[Numpy-discussion] Proposal for matrix_rank function in numpy
josef.pktd at gmail.com
josef.pktd at gmail.com
Tue Dec 15 12:12:37 EST 2009
On Tue, Dec 15, 2009 at 12:01 PM, Matthew Brett <matthew.brett at gmail.com> wrote:
> Hi,
>
> Following on from the occasional discussion on the list, can I propose
> a small matrix_rank function for inclusion in numpy/linalg?
>
> I suggest it because it seems rather a basic need for linear algebra,
> and it's very small and simple...
>
> I've appended an implementation with some doctests in the hope that it
> will be acceptable,
>
> Robert - I hope you don't mind me quoting you in the notes.
>
> Thanks a lot,
>
> Matthew
>
>
> def 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`.
>
> 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``.
>
> Examples
> --------
> >>> matrix_rank(np.eye(4)) # Full rank matrix
> 4
> >>> matrix_rank(np.c_[np.eye(4),np.eye(4)]) # Rank deficient matrix
> 4
> >>> 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
>
> Notes
> -----
> Golub and van Loan define "numerical rank deficiency" as using
> tol=eps*S[0] (note that S[0] is the maximum singular value and thus
> the 2-norm of the matrix). There really is not one definition, much
> like there isn't a single definition of the norm of a matrix. For
> example, if your data come from uncertain measurements with
> uncertainties greater than floating point epsilon, choosing a
> tolerance of about the uncertainty is probably a better idea (the
> tolerance may be absolute if the uncertainties are absolute rather
> than relative, even). When floating point roundoff is your concern,
> then "numerical rank deficiency" is a better concept, but exactly
> what the relevant measure of the tolerance is depends on the
> operations you intend to do with your matrix. [RK, numpy mailing
> list]
>
> References
> ----------
> Matrix Computations by Golub and van Loan
> '''
> M = np.asarray(M)
> if M.ndim > 2:
> raise TypeError('array should have 2 or fewer dimensions')
> if M.ndim < 2:
> return int(not np.all(M==0))
> S = npl.svd(M, compute_uv=False)
> if tol is None:
> tol = S.max() * np.finfo(S.dtype).eps
> return np.sum(S > tol)
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>
This was missing from numpy compared to matlab and gauss.
If we put it in linalg next to np.linalg.cond, then we could shorten
the name to `rank`, since the meaning of np.linalg.rank should be
pretty obvious.
Josef
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