Gerschgorin circles: intersection of union of circles

Good Morning, The following program creates the Gerschgorin's Circle for matrix A and plot them (in the first figure) and for the matrix A^T (in the second figure). In the third figure the program should find the INTERSECTION of the union of the circles of the matrix A and the union of circles or matrix A^T. In symbol

where

and Hi it's the same but for columns.

For example for the matrix

-2 0 -6

2 -2 0

0 2 -2

The program doesn't work because in the third figure it should plot a circles with center -2 and radius 6.

The program it's the following:

function gerschgorin(A)

if size(A,1) ~= size(A,2) %Il numero delle righe deve essere uguale al numero delle colonne;

error('La matrice deve essere quadrata.');

return;

end

%FIGURA 1: Calcolo e rappresento i cerchi associati alla matrice A;

figure;

for i=1:size(A,1)

% Il cerchio ha centro in (h,k) dove h equivale alla parte reale di A(i,i)

%(i-esimo elemento della diagonale di A) e k che equivale alla parte immaginaria di A(i,i);

h=real(A(i,i)); k=imag(A(i,i));

% Calcolo il raggio del cerchio;

r=0;

for j=1:size(A,1)

if i ~= j

r=r+(norm(A(i,j)));

end

end

t=0:0.01:2*pi; %I valori di t variano da 0 a 2pigreco con passo 0.01;

plot( r*cos(t)+h, r*sin(t)+k ,'-'); %Rappresento i cerchi con le coordinate polari;

hold on

c=plot( h, k,'bo'); %Rappresento i centri dei cerchi con dei pallini blu;

title('Cerchi associati alla matrice A')

end

%Autovalori: calcolo e rapresento gli autovalori con pallini rossi;

ev=eig(A);

for i=1:size(ev)

rev=plot(real(ev(i)),imag(ev(i)),'ro');

end

xline(0); yline(0);

axis equal;

legend([c rev],'Centri','Autovalori')

grid on;

xlabel('Parte Reale'); ylabel('Parte Immaginaria')

%FIGURA 2:Calcolo e rappresento i cerchi associati alla matrice A trasposta;

figure;

for i=1:size(A,1)

h=real(A(i,i)); k=imag(A(i,i));

r=0;

for j=1:size(A,1)

if i ~= j

r=r+(norm(A(j,i)));

end

end

t=0:0.01:2*pi;

plot( r*cos(t)+h, r*sin(t)+k ,'-'); hold on;

c=plot( h, k,'bo');

title('Cerchi associati alla matrice \itA^{T}')

end

%Autovalori

ev=eig(A);

for i=1:size(ev)

rev=plot(real(ev(i)),imag(ev(i)),'ro');

end

xline(0); yline(0);

axis equal;

legend([c rev],'Centri','Autovalori')

grid on;

xlabel('Parte Reale'); ylabel('Parte Immaginaria')

% FIGURA 3: Rappresento l' intersezione tra l'unione dei cerchi associati alla matrice A e l'unione

% dei cerchi associati alla matrice A ^ T (l'intersezione è data dalle aree con il bordo

% rosso)

figure;

for i = 1:size(A,1)

h = real(A(i,i));

k = imag(A(i,i));

r1 = 0;

r2 = 0;

x = [];

y = [];

for j = 1:size(A,1)

if i ~= j

r1 = r1 + abs(A(i,j));

r2 = r2 + abs(A(j,i));

x1 = r1*cos(t) + h;

y1 = r1*sin(t) + k;

x2 = r2*cos(t) + h;

y2 = r2*sin(t) + k;

x = [x, x1];

y = [y, y1];

end

end

polyout1 = polyshape(x, y);

polyout2 = polyshape(x2, y2);

intersection = intersect(polyout1, polyout2);

plot(intersection, 'EdgeColor', 'red');

hold on;

c = plot(h, k, 'bo');

title ("INtersezione dei due insiemi di cerchi");

end

%Autovalori (Calcolo e rappresentazione con pallini rossi)

ev=eig(A);

for i=1:size(ev)

rev=plot(real(ev(i)),imag(ev(i)),'ro');

end

xline(0); yline(0);

axis equal;

grid on;

xlabel('Parte Reale'); ylabel('Parte Immaginaria')

legend([c rev],'Centri','Autovalori')

end

THANK YOU for your anwers.

Answer 1

[Adeline]

I understand you are trying to execute a program to create the Gerschgorin's Circle for the example matrix A.

The mentioned code has three sections where section-1 generates three circles plotted in figure 1 with their centers at -2 and radius as 6, 2 and 2, respectively. Similarly, section-2 plots three circles with their centers at -2 and radius as 2, 2 and 6, respectively.

In the third section, the objective is to unite the three circles from section-1 to form one polyshape. Similarly, the three circles in section-2 should form another polyshape. The intersection of these polyshapes would result in a single circle with center 2 and radius 6.

However, the mentioned code intersects the first circle generated in section-1 with the first circle generated in section-2. It repeats these steps for the next two circles resulting in three circles with centre at -2 and radius as 2.