Question

Following is a part of a code I am writing in matlab, here I want to carry out the following simple mathematical operation [A][X]=[B], where [X] is unknown. In my case I have length of k around 1600000. So all I want is to get the values for g1,g2 and g3 for each element on the array. I have tried the following

k31 = omega3./(d)      
k32 = omega3_2./(d)    
A   = [2,1,5;-2,-1,-5];    
X   = [g1;g2;g3];

for ii = 1:length(k31)    
    B = [k31(ii); k32(ii)];    
    X = pinv(A).*B;    
end

display(g1,g2,g3)

I am using pseudo-inverse so basically I can get a solution for each X and I have made some edit there….and x is unknown, MATHEMATICALLY it can be done, but am not able to code it

Also how do I plot the values of g1 g2 g3 with x and y as follows scatter(x(1:end-1), y(1:end-1), 5, g1); scatter(x(1:end-1), y(1:end-1), 5, g2) and scatter(x(1:end-1), y(1:end-1), 5, g3)

Answer 1

In my opinion you are better off creating a pseudo inverse from the singular value decomposition. Moore-Penrose pseudo inverse will work, but I think it gives some weird results sometimes. The least squares can be unstable, especially since the rank of your example matrix is not full.

Also; don’t calculate the inverse for every iteration!

Here is an example, where rank-deficiency of A is taken care of:

A   = [2,1,5;-2,-1,-5];

% pseudo inverse of A
[m,n]   = size(A);
% get SVD
[U,S,V] = svd(A);
% check rank
r       = rank(S);
SR      = S(1:r,1:r);
% make complete if rank is not full
SRc     = [SR^-1 zeros(r,m-r);zeros(n-r,r) zeros(n-r,m-r)];
% create inverse
A_inv   = V*SRc*U.';

X=[];    
for i = 1:1600
    % this actually takes most of the time
    k31 = rand(1000, 1);
    k32 = rand(1000, 1);
    B   = [k31';k32'];
    % X is overwritten on every loop...
    X   = [X A_inv*B];
end

N_xy = 1000;
x    = rand(N_xy,1);
y    = rand(N_xy,1);
g1   = X(1,1:N_xy);

figure(1), clf
scatter(x,y,5,g1)

I didn’t plot all 1600000 points since that can’t be what you want