cond

→Tensor

cond(A: Tensor, p: int | float | str | None = None)

source edit

Compute the condition number of a matrix.

The condition number under norm $\|\cdot\|_p$ is

\kappa_p(A) \,=\, \|A\|_p \,\|A^{-1}\|_p.

For the spectral ( $p = 2$ ) norm this simplifies to the ratio of the largest to smallest singular value:

\kappa_2(A) \,=\, \sigma_{\max}(A) \,/\, \sigma_{\min}(A).

The condition number quantifies how sensitive the solution of $Ax = b$ is to perturbations in $A$ or $b$ .

Parameters

ATensor

Input matrix of shape (*, m, n).

p(int, float, str or None)= None

Norm order: None (default) or 2 for the spectral norm, -2 for its reciprocal

\sigma_{\min} / \sigma_{\max}

, "nuc", or one of the matrix_norm orders "fro", 1, -1, inf, -inf.

Returns

Tensor

Condition number, shape (*,).

Raises

ValueError

For an unknown p, an input with fewer than two dimensions, or a non-square input under any order but

\pm 2

NotImplementedError

For a complex A: every order needs its singular values or its inverse, and neither has a complex kernel yet.

Notes

A condition number near $1/\varepsilon_{\mathrm{mach}}$ indicates numerical singularity. Non-spectral orders require an explicit inverse, so prefer p = 2 for rank-deficient or rectangular matrices.

The spectral orders and "nuc" are read off the singular values; the others are $\|A\|_p \|A^{-1}\|_p$ with the inverse from inv_ex, so a singular matrix has condition number $\infty$ rather than an error, each matrix of a batch on its own. A matrix with no entries has condition number 0.

Examples

>>> import lucid
>>> from lucid.linalg import cond
>>> cond(lucid.tensor([[1.0, 0.0], [0.0, 1e-6]]))
tensor(1e+06)
>>> cond(lucid.tensor([[1.0, 2.0], [2.0, 4.0]]), "fro")
tensor(inf)