Solving a System by Iteration

• There is another way to attack a system of nonlinear equations. Consider this pair of equations:
  
  $$e^x-y=0$$
  
  
  
  
  $$xy-e^x=0$$
  
  

• We know how to solve a single nonlinear equation by fixed-point iterations -we rearrange it to solve for the variable in a way that successive computations may reach a solution.
  
  $$x=ln(y)$$
  
  
  
  
  $$y=e^x/x$$
  
  
  To start, we guess at a value for $y$, say, $y = 2$. See Table 3.
  

  Table 3: An example for solving a system by iteration

  y-value	x-value
  2	0.69315
  2.88539	1.05966
  2.72294	1.00171
  2.71829	1.00000
  2.71828	1.00000

  
  which are precisely the correct results.
• Here is another example for the pair of equations whose plot is Fig. 5.
  
  $$x^2 + y^2 = 4$$
  
  
  
  
  $$e^x + y = 1$$
  
  
  rearrangement;
  
  $$y=-\sqrt{(4-x^2)} x=ln(1-y)$$
  
  
  and begin with $x=1.0$, the successive values for y and x are: See Table 4.
  

  Table 4: Another example for solving a system by iteration

  y-value	x-value
  -1.7291	1.0051
  -1.72975	1.00398
  -1.72961	1.00421
  -1.72964	1.00416
  -1.72963	1.00417

  
  and we are converging to the solution in an oscillatory manner.