MATLAB-CODES.rar_linear algebra

%% Q11: Test the convergence of Jacobi, Gauss Sediel, and SOR Iterative methods for Linear system Ax=b % where its coefficient matrix A given as the following : A=[4 -1 0 0;-1 4 -1 0;0 -1 4 -1;0 0 -1 4] %% Convert A to be A=L+D+U [L,D,U]=LDU_M(A); %% Get the Jacobi and Gauss-Sediel Itartion matrix T_J=Jacobi_Itr_M(L,D,U) T_G=Gauss_Sed_Itr_M(L,D,U) %% Find the the Spectral radius of the three methods Roh_TJ=Spectral_Radius_M(T_J); Roh_TG=Spectral_Radius_M(T_G); W=2/(1+sqrt(1-(Roh_TJ)^2)); % T_W=SOR_Itr_M(L,D,U,W) % Roh_TW=Spectral_Radius_M(T_W) Roh_TW=abs(w-1); %% Test the convergence Condition if Roh_TJ<1 sprintf('The spectral raduis of Jacobi Iteration Matrix=%0.4g <1',Roh_TJ) disp('The system "Converge" with Jacobi Iterative method'); else disp('The system "Not Converge" with Jacobi Iterative method'); end if Roh_TG<1 sprintf('The spectral raduis of Gauss Sediel Iteration Matrix=%0.4g <1',Roh_TG) disp('The system "Converge" with Gauss Sediel Iterative method'); else disp('The system "Not Converge" with Gauss Sediel Iterative method'); end if Roh_TW<1 sprintf('The spectral raduis of SOR Iteration Matrix=%0.4g <1',Roh_TW) disp('The system "Converge" with SOR Iterative method'); else disp('The system "Not Converge" with SOR Iterative method'); end %% Comapre between these Three methods if (Roh_TW<Roh_TJ && Roh_TJ<Roh_TG) disp('Roh_TW < Roh_TJ < Roh_TG'); disp('Then The SOR faster than Jacobi that faster than Gauss-Sed in convergence'); elseif (Roh_TW<Roh_TG && Roh_TG<Roh_TJ) disp('Roh_TW < Roh_TG < Roh_TJ'); disp('Then The SOR faster than Gauss-Sed that faster than Jacbi in convergence'); elseif (Roh_TJ<Roh_TW && Roh_TW<Roh_TG) disp('Roh_TJ < Roh_TW < Roh_TG'); disp('Then The Jacobi faster than SOR that faster than Gauss-Sed in convergence'); elseif (Roh_TJ<Roh_TG && Roh_TG<Roh_TW) disp('Roh_TJ < Roh_TG < Roh_TW'); disp('Then The Jacobi faster than Gauss-Sed that faster than SOR in convergence'); elseif (Roh_TG<Roh_TW && Roh_TW<Roh_TJ) disp('Roh_TG < Roh_TW < Roh_TJ'); disp('Then The Gauss-Sed faster than SOR that faster than Jacbi in convergence'); elseif (Roh_TG<Roh_TJ && Roh_TJ<Roh_TW) disp('Roh_TG < Roh_TJ < Roh_TW'); disp('Then The Gauss-Sed faster than Jacbi that faster than SOR in convergence'); end