MATLAB: Eig() shows positive eigenvalues while chol() failed

eig chol

Hi guys,

I was LU-decomposing a supposed-to-be positive-definite matrix using "chol" in a loop. However, occasionally it failed with the warning

 " Error using chol
   Matrix must be positive definite. "

eig() shows that the smallest eigenvalue is 1.8*10^(-4) which I think should be well above zero. BTW, the largest one is 7.7*10^9.

How can I get around this dilemma? Your suggestions are greatly appreciated. Many thanks.

Best Answer

I wrote nearestSPD (download it from the file exchange) for this purpose. It will tweak the values of your matrix slightly so that chol should always succeed. Of course, if the required tweak is necessarily large, it will still make a minimal change so that the result is as close as possible, yet will be positive definite.

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MATLAB: Assemblt of global stiffness matrix

A determinant is NEVER a good measure of whether a matrix is singular. And as matrices get large, the determinant becomes more than useless.

For example,

det(eye(1000)) == 1

however,

det(0.1*eye(1000)) == ???

When computed in floating point in MATLAB, the latter will yield zero, which I imagine you know not to be true. However, both matrices are equally non-singular. As easily, I could have modified the problem to yield inf by a different scale factor.

Answer? Don't use the determinant for anything of importance. It will only guide you to a poor conclusion.

Of course, this does not imply that your matrix has been created correctly or incorrectly. However, your test of that using the determinant is a useless one. Instead, you might look at the rank, or the eigenvalues. A valid stiffness matrix will be at least positive semi-definite. Don't forget that even for a singular matrix, eig can return negative eigenvalues on the order of -eps.

MATLAB: Chol() say matrix is not positive defnite even though all eigenvalues are positive

You're not factorizing the matrix you think you're factorizing. From the documentation for the chol function: "If A is nonsymmetric , then chol treats the matrix as symmetric and uses only the diagonal and upper triangle of A."

Is your Qd matrix symmetric?

issymmetric(Qd) % false

So what matrix are you actually trying to factorize?

A = triu(Qd) + triu(Qd, 1).'

That matrix is symmetric but is not positive definite.

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MATLAB: Chol() say matrix is not positive defnite even though all eigenvalues are positive

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