Gerschgorin circles: intersection of union of circles
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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
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/1409714/image.jpeg)
where
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/1409719/image.jpeg)
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.
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Antworten (1)
Adeline
am 25 Aug. 2023
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.
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