Solving a System by Iteration

• There is another way to attack a system of nonlinear equations.
• Consider this pair of equations:
  equations;
  

  $e^x-y=0$,

  $xy-e^x=0$

  rearrangement;
  

  $x=ln(y)$,

  $y=e^x/x$

• 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.

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

• To start, we guess at a value for $y$, say, $y = 2$. See Table 3.
• Final values are precisely the correct results.

$\bullet$ Example: Another example for the pair of equations whose plot is Fig. 7.

equations;

$x^2 + y^2 = 4$,

$e^x + y = 1$

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

$\bullet$ We are converging to the solution in an oscillatory manner.