SOR method

After calculating the new value X by the Gauss-seidel method, that result is modified for a much more praised result.

It´s general formula is:

Where:

w : Factor of relaxation

w: < 1 Subrelaxation

> 1 Overrelaxation

If w = 1 then it´s the general formula of Gauss-Seidel

Notes:

It´s to be known that we don't take w = 0 because it would´t find better results on the initial approximation,

nor we take w ≥ 2 because this causes diversion.

CODE

clc

clear all

format long

a=input('Ingrese la matriz de coeficientes:\n ');

a

b=input('\nIngrese los términos independientes:\n ');

x=input('\nIngrese el vector con las aproximacimaciones Iniciales:\n ');

iter=input('\nIngrese el número máximo de iteraciones:\n ');

tol=input('\nIngrese la tolerancia:\n ');

w=input('\nIngrese el landa de la relajacion:\n ');

k=norm(a)*norm(a^-1);

disp('condicional=')

disp(k)

determinante=det(a);

if determinante==0

disp('El determinante es cero, el problema no tiene solución única')

end

n=length(b);

d=diag(diag(a));

l=d-tril(a);

u=d-triu(a);

fprintf('\n SOLUCION:\n')

fprintf('\nLa matriz de transicion de gauss seidel:\n')

Tw=((d-w*l)^-1)*((1-w)*d+w*u);

disp(Tw)

re=max(abs(eig(Tw)))

if re>1

disp('Radio Espectral mayor que 1')

disp('el método no converge')

return

end

fprintf('\nEl vector constante es::\n')

Cw=w*(d-w*l)^-1*b;

disp(Cw)

i=0;

err=tol+1;

z=[i,x(1),x(2),x(3),x(4),err];

while err>tol & i<iter

xi=Tw*x+Cw;

%err=max(sqrt(((xi(1)-x(1))^2)+((xi(2)-x(2))^2)+((xi(3)-x(3))^2)));

%err=norm(xi-x);

%err=max(abs(xi-x));

err=norm(xi-x)/norm(xi);

x=xi;

i=i+1;

z(i,1)=i;

z(i,2)=x(1);

z(i,3)=x(2);

z(i,4)=x(3);

z(i,5)=x(4);

z(i,6)=err;

end

%fprintf('\nTABLA:\n\n n x1 x2 x3 Error\n\n ')

disp(z)