def test_real():
C = ComplexNumbers()
R = RealNumbers()
Z = Integers()
# __contains__
assert -1 in R
assert 1 in R
assert 0 in R
assert -1.0 in R
assert 1.0 in R
assert 0.0 in R
assert 2j not in R
assert 2 + 2j not in R
assert 'a' not in R
# contains_set
assert not R.contains_set(C)
assert R.contains_set(R)
assert C.contains_set(Z)
# __eq__
assert R != C
assert R == R
assert R != Z
# element
assert C.element() == float(0.0)
assert C.element(1) == float(1.0)
def _call(self, vf, out):
"""Implement ``self(vf, out)``."""
if self.domain.field == ComplexNumbers():
out.multiply(vf[0], self._vecfield[0].conj())
else:
out.multiply(vf[0], self._vecfield[0])
if self.is_weighted:
out *= self.weights[0]
if self.domain.size == 1:
return
tmp = self.range.element()
for vfi, gi, wi in zip(vf[1:], self.vecfield[1:],
self.weights[1:]):
if self.domain.field == ComplexNumbers():
tmp.multiply(vfi, gi.conj())
else:
tmp.multiply(vfi, gi)
if self.is_weighted:
tmp *= wi
out += tmp
def test_real():
C = ComplexNumbers()
R = RealNumbers()
Z = Integers()
# __contains__
assert -1 in R
assert 1 in R
assert 0 in R
assert -1.0 in R
assert 1.0 in R
assert 0.0 in R
assert 2j not in R
assert 2 + 2j not in R
assert 'a' not in R
# contains_set
assert not R.contains_set(C)
assert R.contains_set(R)
assert C.contains_set(Z)
# __eq__
assert R != C
assert R == R
assert R != Z
# element
assert C.element() == float(0.0)
assert C.element(1) == float(1.0)
def default_dtype(field=None):
"""Return the default of `Fn` data type for a given field.
Parameters
----------
field : `Field`
Set of numbers to be represented by a data type.
Currently supported : `RealNumbers`, `ComplexNumbers`
Returns
-------
dtype : `type`
Numpy data type specifier. The returned defaults are:
``RealNumbers()`` : ``np.dtype('float32')``
``ComplexNumbers()`` : `NotImplemented`
"""
if field is None or field == RealNumbers():
return np.dtype('float32')
elif field == ComplexNumbers():
return NotImplemented
else:
raise ValueError('no default data type defined for field {}'
''.format(field))
def test_integers():
C = ComplexNumbers()
R = RealNumbers()
Z = Integers()
# __contains__
assert -1 in Z
assert 1 in Z
assert 0 in Z
assert -1.0 not in Z
assert 1.0 not in Z
assert 0.0 not in Z
assert 2j not in Z
assert 2 + 2j not in Z
assert 'a' not in Z
# contains_set
assert not Z.contains_set(C)
assert not Z.contains_set(R)
assert Z.contains_set(Z)
# __eq__
assert Z != C
assert Z != R
assert Z == Z
# element
assert Z.element() == int(0)
assert Z.element(1) == int(1)
assert Z.element(1.5) == int(1.5)
def test_complex():
C = ComplexNumbers()
R = RealNumbers()
Z = Integers()
# __contains__
assert -1 in C
assert 1 in C
assert 0 in C
assert -1.0 in C
assert 1.0 in C
assert 0.0 in C
assert 2j in C
assert 2 + 2j in C
assert 'a' not in C
# contains_set
assert C.contains_set(C)
assert C.contains_set(R)
assert C.contains_set(Z)
# __eq__
assert C == C
assert C != R
assert C != Z
# element
assert C.element() == complex(0.0, 0.0)
assert C.element(1) == complex(1.0, 0.0)
assert C.element(1 + 2j) == complex(1.0, 2.0)
def test_complex():
C = ComplexNumbers()
R = RealNumbers()
Z = Integers()
# __contains__
assert -1 in C
assert 1 in C
assert 0 in C
assert -1.0 in C
assert 1.0 in C
assert 0.0 in C
assert 2j in C
assert 2 + 2j in C
assert 'a' not in C
# contains_set
assert C.contains_set(C)
assert C.contains_set(R)
assert C.contains_set(Z)
# __eq__
assert C == C
assert C != R
assert C != Z
# element
assert C.element() == complex(0.0, 0.0)
assert C.element(1) == complex(1.0, 0.0)
assert C.element(1 + 2j) == complex(1.0, 2.0)
def __init__(self, shape, dtype):
"""Initialize a new instance.
Parameters
----------
shape : nonnegative int or sequence of nonnegative ints
Number of entries of type ``dtype`` per axis in this space. A
single integer results in a space with rank 1, i.e., 1 axis.
dtype :
Data type of elements in this space. Can be provided
in any way the `numpy.dtype` constructor understands, e.g.
as built-in type or as a string.
For a data type with a ``dtype.shape``, these extra dimensions
are added *to the left* of ``shape``.
"""
# Handle shape and dtype, taking care also of dtypes with shape
try:
shape, shape_in = tuple(safe_int_conv(s) for s in shape), shape
except TypeError:
shape, shape_in = (safe_int_conv(shape), ), shape
if any(s < 0 for s in shape):
raise ValueError('`shape` must have only nonnegative entries, got '
'{}'.format(shape_in))
dtype = np.dtype(dtype)
# We choose this order in contrast to Numpy, since we usually want
# to represent discretizations of vector- or tensor-valued functions,
# i.e., if dtype.shape == (3,) we expect f[0] to have shape `shape`.
self.__shape = dtype.shape + shape
self.__dtype = dtype.base
if is_real_dtype(self.dtype):
# real includes non-floating-point like integers
field = RealNumbers()
self.__real_dtype = self.dtype
self.__real_space = self
self.__complex_dtype = TYPE_MAP_R2C.get(self.dtype, None)
self.__complex_space = None # Set in first call of astype
elif is_complex_floating_dtype(self.dtype):
field = ComplexNumbers()
self.__real_dtype = TYPE_MAP_C2R[self.dtype]
self.__real_space = None # Set in first call of astype
self.__complex_dtype = self.dtype
self.__complex_space = self
else:
field = None
LinearSpace.__init__(self, field)
def derivative(self, vf):
"""Derivative of the point-wise norm operator at ``vf``.
The derivative at ``F`` of the point-wise norm operator ``N``
with finite exponent ``p`` and weights ``w`` is the pointwise
inner product with the vector field
``x --> N(F)(x)^(1-p) * [ F_j(x) * |F_j(x)|^(p-2) ]_j``.
Note that this is not well-defined for ``F = 0``. If ``p < 2``,
any zero component will result in a singularity.
Parameters
----------
vf : domain `element-like`
Vector field ``F`` at which to evaluate the derivative
Returns
-------
deriv : `PointwiseInner`
Derivative operator at the given point ``vf``
Raises
------
NotImplementedError
* if the vector field space is complex, since the derivative
is not linear in that case
* if the exponent is ``inf``
"""
if self.domain.field == ComplexNumbers():
raise NotImplementedError('operator not Frechet-differentiable '
'on a complex space')
if self.exponent == float('inf'):
raise NotImplementedError('operator not Frechet-differentiable '
'for exponent = inf')
vf = self.domain.element(vf)
vf_pwnorm_fac = self(vf)
vf_pwnorm_fac **= (self.exponent - 1)
inner_vf = vf.copy()
for gi in inner_vf:
gi /= vf_pwnorm_fac * gi ** (self.exponent - 2)
return PointwiseInner(self.domain, inner_vf, weight=self.weights)
def __str__(self):
"""Return ``str(self)``."""
inner_str = '{}'.format(self.domain)
dtype_str = dtype_repr(self.out_dtype)
if self.field == RealNumbers():
if self.out_dtype == np.dtype('float64'):
pass
else:
inner_str += ', out_dtype={}'.format(dtype_str)
elif self.field == ComplexNumbers():
if self.out_dtype == np.dtype('complex128'):
inner_str += ', field={!r}'.format(self.field)
else:
inner_str += ', out_dtype={}'.format(dtype_str)
else: # different field, name explicitly
inner_str += ', field={!r}'.format(self.field)
inner_str += ', out_dtype={}'.format(dtype_str)
return '{}({})'.format(self.__class__.__name__, inner_str)
def __init__(self, size, dtype):
"""Initialize a new instance.
Parameters
----------
size : `int`
The number of dimensions of the space
dtype : `object`
The data type of the storage array. Can be provided in any
way the `numpy.dtype` function understands, most notably
as built-in type, as one of NumPy's internal datatype
objects or as string.
Only scalar data types (numbers) are allowed.
"""
NtuplesBase.__init__(self, size, dtype)
if not is_scalar_dtype(self.dtype):
raise TypeError('{!r} is not a scalar data type'.format(dtype))
if is_real_dtype(self.dtype):
field = RealNumbers()
self._is_real = True
self._real_dtype = self.dtype
self._real_space = self
self._complex_dtype = _TYPE_MAP_R2C.get(self.dtype, None)
self._complex_space = None # Set in first call of astype
else:
field = ComplexNumbers()
self._is_real = False
self._real_dtype = _TYPE_MAP_C2R[self.dtype]
self._real_space = None # Set in first call of astype
self._complex_dtype = self.dtype
self._complex_space = self
self._is_floating = is_floating_dtype(self.dtype)
LinearSpace.__init__(self, field)
def __init__(self, domain, field=None, out_dtype=None):
"""Initialize a new instance.
Parameters
----------
domain : `Set`
The domain of the functions
field : `Field`, optional
The range of the functions, usually the `RealNumbers` or
`ComplexNumbers`. If not given, the field is either inferred
from ``out_dtype``, or, if the latter is also `None`, set
to ``RealNumbers()``.
out_dtype : optional
Data type of the return value of a function in this space.
Can be given in any way `numpy.dtype` understands, e.g. as
string ('float64') or data type (`float`).
By default, 'float64' is used for real and 'complex128'
for complex spaces.
"""
if not isinstance(domain, Set):
raise TypeError('`domain` {!r} not a Set instance'.format(domain))
if field is not None and not isinstance(field, Field):
raise TypeError('`field` {!r} not a `Field` instance'
''.format(field))
# Data type: check if consistent with field, take default for None
dtype, dtype_in = np.dtype(out_dtype), out_dtype
# Default for both None
if field is None and out_dtype is None:
field = RealNumbers()
out_dtype = np.dtype('float64')
# field None, dtype given -> infer field
elif field is None:
if is_real_dtype(dtype):
field = RealNumbers()
elif is_complex_floating_dtype(dtype):
field = ComplexNumbers()
else:
raise ValueError('{} is not a scalar data type'
''.format(dtype_in))
# field given -> infer dtype if not given, else check consistency
elif field == RealNumbers():
if out_dtype is None:
out_dtype = np.dtype('float64')
elif not is_real_dtype(dtype):
raise ValueError('{} is not a real data type'
''.format(dtype_in))
elif field == ComplexNumbers():
if out_dtype is None:
out_dtype = np.dtype('complex128')
elif not is_complex_floating_dtype(dtype):
raise ValueError('{} is not a complex data type'
''.format(dtype_in))
# Else: keep out_dtype=None, which results in lazy dtype determination
LinearSpace.__init__(self, field)
FunctionSet.__init__(self, domain, field, out_dtype)
# Init cache attributes for real / complex variants
if self.field == RealNumbers():
self._real_out_dtype = self.out_dtype
self._real_space = self
self._complex_out_dtype = _TYPE_MAP_R2C.get(self.out_dtype,
np.dtype(object))
self._complex_space = None
elif self.field == ComplexNumbers():
self._real_out_dtype = _TYPE_MAP_C2R[self.out_dtype]
self._real_space = None
self._complex_out_dtype = self.out_dtype
self._complex_space = self
else:
self._real_out_dtype = None
self._real_space = None
self._complex_out_dtype = None
self._complex_space = None
def __init__(self, vfspace, vecfield, weight=None):
"""Initialize a new instance.
Parameters
----------
vfspace : `ProductSpace`
Space of vector fields on which the operator acts.
It has to be a product space of identical spaces, i.e. a
power space.
vecfield : domain `element-like`
Vector field with which to calculate the point-wise inner
product of an input vector field
weight : `array-like` or `float`, optional
Weighting array or constant for the norm. If an array is
given, its length must be equal to ``domain.size``, and
all entries must be positive. A provided constant must be
positive.
By default, the weights are is taken from
``domain.weighting``. Note that this excludes unusual
weightings with custom inner product, norm or dist.
Examples
--------
We make a tiny vector field space in 2D and create the
point-wise inner product operator with a fixed vector field.
The operator maps a vector field to a scalar function:
>>> import odl
>>> spc = odl.uniform_discr([-1, -1], [1, 1], (1, 2))
>>> vfspace = odl.ProductSpace(spc, 2)
>>> fixed_vf = np.array([[[0, 1]],
... [[1, -1]]])
>>> pw_inner = PointwiseInner(vfspace, fixed_vf)
>>> pw_inner.range == spc
True
Now we can calculate the inner product in each point:
>>> x = vfspace.element([[[1, -4]],
... [[0, 3]]])
>>> print(pw_inner(x))
[[0.0, -7.0]]
"""
if not isinstance(vfspace, ProductSpace):
raise TypeError('`vfsoace` {!r} is not a ProductSpace '
'instance'.format(vfspace))
super().__init__(domain=vfspace, range=vfspace[0], linear=True)
# Bail out if the space is complex but we cannot take the complex
# conjugate.
if (self.domain.field == ComplexNumbers() and
not hasattr(self.base_space.element_type, 'conj')):
raise NotImplementedError(
'base space element type {!r} does not implement conj() '
'method required for complex inner products'
''.format(self.base_space.element_type))
self._vecfield = self.domain.element(vecfield)
# Handle weighting, including sanity checks
if weight is None:
if hasattr(self.domain.weighting, 'vector'):
self._weights = self.domain.weighting.vector
elif hasattr(self.domain.weighting, 'const'):
self._weights = (self.domain.weighting.const *
np.ones(len(self.domain)))
else:
raise ValueError('weighting scheme {!r} of the domain does '
'not define a weighting vector or constant'
''.format(self.domain.weighting))
elif np.isscalar(weight):
if weight <= 0:
raise ValueError('weighting constant must be positive, got '
'{}'.format(weight))
self._weights = float(weight) * np.ones(self.domain.size)
else:
self._weights = np.asarray(weight, dtype='float64')
if (not np.all(self.weights > 0) or
not np.all(np.isfinite(self.weights))):
raise ValueError('weighting array {} contains invalid '
'entries'.format(weight))
self._is_weighted = not np.array_equiv(self.weights, 1.0)
