algorithms.utils.matrices

Module: algorithms.utils.matrices

Utilities for working with matrices

Functions

nipy.algorithms.utils.matrices.full_rank(X, r=None)

Return full-rank matrix whose column span is the same as X

Uses an SVD decomposition.

If the rank of X is known it can be specified by r – no check is made to ensure that this really is the rank of X.

Parameters:

X : array-like

2D array which may not be of full rank.

r : None or int

Known rank of X. r=None results in standard matrix rank calculation. We do not check r is really the rank of X; it is to speed up calculations when the rank is already known.

Returns:

fX : array

Full-rank matrix with column span matching that of X

nipy.algorithms.utils.matrices.matrix_rank(M, tol=None)

Return rank of matrix using SVD method

Rank of the array is the number of SVD singular values of the array that are greater than tol.

This version of matrix rank is very similar to the numpy.linalg version except for the use of:

  • scipy.linalg.svd istead of numpy.linalg.svd.
  • the MATLAB algorithm for default tolerance calculation

matrix_rank appeared in numpy.linalg in December 2009, first available in numpy 1.5.0.

Parameters:

M : array-like

array of <=2 dimensions

tol : {None, float}

threshold below which SVD values are considered zero. If tol is None, and S is an array with singular values for M, and eps is the epsilon value for datatype of S, then tol set to S.max() * eps * max(M.shape).

Notes

We check for numerical rank deficiency by using tol=max(M.shape) * eps * S[0] (where S[0] is the maximum singular value and thus the 2-norm of the matrix). This is one tolerance threshold for rank deficiency, and the default algorithm used by MATLAB [#2]_. When floating point roundoff is the main concern, then “numerical rank deficiency” is a reasonable choice. In some cases you may prefer other definitions. 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

[2]
    1. Golub and C. F. Van Loan, _Matrix Computations_.

Baltimore: Johns Hopkins University Press, 1996. .. [#2] http://www.mathworks.com/help/techdoc/ref/rank.html

Examples

>>> 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.zeros((4,4))) # All zeros - zero rank
0
>>> matrix_rank(np.ones((4,))) # 1 dimension - rank 1 unless all 0
1
>>> matrix_rank(np.zeros((4,)))
0
>>> matrix_rank([1]) # accepts array-like
1
nipy.algorithms.utils.matrices.pos_recipr(X)

Return element-wise reciprocal of array, setting X>=0 to 0

Return the reciprocal of an array, setting all entries less than or equal to 0 to 0. Therefore, it presumes that X should be positive in general.

Parameters:

X : array-like

Returns:

rX : array

array of same shape as X, dtype np.float, with values set to 1/X where X > 0, 0 otherwise

nipy.algorithms.utils.matrices.recipr0(X)

Return element-wise reciprocal of array, `X`==0 -> 0

Return the reciprocal of an array, setting all entries equal to 0 as 0. It does not assume that X should be positive in general.

Parameters:X : array-like
Returns:rX : array