numpy.linalg.cond

numpy.linalg.cond(x, p=None)

Compute the condition number of a matrix.

This function is capable of returning the condition number using one of seven different norms, depending on the value of p (see Parameters below).

Parameters:

x : (..., M, N) array_like

The matrix whose condition number is sought.

p : {None, 1, -1, 2, -2, inf, -inf, ‘fro’}, optional

Order of the norm:

p	norm for matrices
None	2-norm, computed directly using the SVD
‘fro’	Frobenius norm
inf	max(sum(abs(x), axis=1))
-inf	min(sum(abs(x), axis=1))
1	max(sum(abs(x), axis=0))
-1	min(sum(abs(x), axis=0))
2	2-norm (largest sing. value)
-2	smallest singular value

inf means the numpy.inf object, and the Frobenius norm is the root-of-sum-of-squares norm.

Returns:

c : {float, inf}

The condition number of the matrix. May be infinite.

See also

numpy.linalg.norm

Notes

The condition number of x is defined as the norm of x times the norm of the inverse of x [R40]; the norm can be the usual L2-norm (root-of-sum-of-squares) or one of a number of other matrix norms.

References

[R40]

(1, 2) G. Strang, Linear Algebra and Its Applications, Orlando, FL, Academic Press, Inc., 1980, pg. 285.

Examples

>>> from numpy import linalg as LA
>>> a = np.array([[1, 0, -1], [0, 1, 0], [1, 0, 1]])
>>> a
array([[ 1,  0, -1],
       [ 0,  1,  0],
       [ 1,  0,  1]])
>>> LA.cond(a)
1.4142135623730951
>>> LA.cond(a, 'fro')
3.1622776601683795
>>> LA.cond(a, np.inf)
2.0
>>> LA.cond(a, -np.inf)
1.0
>>> LA.cond(a, 1)
2.0
>>> LA.cond(a, -1)
1.0
>>> LA.cond(a, 2)
1.4142135623730951
>>> LA.cond(a, -2)
0.70710678118654746
>>> min(LA.svd(a, compute_uv=0))*min(LA.svd(LA.inv(a), compute_uv=0))
0.70710678118654746