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Finding the nearest positive definite matrix

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This tool saves your covariance matrices, turning them into something that really does have the property you will need. That is, when you are trying to use a covariance matrix in a tool like mvnrnd, it makes no sense if your matrix is not positive definite. So mvnrnd will fail in that case.

But sometimes, it appears that users end up with matrices that are NOT symmetric and positive definite (commonly abbreviated as SPD) and they still wish to use them to generate random numbers, often in a tool like mvnrnd. A solution is to find the NEAREST matrix (minimizing the Frobenius norm of the difference) that has the desired property of being SPD.

I see the question come up every once in a while, so I looked in the file exchange to see what is up there. All I found was nearest_posdef. While this usually almost works, it could be better. It actually failed completely on most of my test cases, and it was not as fast as I would like, using an optimization. In fact, in the comments to nearest_posdef, a logical alternative was posed. That alternative too has its failures, so I wrote nearestSPD, partly based on what I found in the work of Nick Higham.

nearestSPD works on any matrix, and it is reasonably fast. As a test, randn generates a matrix that is not symmetric nor is it at all positive definite in general.
U = randn(100);

nearestSPD will be able to convert U into something that is indeed SPD, and for a 100 by 100 matrix, do it quickly enough.

tic,Uj = nearestSPD(U);toc
Elapsed time is 0.008964 seconds.

The ultimate test of course, is to use chol. If chol returns a second argument that is zero, then MATLAB (and mvnrnd) will be happy!
[R,p] = chol(Uj);
p =

Comments and Ratings (19)


Jesse (view profile)

Hi John,

I had a read of this code and it's quite a nice little code and well put together, upon reading some of the comments, I came to see why it might be taking so long for some people and they reason I came to is your use of min(eig). In certain pathological cases, such as when an eigenvalue is exactly equal to zero (consider x=[1,2,0;3,4,0;0,0,0]) the algorithm fails as it will repeatedly try to add zero to fix the matrix, thus failing. I suggest adding a small check max(tol,min(eig)) instead of min(eig), where tol is some arbitrary small number like 10^-12 times the maximum singular value or some other property. Cheers, Jesse.


Mayssa (view profile)

After running the code I still had negative eigenvalues, obvisiouly, the chol test wasn't efficient with my matrix. Instead, I tried this test (p = any(eig(Ahat)<0); worked much better.
Thanks for the code !

John D'Errico

John D'Errico (view profile)

Nil suggests that his algorithm is simpler. Yet, it will often fail.

A0 = rand(10);
B = (A0 + A0')/2;
[U,Sigma] = eig(B);
Ahat = U*max(Sigma,0)*U';
Error using chol
Matrix must be positive definite.

Instead, the result from nearestSPD will survive the chol test with no error.

L = chol(nearestSPD(A0));

Nil Garcia

In my opinion the code seems unnecessary lengthy. This should be enough:
B = (A + A')/2;
[U,Sigma] = eig(B);
Ahat = U*max(Sigma,0)*U';

dhaba india


820408 (view profile)

In my simulations, A is a covariance matrix that is not really PSD because of floating point precision. I used B=nearestPSD(A) to fix this problem but for eig(B), I am getting negative eigen values that are very close to zero. Can you please comment on why EVD test is failing

Andrea Libri

John D'Errico

John D'Errico (view profile)

Please send me an example case that has this problem.

Yi He

Yi He (view profile)

Thanks John. But when I run some 125*125 covariance matrices, the progress stands at ' mineig = min(eig(Ahat));' for pretty long time (actually almost over 10 hours). What could I do with this issue?
Thanks again.


Meng (view profile)

Thank you. Very useful!

John D'Errico

John D'Errico (view profile)

I actually nudge the matrix just a bit at the very end if necessary. The code should ensure that chol applied to the result will always yield a valid factorization, and that is essentially the test in MATLAB to be truly SPD. So while the Higham algorithm will ensure positive semi-definite, if chol should always work, then it will indeed be positive definite.

Ioannis P

Could you please explain if this code is giving a positive definite or a semi-positive definite matrix? You have written the following:

"From Higham: "The nearest symmetric positive semidefinite matrix in the
Frobenius norm to an arbitrary real matrix A is shown to be (B + H)/2,
where H is the symmetric polar factor of B=(A + A')/2."
arguments: (input)
A - square matrix, which will be converted to the nearest Symmetric
Positive Definite Matrix."

Could you please clarify this? Thanks!


Ahmed (view profile)


Eric (view profile)

Kudos to you, John, mostly for calling attention to Higham's paper. Trying to use the other files you mentioned was driving me crazy, because of their high probability of failure.

John D'Errico

John D'Errico (view profile)

Sorry about that. Frobenius norm is minimized.

Petr Posik

Hi John. I miss in the description how the "nearness" of the 2 matrices, U and Uj, is measured. Could you comment on that? Thanks, Petr



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