Page 219 - MATLAB an introduction with applications

204 ——— MATLAB: An Introduction with Applications

 n 
i ∑
x = 1  b − A x  i = 1, 2, …, n
i
A ii    j= 1 ij j   
ji ≠
Hence, the iterative scheme is
 n 
i ∑
x ← 1  b − A x j 
ij
i
A ii    j= 1    i = 1, 2, …, n
ji ≠
We start by choosing the starting vector x. The procedure for Gauss-Seidel algorithm is summarized here
with relaxation:
(k)
th
(a) conducting k iterations with ω = 1 (or k = 10). After the k iteration record ∆x .
(b) carryout additional p iterations (p ≥ 1) and record ∆x (k + p) after the last iteration.
(c) perform all subsequent iterations with ω = ω , where
opt
2
ω opt =
+
∆
1+ 1− ∆ (kp ) / x ( ) k ) 1/ p
( x
4.7 GAUSS-JORDAN METHOD

Let us consider a system of linear algebraic equations, in the matrix form
[A]{x} = {b}
where, for simplification, [A] is of order 3 × 3. The augmented matrix is

a  a a b 1 
 a 11 a 12 a 13 b 
 21 22 23 2  ...(4.6)
3
 a  31 a 32 a 33 b 
The solution of equation (4.6) is
–1
[I]{x} = [A] {b}
where [I] is the identity matrix. The augmented matrix is
 10 0 α 1 
 01 0 α 
 2  ...(4.7)
  001 α 3 

In the Gauss-Jordan method the augmented matrix (4.6) is converted to the augmented matrix (4.7) by a
series of operations similar to the Gaussian elimination method. In the Gaussian elimination method an
upper triangular matrix is derived while in the Gauss-Jordan method an identity matrix is derived.