Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 9.735910e-18.

I am solving the laplace equation in spherical coordinates with a cell (spherical capacitor) at the origin. The equation becomes,

After discretizing using the finite difference method and applying the appropriate boundary conditions, I am getting the correct answer off by ~ 0.3% , I am also recieving the warning message that the Matrix is close to singular. Is RCOND = 9.735910e-18 small enough that I can ignore this warning? How does this warning effect my solution?

close all
clear all
clc
%% setting up A matrix
a = 50e-6; % cell radius
dth = pi/128; % angle step size
dp = 3*a/60; % radius step size
dt = 1e-8; % time step
angles = dth/2:dth:2*pi-dth/2; % angle values
radii = 0:dp:3*a; % radii values

E = 40e3 ; % applid field strength 
C = 1e-2; % Capacitance
g = 2; % Conductance
si = 0.455; % intracellular conductivity
se = 5; %extracellular conductivity
Vrest = -0.08;
% constructing 'A' Matrix
A = zeros(15616,15872);
k = 256; 
for i = 1:60
for j = 1:256
    if j == 1 && i ~= 21 && i~=20 && i~=60
    A(k*(i-1)+j,k*i+j) = -2/dp^2 - 2/(((i*dp)^2)*(dth^2)); %U(i,j)
    A(k*(i-1)+j,i*k+1+j) = 1/(((i*dp)^2)*(dth^2)) + cot((2*pi/256)*(j-1+0.5))/(((i*dp)^2)*2*dth) ; %U(i,j+1)
    A(k*(i-1)+j,(i+1)*k) = 1/(((i*dp)^2)*(dth^2)) - cot((2*pi/256)*(j-1+0.5))/(((i*dp)^2)*2*dth); %U(i,j-1)
    A(k*(i-1)+j,(i-1)*k + j) = 1/dp^2 - 2/(2*i*dp*dp); %U(i-1,j)
    A(k*(i-1)+j,k*(i+1)+j) = 1/dp^2 + 2/(2*i*dp*dp);  %U(i+1,j)
    elseif j == k && i ~= 21 && i~=20 && i~=60
    A(k*(i-1)+j,k*i+j) = -2/dp^2 - 2/(((i*dp)^2)*(dth^2));
    A(k*(i-1)+j,i*k+1) = 1/(((i*dp)^2)*(dth^2)) + cot((2*pi/256)*(j-1+0.5))/(((i*dp)^2)*2*dth) ;
    A(k*(i-1)+j,i*k-1+j) = 1/(((i*dp)^2)*(dth^2)) - cot((2*pi/256)*(j-1+0.5))/(((i*dp)^2)*2*dth);
    A(k*(i-1)+j,(i-1)*k+j) = 1/dp^2 - 2/(2*i*dp*dp);
    A(k*(i-1)+j,k*(i+1)+j) = 1/dp^2 + 2/(2*i*dp*dp);  
    elseif  i ~= 21 && i~=20 && i~=60
    A(k*(i-1)+j,k*i+j) = -2/dp^2 - 2/(((i*dp)^2)*(dth^2));
    A(k*(i-1)+j,i*k+1+j) = 1/(((i*dp)^2)*(dth^2)) + cot((2*pi/256)*(j-1+0.5))/(((i*dp)^2)*2*dth) ;
    A(k*(i-1)+j,i*k-1+j) = 1/(((i*dp)^2)*(dth^2)) - cot((2*pi/256)*(j-1+0.5))/(((i*dp)^2)*2*dth);
    A(k*(i-1)+j,(i-1)*k+j) = 1/dp^2 - 2/(2*i*dp*dp);
    A(k*(i-1)+j,k*(i+1)+j) = 1/dp^2 + 2/(2*i*dp*dp);  
    elseif i == 21
    A(k*(i-1)+j,k*i+j) = 3*se/(2*dp); %U(i,j)
    A(k*(i-1)+j,k*(i+1)+j) = -4*se/(2*dp); %U(i+1,j)  
    A(k*(i-1)+j,k*(i+2)+j) = se/(2*dp) ; %U(i+2,j)
    A(k*(i-1)+j,15616+j) = -C/dt -g; %U(Vm,j)
    elseif i == 20
    A(k*(i-1)+j,k*i+j) = -3*si/(2*dp); %U(i,j)
    A(k*(i-1)+j,k*(i-1)+j) = 4*si/(2*dp); %U(i-1,j)  
    A(k*(i-1)+j,k*(i-2)+j) = -si/(2*dp) ; %U(i-2,j)
    A(k*(i-1)+j,15616+j) = -C/dt -g; %U(Vm,j)    
    elseif i == 60
    A(k*(i)+j,k*(i+1)+j) = 1; %U(i,j)
    A(k*(i)+j,k*20+j) = -1; %U(20,j)
    A(k*(i)+j,k*21+j) = 1; %U(21,j)
    end  
end 
end
% origin is average of surounding disk
A = [zeros(256,15872) ; A ] ;
A(1:256,1:256) = diag(-256*ones(1,256));
A(1:256,257:512) = ones(256,256);
A(15361:end-256,15361:end-256) = diag(ones(1,256)); % Boundary Condition E_app
b = zeros(15872,1);
b(15361:end-256) = -E*3*a*cos(angles); % Boundary Condition E_app
b(5121:5376) = -1*(Vrest*(C/dt) + g*Vrest);
b(5377:5632) = -1*(Vrest*(C/dt) + g*Vrest);
dA = decomposition(A);
x = dA\b;
X = transpose(reshape(x,[256,62]));
Vm = zeros(1000,256);
Vm(1,:) = -0.08;
for t = 1:1000
    Vm(t+1,:) = X(62,:);
    b(5121:5376) = -1*((transpose(X(62,:)))*(C/dt) + g*Vrest);
    b(5377:5632) = -1*((transpose(X(62,:)))*(C/dt) + g*Vrest);
    x = dA\b; 
    X = transpose(reshape(x,[256,62]));  
end 
imagesc(X(1:end-1,:))

[John D'Errico]

Small reciprocal condition numbers are NOT good. It is NOT small enough to ignore. In fact, it is too large.

[Anthony Gurunian]

Ok, that's my bad, I mean is it large enough to ignore? I was doing this earlier with polar coordinates, and I wasn't getting a warning. I went back to check what the RCOND was, turns out it was 3.07e-12. How is error related to RCOND, and can I do anything to bring RCOND closer to 1?

Answer 1

[Christine Tobler]

I tried plotting the following two things:

>> figure; semilogy(max(abs(A), [], 1))
>> figure; semilogy(max(abs(A), [], 2))

From these, it looks like the first 200 columns and the last few hundreds rows of A have much smaller scale than the others. This will usually lead to a bad condition number - if a row or column of A was exactly zero, that matrix A would be singular.

If it makes sense for your problem to change the scaling on those rows and/or columns, that is likely to improve the conditioning of A. Possibly changing the scaling of the boundary conditions would solve the problem.

Keep in mind this isn't guaranteed: If a matrix has some rows or columns with much smaller scale (about 1e-15) than others, this means it has a bad condition number - but if all rows and columns have similar scaling, that's not enough to make it well conditioned in general (think of the matrix of all ones for example, which is singular).

[Anthony Gurunian]

I see. I guess it's a property of the system then. The equations have a different form at those locations. I am getting the right answer, so I'm not too woried about it. Thanks for the help!