| ¥£ j qz y C§ 5

„ 5| | 5£ |

¡

¢ ¡

©¡

¨§ £ "!

We begin with the decomposition

r±0 i ¯®

² ² $¬ §

# ()& &

%

'

²§ ²

in which is the diagonal of , its strict lower part, and its strict upper part, as

# 1

% 1

'

§

illustrated in Figure 4.1. It is always assumed that the diagonal entries of are all nonzero.

-F

D

-E

CDB@85¤4 2

A 9 76 3

Initial partitioning of matrix A.

The Jacobi iteration determines the -th component of the next approximation so as

E

to annihilate the -th component of the residual vector. In the following, denotes the P H GR

E QI¤F

© ¨

RS

-th component of the iterate and the -th component of the right-hand side . Thus,

E E

H

writing T

r±a i ¯®

X

tA¨

©§ ² ¬

¡Y` R

(

T

"VH

WU

RX b b

in which represents the -th component of the vector , yields

E

« ² ¬ ePW"QUIHG R F dRR c gf

( R S xG u F vR c rg q

w VI u

PH

iph

sth

or

« ²RS‚ ± ¯ i ¯®

gf

…GpF vR c rg

¬ rP"QUDHGpR F vE

¬ r¤

q

„ QI u u

W dR y c PH ‘ V(

(

R y

h

i

sth

This is a component-wise form of the Jacobi iteration. All components of the next iterate

©

can be grouped into the vector . The above notation can be used to rewrite the Jacobi

"QH

WU

iteration (4.4) in vector form as T

r±— i ¯®

©X

¬ ’QUH © ¨

' w % W ”#` – W •# w

“ “

W H

Similarly, the Gauss-Seidel iteration corrects the -th component of the current ap- E

¬AE r¤V(™˜( y

proximate solution, in the order , again to annihilate the -th component of

( E

the residual. However, this time the approximate solution is updated immediately after the

r¤Q(™˜( y E

¬

new component is determined. The newly computed components , can HR

PVIGF (

be changed within a working vector which is rede¬ned at each relaxation step. Thus, since

³

£ q¦ ¡k q`¤¨¤¦tst ¢ V£¤

s©| | §£¥ j ¦

£¡¨ !p

§

¢

vE

¬

the order is , the result at the -th step is

¡™(

(˜ E

y R

« f ² ePW"QUIHG R F dRR c ² Pe’QUIHG u F vuR c W “ f ² R S a± i ¯®

$ QIG u F u R c ¬

(Y

W PH

u

u

R

W "U

W

which leads to the iteration,

R

« f ² rPW"QUDHGpu F u R c W “ f ² ‚ a± i ¯®

„ R S wxGpF u R c

¬ Pe"QUIHGpR F "(

¬E r¤

PVI u

W IR y c H ‘ (

(

R y

u

Ru

W "U

W

The de¬ning equation (4.6) can be written as

&² W"VUH ©˜% w ¨

©# ¬ H '

©

w ¡Y

(

"VH

WU

which leads immediately to the vector form of the Gauss-Seidel iteration T T

a± i ¯®