Behaviour of backslash operator for non-square matrices least-squares fitting

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Matt O'Donnell 2020-2-24

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评论： Matt O'Donnell 2020-2-24

I am currently trying to reproduce a set of results from a pre-existing project and can't get to the bottom of the following difference.

Phi = design matrix of input data size(29507x97)

Ref = reference values to fit to, six data sets, size(29507x6)

Phi = rand(29507,97);
Ref = rand(29507,6);
c1 = Phi\Ref;
for ii = 1:6
    c2(:,ii) = Phi\Ref(:,ii);
end
all(all(c1==c2))

I would have expected c1 to give identical results to c2. I can't find detailed information on what the "\" algorithm is doing differently between these case.

Any help appreciated!

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Answer 1

the cyclist 2020-2-24

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https://ww2.mathworks.cn/matlabcentral/answers/507282-behaviour-of-backslash-operator-for-non-square-matrices-least-squares-fitting#answer_417137

在 MATLAB Online 中打开

They are equal, to within floating-point precision. Notice that

max(abs(c1(:)-c2(:)))

is around 1e-16.

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Matt J 2020-2-24

编辑：Matt J 2020-2-24

Also, internal parallelization of mldivide and other linear algebra operations is different depending on the size of the inputs. So, when you perform the mldivide column-by-column, the inputs are divided up into parallel chunks in a different way, leading to floating point differences.

Matt O'Donnell 2020-2-24

@the cyclist Yes. I agree with that as the likely root cause.

I thought it would be doing the same least squares calculation on a column by column basis as each column should be treated independently from each other in terms of doing the expected calculation.i.e. column 1 should never know anything about the other 5 columns's calculations mathematically speaking. But as @Matt J has suggested it is doing something across the columns, or even splitting each column differently, which leads to the difference/calculation order effect as suggested.

Many thanks to both of you for helping to get my head around this.

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