Speeding up lsqlin to find the base of a matrix

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Tintin Milou 2022-8-4

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评论： Bruno Luong 2022-8-7

采纳的回答： Bruno Luong

Hello,

I have to solve the following problem over and over again for slightly different values of mu.

n = lsqlin(mu-eye(J),zeros(J,1),[],[],ones(1,J),1,[],[],[]);

The matrix mu has columns sum up to 1, values are between zero and 1 and the diagonal elements are typically above 0.9. There are only very few 0 elements in mu (although many of them are 'close' to 0, e.g. 1e-4). J is equal to 400.

You can do the same calculation using

n = null(mu-eye(J),1e-10);

n = n/sum(n);

but that's not any faster. Are there any ideas on how to speed that up? Since the solution n does not change much in the various calls, I thought about providing an initial guess, but the interior algorithm does not accept any initial guesses.

Here's a sample matrix (with J=5)

0.980472260884484 0.0169062020634941 1.31828882462499e-05 0.00712276192859210 0.00734667253541008

0.0122127547484456 0.972782440495672 1.15814051875989e-05 0.00808693210831310 0.00830831290909974

3.60311943991737e-08 4.68217920214649e-08 0.999953080612125 1.05940998862873e-07 3.71871063642989e-05

0.00445881693357994 0.00660554154798086 1.18490160022582e-05 0.976444331730883 0.0148592752450766

0.00285613140229620 0.00370576907106219 1.03060784389177e-05 0.00834586829121426 0.969448552204049

Thanks!

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John D'Errico 2022-8-4

There are at least a couple of other ways I could describe to solve for the null space of a matrix. It would be easier if you would provide a sample matrix to play with, and compare methods, without needing to describe in detail how to solve it for each method.

Tintin Milou 2022-8-4

The different mu are not available at the same time. I'll provide a sample matrix to play with.

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采纳的回答

Bruno Luong 2022-8-4

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Can you try this:

A = mu-eye(size(A));
[Q,R,p] = qr(A,'vector');
n = [R(1:end-1,1:end-1)\R(1:end-1,end); -1];
n(p) = n/sum(n)

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Tintin Milou 2022-8-7

Update: Before calling fsolve for the outer loop, I now add an fsolve on a small-scale problem to get a reasonable initial guess for the complete problem. With this improved initial guess, the code runs more smoothly and the initial suggestion by Bruno Luong is the best solution I have found. Thanks again!