How to vectorize A\B operation on slices of 3D matrices?

Dear Matlab Users,

I am trying to solve a few separate sets of Ax = B equations with A\B or mldivide(A,B). A and B store equations components in their corresponding slices.

This is feasible with a for loop. Could it be done faster in some other way, with vectorization?

num_slices = 5;
A = rand(6,6, num_slices);
x = zeros(num_slices,6);
B = rand(6,1,num_slices);

% Working solution
for ii=1:num_slices
    x(ii,:) =(A(:,:,ii)\B(:,:,ii))';
end

% Failed vectorization attempt
x_alternative =(A\B)'; % error here

The error I get is obvious as I do not use the "\" operation correctly:

Arguments must be 2-D, or at least one argument must be scalar. Use LDIVIDE (.\) for elementwise left division.

I would appreciate your help and suggestions in this matter.

EDIT:

I know about the option to flatten everything to sparse matices as suggested a few years ago here:

https://stackoverflow.com/questions/6344800/solving-multiple-linear-systems-using-vectorization

It would be great to learn if anything changed significantly from that time. Thank you!

Regards,

Jakub Kaminski

[Jan]

There is a tool in the file exchange which does exactly what you want in parallel. I searched for it for half an hour now, without success.

I've found it:

• ndfun - unfortunately I cannot find it in the FileExchange anymore?!
• But this works too: https://www.mathworks.com/matlabcentral/fileexchange/37515-mmx

[James Tursa]

The ndfun code used to be here, but the links are now broken:

https://www.mit.edu/~pwb/matlab/

But I did find the source code here:

https://www.mit.edu/~pwb/matlab/ndfun/ndfun.m

https://www.mit.edu/~pwb/matlab/ndfun/ndfun.c

Answer 1

[Bruno Luong]

Yes there is new news since I post this MultipleQR fex (MEX required, OpenMP multithreading), for 6 x 6 this is the speed up on my LAPTOP, R2021a to solve one miillion linear systems A*x = y:

>> TestMultipleQRSolve
size(A) = [6 6 1000000]
size(y) = [6 1 1000000]
MultipleQRSolve time   = 0.618821 [s]
Matlab loop time       = 8.42263 [s]

For pure MATLAB sparse trick, you might be interested in my implementation of Multiple same size linear solver

[Bruno Luong]

I complete with the timings of the three methods

>> TestMultipleQRSolve
size(A) = [6 6 1000000]
size(y) = [6 1 1000000]
MultipleQRSolve time   = 0.596974 [s]
Matlab loop time       = 7.98282 [s]
SliceMultiSolver time  = 2.56455 [s] % sparse trick

Answer 2

[Jan]

Do you have a compiler installed? Then use:

https://www.mathworks.com/matlabcentral/fileexchange/37515-mmx

[Bruno Luong]

Benchmark with mmx('backslash', ...)

>> TestMultipleQRSolve
size(A) = [6 6 1000000]
size(y) = [6 1 1000000]
MultipleQRSolve time   = 0.528698 [s]
Matlab loop time       = 5.93315 [s]
SliceMultiSolver time  = 1.8415 [s]
mmx(...) time          = 0.369092 [s]