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"""Abstract linear algebra library.
This module defines a class hierarchy that implements a kind of "lazy"
matrix representation, called the ``LinearOperator``. It can be used to do
linear algebra with extremely large sparse or structured matrices, without
representing those explicitly in memory. Such matrices can be added,
multiplied, transposed, etc.
As a motivating example, suppose you want have a matrix where almost all of
the elements have the value one. The standard sparse matrix representation
skips the storage of zeros, but not ones. By contrast, a LinearOperator is
able to represent such matrices efficiently. First, we need a compact way to
represent an all-ones matrix::
>>> import numpy as np
>>> class Ones(LinearOperator):
... def __init__(self, shape):
... super(Ones, self).__init__(dtype=None, shape=shape)
... def _matvec(self, x):
... return np.repeat(x.sum(), self.shape[0])
Instances of this class emulate ``np.ones(shape)``, but using a constant
amount of storage, independent of ``shape``. The ``_matvec`` method specifies
how this linear operator multiplies with (operates on) a vector. We can now
add this operator to a sparse matrix that stores only offsets from one::
>>> from scipy.sparse import csr_matrix
>>> offsets = csr_matrix([[1, 0, 2], [0, -1, 0], [0, 0, 3]])
>>> A = aslinearoperator(offsets) + Ones(offsets.shape)
>>> A.dot([1, 2, 3])
array([13, 4, 15])
The result is the same as that given by its dense, explicitly-stored
counterpart::
>>> (np.ones(A.shape, A.dtype) + offsets.toarray()).dot([1, 2, 3])
array([13, 4, 15])
Several algorithms in the ``scipy.sparse`` library are able to operate on
``LinearOperator`` instances.
"""
from __future__ import division, print_function, absolute_import
import numpy as np
from scipy.sparse import isspmatrix
from scipy.sparse.sputils import isshape, isintlike
__all__ = ['LinearOperator', 'aslinearoperator']
class LinearOperator(object):
"""Common interface for performing matrix vector products
Many iterative methods (e.g. cg, gmres) do not need to know the
individual entries of a matrix to solve a linear system A*x=b.
Such solvers only require the computation of matrix vector
products, A*v where v is a dense vector. This class serves as
an abstract interface between iterative solvers and matrix-like
objects.
To construct a concrete LinearOperator, either pass appropriate
callables to the constructor of this class, or subclass it.
A subclass must implement either one of the methods ``_matvec``
and ``_matmat``, and the attributes/properties ``shape`` (pair of
integers) and ``dtype`` (may be None). It may call the ``__init__``
on this class to have these attributes validated. Implementing
``_matvec`` automatically implements ``_matmat`` (using a naive
algorithm) and vice-versa.
Optionally, a subclass may implement ``_rmatvec`` or ``_adjoint``
to implement the Hermitian adjoint (conjugate transpose). As with
``_matvec`` and ``_matmat``, implementing either ``_rmatvec`` or
``_adjoint`` implements the other automatically. Implementing
``_adjoint`` is preferable; ``_rmatvec`` is mostly there for
backwards compatibility.
Parameters
----------
shape : tuple
Matrix dimensions (M,N).
matvec : callable f(v)
Returns returns A * v.
rmatvec : callable f(v)
Returns A^H * v, where A^H is the conjugate transpose of A.
matmat : callable f(V)
Returns A * V, where V is a dense matrix with dimensions (N,K).
dtype : dtype
Data type of the matrix.
Attributes
----------
args : tuple
For linear operators describing products etc. of other linear
operators, the operands of the binary operation.
See Also
--------
aslinearoperator : Construct LinearOperators
Notes
-----
The user-defined matvec() function must properly handle the case
where v has shape (N,) as well as the (N,1) case. The shape of
the return type is handled internally by LinearOperator.
LinearOperator instances can also be multiplied, added with each
other and exponentiated, all lazily: the result of these operations
is always a new, composite LinearOperator, that defers linear
operations to the original operators and combines the results.
Examples
--------
>>> import numpy as np
>>> from scipy.sparse.linalg import LinearOperator
>>> def mv(v):
... return np.array([2*v[0], 3*v[1]])
...
>>> A = LinearOperator((2,2), matvec=mv)
>>> A
<2x2 _CustomLinearOperator with dtype=float64>
>>> A.matvec(np.ones(2))
array([ 2., 3.])
>>> A * np.ones(2)
array([ 2., 3.])
"""
def __new__(cls, *args, **kwargs):
if cls is LinearOperator:
# Operate as _CustomLinearOperator factory.
return super(LinearOperator, cls).__new__(_CustomLinearOperator)
else:
obj = super(LinearOperator, cls).__new__(cls)
if (type(obj)._matvec == LinearOperator._matvec
and type(obj)._matmat == LinearOperator._matmat):
raise TypeError("LinearOperator subclass should implement"
" at least one of _matvec and _matmat.")
return obj
def __init__(self, dtype, shape):
"""Initialize this LinearOperator.
To be called by subclasses. ``dtype`` may be None; ``shape`` should
be convertible to a length-2 tuple.
"""
if dtype is not None:
dtype = np.dtype(dtype)
shape = tuple(shape)
if not isshape(shape):
raise ValueError("invalid shape %r (must be 2-d)" % (shape,))
self.dtype = dtype
self.shape = shape
def _init_dtype(self):
"""Called from subclasses at the end of the __init__ routine.
"""
if self.dtype is None:
v = np.zeros(self.shape[-1])
self.dtype = np.asarray(self.matvec(v)).dtype
def _matmat(self, X):
"""Default matrix-matrix multiplication handler.
Falls back on the user-defined _matvec method, so defining that will
define matrix multiplication (though in a very suboptimal way).
"""
return np.hstack([self.matvec(col.reshape(-1,1)) for col in X.T])
def _matvec(self, x):
"""Default matrix-vector multiplication handler.
If self is a linear operator of shape (M, N), then this method will
be called on a shape (N,) or (N, 1) ndarray, and should return a
shape (M,) or (M, 1) ndarray.
This default implementation falls back on _matmat, so defining that
will define matrix-vector multiplication as well.
"""
return self.matmat(x.reshape(-1, 1))
def matvec(self, x):
"""Matrix-vector multiplication.
Performs the operation y=A*x where A is an MxN linear
operator and x is a column vector or 1-d array.
Parameters
----------
x : {matrix, ndarray}
An array with shape (N,) or (N,1).
Returns
-------
y : {matrix, ndarray}
A matrix or ndarray with shape (M,) or (M,1) depending
on the type and shape of the x argument.
Notes
-----
This matvec wraps the user-specified matvec routine or overridden
_matvec method to ensure that y has the correct shape and type.
"""
x = np.asanyarray(x)
M,N = self.shape
if x.shape != (N,) and x.shape != (N,1):
raise ValueError('dimension mismatch')
y = self._matvec(x)
if isinstance(x, np.matrix):
y = np.asmatrix(y)
else:
y = np.asarray(y)
if x.ndim == 1:
y = y.reshape(M)
elif x.ndim == 2:
y = y.reshape(M,1)
else:
raise ValueError('invalid shape returned by user-defined matvec()')
return y
def rmatvec(self, x):
"""Adjoint matrix-vector multiplication.
Performs the operation y = A^H * x where A is an MxN linear
operator and x is a column vector or 1-d array.
Parameters
----------
x : {matrix, ndarray}
An array with shape (M,) or (M,1).
Returns
-------
y : {matrix, ndarray}
A matrix or ndarray with shape (N,) or (N,1) depending
on the type and shape of the x argument.
Notes
-----
This rmatvec wraps the user-specified rmatvec routine or overridden
_rmatvec method to ensure that y has the correct shape and type.
"""
x = np.asanyarray(x)
M,N = self.shape
if x.shape != (M,) and x.shape != (M,1):
raise ValueError('dimension mismatch')
y = self._rmatvec(x)
if isinstance(x, np.matrix):
y = np.asmatrix(y)
else:
y = np.asarray(y)
if x.ndim == 1:
y = y.reshape(N)
elif x.ndim == 2:
y = y.reshape(N,1)
else:
raise ValueError('invalid shape returned by user-defined rmatvec()')
return y
def _rmatvec(self, x):
"""Default implementation of _rmatvec; defers to adjoint."""
if type(self)._adjoint == LinearOperator._adjoint:
# _adjoint not overridden, prevent infinite recursion
raise NotImplementedError
else:
return self.H.matvec(x)
def matmat(self, X):
"""Matrix-matrix multiplication.
Performs the operation y=A*X where A is an MxN linear
operator and X dense N*K matrix or ndarray.
Parameters
----------
X : {matrix, ndarray}
An array with shape (N,K).
Returns
-------
Y : {matrix, ndarray}
A matrix or ndarray with shape (M,K) depending on
the type of the X argument.
Notes
-----
This matmat wraps any user-specified matmat routine or overridden
_matmat method to ensure that y has the correct type.
"""
X = np.asanyarray(X)
if X.ndim != 2:
raise ValueError('expected 2-d ndarray or matrix, not %d-d'
% X.ndim)
M,N = self.shape
if X.shape[0] != N:
raise ValueError('dimension mismatch: %r, %r'
% (self.shape, X.shape))
Y = self._matmat(X)
if isinstance(Y, np.matrix):
Y = np.asmatrix(Y)
return Y
def __call__(self, x):
return self*x
def __mul__(self, x):
return self.dot(x)
def dot(self, x):
"""Matrix-matrix or matrix-vector multiplication.
Parameters
----------
x : array_like
1-d or 2-d array, representing a vector or matrix.
Returns
-------
Ax : array
1-d or 2-d array (depending on the shape of x) that represents
the result of applying this linear operator on x.
"""
if isinstance(x, LinearOperator):
return _ProductLinearOperator(self, x)
elif np.isscalar(x):
return _ScaledLinearOperator(self, x)
else:
x = np.asarray(x)
if x.ndim == 1 or x.ndim == 2 and x.shape[1] == 1:
return self.matvec(x)
elif x.ndim == 2:
return self.matmat(x)
else:
raise ValueError('expected 1-d or 2-d array or matrix, got %r'
% x)
def __matmul__(self, other):
if np.isscalar(other):
raise ValueError("Scalar operands are not allowed, "
"use '*' instead")
return self.__mul__(other)
def __rmatmul__(self, other):
if np.isscalar(other):
raise ValueError("Scalar operands are not allowed, "
"use '*' instead")
return self.__rmul__(other)
def __rmul__(self, x):
if np.isscalar(x):
return _ScaledLinearOperator(self, x)
else:
return NotImplemented
def __pow__(self, p):
if np.isscalar(p):
return _PowerLinearOperator(self, p)
else:
return NotImplemented
def __add__(self, x):
if isinstance(x, LinearOperator):
return _SumLinearOperator(self, x)
else:
return NotImplemented
def __neg__(self):
return _ScaledLinearOperator(self, -1)
def __sub__(self, x):
return self.__add__(-x)
def __repr__(self):
M,N = self.shape
if self.dtype is None:
dt = 'unspecified dtype'
else:
dt = 'dtype=' + str(self.dtype)
return '<%dx%d %s with %s>' % (M, N, self.__class__.__name__, dt)
def adjoint(self):
"""Hermitian adjoint.
Returns the Hermitian adjoint of self, aka the Hermitian
conjugate or Hermitian transpose. For a complex matrix, the
Hermitian adjoint is equal to the conjugate transpose.
Can be abbreviated self.H instead of self.adjoint().
Returns
-------
A_H : LinearOperator
Hermitian adjoint of self.
"""
return self._adjoint()
H = property(adjoint)
def transpose(self):
"""Transpose this linear operator.
Returns a LinearOperator that represents the transpose of this one.
Can be abbreviated self.T instead of self.transpose().
"""
return self._transpose()
T = property(transpose)
def _adjoint(self):
"""Default implementation of _adjoint; defers to rmatvec."""
shape = (self.shape[1], self.shape[0])
return _CustomLinearOperator(shape, matvec=self.rmatvec,
rmatvec=self.matvec,
dtype=self.dtype)
class _CustomLinearOperator(LinearOperator):
"""Linear operator defined in terms of user-specified operations."""
def __init__(self, shape, matvec, rmatvec=None, matmat=None, dtype=None):
super(_CustomLinearOperator, self).__init__(dtype, shape)
self.args = ()
self.__matvec_impl = matvec
self.__rmatvec_impl = rmatvec
self.__matmat_impl = matmat
self._init_dtype()
def _matmat(self, X):
if self.__matmat_impl is not None:
return self.__matmat_impl(X)
else:
return super(_CustomLinearOperator, self)._matmat(X)
def _matvec(self, x):
return self.__matvec_impl(x)
def _rmatvec(self, x):
func = self.__rmatvec_impl
if func is None:
raise NotImplementedError("rmatvec is not defined")
return self.__rmatvec_impl(x)
def _adjoint(self):
return _CustomLinearOperator(shape=(self.shape[1], self.shape[0]),
matvec=self.__rmatvec_impl,
rmatvec=self.__matvec_impl,
dtype=self.dtype)
def _get_dtype(operators, dtypes=None):
if dtypes is None:
dtypes = []
for obj in operators:
if obj is not None and hasattr(obj, 'dtype'):
dtypes.append(obj.dtype)
return np.find_common_type(dtypes, [])
class _SumLinearOperator(LinearOperator):
def __init__(self, A, B):
if not isinstance(A, LinearOperator) or \
not isinstance(B, LinearOperator):
raise ValueError('both operands have to be a LinearOperator')
if A.shape != B.shape:
raise ValueError('cannot add %r and %r: shape mismatch'
% (A, B))
self.args = (A, B)
super(_SumLinearOperator, self).__init__(_get_dtype([A, B]), A.shape)
def _matvec(self, x):
return self.args[0].matvec(x) + self.args[1].matvec(x)
def _rmatvec(self, x):
return self.args[0].rmatvec(x) + self.args[1].rmatvec(x)
def _matmat(self, x):
return self.args[0].matmat(x) + self.args[1].matmat(x)
def _adjoint(self):
A, B = self.args
return A.H + B.H
class _ProductLinearOperator(LinearOperator):
def __init__(self, A, B):
if not isinstance(A, LinearOperator) or \
not isinstance(B, LinearOperator):
raise ValueError('both operands have to be a LinearOperator')
if A.shape[1] != B.shape[0]:
raise ValueError('cannot multiply %r and %r: shape mismatch'
% (A, B))
super(_ProductLinearOperator, self).__init__(_get_dtype([A, B]),
(A.shape[0], B.shape[1]))
self.args = (A, B)
def _matvec(self, x):
return self.args[0].matvec(self.args[1].matvec(x))
def _rmatvec(self, x):
return self.args[1].rmatvec(self.args[0].rmatvec(x))
def _matmat(self, x):
return self.args[0].matmat(self.args[1].matmat(x))
def _adjoint(self):
A, B = self.args
return B.H * A.H
class _ScaledLinearOperator(LinearOperator):
def __init__(self, A, alpha):
if not isinstance(A, LinearOperator):
raise ValueError('LinearOperator expected as A')
if not np.isscalar(alpha):
raise ValueError('scalar expected as alpha')
dtype = _get_dtype([A], [type(alpha)])
super(_ScaledLinearOperator, self).__init__(dtype, A.shape)
self.args = (A, alpha)
def _matvec(self, x):
return self.args[1] * self.args[0].matvec(x)
def _rmatvec(self, x):
return np.conj(self.args[1]) * self.args[0].rmatvec(x)
def _matmat(self, x):
return self.args[1] * self.args[0].matmat(x)
def _adjoint(self):
A, alpha = self.args
return A.H * alpha
class _PowerLinearOperator(LinearOperator):
def __init__(self, A, p):
if not isinstance(A, LinearOperator):
raise ValueError('LinearOperator expected as A')
if A.shape[0] != A.shape[1]:
raise ValueError('square LinearOperator expected, got %r' % A)
if not isintlike(p) or p < 0:
raise ValueError('non-negative integer expected as p')
super(_PowerLinearOperator, self).__init__(_get_dtype([A]), A.shape)
self.args = (A, p)
def _power(self, fun, x):
res = np.array(x, copy=True)
for i in range(self.args[1]):
res = fun(res)
return res
def _matvec(self, x):
return self._power(self.args[0].matvec, x)
def _rmatvec(self, x):
return self._power(self.args[0].rmatvec, x)
def _matmat(self, x):
return self._power(self.args[0].matmat, x)
def _adjoint(self):
A, p = self.args
return A.H ** p
class MatrixLinearOperator(LinearOperator):
def __init__(self, A):
super(MatrixLinearOperator, self).__init__(A.dtype, A.shape)
self.A = A
self.__adj = None
self.args = (A,)
def _matmat(self, X):
return self.A.dot(X)
def _adjoint(self):
if self.__adj is None:
self.__adj = _AdjointMatrixOperator(self)
return self.__adj
class _AdjointMatrixOperator(MatrixLinearOperator):
def __init__(self, adjoint):
self.A = adjoint.A.T.conj()
self.__adjoint = adjoint
self.args = (adjoint,)
self.shape = adjoint.shape[1], adjoint.shape[0]
@property
def dtype(self):
return self.__adjoint.dtype
def _adjoint(self):
return self.__adjoint
class IdentityOperator(LinearOperator):
def __init__(self, shape, dtype=None):
super(IdentityOperator, self).__init__(dtype, shape)
def _matvec(self, x):
return x
def _rmatvec(self, x):
return x
def _matmat(self, x):
return x
def _adjoint(self):
return self
def aslinearoperator(A):
"""Return A as a LinearOperator.
'A' may be any of the following types:
- ndarray
- matrix
- sparse matrix (e.g. csr_matrix, lil_matrix, etc.)
- LinearOperator
- An object with .shape and .matvec attributes
See the LinearOperator documentation for additional information.
Examples
--------
>>> from scipy.sparse.linalg import aslinearoperator
>>> M = np.array([[1,2,3],[4,5,6]], dtype=np.int32)
>>> aslinearoperator(M)
<2x3 MatrixLinearOperator with dtype=int32>
"""
if isinstance(A, LinearOperator):
return A
elif isinstance(A, np.ndarray) or isinstance(A, np.matrix):
if A.ndim > 2:
raise ValueError('array must have ndim <= 2')
A = np.atleast_2d(np.asarray(A))
return MatrixLinearOperator(A)
elif isspmatrix(A):
return MatrixLinearOperator(A)
else:
if hasattr(A, 'shape') and hasattr(A, 'matvec'):
rmatvec = None
dtype = None
if hasattr(A, 'rmatvec'):
rmatvec = A.rmatvec
if hasattr(A, 'dtype'):
dtype = A.dtype
return LinearOperator(A.shape, A.matvec,
rmatvec=rmatvec, dtype=dtype)
else:
raise TypeError('type not understood')