Fixed-Point Iteration Method
Let
f1(x, y, . . . , z) = 0
f2(x, y, . . ., z) = 0
.  .  .
.  .  .
.  .  .
fn(x, y, . . ., z) = 0 
are n Transcendental equations in n independent variables x, y, . . ., z. Then by starting with some initial approximation (x0, y0, . . ., z0) generating a sequence {(xi, yi, . . ., zi)} using


xi+1 = g1(xi, yi , . . . ,zi)  from the first equation
yi+1 = g2(xi, yi , . . . ,zi)  from the second equation
.  .  .
.  .  .
.  .  .
zi+1 = gn(xi, yi , . . . ,zi) from the last equation

which converges to (s, t , . . ., u) is called the fixed point iteration to solve system of non-linear equations.

 
Condition for Convergence :
The above fixed point iteration scheme converges only if
| g i |
| g i
|
| g i   | 
  + 
    +   .  .  .    +
x
y
z
at (s, t, . . ., u) must be less than one for all i = 1, 2, . . ., n.
Example:
system of equations- fixed point iteration
Solve for  x  and y  if  x 2 - y = 0     and   8x - 4x2 +32 - 9y2 = 0   .
Let   xi+1 = g1(xi,  yi) = (2xi + x2i - y)/2                                                
yi+1 = g2(x   i, yi) = (2xi - x2i + 8)/9  +  (4yi - y2i)/4

Let the initial approximation is (-1, 1)
 
 
i
0
1
2
3
4
5
6
7
8
9
10
xi
-1
-1
-1.153
-1.153
-1.206
-1.181
-1.169
-1.172
-1.175
-1.174
-1.174
yi
1
1.306
1.435
1.435
1.405
1.371
1.373
1.379
1.379
1.375
1.375

 
Solution of Transcendental Equations | Solution of Linear System of Algebraic Equations | Interpolation & Curve Fitting
Numerical Differentiation & Integration | Numerical Solution of Ordinary Differential Equations
Numerical Solution of Partial Differential Equations