Understanding linear systems to write computer programs to manually determine the Eigen Value and Spectral Radius of the given system of equations

Understanding linear systems to write computer programs to manually determine the Eigen Value and Spectral Radius of the given system of equations

## Matlab code (main code with functions code)

clear all;
close all;
clc;
n=10;
method =input('Enter the method no.')
%%
%%solving linear system and solving using itrative methods
%Diagonal matrix

D = [5 0 0;0 9 0;0 0 -7];
%Upper diagonal matrix
U = [0 1 2;0 0 4;0 0 0];
%Lower diagonal matrix
L = [0 0 0;-3 0 0;1 2 0];
B=[10 -14 33]';%RHS Residual

A =D+L+U;%Coefficient matrix

%Diagonal magnification
E =linspace(0.5,2,10);
x=[0 0 0 ]';
Dold = D;
xold=x;

Tjac=inv(D)*(L+U);
Eigen_value_jac=eig(Tjac);
rho_jac_sor = max(abs(real(Eigen_value_jac)));

for i=1:length(E)

D = E(i)*Dold;
%Diagonal magnification
  %Jacobi method
  if method==1
      Tjac=inv(D)*(L+U);
      [jac_it(i),x,error]=jacobi_iter(D,L,U,B,x);
      
      Eigen_value_jac=eig(Tjac);
      rho_jac(i) = max(abs(real(Eigen_value_jac)));
  end
  
  %Gauss seidel method
  if method==2
      Tgs=inv(D+L)*(U);
      [GS_it(i),x,error]=GS_iter(D,L,U,B,x);
      
      Eigen_value_gs=eig(Tgs);
      rho_gs(i) = max(abs(real(Eigen_value_gs)));
  end
   
  %Sucessive over relaxation method
    if method==3
      w=0.95;  
      Tsor=(inv(D+w*L)*((w-1)*D+(w*U)));
      [sor_it(i),x,error]=sor_iter(D,L,U,B,x);
      
      Eigen_value_sor=eig(Tsor);
      rho_sor(i) = max(abs(real(Eigen_value_sor)));
  end
   
  
end

if method==1
figure(1)
subplot(3,1,1)
plot(E,rho_jac);
xlabel('Diagonal magnification');
ylabel('Spectral radius');
title('Jacobi iterative solver');
legend('Diagonal magnification vs spectral radius');
hold on
subplot(3,1,2)
loglog(rho_jac,jac_it,'r');
xlabel('Spectral radius');
ylabel('No.of iteration');
legend('Spectral radius vs No.of iteration','location','northwest');
end

if method==2
figure(1)
subplot(3,1,1)
plot(E,rho_gs);
xlabel('Diagonal magnification');
ylabel('Spectral radius');
title('Gauss seidel method');
legend('Diagonal magnification vs Spectral radius');
hold on
subplot(3,1,2)
loglog(rho_gs,GS_it,'r');
xlabel('Spectral radius');
ylabel('No.of iteration');
legend('Spectral radius vs No.of iteration','location','northwest');
end

if method==3
figure(1)
subplot(3,1,1)
plot(E,rho_sor);
xlabel('Diagonal magnification');
ylabel('Spectral radius');
title('Sucessive over relaxation');
legend('Diagonal magnification vs Spectral radius');
hold on
subplot(3,1,2)
loglog(rho_sor,sor_it,'r');
xlabel('Spectral radius');
ylabel('No.of iteration');
legend('Spectral radius vs No.of iteration','location','northwest');
end


------------------------------------------------------------------------------------------
%% function jacobi
function [jac_it,x,error]=jacobi_iter(D,L,U,B,x)
x=[0 0 0]';
jac_it=0;
error=9e9;
tol=1e-4;
xold = x;
    while error>tol
        jac_it=jac_it+1;
        %Updated value
        x = (inv(D)*(L+U)*xold) +inv(D)*B;
        error=max(abs(xold-x));%error
        
        xold = x;
      
    end
    
end

----------------------------------------------------------------------------------------------
%% function Gauss seidel
function [GS_it,x,error]=GS_iter(D,L,U,B,x)
x=[0 0 0]';
GS_it=0;
error=9e9;
tol=1e-4;
xold = x;
    while error>tol
        GS_it=GS_it+1;
        %Updated value
        x = (inv(D+L)*(U)*xold)+inv(D+L)*B;
        error=max(abs(xold-x));%error
        
        xold = x;
      
    end
    
end

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%% function Successive Over Relaxation
function [sor_it,x,error]=sor_iter(D,L,U,B,x,w)
x=[0 0 0]';
sor_it=0;
error=9e9;
tol=1e-4;
xold = x;
    while error>tol
        sor_it=sor_it+1;
        w=0.75;
        %Updated value
        x = (inv(D+L*w)*((w-1)*D+(w*U))*xold)+inv(D+w*L)*B;
        error=max(abs(xold-x));%error
        
        xold = x;
      
    end
    
end

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Results :-

Plots of jacobi method:

Plots of Gauss seidel method:-

Plots of Successive Over Relaxation:-