def __init__(self, c_or_r, r=False, variable=None):
if isinstance(c_or_r, poly1d):
self._variable = c_or_r._variable
self._coeffs = c_or_r._coeffs
if set(c_or_r.__dict__) - set(self.__dict__):
msg = ("In the future extra properties will not be copied "
"across when constructing one poly1d from another")
warnings.warn(msg, FutureWarning, stacklevel=2)
self.__dict__.update(c_or_r.__dict__)
if variable is not None:
self._variable = variable
return
if r:
c_or_r = poly(c_or_r)
c_or_r = atleast_1d(c_or_r)
if c_or_r.ndim > 1:
raise ValueError("Polynomial must be 1d only.")
c_or_r = trim_zeros(c_or_r, trim='f')
if len(c_or_r) == 0:
c_or_r = NX.array([0.])
self._coeffs = c_or_r
if variable is None:
variable = 'x'
self._variable = variable
def __new__(subtype, data, dtype=None, copy=True):
warnings.warn(
'the matrix subclass is not the recommended way to '
'represent matrices or deal with linear algebra (see '
'https://docs.scipy.org/doc/numpy/user/'
'numpy-for-matlab-users.html). '
'Please adjust your code to use regular ndarray.',
PendingDeprecationWarning,
stacklevel=2)
if isinstance(data, matrix):
dtype2 = data.dtype
if (dtype is None):
dtype = dtype2
if (dtype2 == dtype) and (not copy):
return data
return data.astype(dtype)
if isinstance(data, N.ndarray):
if dtype is None:
intype = data.dtype
else:
intype = N.dtype(dtype)
new = data.view(subtype)
if intype != data.dtype:
return new.astype(intype)
if copy: return new.copy()
else: return new
if isinstance(data, str):
data = _convert_from_string(data)
# now convert data to an array
arr = N.array(data, dtype=dtype, copy=copy)
ndim = arr.ndim
shape = arr.shape
if (ndim > 2):
raise ValueError("matrix must be 2-dimensional")
elif ndim == 0:
shape = (1, 1)
elif ndim == 1:
shape = (1, shape[0])
order = 'C'
if (ndim == 2) and arr.flags.fortran:
order = 'F'
if not (order or arr.flags.contiguous):
arr = arr.copy()
ret = N.ndarray.__new__(subtype,
shape,
arr.dtype,
buffer=arr,
order=order)
return ret
def column_stack(tup):
"""
Stack 1-D arrays as columns into a 2-D array.
Take a sequence of 1-D arrays and stack them as columns
to make a single 2-D array. 2-D arrays are stacked as-is,
just like with `hstack`. 1-D arrays are turned into 2-D columns
first.
Parameters
----------
tup : sequence of 1-D or 2-D arrays.
Arrays to stack. All of them must have the same first dimension.
Returns
-------
stacked : 2-D array
The array formed by stacking the given arrays.
See Also
--------
stack, hstack, vstack, concatenate
Examples
--------
>>> a = np.array((1,2,3))
>>> b = np.array((2,3,4))
>>> np.column_stack((a,b))
array([[1, 2],
[2, 3],
[3, 4]])
"""
arrays = []
for v in tup:
arr = array(v, copy=False, subok=True)
if arr.ndim < 2:
arr = array(arr, copy=False, subok=True, ndmin=2).T
arrays.append(arr)
return _nx.concatenate(arrays, 1)
def array_split(ary, indices_or_sections, axis=0):
"""
Split an array into multiple sub-arrays.
Please refer to the ``split`` documentation. The only difference
between these functions is that ``array_split`` allows
`indices_or_sections` to be an integer that does *not* equally
divide the axis. For an array of length l that should be split
into n sections, it returns l % n sub-arrays of size l//n + 1
and the rest of size l//n.
See Also
--------
split : Split array into multiple sub-arrays of equal size.
Examples
--------
>>> x = np.arange(8.0)
>>> np.array_split(x, 3)
[array([ 0., 1., 2.]), array([ 3., 4., 5.]), array([ 6., 7.])]
>>> x = np.arange(7.0)
>>> np.array_split(x, 3)
[array([ 0., 1., 2.]), array([ 3., 4.]), array([ 5., 6.])]
"""
try:
Ntotal = ary.shape[axis]
except AttributeError:
Ntotal = len(ary)
try:
# handle scalar case.
Nsections = len(indices_or_sections) + 1
div_points = [0] + list(indices_or_sections) + [Ntotal]
except TypeError:
# indices_or_sections is a scalar, not an array.
Nsections = int(indices_or_sections)
if Nsections <= 0:
raise ValueError('number sections must be larger than 0.')
Neach_section, extras = divmod(Ntotal, Nsections)
section_sizes = ([0] + extras * [Neach_section + 1] +
(Nsections - extras) * [Neach_section])
div_points = _nx.array(section_sizes).cumsum()
sub_arys = []
sary = _nx.swapaxes(ary, axis, 0)
for i in range(Nsections):
st = div_points[i]
end = div_points[i + 1]
sub_arys.append(_nx.swapaxes(sary[st:end], axis, 0))
return sub_arys
def kron(a, b):
"""
Kronecker product of two arrays.
Computes the Kronecker product, a composite array made of blocks of the
second array scaled by the first.
Parameters
----------
a, b : array_like
Returns
-------
out : ndarray
See Also
--------
outer : The outer product
Notes
-----
The function assumes that the number of dimensions of `a` and `b`
are the same, if necessary prepending the smallest with ones.
If `a.shape = (r0,r1,..,rN)` and `b.shape = (s0,s1,...,sN)`,
the Kronecker product has shape `(r0*s0, r1*s1, ..., rN*SN)`.
The elements are products of elements from `a` and `b`, organized
explicitly by::
kron(a,b)[k0,k1,...,kN] = a[i0,i1,...,iN] * b[j0,j1,...,jN]
where::
kt = it * st + jt, t = 0,...,N
In the common 2-D case (N=1), the block structure can be visualized::
[[ a[0,0]*b, a[0,1]*b, ... , a[0,-1]*b ],
[ ... ... ],
[ a[-1,0]*b, a[-1,1]*b, ... , a[-1,-1]*b ]]
Examples
--------
>>> np.kron([1,10,100], [5,6,7])
array([ 5, 6, 7, 50, 60, 70, 500, 600, 700])
>>> np.kron([5,6,7], [1,10,100])
array([ 5, 50, 500, 6, 60, 600, 7, 70, 700])
>>> np.kron(np.eye(2), np.ones((2,2)))
array([[ 1., 1., 0., 0.],
[ 1., 1., 0., 0.],
[ 0., 0., 1., 1.],
[ 0., 0., 1., 1.]])
>>> a = np.arange(100).reshape((2,5,2,5))
>>> b = np.arange(24).reshape((2,3,4))
>>> c = np.kron(a,b)
>>> c.shape
(2, 10, 6, 20)
>>> I = (1,3,0,2)
>>> J = (0,2,1)
>>> J1 = (0,) + J # extend to ndim=4
>>> S1 = (1,) + b.shape
>>> K = tuple(np.array(I) * np.array(S1) + np.array(J1))
>>> c[K] == a[I]*b[J]
True
"""
b = asanyarray(b)
a = array(a, copy=False, subok=True, ndmin=b.ndim)
ndb, nda = b.ndim, a.ndim
if (nda == 0 or ndb == 0):
return _nx.multiply(a, b)
as_ = a.shape
bs = b.shape
if not a.flags.contiguous:
a = reshape(a, as_)
if not b.flags.contiguous:
b = reshape(b, bs)
nd = ndb
if (ndb != nda):
if (ndb > nda):
as_ = (1, ) * (ndb - nda) + as_
else:
bs = (1, ) * (nda - ndb) + bs
nd = nda
result = outer(a, b).reshape(as_ + bs)
axis = nd - 1
for _ in range(nd):
result = concatenate(result, axis=axis)
wrapper = get_array_prepare(a, b)
if wrapper is not None:
result = wrapper(result)
wrapper = get_array_wrap(a, b)
if wrapper is not None:
result = wrapper(result)
return result
def apply_over_axes(func, a, axes):
"""
Apply a function repeatedly over multiple axes.
`func` is called as `res = func(a, axis)`, where `axis` is the first
element of `axes`. The result `res` of the function call must have
either the same dimensions as `a` or one less dimension. If `res`
has one less dimension than `a`, a dimension is inserted before
`axis`. The call to `func` is then repeated for each axis in `axes`,
with `res` as the first argument.
Parameters
----------
func : function
This function must take two arguments, `func(a, axis)`.
a : array_like
Input array.
axes : array_like
Axes over which `func` is applied; the elements must be integers.
Returns
-------
apply_over_axis : ndarray
The output array. The number of dimensions is the same as `a`,
but the shape can be different. This depends on whether `func`
changes the shape of its output with respect to its input.
See Also
--------
apply_along_axis :
Apply a function to 1-D slices of an array along the given axis.
Notes
------
This function is equivalent to tuple axis arguments to reorderable ufuncs
with keepdims=True. Tuple axis arguments to ufuncs have been available since
version 1.7.0.
Examples
--------
>>> a = np.arange(24).reshape(2,3,4)
>>> a
array([[[ 0, 1, 2, 3],
[ 4, 5, 6, 7],
[ 8, 9, 10, 11]],
[[12, 13, 14, 15],
[16, 17, 18, 19],
[20, 21, 22, 23]]])
Sum over axes 0 and 2. The result has same number of dimensions
as the original array:
>>> np.apply_over_axes(np.sum, a, [0,2])
array([[[ 60],
[ 92],
[124]]])
Tuple axis arguments to ufuncs are equivalent:
>>> np.sum(a, axis=(0,2), keepdims=True)
array([[[ 60],
[ 92],
[124]]])
"""
val = asarray(a)
N = a.ndim
if array(axes).ndim == 0:
axes = (axes, )
for axis in axes:
if axis < 0:
axis = N + axis
args = (val, axis)
res = func(*args)
if res.ndim == val.ndim:
val = res
else:
res = expand_dims(res, axis)
if res.ndim == val.ndim:
val = res
else:
raise ValueError("function is not returning "
"an array of the correct shape")
return val
def tile(A, reps):
"""
Construct an array by repeating A the number of times given by reps.
If `reps` has length ``d``, the result will have dimension of
``max(d, A.ndim)``.
If ``A.ndim < d``, `A` is promoted to be d-dimensional by prepending new
axes. So a shape (3,) array is promoted to (1, 3) for 2-D replication,
or shape (1, 1, 3) for 3-D replication. If this is not the desired
behavior, promote `A` to d-dimensions manually before calling this
function.
If ``A.ndim > d``, `reps` is promoted to `A`.ndim by pre-pending 1's to it.
Thus for an `A` of shape (2, 3, 4, 5), a `reps` of (2, 2) is treated as
(1, 1, 2, 2).
Note : Although tile may be used for broadcasting, it is strongly
recommended to use numpy1's broadcasting operations and functions.
Parameters
----------
A : array_like
The input array.
reps : array_like
The number of repetitions of `A` along each axis.
Returns
-------
c : ndarray
The tiled output array.
See Also
--------
repeat : Repeat elements of an array.
broadcast_to : Broadcast an array to a new shape
Examples
--------
>>> a = np.array([0, 1, 2])
>>> np.tile(a, 2)
array([0, 1, 2, 0, 1, 2])
>>> np.tile(a, (2, 2))
array([[0, 1, 2, 0, 1, 2],
[0, 1, 2, 0, 1, 2]])
>>> np.tile(a, (2, 1, 2))
array([[[0, 1, 2, 0, 1, 2]],
[[0, 1, 2, 0, 1, 2]]])
>>> b = np.array([[1, 2], [3, 4]])
>>> np.tile(b, 2)
array([[1, 2, 1, 2],
[3, 4, 3, 4]])
>>> np.tile(b, (2, 1))
array([[1, 2],
[3, 4],
[1, 2],
[3, 4]])
>>> c = np.array([1,2,3,4])
>>> np.tile(c,(4,1))
array([[1, 2, 3, 4],
[1, 2, 3, 4],
[1, 2, 3, 4],
[1, 2, 3, 4]])
"""
try:
tup = tuple(reps)
except TypeError:
tup = (reps, )
d = len(tup)
if all(x == 1 for x in tup) and isinstance(A, _nx.ndarray):
# Fixes the problem that the function does not make a copy if A is a
# numpy1 array and the repetitions are 1 in all dimensions
return _nx.array(A, copy=True, subok=True, ndmin=d)
else:
# Note that no copy of zero-sized arrays is made. However since they
# have no data there is no risk of an inadvertent overwrite.
c = _nx.array(A, copy=False, subok=True, ndmin=d)
if (d < c.ndim):
tup = (1, ) * (c.ndim - d) + tup
shape_out = tuple(s * t for s, t in zip(c.shape, tup))
n = c.size
if n > 0:
for dim_in, nrep in zip(c.shape, tup):
if nrep != 1:
c = c.reshape(-1, n).repeat(nrep, 0)
n //= dim_in
return c.reshape(shape_out)
def __getitem__(self, key):
# handle matrix builder syntax
if isinstance(key, str):
frame = sys._getframe().f_back
mymat = matrixlib.bmat(key, frame.f_globals, frame.f_locals)
return mymat
if not isinstance(key, tuple):
key = (key, )
# copy attributes, since they can be overridden in the first argument
trans1d = self.trans1d
ndmin = self.ndmin
matrix = self.matrix
axis = self.axis
objs = []
scalars = []
arraytypes = []
scalartypes = []
for k, item in enumerate(key):
scalar = False
if isinstance(item, slice):
step = item.step
start = item.start
stop = item.stop
if start is None:
start = 0
if step is None:
step = 1
if isinstance(step, complex):
size = int(abs(step))
newobj = function_base.linspace(start, stop, num=size)
else:
newobj = _nx.arange(start, stop, step)
if ndmin > 1:
newobj = array(newobj, copy=False, ndmin=ndmin)
if trans1d != -1:
newobj = newobj.swapaxes(-1, trans1d)
elif isinstance(item, str):
if k != 0:
raise ValueError("special directives must be the "
"first entry.")
if item in ('r', 'c'):
matrix = True
col = (item == 'c')
continue
if ',' in item:
vec = item.split(',')
try:
axis, ndmin = [int(x) for x in vec[:2]]
if len(vec) == 3:
trans1d = int(vec[2])
continue
except Exception:
raise ValueError("unknown special directive")
try:
axis = int(item)
continue
except (ValueError, TypeError):
raise ValueError("unknown special directive")
elif type(item) in ScalarType:
newobj = array(item, ndmin=ndmin)
scalars.append(len(objs))
scalar = True
scalartypes.append(newobj.dtype)
else:
item_ndim = ndim(item)
newobj = array(item, copy=False, subok=True, ndmin=ndmin)
if trans1d != -1 and item_ndim < ndmin:
k2 = ndmin - item_ndim
k1 = trans1d
if k1 < 0:
k1 += k2 + 1
defaxes = list(range(ndmin))
axes = defaxes[:k1] + defaxes[k2:] + defaxes[k1:k2]
newobj = newobj.transpose(axes)
objs.append(newobj)
if not scalar and isinstance(newobj, _nx.ndarray):
arraytypes.append(newobj.dtype)
# Ensure that scalars won't up-cast unless warranted
final_dtype = find_common_type(arraytypes, scalartypes)
if final_dtype is not None:
for k in scalars:
objs[k] = objs[k].astype(final_dtype)
res = self.concatenate(tuple(objs), axis=axis)
if matrix:
oldndim = res.ndim
res = self.makemat(res)
if oldndim == 1 and col:
res = res.T
return res
def roots(p):
"""
Return the roots of a polynomial with coefficients given in p.
The values in the rank-1 array `p` are coefficients of a polynomial.
If the length of `p` is n+1 then the polynomial is described by::
p[0] * x**n + p[1] * x**(n-1) + ... + p[n-1]*x + p[n]
Parameters
----------
p : array_like
Rank-1 array of polynomial coefficients.
Returns
-------
out : ndarray
An array containing the roots of the polynomial.
Raises
------
ValueError
When `p` cannot be converted to a rank-1 array.
See also
--------
poly : Find the coefficients of a polynomial with a given sequence
of roots.
polyval : Compute polynomial values.
polyfit : Least squares polynomial fit.
poly1d : A one-dimensional polynomial class.
Notes
-----
The algorithm relies on computing the eigenvalues of the
companion matrix [1]_.
References
----------
.. [1] R. A. Horn & C. R. Johnson, *Matrix Analysis*. Cambridge, UK:
Cambridge University Press, 1999, pp. 146-7.
Examples
--------
>>> coeff = [3.2, 2, 1]
>>> np.roots(coeff)
array([-0.3125+0.46351241j, -0.3125-0.46351241j])
"""
# If input is scalar, this makes it an array
p = atleast_1d(p)
if p.ndim != 1:
raise ValueError("Input must be a rank-1 array.")
# find non-zero array entries
non_zero = NX.nonzero(NX.ravel(p))[0]
# Return an empty array if polynomial is all zeros
if len(non_zero) == 0:
return NX.array([])
# find the number of trailing zeros -- this is the number of roots at 0.
trailing_zeros = len(p) - non_zero[-1] - 1
# strip leading and trailing zeros
p = p[int(non_zero[0]):int(non_zero[-1]) + 1]
# casting: if incoming array isn't floating point, make it floating point.
if not issubclass(p.dtype.type, (NX.floating, NX.complexfloating)):
p = p.astype(float)
N = len(p)
if N > 1:
# build companion matrix and find its eigenvalues (the roots)
A = diag(NX.ones((N - 2, ), p.dtype), -1)
A[0, :] = -p[1:] / p[0]
roots = eigvals(A)
else:
roots = NX.array([])
# tack any zeros onto the back of the array
roots = hstack((roots, NX.zeros(trailing_zeros, roots.dtype)))
return roots
def nan_to_num(x, copy=True):
"""
Replace NaN with zero and infinity with large finite numbers.
If `x` is inexact, NaN is replaced by zero, and infinity and -infinity
replaced by the respectively largest and most negative finite floating
point values representable by ``x.dtype``.
For complex dtypes, the above is applied to each of the real and
imaginary components of `x` separately.
If `x` is not inexact, then no replacements are made.
Parameters
----------
x : scalar or array_like
Input data.
copy : bool, optional
Whether to create a copy of `x` (True) or to replace values
in-place (False). The in-place operation only occurs if
casting to an array does not require a copy.
Default is True.
.. versionadded:: 1.13
Returns
-------
out : ndarray
`x`, with the non-finite values replaced. If `copy` is False, this may
be `x` itself.
See Also
--------
isinf : Shows which elements are positive or negative infinity.
isneginf : Shows which elements are negative infinity.
isposinf : Shows which elements are positive infinity.
isnan : Shows which elements are Not a Number (NaN).
isfinite : Shows which elements are finite (not NaN, not infinity)
Notes
-----
NumPy uses the IEEE Standard for Binary Floating-Point for Arithmetic
(IEEE 754). This means that Not a Number is not equivalent to infinity.
Examples
--------
>>> np.nan_to_num(np.inf)
1.7976931348623157e+308
>>> np.nan_to_num(-np.inf)
-1.7976931348623157e+308
>>> np.nan_to_num(np.nan)
0.0
>>> x = np.array([np.inf, -np.inf, np.nan, -128, 128])
>>> np.nan_to_num(x)
array([ 1.79769313e+308, -1.79769313e+308, 0.00000000e+000,
-1.28000000e+002, 1.28000000e+002])
>>> y = np.array([complex(np.inf, np.nan), np.nan, complex(np.nan, np.inf)])
>>> np.nan_to_num(y)
array([ 1.79769313e+308 +0.00000000e+000j,
0.00000000e+000 +0.00000000e+000j,
0.00000000e+000 +1.79769313e+308j])
"""
x = _nx.array(x, subok=True, copy=copy)
xtype = x.dtype.type
isscalar = (x.ndim == 0)
if not issubclass(xtype, _nx.inexact):
return x[()] if isscalar else x
iscomplex = issubclass(xtype, _nx.complexfloating)
dest = (x.real, x.imag) if iscomplex else (x, )
maxf, minf = _getmaxmin(x.real.dtype)
for d in dest:
_nx.copyto(d, 0.0, where=isnan(d))
_nx.copyto(d, maxf, where=isposinf(d))
_nx.copyto(d, minf, where=isneginf(d))
return x[()] if isscalar else x