flopscope.numpy.linalg.cond

fnp.linalg.cond(x, p=None)[flopscope source][numpy source]

Compute the condition number of a matrix.

Adapted from NumPy docs np.linalg.cond

Arealinalg

Typecustom

Cost

4×

m \cdot n \cdot \min(m,n)

Flopscope Context

Condition number. Cost: m*n*min(m,n) (via SVD).

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 used in the condition number computation:

table





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 flops.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.

Notes

The condition number of x is defined as the norm of x times the norm of the inverse of x [1]_; the norm can be the usual L2-norm (root-of-sum-of-squares) or one of a number of other matrix norms.

References

footnote

1

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

Examples

>>> import flopscope.numpy as fnp
>>> from numpy import linalg as LA
>>> a = flops.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, flops.inf)
2.0
>>> LA.cond(a, -flops.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 # may vary
>>> (min(LA.svd(a, compute_uv=False)) *
... min(LA.svd(LA.inv(a), compute_uv=False)))
0.70710678118654746 # may vary