## Iterative solution of a system of linear equations and an analysis of spectral radius of a matrix

UNDERSTANDING LINEAR SYSTEMS
(ANALYSIS OF VARIOUS ITERATIVE SCHEMES TO SOLVE A SYSTEM OF LINEAR EQUATIONS TO FIND THE EIGEN VALUES AND SPECTRAL RADIUS)

(A) PROBLEM STATEMENT:

Given coefficient matrix:

A = [[5,1,2],[-3,9,4],[1,2,-7]]

Given Solution Matrix:

X = [[x1],[x2],[x3]]

Given RHS Matrix:

B = [,[-14],]

We will be solving the system of linear equations using various iterative methods such as:

i. Gauss Jacobi method

ii. Gauss Siedel method

iii. Successive Over Relaxation (SOR) method

Theory:

For a general system of 3 linear equations in 3 variables we have:

[[a11,a12,a13],[a21,a22,a23],[a31,a32,a33]][[x1],[x2],[x3]]=[[b1],[b2],[b3]]

or [A][X]=[B] The solution is given by [x]=[A]^-1[B]

The Eigen Values of the matrix is given by the equation: |A-lambdaI|=0

|([a11,a12,a13],[a21,a22,a23],[a31,a32,a33])-lambda([1,0,0],[0,1,0],[0,0,1])|=0

This equation is known as the the characteristic polynomial of the matrix and the roots are the eigen values lambda of the matrix:

alambda^3+blambda^2+clambda+d=0

The maximum value of the Eigen Values (roots of the eqn.) is known as the Spectral Radius rho of the matrix.

(B) MATLAB PROGRAMS:

1. Main Syntax of the program:

%% MATLAB Program to compute the Eigen Values and Spectral Radius of a 3x3 Matrix
clear all;
close all;
clc;
%Given Matrix:
A = [5 1 2; -3 9 4; 1 2 -7];
% Defining the Coefficient Matrix:
D = [ A(1,1) 0 0; 0 A(2,2) 0; 0 0 A(3,3) ]; %Diagonal elements
L = [ 0 0 0; A(2,1) 0 0; A(3,1) A(3,2) 0 ]; % Lower triangular elements
U = [ 0 A(1,2) A(1,3); 0 0 A(2,3); 0 0 0 ]; % Upper triangular elements

B = [10 ; -14; 33];
m = 1; %Diagonal Magnification Parameter
% Computation of Eigen Values:
% Solving the equation |A-xI|=0
% Characteristic equation for the system: -x^3+7x^2+60x-402=0

coeff = [-1 7 60 -402];
Eigen_value = roots(coeff);
% Checking the Eigen Values of the Matrix
Eigen_actual = eig(A);
%Checking the spectral radius of A:

%% Solving the matrix by Iterative methods:
% Analytical Solution as reference:
exact_sol = AB;
% Solution by Gauss Jacobi
[iter_jac,x_jac,eig_jacobi,rho_jacobi] = jacobi_iter(D,L,U,B,m);
% Solution by Gauss Siedel
[iter_gs,x_gs,eig_gs,rho_gs] = siedel_iter(D,L,U,B,m);
% Solution by Successive Over Relaxation:
[iter_sor,x_sor,eig_sor,rho_sor] = sor_iter(D,L,U,B,m);

%% Solution of the system:
% Saving a copy of the solution Matrix before changing the diagonal terms
x_jac_old = x_jac; %Solution by Jacobi Method
x_gs_old = x_gs; %Solution by Gauss Siedel method
x_sor_old = x_sor; %Solution by SOR method

% Saving a copy of the spectral radius before changing the diagonal terms
iter_jac_old = iter_jac;
iter_gs_old = iter_gs;
iter_sor_old = iter_sor;

% Saving the values of eigen values before changing the diagonal terms:
eig_jac_old = eig_jacobi;
eig_gs_old = eig_gs;
eig_sor_old = eig_sor;

%% Setting the spectral radius to 1 by modifying diagonal elements:
for k = 1:3
if k == 1
rho_jac_old = rho_jacobi;
m = 0.4602;
[iter_jac,x_jac,eig_jacobi,rho_jacobi] = jacobi_iter(D,L,U,B,m);
end
if k == 2
rho_gs_old = rho_gs;
m = 0.5552;
[iter_gs,x_gs,eig_gs,rho_gs] = siedel_iter(D,L,U,B,m);
end
if k == 3
rho_sor_old = rho_sor;
m = 0.5337;
[iter_sor,x_sor,eig_sor,rho_sor] = sor_iter(D,L,U,B,m);
end
end


2. User Defined Function: Gauss Jacobi Method

function [iter_jac,x_jac,eig_jacobi,rho_jacobi] = jacobi_iter(D,L,U,B,m)
%% Solving the Linear system using Gauss Jacobi Method:
error = 1e9;
tolerance = 1e-5;
iter_count = 0;
D = m*D;
%Iteration Matrix for Jacobi method:
T_jac = inv(D)*(L+U);

x_old = [ 0 ; 0 ; 0 ];
while (error > tolerance)
x = -T_jac*x_old + inv(D)*B;
error = max(abs(x-x_old));
x_old = x;
iter_count = iter_count + 1;
end
iter_jac = iter_count;
x_jac = x';
I = eye(3);
syms f(X);
f(X) = det(T_jac-X*I);
Xjac = double(coeffs(f,X,'All'));
eig_jacobi = roots(Xjac);
rho_jacobi = max(real(abs(eig_jacobi)));
end



3. User Defined Function: Gauss Siedel Method

function [iter_gs,x_gs,eig_gs,rho_gs] = siedel_iter(D,L,U,B,m)
%% Solving the Matrix by Gauss Siedel method
error = 1e9;
tolerance = 1e-5;
iter_count = 0;

D = m*D;
% Iteration Matrix for Gauss Siedel
T_gs = inv(D+L)*U;
x_old = [ 0; 0; 0 ];

while error > tolerance
x = -T_gs*x_old + inv(D+L)*B;
error = max(abs(x-x_old));
x_old = x;
iter_count = iter_count + 1;
end
iter_gs = iter_count;
x_gs = x';
I = eye(3);
syms f(X);
f(X) = det(T_gs-X*I);
Xgs = double(coeffs(f,X,'All'));
eig_gs = roots(Xgs);
rho_gs = max(abs(real(eig_gs)));
end


4. User Defined Function: Successive Over Relaxation (SOR) method

function [iter_sor,x_sor,eig_sor,rho_sor] = sor_iter(D,L,U,B,m)
%% Solving the system by Succsssive Over Relaxation method:
error = 1e9;
tolerance = 1e-5;
iter_count = 0;
w = 0.9;
D = m*D;
%Iteration Matrix for SOR Method:
T_sor= inv(L+D)*(U-(L+D)*(1-w));

x_old = [0 ; 0; 0 ];
while error > tolerance
x = x_old - inv(D+w*L)*((D+w*L) - (1-w)*D +w*U)*x_old + w*inv(D+w*L)*B;
error = max(abs(x - x_old));
x_old = x;
iter_count = iter_count + 1;
end
iter_sor = iter_count;
x_sor = x';
I = eye(3);
syms f(X);
f(X) = det(T_sor-X*I);
Xsor = double(coeffs(f,X,'All'));
eig_sor = roots(Xsor);
rho_sor = max(real(abs(eig_sor)));
end



To the solve system iteratively we need to split the coefficient matrix [A] into the lower diagonal matrix [L], the upper diagonal matrix [U] and the diagonal matrix [D] such that

[A] = [L] +[D] +[U]

where L = [[0,0,0],[-3,0,0],[1,2,0]]

D = [[5,0,0],[0,9,0],[0,0,-7]]

U = [[0,1,2],[0,0,4],[0,0,0]]

the following user defined functions solve the system iteratively by computing the iteration matrix for each method. It is updated with each iteration until convergence is achieved.

Iteration Matrices for different iterative schemes: ([T] = iteration matrix)

i. Gauss Jacobi method: [T] = [D]^-1[L+U]

ii. Gauss Siedel method:  [T] = [L+D]^-1[U]

iii. Successive Over Relaxation method: [T] = [L+D]^-1([U]-(1-omega)[L+D])

(C) RESULTS AND DISCUSSIONS:

1. SOLUTION OF THE SYSTEM:

The exact solution of the system is given by [X]=[A]^-1[B]

[[x1],[x2],[x3]]=[[3.298507462686567],[1.268656716417911],[-3.880597014925373]]

The solution by iterative methods are given below:

i. Gauss Jacobi Method:

[[x1],[x2],[x3]]=[[3.298505536984095],[1.268657882614232],[-3.880594095875341]]

The number of iterations taking place: 21

ii. Gauss Siedel Method:

[[x1],[x2],[x3]]=[[3.298508561190570],[1.268657661889412],[-3.880596587861515]]

The number of iterations taking place: 14

iii. Successive Over Relaxation (SOR) method:

[[x1],[x2],[x3]]=[[3.298507986220588],[1.268656460583268],[-3.880596966137974]]

The number of iterations taking place: 08

For the SOR method we have used the relaxation parameter omega=0.9 i.e. under-relaxed condition. It is because keeping the system underrelaxed gives us the lowest number of  iterations as compared to omega>1 (over-relaxed condition). In that case the system becomes divergent and convergence is achieved after a far larger number of iterations as compared to the under-relaxed condition. The system changes to Gauss-Siedel method at omega=1.

2. EIGEN VALUES OF THE SYSTEM:

i. Gauss Jacobi Method:

[[lambda1],[lambda2],[lambda3]]=[[-0.0487 + 0.5078i],[-0.0487 - 0.5078i],[0.0975]]

ii. Gauss Siedel Method:

[[lambda1],[lambda2],[lambda3]]=[,[0.3276],[-0.0387]]

iii. Successive Over Relaxation Method:

[[lambda1],[lambda2],[lambda3]]=[[0.2276],[-0.1387],[-0.1]]

3. SPECTRAL RADIUS OF THE SYSTEM:

By exact method we have rho = 8.490

Solving the system iteratively we have the values of spectral radii by different methods:

i. Gauss Jacobi Method: rho = 0.5102

ii. Gauss Siedel Method: rho = 0.3276

iii. Successive Over Relaxation Method: rho = 0.2276

The SOR method gives the lowest value of rho with omega = 0.9

(D) CHANGING THE SPECTRAL RADIUS TO 1.1 BY MODIFYING THE DIAGONAL ELEMENTS:

In the following program we define m as the Diagonal Magnification Parameter. By changing it suitably we can set the spectral radius to 1.1 each time for different iterative schemes.

i. Gauss Jacobi Method:

Changing m to 0.4602 gives us a value of rho=1.1087 and the matrix becomes unsolvable by the above method.

ii. Gauss Siedel Method:

Changing m to 0.5552 gives us a value of rho=1.1035 and the matrix becomes unsolvable by the above method.

iii. Successive Over Relaxation Method:

Changing m to 0.5337 gives us a value of rho=1.1007 but surprisingly the matrix is solved by the SOR method.

(E) EFFECT OF DIAGONAL MAGNIFICATION ON SPECTRAL RADIUS AND NUMBER OF ITERATIONS:

Diagonal magnification refers to multiplying the diagonal elements with a constant to change the spectral radius. Here we have used a range of numbers to observe the same.

1. MATLAB CODE:

%% MATLAB Program to observe the effect of Diagonal Magnification on various Iterative solvers;
clear all;
close all;
clc;
%Given Matrix:
A = [5 1 2; -3 9 4; 1 2 -7];
% Defining the Coefficient Matrix:
D = [ A(1,1) 0 0; 0 A(2,2) 0; 0 0 A(3,3) ]; % Diagonal elements
L = [ 0 0 0; A(2,1) 0 0; A(3,1) A(3,2) 0 ]; % Lower triangular elements
U = [ 0 A(1,2) A(1,3); 0 0 A(2,3); 0 0 0 ]; % Upper triangular elements

B = [10 ; -14; 33];

m = linspace(5,0.2,10);
% Computation of Eigen Values:
% Solving the equation |A-xI|=0
% Characteristic equation for the system: -x^3+7x^2+60x-402=0

coeff = [-1 7 60 -402];
Eigen_value = roots(coeff);
% Checking the Eigen Values of the Matrix
Eigen_actual = eig(A);
%Checking the spectral radius of A:

%% Solution of the matrix by Iterative methods:
% Analytical Solution as reference:
exact_sol = AB;

for i = 1:length(m)
% Solution by Gauss Jacobi
[iter_jac(i),x_jac,eig_jacobi,rho_jacobi(i)] = jacobi_iter(D,L,U,B,m(i));
% Solution by Gauss Siedel
[iter_gs(i),x_gs,eig_gs,rho_gs(i)] = siedel_iter(D,L,U,B,m(i));
% Solution by Successive Over Relaxation:
[iter_sor(i),x_sor,eig_sor,rho_sor(i)] = sor_iter(D,L,U,B,m(i));
end

%% Plotting the solution:
% Plotting the Iteration Numbers vs SPectral Radii
fig_1 = figure(1);
hold on;
plot(rho_jacobi,iter_jac);
plot(rho_gs,iter_gs);
plot(rho_sor,iter_sor);
legend('Gauss Jacobi','Gauss Siedel','SOR');
grid minor;
ylabel('Number of Iterations: ');
title('Plot of Number of Iterations vs Spectral Radius (rho) for Iterative Schemes');

% Plotting the Diagonal Magnification Values vs Spectral Radii
fig_2 = figure(2);
hold on;
plot(m,rho_jacobi,'--m');
plot(m,rho_gs,'*b');
plot(m,rho_sor,'-.g');
legend('Gauss Jacobi','Gauss Siedel','SOR');
grid on;
xlabel('Diagonal Magnification Parameter');
title('Plot of the Spectral Radius: (rho) value vs a range of diagonal magnification values');



2. RESULTS AND DISCUSSIONS:

(i) Plotting the Iteration Numbers vs Spectral Radii: As we increase the spectral radius, the number of iterations also increases. The slope of the curve for the Jacobi method is much more compared to Gauss Siedel and SOR, hence requiring a larger number of iterations for convergence.

(ii) Plotting the Spectral Radii vs Diagonal Magnification Values: The spectral radius decreases more rapidly for SOR and Gauss Siedel Method as compared to  Jacobi method for the same increase in Diagonal Magnification Parameter.

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