Matlab - fixed point iterations for general equation (nonlinear)

fixed point iterations:

find solution of X = F(X) as a limit of a sequence X_i+1 = F(X_i), i = 1, 2, ... (X can be a nuber, a vector, a matrix ...)

fixed point iterations for real equation:

ex. 1: x = sin(x) + 1

graphical solution:

>> x=[-3:0.2:3];
>> plot(x, sin(x) + 1, x, x)

numerical solution:

>> x = 6;           % choose an initial value
>> x = sin(x) + 1   % repeat, until x stays unchanged

ex. 2: x = 2 - cos(x/2)

>> x = 3;            % choose an initial value
>> x = 2 - cos(x/2)  % repeat, until x stays unchanged

graphical solution:

>> x=[-3:0.2:3];
>> plot(x, 2 - cos(x/2), x, x)

ex. 3: x = 2 - 3*cos(x)

>> x = 3;            % choose an initial value
>> x = 2 - 3*cos(x)  % does not converge

check graphically that the solution exists:

>> x=[-3:0.2:3];
>> plot(x, 2 - 3*cos(x), x, x)

fixed point iterations for vector equation:  X = F(X),  X = ( x1, x2 )^T

ex. 4:   F(X) = ( cos((x1-x2)/4), sin(x1+x2)/3 )^T

>> X = [0 ; 0];                                  % choose an initial value
>> X = [cos((X(1)-X(2))/4); sin(X(1)-X(2))/3 ]   % repeat until convergence

ex. 5:   F(X) = ( cos(4·(x1-x2)), 3·sin(x1+x2) )^T

>> X = [0 ; 0];                                  % choose an initial value
>> X = [cos(4*(X(1)-X(2))); 3*sin(X(1)-X(2)) ]   % does not converge

ex. 6 - linear equation:   F(X) = ( p·(a·x1+b·x2) + v1 , p·(c·x1 + d·x2) + v2)^T

>> a=2; b=3; c=1; d=2;  p = 0.1; v = [3 ; 6];    % choose the parameters
>> X = [0 ; 0];                                  % choose an initial value
>>  % write the example in matrix form:
>> A = [a b ;c d];
>> X = p*A*X + v          % repeat; the convergence depends on value of p
>> max(abs(eig(p*A)))     % spectral radius of the matrix p*A