equality establish convergence of the algorithm?

2 !2

` 9 G3

C ! C

In Gastinel™s method, the vector is selected in such a way that , e.g., by

v

1 §

C £C ! X£ P3TR IHQ1

P 9

de¬ning the components of to be , where is the current

!

residual vector. Show that

Wv

5 3 2 & " 2 32 "2

’

9eq3

7Q¤

fq3

9

in which is the spectral condition number of . Does this prove that the algorithm

7

converges?

•” Compare the cost of one step of this method with that of cyclic Gauss-Seidel (see Example

R ¢

£ £ 0FR ¨¦1

© §¥

5.1) and that of “optimal” Gauss-Seidel where at each step and is a

component of largest magnitude in the current residual vector.

¢

3 In Section 5.3.3, it was shown that taking a one-dimensional projection technique with

I£

SR ¨¦1

© §¥ ! FR T¦1

© §¥ !

X X i

and is mathematically equivalent to using the usual steepest

¡

§X

X

descent algorithm applied to the normal equations . Show that an orthogonal pro-

¢

§ X ¡ X

jection method for using a subspace is mathematically equivalent to applying

¢ ¢

£ § ¡

a projection method onto , orthogonally to for solving the system .

4 Consider the matrix

1@

U %V

U"

% A

V %

µ£ „ ¢

|5¥ qzl 5 ¢ ¤ ¤¥

„ 5| | §

¢tC

9 ¡

§ "!

Find a rectangle or square in the complex plane which contains all the eigenvalues of ,

without computing the eigenvalues.

` Is the Minimal Residual iteration guaranteed to converge for a linear system with the ma-

trix ?

5 Consider the linear system

§

p r

v v v

§

p

p p

in which and are both nonsingular matrices of size each.

v

¢

h£

¢ BCBB v