Feature request: ispossemidef function

[ignace-computing]

Currently, there exists a function isposdef.
I believe that it would be good to create a function ispossemidef, checking whether a matrix is positive semi-definite.
Any ideas on what would be a good way to implement such a method?

[stevengj]

This is quite difficult to check reliably. The basic issue is that roundoff errors make it virtually impossible to distinguish a matrix with zero eigenvalues from one with slightly negative eigenvalues.

In principle, we could do something similar to JuliaLang/julia#35057, i.e. checking that the eigenvalues are $\ge -\lambda_{\max} \epsilon$ for some relative tolerance $\epsilon$. Note that this would be much slower than the Cholesky-based approach of isposdef.

See also many previous discussions of this and related topics:

• What is the fastest way to check if a hermitian matrix is positive semi-definite?
• Verify a matrix is positive semi-definite
• Cholesky Decomposition of a Sparse Symmetric Positive Semidefinite (SPSD) Singular Matrix
• Cholesky decomposition of low-rank positive-semidefinite matrix
• Random draws of multivariate normal with positive semi-definite covariance matrix

[araujoms]

I would simply do isposdef(x+ϵ*I) for some tolerance ϵ. This is both too simple and too fragile to include in LinearAlgebra, though.