©!¨

Verify that the eigenvalues of are given by

@285642)0(&$ ¡"

#

5 97 31 ' % ¤ BCBB

AAA

where

7 D

F¤ "

E#

and that an eigenvector associated with each is

X V 9 7 3 P A A 9 7 5 % 3 R 9 7 R P HI # G

A YW285 ¤ TR6Q1 BABBC 280U4SPQ1 28563TSQ1

Under what condition on does this matrix become positive de¬nite?

`

%

Now take . How does this matrix relate to the matrices seen in Chapter 2 for one-

dimensional problems?

a Will the Jacobi iteration converge for this matrix? If so, what will its convergence factor

be?

Wa

a

Will the Gauss-Seidel iteration converge for this matrix? If so, what will its convergence

factor be?

UWa

aa

For which values of will the SOR iteration converge?

b

2 Prove that the iteration matrix of SSOR, as de¬ned by (4.13), can be expressed as

dec

h

g

A QTu9 sb ypq3xwTu9 sb Ipq09 b iU3 b

%

3

pv t r

v t fc

d

Deduce the expression (4.27) for the preconditioning matrix associated with the SSOR iteration.

p

3 Let be a matrix with a positive diagonal .

Obtain an expression equivalent to that of (4.13) for but which involves the matrices

dc

„‚

… !‚

…

uW„t p uW„t p WvTt p r WTt p

v

v v

and .

` Show that

!‚

!‚ 3 „t 9 „‚

!‚ 09 g 9

WTt p g

gv 9 g 9 b “‘3

d c Wv p

v b 3 Tt 9

bv b3

b ’‘3 E

b3

E

•”

Now assume that in addition to having a positive diagonal, is symmetric. Prove that the

eigenvalues of the SSOR iteration matrix are real and nonnegative. dec

µ£ „ ¢

|5¥ j qz A C¡y 5

„ 5| | 5£§|

¢ ¡

£C @¡

¨§ £ "!

4 Let p

r

v

p ¡r

„

..

p .

.. ..

‚

¡r

. . ¢

¢ ¢p

¤p

£

where the blocks are nonsingular matrices which are not necessarily diagonal.

What are the block Jacobi and block Gauss-Seidel iteration matrices?

` Show a result similar to that in Proposition 4.3 for the Jacobi iteration matrix.

•”

Show also that for (1) the block Gauss-Seidel and block Jacobi iterations either both

b

converge or both diverge, and (2) when they both converge, then the block Gauss-Seidel

iteration is (asymptotically) twice as fast as the block Jacobi iteration.

¥ ¨v„t ¦

§ §

5 According to formula (4.23), the vector in iteration (4.22) should be equal to , where

is the right-hand side and is given in (4.52). Yet, formula (4.51) gives a different expression

¦

¥

for . Reconcile the two results, i.e., show that the expression (4.51) can also be rewritten as

© E 3 Tt 9 g © E % ¥ g

3 © A § Tt 9

v v

6 Show that a matrix has Property A if and only if there is a permutation matrix such that

Tt

v is consistently ordered.

7 Consider a matrix which is consistently ordered. Show that the asymptotic convergence rate

for Gauss-Seidel is double that of the Jacobi iteration.

8 A matrix of the form

%

I

r

"

is called a three-cyclic matrix.

What are the eigenvalues of ? (Express them in terms of eigenvalues of a certain matrix

which depends on , , and .)

r

`

Ep p

Assume that a matrix has the form , where is a nonsingular diagonal matrix,

and is three-cyclic. How can the eigenvalues of the Jacobi iteration matrix be related to

those of the Gauss-Seidel iteration matrix? How does the asymptotic convergence rate of the