y

Av ˜¤

¥¦ ¥¡ 9

A

' 1P ¥

dR c

§ ¬ ’E

¬

R

Let be a consistently ordered matrix such that for Y

¥

#

W¤V(£¢ ¡( y ¬

, and let . Then if is a nonzero eigenvalue of the SOR iteration matrix ©

Y

, any scalar such that

T

0

a± ¯ i ¯®

£X

£ £#

#w© ² ¬

! ©

y

is an eigenvalue of the Jacobi iteration matrix . Conversely, if is an eigenvalue of the ¢ £¡

¢

Jacobi matrix and if a scalar satis¬es (4.46), then is an eigenvalue of .

¢ © © 0

µ£ „ ¢

|5¥ j qz A C¡y 5

„ 5| | 5£§|

¡C ¢ ¡

C @¡

¨§ £ "!

£ ™£¤¢¡6

A

Denote by and by , so that

£ ¤

%W “ # ' W “ T# T T

X X X#

# #

² ¬ ²

£¡

¢ w

£ ¤

W“

0 y

and the Jacobi iteration matrix is merely . Writing that is an eigenvalue yields

w

£ ¤T ©

T T

X XX

£¤¢

¡ # # Y$ #

² ² ² ¬

w

£ ¤

© W“

y

which is equivalent to TT T T

²X£# ² ¬ XeX X # ²

¤¥

£¡ # w

¤

© $

Y

y

or eT

T T

X rX

X

¦¡¥

£ #w© #

² ² ¬

w

§©

£ ¤ Y`

y

¥

$# ¬

Since , this can be rewritten as

Y T

# w © £¦¥

² y ² X

¡

¬

w

§©

£ ¤ $

(Y

# T

X

¤T

#w©

# ² §

which means that is an eigenvalue of . Since is consistently £¥¢ w

£© ¥¢

£ ¤¢

£ X

¤ w X y§© T ¥¢

ordered, the eigenvalues of which are equal to are the W“© w

£ £ ¤

W© W©

£

same as those of , where is the Jacobi iteration matrix. The proof

¤ w £ W© w

£ ¤

follows immediately.

#

This theorem allows us to compute an optimal value for , which can be shown to be

equal to

r± ¯ i ¯®

T

˜

¬ $X" # £X

²

¨w

§ ¢

y# y # ¬

A typical SOR procedure starts with some , for example, , then proceeds with a

T# y

number of SOR steps with this . The convergence rate for the resulting iterates is esti- X #

mated providing an estimate for using Theorem 4.7. A better is then obtained from

¢ #

the formula (4.47), and the iteration restarted. Further re¬nements of the optimal are

calculated and retro¬tted in this manner as the algorithm progresses.

™¦Q¢m•“ •¢gmc¢©

— © “©™ • “©™

¡

The Alternating Direction Implicit (ADI) method was introduced in the mid-1950s by

Peaceman and Rachford [162] speci¬cally for solving equations arising from ¬nite dif-

ference discretizations of elliptic and parabolic Partial Differential Equations. Consider a

partial differential equation of elliptic type T T

T T T

Xb Xb

© © r± ¯ i ¯®

£ £

Xb Xb Xb

"

(

(