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Is there any method to accelerate many small matrix and vector's "mldivide" (4*4)?

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I am trying to solve many(4000000) mldivide evaluation. All of them are in the form of "x = A\b". A is a 4*4 matrix, and b is 4*1 vector. e.g.
num = 4000000;
A = rand(4,4,num);
b = rand(4,num);
x = zeros(4,num);
for i=1:num
x(:,i) = A(:,:,i)\b(:,i);
end
Should I use the parfor? Or do it in the vectorized way? Any suggestion would be appreciated.
----------------------------------------------------------------------------------------------------------------------------
The following is my time profile of the answers. My computer is somehow old. only 2GZ cpu.
num = 4000000;
A = rand(4,4,num);
b = rand(4,num);
%% method1 : for loop (slowest)
tic
x1 = zeros(4,num);
for i=1:num
x1(:,i) = A(:,:,i)\b(:,i);
end
toc % Elapsed time is 61.896299 seconds.
%% method2 : parfor with 10 workers
% parpool(10);
tic
x2 = zeros(4,num);
parfor i=1:num
x2(:,i) = A(:,:,i)\b(:,i);
end
toc % Elapsed time is 6.148422 seconds.
%% method 3 (@Walter Roberson)
tic
syms tA [4 4]
syms tb [4 1];
tx = tA\tb;
X = matlabFunction(tx, 'vars', {[tA(:); tb(:)]});
A3 = reshape(A,16,[]);
x3 = X(reshape([A3;b], 20, []));
toc % Elapsed time is 2.351828 seconds.
%% method 4 (@ Bruno Luong)
tic
x4 = MultiSolver(A, b);
toc % Elapsed time is 9.759959 seconds.
%% the difference ratio between answers
max(max(abs((x1-x2)./x1))) % 0
max(max(abs((x1-x3)./x1))) % 5.9513e-09
max(max(abs((x1-x4)./x1))) % 2.9376e-10
If any problem or suggestions, please leave a message.

  1 Comment

Walter Roberson
Walter Roberson on 23 Sep 2020
Note that once the function X had been created, it could be re-used; there is definitely set-up cost that could be amortized over repeated use.

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Accepted Answer

Walter Roberson
Walter Roberson on 23 Sep 2020
syms tA [4 4]
syms tb [4 1];
tx = tA\tb;
X = matlabFunction(tx, 'vars', {[tA(:); tb(:)]});
x = X(reshape([a,b], 20, []));
This creates a vectorized function X, that accepts a 20 x N matrix and returns a 4 x N matrix.

  2 Comments

wei zhang
wei zhang on 23 Sep 2020
I have a further problem when using your code, please help me.
I create a function like below.
function X = test1
syms tA [4 4]
syms tb [4 1]
tx = tA\tb ;
X = matlabFunction(tx, 'vars', {[tA(:); tb(:)]});
end
I always got a red wavy line below " tA\tb" in the function editor, which says "the variable "tA" ("tb") might be used before it is defined". Is it necessary to correct it? If yes, how to correct it?
Walter Roberson
Walter Roberson on 23 Sep 2020
function X = test1
tA = sym('tA', [4 4]);
tb = sym('tB', [4 1]);
tx = tA\tb ;
X = matlabFunction(tx, 'vars', {[tA(:); tb(:)]});
end
will shut up the warning.

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More Answers (1)

Bruno Luong
Bruno Luong on 23 Sep 2020
Edited: Bruno Luong on 23 Sep 2020
Use my FEX file of MultiSolver
num = 4000000;
A = rand(4,4,num);
b = rand(4,num);
x = zeros(4,num);
tic
for i=1:num
x(:,i) = A(:,:,i)\b(:,i);
end
toc % Elapsed time is 19.004386 seconds.
% https://www.mathworks.com/matlabcentral/fileexchange/24260-multiple-same-size-linear-solver
tic
X = MultiSolver(A, b);
toc % Elapsed time is 4.601745 seconds.

  4 Comments

Show 1 older comment
Bruno Luong
Bruno Luong on 23 Sep 2020
Just wonder how much tic/toc run on the example with Walter symbolic solution? I don't have the tbx to test.
wei zhang
wei zhang on 23 Sep 2020
I had submitted a time profile with answers. Please check it in the edited question part. If you have any suggestions, please let me know.

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