freealg.FreeForm.norm

FreeForm.norm(size=None, order=2, seed=None, **kwargs)

Estimate the Schatten norm of a Hermitian matrix.

This function estimates the norm of the freeform or a larger matrix containing it using free decompression.

Parameters:

sizeint, default=None

The size of the matrix containing \(\mathbf{A}\) to estimate eigenvalues of. If None, returns estimates of the eigenvalues of \(\mathbf{A}\) itself.

order{float, ''inf, '-inf', 'fro', 'nuc'}, default=2

Order of the norm.

• float \(p\): Schatten p-norm.

• 'inf': Largest absolute eigenvalue

\(\max \vert \lambda_i \vert)\) * '-inf': Smallest absolute eigenvalue \(\min \vert \lambda_i \vert)\) * 'fro': Frobenius norm corresponding to \(p=2\) * 'nuc': Nuclear (or trace) norm corresponding to \(p=1\)

Returns:

normfloat

matrix norm

See also

eigh

cond

slogdet

trace

Notes

Thes Schatten \(p\)-norm is defined by

\[\Vert \mathbf{A} \Vert_p = \left( \sum_{i=1}^N \vert \lambda_i \vert^p \right)^{1/p}.\]

Examples

>>> ...