def _null_space2(numpy, A, eps):
# (INTERNAL) Return the nullspace and rank of matrix A
# @see: <https://SciPy-Cookbook.ReadTheDocs.io/items/RankNullspace.html>,
# <https://NumPy.org/doc/stable/reference/generated/numpy.linalg.svd.html>,
# <https://StackOverflow.com/questions/19820921>,
# <https://StackOverflow.com/questions/2992947> and
# <https://StackOverflow.com/questions/5889142>
A = numpy.array(A)
m = max(numpy.shape(A))
if m != 4: # for this usage
raise _AssertionError(shape=m, txt=modulename(_null_space2, True))
# if needed, square A, pad with zeros
A = numpy.resize(A, m * m).reshape(m, m)
# try: # no numpy.linalg.null_space <https://docs.SciPy.org/doc/>
# return scipy.linalg.null_space(A) # XXX no scipy.linalg?
# except AttributeError:
# pass
_, s, v = numpy.linalg.svd(A)
t = max(eps, eps * s[0]) # tol, s[0] is largest singular
r = numpy.sum(s > t) # rank
if r == 3: # get null_space
n = numpy.transpose(v[r:])
s = numpy.shape(n)
if s != (m, 1): # bad null_space shape
raise _AssertionError(shape=s, txt=modulename(_null_space2, True))
e = float(numpy.max(numpy.abs(numpy.dot(A, n))))
if e > t: # residual not near-zero
raise _AssertionError(eps=e, txt=modulename(_null_space2, True))
else: # coincident, colinear, concentric centers, ambiguous, etc.
n = None
# del A, s, vh # release numpy
return n, r
def _Error(clas, arg, name=NN, Error=UnitError, txt=_invalid_):
'''(INTERNAL) Return an error with explanation.
@arg clas: The C{units} class or sub-class.
@arg arg: The original C{unit} value.
@kwarg name: The instance name (C{str}).
@kwarg Error: Optional error, overriding the default
L{UnitError}.
@kwarg txt: Optional explanation of the error (C{str}).
@returns: An B{C{Error}} instance.
'''
n = name if name else modulename(clas).lstrip(_UNDER_)
return Error(n, arg, txt=txt)
def _Error(clas, arg, name='', Error=UnitError, txt='invalid'):
'''(INTERNAL) Return an error with explanation.
@arg clas: The C{units} class or sub-class.
@arg arg: The original C{unit} value.
@kwarg name: The instance name (C{str}).
@kwarg Error: Optional error, overriding the default
L{UnitError}.
@kwarg txt: Explanation of the error (C{str}).
@returns: An B{C{Error}} instance.
'''
n = (', name=%r' % (name, )) if name else ''
return Error('%s(%r%s) %s' % (modulename(clas), arg, n, txt))