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ITERATIVE METHODS TO SOLVE EQUATIONS 103
The following example illustrates that the Jacobi iteration and the Gauss–Seidel
iteration can also be used for solving a system of nonlinear equations, although there
is no guarantee that it will work for every nonlinear equation.

Example 2.5. Gauss–Seidel Iteration for Solving a Set of Nonlinear Equations.
We are going to use the Gauss–Seidel iteration to solve a system of nonlinear
equations as
2
2
x + 10x 1 + 2x − 13 = 0
1 2
(E2.5.1)
2
3
2x − x + 5x 2 − 6 = 0
1 2
In order to do so, we convert these equations into the following form, which
suits the Gauss–Seidel scheme.
2 2
(13 − x − 2x )/10
x 1 1 2
= 3 2 (E2.5.2)
x 2 (6 − 2x + x )/5
1 2
We make the MATLAB program “nm2e05.m”, which uses the Gauss–Seidel
iteration to solve these equations. Interested readers are recommended to run
this program to see that this simple iteration yields the solution within the given
tolerance of error in just six steps. How marvelous it is to solve the system of
nonlinear equations without any special algorithm!

(cf) Due to its remarkable capability to deal with a system of nonlinear equations, the
Gauss–Seidel iterative method plays an important role in solving partial differential
equations (see Chapter 9).

%nm2e05.m
% use Gauss–Seidel iteration to solve a set of nonlinear equations
clear
kmax = 100; tol = 1e-6;
x = zeros(2,1); %initial value
for k = 1:kmax
xp = x; % to remember the previous solution
x(1) = (13 - x(1)^2 - 2*x(2)^2)/10; % (E2.5.2)
x(2) = (6 - x(1)^3)/5;
if norm(x - xp)/(norm(xp) + eps)<tol, break; end
end
k, x

2.5.3 The Convergence of Jacobi and Gauss–Seidel Iterations

Jacobi and Gauss–Seidel iterations have a very simple computational structure
because they do not need any matrix inversion. So, it may be of practical use, if
only the convergence is guaranteed. However, everything cannot always be fine,