Linear Algebra Error: Matrix is close to singular or badly scaled.

Hi,

I have a simple localization code that calculates the x,y,z coordinates for a wave heard at 4 receivers. The equations are set up using matrices but it is giving me an error I don't understand. I checked if the matrix reaches singularity and it does not. So I am not sure what the error is pointing to. I am attaching the code here:

% Sensor/receiver locations/ vertices of a tetrahedron
% si(xi,yi,zi)
r1 = [0.125 0.125 0.125];
r2 = [0.25 0.125 0.125];
r3 = [0.1875 0.25 0.125];
r4 = [0.1875 0.1875 0.25];

% Calculate the Time of Arrival at the receivers 
c = 343; % m/s
t = zeros(1, 4);
sound_loc = [10 20 10]; % Replace x, y, and z with your sound location coordinates
vertices = [r1; r2; r3; r4]; % Replace vertex1 to vertex4 with your 3D vertices

for j = 1:4
    t(j) = pdist([sound_loc; vertices(j, :)], 'euclidean') / c;
end

% Setting up matrices to solve for location of source

A = [2*r1(1) 2*r1(2) 2*r1(3) 2*c*t(1) -1; 2*r2(1) 2*r2(2) 2*r2(3) 2*c*t(2) -1; 2*r3(1) 2*r3(2) 2*r3(3) 2*c*t(3) -1; 2*r4(1) 2*r4(2) 2*r4(3) 2*c*t(4) -1 ];
%u = [x y z -c*t r^2-c^2*t^2];
r1_sq = sqrt(r1(1)^2 + r1(2)^2 + r1(3)^2);
r2_sq = sqrt(r2(1)^2 + r2(2)^2 + r2(3)^2);
r3_sq = sqrt(r3(1)^2 + r3(2)^2 + r3(3)^2);
r4_sq = sqrt(r4(1)^2 + r4(2)^2 + r4(3)^2);
b = [r1_sq-c^2*t(1)^2; r2_sq-c^2*t(2)^2; r3_sq-c^2*t(3)^2; r4_sq-c^2*t(4)^2 ];

% Calculating u : u = [x y z -c*t r^2-c^2*t^2];
% u = (A^T*A)^-1*A^T*b
t = 0;
u1 = (transpose(A)*b);
u2 = (transpose(A)*A);
u =  inv(u2) * u1;

[Star Strider]

The last value in ‘u1’ and the last row of ‘u2’ are zero, throwing the Warning.

What do you want to calculate?

% Sensor/receiver locations/ vertices of a tetrahedron
% si(xi,yi,zi)
r1 = [0.125 0.125 0.125];
r2 = [0.25 0.125 0.125];
r3 = [0.1875 0.25 0.125];
r4 = [0.1875 0.1875 0.25];

% Calculate the Time of Arrival at the receivers 
c = 343; % m/s
t = zeros(1, 4);
sound_loc = [10 20 10]; % Replace x, y, and z with your sound location coordinates
vertices = [r1; r2; r3; r4]; % Replace vertex1 to vertex4 with your 3D vertices

for j = 1:4
    t(j) = pdist([sound_loc; vertices(j, :)], 'euclidean') / c;
end

% Setting up matrices to solve for location of source

A = [2*r1(1) 2*r1(2) 2*r1(3) 2*c*t(1) -1; 2*r2(1) 2*r2(2) 2*r2(3) 2*c*t(2) -1; 2*r3(1) 2*r3(2) 2*r3(3) 2*c*t(3) -1; 2*r4(1) 2*r4(2) 2*r4(3) 2*c*t(4) -1 ];
%u = [x y z -c*t r^2-c^2*t^2];
r1_sq = sqrt(r1(1)^2 + r1(2)^2 + r1(3)^2);
r2_sq = sqrt(r2(1)^2 + r2(2)^2 + r2(3)^2);
r3_sq = sqrt(r3(1)^2 + r3(2)^2 + r3(3)^2);
r4_sq = sqrt(r4(1)^2 + r4(2)^2 + r4(3)^2);
b = [r1_sq-c^2*t(1)^2; r2_sq-c^2*t(2)^2; r3_sq-c^2*t(3)^2; r4_sq-c^2*t(4)^2 ];

% Calculating u : u = [x y z -c*t r^2-c^2*t^2];
% u = (A^T*A)^-1*A^T*b
t = 0;

u1 = A\b

u1 = 5×1

5.2191 10.1025 5.1241 -12.2462 0

u2 = A\A

u2 = 5×5

1.0000 0.0000 0.0000 0.0000 -0.0083 -0.0000 1.0000 0.0000 0.0000 -0.0167 0.0000 0.0000 1.0000 0.0000 -0.0083 -0.0000 -0.0000 -0.0000 1.0000 -0.0204 0 0 0 0 0

u =  u2 \ u1

Warning: Matrix is singular to working precision.

u = 5×1

NaN NaN NaN NaN NaN

.

[Bruno Luong]

@Star Strider "u2 = A\A"

where it comes from??? what you want to show here?

[Shrishti Yadav]

The original equation was: Au = b with A as an mxn matrix and u is nx1 and b is nx1

so taking the transpose on both sides and then get only u on the left hand side, we get the following after multiplying both sides by the inverse of the transpose of A * A:

u = inv(A^TA)*A^T*b

So it should be ok in terms of singularity. The goal was to calculate u correctly. that was it. i didn't know if the way i am multiplying it out makes an error.

Answer 1

[Star Strider]

I am not certain what you want to do.

Consider using the pinv function.

Answer 2

[Bruno Luong]

@Shrishti Yadav "I checked if the matrix reaches singularity and it does not. "

Your A matrix has size 4 x 5, the the rank is maximum 4.

The matrix u2 = A'*A is 5 x 5 with maximum rank <= 4 so it must be singular, despite what you claim.

In short you try to solve for 5 unknown with 4 equations. The system is then underdetermined and MATLAB warns you that.