rank deficiency. The threshold may declare a matrix
`A` rank deficient even if the linear combination of some columns of `A`
is not exactly equal to another column of `A` but only numerically very
close to another column of `A`.

We chose our default threshold because it is in wide use. Other thresholds
are possible.  For example, elsewhere in the 2007 edition of *Numerical
recipes* there is an alternative threshold of ``S.max() *
np.finfo(A.dtype).eps / 2. * np.sqrt(m + n + 1.)``. The authors describe
this threshold as being based on "expected roundoff error" (p 71).

The thresholds above deal with floating point roundoff error in the
calculation of the SVD.  However, you may have more information about
the sources of error in `A` that would make you consider other tolerance
values to detect *effective* rank deficiency. The most useful measure
of the tolerance depends on the operations you intend to use on your
matrix. For example, if your data come from uncertain measurements with
uncertainties greater than floating point epsilon, choosing a tolerance
near that uncertainty may be preferable. The tolerance may be absolute
if the uncertainties are absolute rather than relative.

References
----------
.. [1] MATLAB reference documentation, "Rank"
       https://www.mathworks.com/help/techdoc/ref/rank.html
.. [2] W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery,
       "Numerical Recipes (3rd edition)", Cambridge University Press, 2007,
       page 795.

Examples
--------
>>> import numpy as np
>>> from numpy.linalg import matrix_rank
>>> matrix_rank(np.eye(4)) # Full rank matrix
4
>>> I=np.eye(4); I[-1,-1] = 0. # rank deficient matrix
>>> matrix_rank(I)
3
>>> matrix_rank(np.ones((4,))) # 1 dimension - rank 1 unless all 0
1
>>> matrix_rank(np.zeros((4,)))
0
Nz#`tol` and `rtol` can't be both set.rš