cond

→Tensor

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

source

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) and 2 use the spectral norm via SVD; -2 returns the reciprocal

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

. Other orders dispatch to norm.

Returns

Tensor

Condition number, shape (*,).

Notes

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

Examples

>>> import lucid
>>> from lucid.linalg import cond
>>> cond(lucid.tensor([[1.0, 0.0], [0.0, 1e-6]]))
Tensor(1000000.0)