Numerical methods - lecture 5
Jiří Zelinka
Autumn 2017
Jiří Zelinka
1/11
Iteration methods for solving system of linear equations
Ax = b —>    x = 7~x + g Iteration process:
x/c+i =Txk + g = 0,1,...
Solution:
x = (E-T)-1g
2/11
Theorem
The sequence {x^}^ determined by the iteration process x = Tx + g converges for every initial iteration
x° g Rn       p(T) < 1. In this case
lim x = x, x = Tx + g
/c—>-oo
Jiří Zelinka
Numerical methods - lecture 5
3/11
Jacobi iteration method
System of linear equations:
Ax = b
/-th equation:
an*i H-----1- 3nXi H-----h a/A?xn = b;
The component x,- is expressed
n
Xi = -
E
j'=i
Xj +
'//
and it is used as the new (k + l)-th iteration
n
xf+1 = -
E
j'=i
'y „/c
7    ' 5
a a ^a
□ iS1
Jiří Zelinka Numerical methods - lecture 5
4/11
Matrix notation
x
X
/c+1
/c+1
/c+1
V
o
^21
^22
3nl
'nn
312
an
0
3n2
'nn
3l„ \
an
32 n 322
0
X-,
Xr
3ll
b2_
322
V 7Z J
5/11
Ax = b,      A = D + L+U, Ax = (D + L + U)x = b
/ au O \
\ o      ann y
Jiří Zelinka
6/11
L =
í o
■
V anl ■ ( O 3i2
U =
Vo
a„,„-i  O /
3ln \
3/7—1, n
O
= -D"1(/.+ Ľ)x+ D_ib.
-i
■k+i _ n-i
= -D~l(L+ U)xk + D^b.
□ iS1
Jiří Zelinka Numerical methods - lecture 5
7/11
x^1 = Tjxk + D~1b:
Tj = -D~\L + U), ty = -f for i^j, ta = 0
3,7
a
Tj =
0
^21
322
V
3nl
]nn
3l2
0
3n2
]nn
3l„ \
3ll 32 n
322
0
D_1b =
3ll
322
V a™ y
Jiří Zelinka
□ [31 Numerical methods - lecture 5
8/11
Gauss-Seidel iteration method
The component of the new iteration is used in the following step:
x
x
x
k+l
k+1
1
an 1
a22 1
a33
(t
(t>i — 3\2X2 ~ a13x3 ~~ a14x4 — • • • > )
321*i+1 — 323x3 ~ a24*4 — • • • ? )
./c+1
Jc+1
a31*i ' * — a32*2 ' ~ ~~ a34*4 — • • • ? )
X
/c+1
1
'//
i-l
E
j'=i
5ľ
7=/+l
X
J
□ iS1
Jiří Zelinka Numerical methods - lecture 5
9/11
Matrix notation:
Ax = b
{D + L+U)x {D + L)x
= -Ux + b
=  -(D + L)~1Ux + {D + L)-lb
-i
TG = -{D + L)-ľU,      xk+1 = TGxk + {D + L)
-i
Theorem: If A is diagonally dominant matrix, i.e.
an
>
E
'u
or
3;;
>
E
7//
'J'
then Jacobi and Gauss-Seidel methods converge.
□ [31
Jiří Zelinka Numerical methods - lecture 5
Relaxation (Succesive over-relaxation (SOR)) method
xk - k-th iteration
x^1 - the following iteration aquired by the Gauss-Seidel metod
u e (0,2) - relaxation parameter
xk+1 = (1 - u)xk + uxkGf
11/11