© ¬ ' W “ &²

©% ² ¨

w

# # % –W “

W"QH

U H

Computing the new approximation in (4.5) requires multiplying by the inverse of the

#

²

diagonal matrix . In (4.8) a triangular system must be solved with , the lower

# %$

§

triangular part of . Thus, the new approximation in a Gauss-Seidel step can be determined

²

either by solving a triangular system with the matrix or from the relation (4.7). # %&

A backward Gauss-Seidel iteration can also be de¬ned as T

a± i ¯®

X

©'² ¨ H $¬

©% –w

# (

’QH

WU

’W¤(‘¤

²

which is equivalent to making the coordinate corrections in the order .A (

(

y y

Symmetric Gauss-Seidel Iteration consists of a forward sweep followed by a backward

sweep.

The Jacobi and the Gauss-Seidel iterations are both of the form T

a±!w°i ¯®

X

© ¬f¨ w H © ¬ ¨ H © ²

§ –w

(

"UQH

W

in which

R±%°w°i ¯®

¬§ ² "

§ ¬ ¬ ²

is a splitting of , with for Jacobi, for forward Gauss-Seidel, # # %

## ¬

²

and for backward Gauss-Seidel. An iterative method of the form (4.10) can

'

be de¬ned for any splitting of the form (4.11) where is nonsingular. Overrelaxation is

based on the splitting T T T

²X X X

¬ §# %# w '# ##

² ²

# (

y

and the corresponding Successive Over Relaxation (SOR) method is given by the recursion

T T

a±0 w°i ¯®

©X h©'# X # ²

%# w ' #% –# w

#

² ¬ ¨

$ &

"UQH

W H

y

The above iteration corresponds to the relaxation sequence T

Ee( QPDHG R F X # ²

w 01)F #

¬ rP"UVIHG R F ¬ r¤

(R ‘ Q™(

( ((˜

W

y y

in which is de¬ned by the expression in the right-hand side of (4.7). A backward SOR

01( R F

sweep can be de¬ned analogously to the backward Gauss-Seidel sweep (4.9).

³ y5{¦¤„ ¢

| ¥£ j qz y C§ 5

„ 5| | 5£ |

¡

¢ ¡

©¡

¨§ £ "!

A Symmetric SOR (SSOR) step consists of the SOR step (4.12) followed by a back-

ward SOR step, T T

X#

¨ ¥£# ¢ w H h©'# XX ## ² y T

w ' # % ¤¢

#

²T © ¬

£ %

$ &

# % "UQH X ' #

W #w

² © ¬ ¨ "QUH h©& # ² y

w%$

#

"QH

WU W

This gives the recurrence

© ™©§¬

©¨ ¦ ’¨ w

(

’UQH

W H

where T T T

w % # W “ XT ' #

XT X # T

²#¬ ² ¥

¨¦ #

r±Qaw°i ¯®

Tw ' # yW

X X# X

“ %# ² ²

T ( T#

#

± ¯ w°i ¯®

Xy

X# X

²# # ¬ w % #$

## %#

² ² ¨

¨ w W “ ' #& – W “

y

Observing that T T T T T

X X

wX%# T # $

X X

w % #$ #&# # %# # '# # %#

² ² ² ²E ¬ g˜ T

² ²

&

W“ W“

y X X

## (W“ % #

²g ² ¬ ²

˜ w #