def tensorinv(a, ind=2):
"""Computes the inverse of a tensor.
This function computes tensor ``a_inv`` from tensor ``a`` such that
``tensordot(a_inv, a, ind) == I``, where ``I`` denotes the identity tensor.
Args:
a (cupy.ndarray):
The tensor such that
``prod(a.shape[:ind]) == prod(a.shape[ind:])``.
ind (int):
The positive number used in ``axes`` option of ``tensordot``.
Returns:
cupy.ndarray:
The inverse of a tensor whose shape is equivalent to
``a.shape[ind:] + a.shape[:ind]``.
.. seealso:: :func:`numpy.linalg.tensorinv`
"""
util._assert_cupy_array(a)
if ind <= 0:
raise ValueError('Invalid ind argument')
oldshape = a.shape
invshape = oldshape[ind:] + oldshape[:ind]
prod = cupy.internal.prod(oldshape[ind:])
a = a.reshape(prod, -1)
a_inv = inv(a)
return a_inv.reshape(*invshape)
def cholesky(a):
"""Cholesky decomposition.
Decompose a given two-dimensional square matrix into ``L * L.T``,
where ``L`` is a lower-triangular matrix and ``.T`` is a conjugate
transpose operator.
Args:
a (cupy.ndarray): The input matrix with dimension ``(N, N)``
Returns:
cupy.ndarray: The lower-triangular matrix.
.. warning::
This function calls one or more cuSOLVER routine(s) which may yield
invalid results if input conditions are not met.
To detect these invalid results, you can set the `linalg`
configuration to a value that is not `ignore` in
:func:`cupyx.errstate` or :func:`cupyx.seterr`.
.. seealso:: :func:`numpy.linalg.cholesky`
"""
# TODO(Saito): Current implementation only accepts two-dimensional arrays
util._assert_cupy_array(a)
util._assert_rank2(a)
util._assert_nd_squareness(a)
if a.dtype.char == 'f' or a.dtype.char == 'd':
dtype = a.dtype.char
else:
dtype = numpy.promote_types(a.dtype.char, 'f').char
x = a.astype(dtype, order='C', copy=True)
n = len(a)
handle = device.get_cusolver_handle()
dev_info = cupy.empty(1, dtype=numpy.int32)
if dtype == 'f':
potrf = cusolver.spotrf
potrf_bufferSize = cusolver.spotrf_bufferSize
elif dtype == 'd':
potrf = cusolver.dpotrf
potrf_bufferSize = cusolver.dpotrf_bufferSize
elif dtype == 'F':
potrf = cusolver.cpotrf
potrf_bufferSize = cusolver.cpotrf_bufferSize
else: # dtype == 'D':
potrf = cusolver.zpotrf
potrf_bufferSize = cusolver.zpotrf_bufferSize
buffersize = potrf_bufferSize(handle, cublas.CUBLAS_FILL_MODE_UPPER, n,
x.data.ptr, n)
workspace = cupy.empty(buffersize, dtype=dtype)
potrf(handle, cublas.CUBLAS_FILL_MODE_UPPER, n, x.data.ptr, n,
workspace.data.ptr, buffersize, dev_info.data.ptr)
cupy.linalg.util._check_cusolver_dev_info_if_synchronization_allowed(
potrf, dev_info)
util._tril(x, k=0)
return x
def lsqr(A, b):
"""Solves linear system with QR decomposition.
Find the solution to a large, sparse, linear system of equations.
The function solves ``Ax = b``. Given two-dimensional matrix ``A`` is
decomposed into ``Q * R``.
Args:
A (cupy.ndarray or cupy.sparse.csr_matrix): The input matrix with
dimension ``(N, N)``
b (cupy.ndarray): Right-hand side vector.
Returns:
ret (tuple): Its length must be ten. It has same type elements
as SciPy. Only the first element, the solution vector ``x``, is
available and other elements are expressed as ``None`` because
the implementation of cuSOLVER is different from the one of SciPy.
You can easily calculate the fourth element by ``norm(b - Ax)``
and the ninth element by ``norm(x)``.
.. seealso:: :func:`scipy.sparse.linalg.lsqr`
"""
if not cuda.cusolver_enabled:
raise RuntimeError('Current cupy only supports cusolver in CUDA 8.0')
if not cupy.sparse.isspmatrix_csr(A):
A = cupy.sparse.csr_matrix(A)
util._assert_nd_squareness(A)
util._assert_cupy_array(b)
m = A.shape[0]
if b.ndim != 1 or len(b) != m:
raise ValueError('b must be 1-d array whose size is same as A')
# Cast to float32 or float64
if A.dtype == 'f' or A.dtype == 'd':
dtype = A.dtype
else:
dtype = numpy.find_common_type((A.dtype, 'f'), ())
handle = device.get_cusolver_sp_handle()
nnz = A.nnz
tol = 1.0
reorder = 1
x = cupy.empty(m, dtype=dtype)
singularity = numpy.empty(1, numpy.int32)
if dtype == 'f':
csrlsvqr = cusolver.scsrlsvqr
else:
csrlsvqr = cusolver.dcsrlsvqr
csrlsvqr(handle, m, nnz, A._descr.descriptor, A.data.data.ptr,
A.indptr.data.ptr, A.indices.data.ptr, b.data.ptr, tol, reorder,
x.data.ptr, singularity.ctypes.data)
# The return type of SciPy is always float64. Therefore, x must be casted.
x = x.astype(numpy.float64)
ret = (x, None, None, None, None, None, None, None, None, None)
return ret
def inv(a):
"""Computes the inverse of a matrix.
This function computes matrix ``a_inv`` from n-dimensional regular matrix
``a`` such that ``dot(a, a_inv) == eye(n)``.
Args:
a (cupy.ndarray): The regular matrix
Returns:
cupy.ndarray: The inverse of a matrix.
.. seealso:: :func:`numpy.linalg.inv`
"""
if not cuda.cusolver_enabled:
raise RuntimeError('Current cupy only supports cusolver in CUDA 8.0')
# to prevent `a` to be overwritten
a = a.copy()
util._assert_cupy_array(a)
util._assert_rank2(a)
util._assert_nd_squareness(a)
if a.dtype.char == 'f' or a.dtype.char == 'd':
dtype = a.dtype.char
else:
dtype = numpy.find_common_type((a.dtype.char, 'f'), ()).char
cusolver_handle = device.get_cusolver_handle()
dev_info = cupy.empty(1, dtype=dtype)
ipiv = cupy.empty((a.shape[0], 1), dtype=dtype)
if dtype == 'f':
getrf = cusolver.sgetrf
getrf_bufferSize = cusolver.sgetrf_bufferSize
getrs = cusolver.sgetrs
else: # dtype == 'd'
getrf = cusolver.dgetrf
getrf_bufferSize = cusolver.dgetrf_bufferSize
getrs = cusolver.dgetrs
m = a.shape[0]
buffersize = getrf_bufferSize(cusolver_handle, m, m, a.data.ptr, m)
workspace = cupy.empty(buffersize, dtype=dtype)
# LU factorization
getrf(cusolver_handle, m, m, a.data.ptr, m, workspace.data.ptr,
ipiv.data.ptr, dev_info.data.ptr)
b = cupy.eye(m, dtype=dtype)
# solve for the inverse
getrs(cusolver_handle, 0, m, m, a.data.ptr, m, ipiv.data.ptr, b.data.ptr,
m, dev_info.data.ptr)
return b
def _batched_inv(a):
assert (a.ndim >= 3)
util._assert_cupy_array(a)
util._assert_nd_squareness(a)
if a.dtype == cupy.float32:
getrf = cupy.cuda.cublas.sgetrfBatched
getri = cupy.cuda.cublas.sgetriBatched
elif a.dtype == cupy.float64:
getrf = cupy.cuda.cublas.dgetrfBatched
getri = cupy.cuda.cublas.dgetriBatched
elif a.dtype == cupy.complex64:
getrf = cupy.cuda.cublas.cgetrfBatched
getri = cupy.cuda.cublas.cgetriBatched
elif a.dtype == cupy.complex128:
getrf = cupy.cuda.cublas.zgetrfBatched
getri = cupy.cuda.cublas.zgetriBatched
else:
msg = ('dtype must be float32, float64, complex64 or complex128'
' (actual: {})'.format(a.dtype))
raise ValueError(msg)
if 0 in a.shape:
return cupy.empty_like(a)
a_shape = a.shape
# copy is necessary to present `a` to be overwritten.
a = a.copy().reshape(-1, a_shape[-2], a_shape[-1])
handle = device.get_cublas_handle()
batch_size = a.shape[0]
n = a.shape[1]
lda = n
step = n * lda * a.itemsize
start = a.data.ptr
stop = start + step * batch_size
a_array = cupy.arange(start, stop, step, dtype=cupy.uintp)
pivot_array = cupy.empty((batch_size, n), dtype=cupy.int32)
info_array = cupy.empty((batch_size, ), dtype=cupy.int32)
getrf(handle, n, a_array.data.ptr, lda, pivot_array.data.ptr,
info_array.data.ptr, batch_size)
cupy.linalg.util._check_cublas_info_array_if_synchronization_allowed(
getrf, info_array)
c = cupy.empty_like(a)
ldc = lda
step = n * ldc * c.itemsize
start = c.data.ptr
stop = start + step * batch_size
c_array = cupy.arange(start, stop, step, dtype=cupy.uintp)
getri(handle, n, a_array.data.ptr, lda, pivot_array.data.ptr,
c_array.data.ptr, ldc, info_array.data.ptr, batch_size)
cupy.linalg.util._check_cublas_info_array_if_synchronization_allowed(
getri, info_array)
return c.reshape(a_shape)
def cholesky(a):
"""Cholesky decomposition.
Decompose a given two-dimensional square matrix into ``L * L.T``,
where ``L`` is a lower-triangular matrix and ``.T`` is a conjugate
transpose operator. Note that in the current implementation ``a`` must be
a real matrix, and only float32 and float64 are supported.
Args:
a (cupy.ndarray): The input matrix with dimension ``(N, N)``
Returns:
cupy.ndarray: The lower-triangular matrix.
.. seealso:: :func:`numpy.linalg.cholesky`
"""
if not cuda.cusolver_enabled:
raise RuntimeError('Current cupy only supports cusolver in CUDA 8.0')
# TODO(Saito): Current implementation only accepts two-dimensional arrays
util._assert_cupy_array(a)
util._assert_rank2(a)
util._assert_nd_squareness(a)
# Cast to float32 or float64
if a.dtype.char == 'f' or a.dtype.char == 'd':
dtype = a.dtype.char
else:
dtype = numpy.find_common_type((a.dtype.char, 'f'), ()).char
x = a.astype(dtype, order='C', copy=True)
n = len(a)
handle = device.get_cusolver_handle()
dev_info = cupy.empty(1, dtype=numpy.int32)
if dtype == 'f':
buffersize = cusolver.spotrf_bufferSize(
handle, cublas.CUBLAS_FILL_MODE_UPPER, n, x.data.ptr, n)
workspace = cupy.empty(buffersize, dtype=numpy.float32)
cusolver.spotrf(
handle, cublas.CUBLAS_FILL_MODE_UPPER, n, x.data.ptr, n,
workspace.data.ptr, buffersize, dev_info.data.ptr)
else: # dtype == 'd'
buffersize = cusolver.dpotrf_bufferSize(
handle, cublas.CUBLAS_FILL_MODE_UPPER, n, x.data.ptr, n)
workspace = cupy.empty(buffersize, dtype=numpy.float64)
cusolver.dpotrf(
handle, cublas.CUBLAS_FILL_MODE_UPPER, n, x.data.ptr, n,
workspace.data.ptr, buffersize, dev_info.data.ptr)
status = int(dev_info[0])
if status > 0:
raise linalg.LinAlgError(
'The leading minor of order {} '
'is not positive definite'.format(status))
elif status < 0:
raise linalg.LinAlgError(
'Parameter error (maybe caused by a bug in cupy.linalg?)')
util._tril(x, k=0)
return x
def solve(a, b):
"""Solves a linear matrix equation.
It computes the exact solution of ``x`` in ``ax = b``,
where ``a`` is a square and full rank matrix.
Args:
a (cupy.ndarray): The matrix with dimension ``(..., M, M)``.
b (cupy.ndarray): The matrix with dimension ``(...,M)`` or
``(..., M, K)``.
Returns:
cupy.ndarray:
The matrix with dimension ``(..., M)`` or ``(..., M, K)``.
.. warning::
This function calls one or more cuSOLVER routine(s) which may yield
invalid results if input conditions are not met.
To detect these invalid results, you can set the `linalg`
configuration to a value that is not `ignore` in
:func:`cupyx.errstate` or :func:`cupyx.seterr`.
.. seealso:: :func:`numpy.linalg.solve`
"""
# NOTE: Since cusolver in CUDA 8.0 does not support gesv,
# we manually solve a linear system with QR decomposition.
# For details, please see the following:
# https://docs.nvidia.com/cuda/cusolver/index.html#qr_examples
util._assert_cupy_array(a, b)
util._assert_nd_squareness(a)
if not ((a.ndim == b.ndim or a.ndim == b.ndim + 1)
and a.shape[:-1] == b.shape[:a.ndim - 1]):
raise ValueError(
'a must have (..., M, M) shape and b must have (..., M) '
'or (..., M, K)')
# Cast to float32 or float64
if a.dtype.char == 'f' or a.dtype.char == 'd':
dtype = a.dtype
else:
dtype = numpy.find_common_type((a.dtype.char, 'f'), ())
cublas_handle = device.get_cublas_handle()
cusolver_handle = device.get_cusolver_handle()
a = a.astype(dtype)
b = b.astype(dtype)
if a.ndim == 2:
return _solve(a, b, cublas_handle, cusolver_handle)
x = cupy.empty_like(b)
shape = a.shape[:-2]
for i in six.moves.range(numpy.prod(shape)):
index = numpy.unravel_index(i, shape)
x[index] = _solve(a[index], b[index], cublas_handle, cusolver_handle)
return x
def solve(a, b):
"""Solves a linear matrix equation.
It computes the exact solution of ``x`` in ``ax = b``,
where ``a`` is a square and full rank matrix.
Args:
a (cupy.ndarray): The matrix with dimension ``(..., M, M)``.
b (cupy.ndarray): The matrix with dimension ``(...,M)`` or
``(..., M, K)``.
Returns:
cupy.ndarray:
The matrix with dimension ``(..., M)`` or ``(..., M, K)``.
.. warning::
This function calls one or more cuSOLVER routine(s) which may yield
invalid results if input conditions are not met.
To detect these invalid results, you can set the `linalg`
configuration to a value that is not `ignore` in
:func:`cupyx.errstate` or :func:`cupyx.seterr`.
.. seealso:: :func:`numpy.linalg.solve`
"""
if a.ndim > 2 and a.shape[-1] <= get_batched_gesv_limit():
# Note: There is a low performance issue in batched_gesv when matrix is
# large, so it is not used in such cases.
return batched_gesv(a, b)
util._assert_cupy_array(a, b)
util._assert_nd_squareness(a)
if not ((a.ndim == b.ndim or a.ndim == b.ndim + 1) and
a.shape[:-1] == b.shape[:a.ndim - 1]):
raise ValueError(
'a must have (..., M, M) shape and b must have (..., M) '
'or (..., M, K)')
# Cast to float32 or float64
if a.dtype.char == 'f' or a.dtype.char == 'd':
dtype = a.dtype
else:
dtype = numpy.promote_types(a.dtype.char, 'f')
a = a.astype(dtype)
b = b.astype(dtype)
if a.ndim == 2:
return cupy.cusolver.gesv(a, b)
x = cupy.empty_like(b)
shape = a.shape[:-2]
for i in range(numpy.prod(shape)):
index = numpy.unravel_index(i, shape)
x[index] = cupy.cusolver.gesv(a[index], b[index])
return x
def lschol(A, b):
"""Solves linear system with cholesky decomposition.
Find the solution to a large, sparse, linear system of equations.
The function solves ``Ax = b``. Given two-dimensional matrix ``A`` is
decomposed into ``L * L^*``.
Args:
A (cupy.ndarray or cupy.sparse.csr_matrix): The input matrix with
dimension ``(N, N)``. Must be positive-definite input matrix.
Only symmetric real matrix is supported currently.
b (cupy.ndarray): Right-hand side vector.
Returns:
ret (cupy.ndarray): The solution vector ``x``.
"""
if not cuda.cusolver_enabled:
raise RuntimeError('Current cupy only supports cusolver in CUDA 8.0')
if not cupy.sparse.isspmatrix_csr(A):
A = cupy.sparse.csr_matrix(A)
util._assert_nd_squareness(A)
util._assert_cupy_array(b)
m = A.shape[0]
if b.ndim != 1 or len(b) != m:
raise ValueError('b must be 1-d array whose size is same as A')
# Cast to float32 or float64
if A.dtype == 'f' or A.dtype == 'd':
dtype = A.dtype
else:
dtype = numpy.find_common_type((A.dtype, 'f'), ())
handle = device.get_cusolver_sp_handle()
nnz = A.nnz
tol = 1.0
reorder = 1
x = cupy.empty(m, dtype=dtype)
singularity = numpy.empty(1, numpy.int32)
if dtype == 'f':
csrlsvchol = cusolver.scsrlsvchol
else:
csrlsvchol = cusolver.dcsrlsvchol
csrlsvchol(handle, m, nnz, A._descr.descriptor, A.data.data.ptr,
A.indptr.data.ptr, A.indices.data.ptr, b.data.ptr, tol, reorder,
x.data.ptr, singularity.ctypes.data)
# The return type of SciPy is always float64.
x = x.astype(numpy.float64)
return x
def solve(a, b):
"""Solves a linear matrix equation.
It computes the exact solution of ``x`` in ``ax = b``,
where ``a`` is a square and full rank matrix.
Args:
a (cupy.ndarray): The matrix with dimension ``(..., M, M)``.
b (cupy.ndarray): The matrix with dimension ``(...,M)`` or
``(..., M, K)``.
Returns:
cupy.ndarray:
The matrix with dimension ``(..., M)`` or ``(..., M, K)``.
.. seealso:: :func:`numpy.linalg.solve`
"""
# NOTE: Since cusolver in CUDA 8.0 does not support gesv,
# we manually solve a linear system with QR decomposition.
# For details, please see the following:
# https://docs.nvidia.com/cuda/cusolver/index.html#qr_examples
if not cuda.cusolver_enabled:
raise RuntimeError('Current cupy only supports cusolver in CUDA 8.0')
util._assert_cupy_array(a, b)
util._assert_nd_squareness(a)
if not ((a.ndim == b.ndim or a.ndim == b.ndim + 1)
and a.shape[:-1] == b.shape[:a.ndim - 1]):
raise ValueError(
'a must have (..., M, M) shape and b must have (..., M) '
'or (..., M, K)')
# Cast to float32 or float64
if a.dtype.char == 'f' or a.dtype.char == 'd':
dtype = a.dtype
else:
dtype = numpy.find_common_type((a.dtype.char, 'f'), ())
cublas_handle = device.get_cublas_handle()
cusolver_handle = device.get_cusolver_handle()
a = a.astype(dtype)
b = b.astype(dtype)
if a.ndim == 2:
return _solve(a, b, cublas_handle, cusolver_handle)
x = cupy.empty_like(b)
shape = a.shape[:-2]
for i in six.moves.range(numpy.prod(shape)):
index = numpy.unravel_index(i, shape)
x[index] = _solve(a[index], b[index], cublas_handle, cusolver_handle)
return x
def inv(a):
'''Computes the inverse of a matrix.
This function computes matrix ``a_inv`` from n-dimensional regular matrix
``a`` such that ``dot(a, a_inv) == eye(n)``.
Args:
a (cupy.ndarray): The regular matrix
Returns:
cupy.ndarray: The inverse of a matrix.
.. seealso:: :func:`numpy.linalg.inv`
'''
if not cuda.cusolver_enabled:
raise RuntimeError('Current cupy only supports cusolver in CUDA 8.0')
util._assert_cupy_array(a)
util._assert_rank2(a)
util._assert_nd_squareness(a)
b = cupy.eye(len(a), dtype=a.dtype)
return solve(a, b)
def tensorinv(a, ind=2):
"""Computes the inverse of a tensor.
This function computes tensor ``a_inv`` from tensor ``a`` such that
``tensordot(a_inv, a, ind) == I``, where ``I`` denotes the identity tensor.
Args:
a (cupy.ndarray):
The tensor such that
``prod(a.shape[:ind]) == prod(a.shape[ind:])``.
ind (int):
The positive number used in ``axes`` option of ``tensordot``.
Returns:
cupy.ndarray:
The inverse of a tensor whose shape is equivalent to
``a.shape[ind:] + a.shape[:ind]``.
.. warning::
This function calls one or more cuSOLVER routine(s) which may yield
invalid results if input conditions are not met.
To detect these invalid results, you can set the `linalg`
configuration to a value that is not `ignore` in
:func:`cupyx.errstate` or :func:`cupyx.seterr`.
.. seealso:: :func:`numpy.linalg.tensorinv`
"""
util._assert_cupy_array(a)
if ind <= 0:
raise ValueError('Invalid ind argument')
oldshape = a.shape
invshape = oldshape[ind:] + oldshape[:ind]
prod = cupy.internal.prod(oldshape[ind:])
a = a.reshape(prod, -1)
a_inv = inv(a)
return a_inv.reshape(*invshape)
def qr(a, mode='reduced'):
'''QR decomposition.
Decompose a given two-dimensional matrix into ``Q * R``, where ``Q``
is an orthonormal and ``R`` is an upper-triangular matrix.
Args:
a (cupy.ndarray): The input matrix.
mode (str): The mode of decomposition. Currently 'reduced',
'complete', 'r', and 'raw' modes are supported. The default mode
is 'reduced', and decompose a matrix ``A = (M, N)`` into ``Q``,
``R`` with dimensions ``(M, K)``, ``(K, N)``, where
``K = min(M, N)``.
.. seealso:: :func:`numpy.linalg.qr`
'''
if not cuda.cusolver_enabled:
raise RuntimeError('Current cupy only supports cusolver in CUDA 8.0')
# TODO(Saito): Current implementation only accepts two-dimensional arrays
util._assert_cupy_array(a)
util._assert_rank2(a)
if mode not in ('reduced', 'complete', 'r', 'raw'):
if mode in ('f', 'full', 'e', 'economic'):
msg = 'The deprecated mode \'{}\' is not supported'.format(mode)
raise ValueError(msg)
else:
raise ValueError('Unrecognized mode \'{}\''.format(mode))
# Cast to float32 or float64
if a.dtype.char == 'f' or a.dtype.char == 'd':
dtype = a.dtype.char
else:
dtype = numpy.find_common_type((a.dtype.char, 'f'), ()).char
m, n = a.shape
x = a.transpose().astype(dtype, order='C', copy=True)
mn = min(m, n)
handle = device.get_cusolver_handle()
dev_info = cupy.empty(1, dtype=numpy.int32)
# compute working space of geqrf and ormqr, and solve R
if dtype == 'f':
buffersize = cusolver.sgeqrf_bufferSize(handle, m, n, x.data.ptr, n)
workspace = cupy.empty(buffersize, dtype=numpy.float32)
tau = cupy.empty(mn, dtype=numpy.float32)
cusolver.sgeqrf(
handle, m, n, x.data.ptr, m,
tau.data.ptr, workspace.data.ptr, buffersize, dev_info.data.ptr)
else: # dtype == 'd'
buffersize = cusolver.dgeqrf_bufferSize(handle, n, m, x.data.ptr, n)
workspace = cupy.empty(buffersize, dtype=numpy.float64)
tau = cupy.empty(mn, dtype=numpy.float64)
cusolver.dgeqrf(
handle, m, n, x.data.ptr, m,
tau.data.ptr, workspace.data.ptr, buffersize, dev_info.data.ptr)
status = int(dev_info[0])
if status < 0:
raise linalg.LinAlgError(
'Parameter error (maybe caused by a bug in cupy.linalg?)')
if mode == 'r':
r = x[:, :mn].transpose()
return util._triu(r)
if mode == 'raw':
if a.dtype.char == 'f':
# The original numpy.linalg.qr returns float64 in raw mode,
# whereas the cusolver returns float32. We agree that the
# following code would be inappropriate, however, in this time
# we explicitly convert them to float64 for compatibility.
return x.astype(numpy.float64), tau.astype(numpy.float64)
return x, tau
if mode == 'complete' and m > n:
mc = m
q = cupy.empty((m, m), dtype)
else:
mc = mn
q = cupy.empty((n, m), dtype)
q[:n] = x
# solve Q
if dtype == 'f':
buffersize = cusolver.sorgqr_bufferSize(
handle, m, mc, mn, q.data.ptr, m, tau.data.ptr)
workspace = cupy.empty(buffersize, dtype=numpy.float32)
cusolver.sorgqr(
handle, m, mc, mn, q.data.ptr, m, tau.data.ptr,
workspace.data.ptr, buffersize, dev_info.data.ptr)
else:
buffersize = cusolver.dorgqr_bufferSize(
handle, m, mc, mn, q.data.ptr, m, tau.data.ptr)
workspace = cupy.empty(buffersize, dtype=numpy.float64)
cusolver.dorgqr(
handle, m, mc, mn, q.data.ptr, m, tau.data.ptr,
workspace.data.ptr, buffersize, dev_info.data.ptr)
q = q[:mc].transpose()
r = x[:, :mc].transpose()
return q, util._triu(r)
def inv_core(a, cholesky=False):
"""Computes the inverse of a matrix.
This function computes matrix ``a_inv`` from n-dimensional regular matrix
``a`` such that ``dot(a, a_inv) == eye(n)``.
Args:
a (cupy.ndarray): The regular matrix
b (Boolean): Use cholesky decomposition
Returns:
cupy.ndarray: The inverse of a matrix.
.. seealso:: :func:`numpy.linalg.inv`
"""
xp = cupy.get_array_module(a)
if xp == numpy:
if cholesky:
warnings.warn(
"Current fast-inv using cholesky doesn't support numpy.ndarray."
)
return numpy.linalg.inv(a)
if not cuda.cusolver_enabled:
raise RuntimeError('Current cupy only supports cusolver in CUDA 8.0')
# to prevent `a` to be overwritten
a = a.copy()
util._assert_cupy_array(a)
util._assert_rank2(a)
util._assert_nd_squareness(a)
if a.dtype.char == 'f' or a.dtype.char == 'd':
dtype = a.dtype.char
else:
dtype = numpy.find_common_type((a.dtype.char, 'f'), ()).char
cusolver_handle = device.get_cusolver_handle()
dev_info = cupy.empty(1, dtype=cupy.int)
m = a.shape[0]
b = cupy.eye(m, dtype=dtype)
if not cholesky:
if dtype == 'f':
getrf = cusolver.sgetrf
getrf_bufferSize = cusolver.sgetrf_bufferSize
getrs = cusolver.sgetrs
else: # dtype == 'd'
getrf = cusolver.dgetrf
getrf_bufferSize = cusolver.dgetrf_bufferSize
getrs = cusolver.dgetrs
buffersize = getrf_bufferSize(cusolver_handle, m, m, a.data.ptr, m)
# TODO(y1r): cache buffer to avoid malloc
workspace = cupy.empty(buffersize, dtype=dtype)
ipiv = cupy.empty((a.shape[0], 1), dtype=dtype)
# LU Decomposition
getrf(cusolver_handle, m, m, a.data.ptr, m, workspace.data.ptr,
ipiv.data.ptr, dev_info.data.ptr)
# TODO(y1r): check dev_info status
# solve for the inverse
getrs(cusolver_handle, 0, m, m, a.data.ptr, m, ipiv.data.ptr,
b.data.ptr, m, dev_info.data.ptr)
# TODO(y1r): check dev_info status
else:
if dtype == 'f':
potrf = cusolver.spotrf
potrf_bufferSize = cusolver.spotrf_bufferSize
potrs = cusolver.spotrs
else: # dtype == 'd'
potrf = cusolver.dpotrf
potrf_bufferSize = cusolver.dpotrf_bufferSize
potrs = cusolver.dpotrs
buffersize = potrf_bufferSize(cusolver_handle,
cublas.CUBLAS_FILL_MODE_UPPER, m,
a.data.ptr, m)
# TODO(y1r): cache buffer to avoid malloc
workspace = cupy.empty(buffersize, dtype=dtype)
# Cholesky Decomposition
potrf(cusolver_handle, cublas.CUBLAS_FILL_MODE_UPPER, m, a.data.ptr, m,
workspace.data.ptr, buffersize, dev_info.data.ptr)
# TODO(y1r): check dev_info status
# solve for the inverse
potrs(cusolver_handle, cublas.CUBLAS_FILL_MODE_UPPER, m, m, a.data.ptr,
m, b.data.ptr, m, dev_info.data.ptr)
# TODO(y1r): check dev_info status
return b
def lstsq(a, b, rcond=1e-15):
"""Return the least-squares solution to a linear matrix equation.
Solves the equation `a x = b` by computing a vector `x` that
minimizes the Euclidean 2-norm `|| b - a x ||^2`. The equation may
be under-, well-, or over- determined (i.e., the number of
linearly independent rows of `a` can be less than, equal to, or
greater than its number of linearly independent columns). If `a`
is square and of full rank, then `x` (but for round-off error) is
the "exact" solution of the equation.
Args:
a (cupy.ndarray): "Coefficient" matrix with dimension ``(M, N)``
b (cupy.ndarray): "Dependent variable" values with dimension ``(M,)``
or ``(M, K)``
rcond (float): Cutoff parameter for small singular values.
For stability it computes the largest singular value denoted by
``s``, and sets all singular values smaller than ``s`` to zero.
Returns:
tuple:
A tuple of ``(x, residuals, rank, s)``. Note ``x`` is the
least-squares solution with shape ``(N,)`` or ``(N, K)`` depending
if ``b`` was two-dimensional. The sums of ``residuals`` is the
squared Euclidean 2-norm for each column in b - a*x. The
``residuals`` is an empty array if the rank of a is < N or M <= N,
but iff b is 1-dimensional, this is a (1,) shape array, Otherwise
the shape is (K,). The ``rank`` of matrix ``a`` is an integer. The
singular values of ``a`` are ``s``.
.. warning::
This function calls one or more cuSOLVER routine(s) which may yield
invalid results if input conditions are not met.
To detect these invalid results, you can set the `linalg`
configuration to a value that is not `ignore` in
:func:`cupyx.errstate` or :func:`cupyx.seterr`.
.. seealso:: :func:`numpy.linalg.lstsq`
"""
util._assert_cupy_array(a, b)
util._assert_rank2(a)
if b.ndim > 2:
raise linalg.LinAlgError('{}-dimensional array given. Array must be at'
' most two-dimensional'.format(b.ndim))
m, n = a.shape[-2:]
m2 = b.shape[0]
if m != m2:
raise linalg.LinAlgError('Incompatible dimensions')
u, s, vt = cupy.linalg.svd(a, full_matrices=False)
# number of singular values and matrix rank
cutoff = rcond * s.max()
s1 = 1 / s
sing_vals = s <= cutoff
s1[sing_vals] = 0
rank = s.size - sing_vals.sum()
if b.ndim == 2:
s1 = cupy.repeat(s1.reshape(-1, 1), b.shape[1], axis=1)
# Solve the least-squares solution
z = core.dot(u.transpose(), b) * s1
x = core.dot(vt.transpose(), z)
# Calculate squared Euclidean 2-norm for each column in b - a*x
if rank != n or m <= n:
resids = cupy.array([], dtype=a.dtype)
elif b.ndim == 2:
e = b - core.dot(a, x)
resids = cupy.sum(cupy.square(e), axis=0)
else:
e = b - cupy.dot(a, x)
resids = cupy.dot(e.T, e).reshape(-1)
return x, resids, rank, s
def inv(a):
"""Computes the inverse of a matrix.
This function computes matrix ``a_inv`` from n-dimensional regular matrix
``a`` such that ``dot(a, a_inv) == eye(n)``.
Args:
a (cupy.ndarray): The regular matrix
Returns:
cupy.ndarray: The inverse of a matrix.
.. warning::
This function calls one or more cuSOLVER routine(s) which may yield
invalid results if input conditions are not met.
To detect these invalid results, you can set the `linalg`
configuration to a value that is not `ignore` in
:func:`cupyx.errstate` or :func:`cupyx.seterr`.
.. seealso:: :func:`numpy.linalg.inv`
"""
if a.ndim >= 3:
return _batched_inv(a)
# to prevent `a` to be overwritten
a = a.copy()
util._assert_cupy_array(a)
util._assert_rank2(a)
util._assert_nd_squareness(a)
# support float32, float64, complex64, and complex128
if a.dtype.char in 'fdFD':
dtype = a.dtype.char
else:
dtype = numpy.find_common_type((a.dtype.char, 'f'), ()).char
cusolver_handle = device.get_cusolver_handle()
dev_info = cupy.empty(1, dtype=numpy.int32)
ipiv = cupy.empty((a.shape[0], 1), dtype=numpy.intc)
if dtype == 'f':
getrf = cusolver.sgetrf
getrf_bufferSize = cusolver.sgetrf_bufferSize
getrs = cusolver.sgetrs
elif dtype == 'd':
getrf = cusolver.dgetrf
getrf_bufferSize = cusolver.dgetrf_bufferSize
getrs = cusolver.dgetrs
elif dtype == 'F':
getrf = cusolver.cgetrf
getrf_bufferSize = cusolver.cgetrf_bufferSize
getrs = cusolver.cgetrs
elif dtype == 'D':
getrf = cusolver.zgetrf
getrf_bufferSize = cusolver.zgetrf_bufferSize
getrs = cusolver.zgetrs
else:
msg = ('dtype must be float32, float64, complex64 or complex128'
' (actual: {})'.format(a.dtype))
raise ValueError(msg)
m = a.shape[0]
buffersize = getrf_bufferSize(cusolver_handle, m, m, a.data.ptr, m)
workspace = cupy.empty(buffersize, dtype=dtype)
# LU factorization
getrf(cusolver_handle, m, m, a.data.ptr, m, workspace.data.ptr,
ipiv.data.ptr, dev_info.data.ptr)
cupy.linalg.util._check_cusolver_dev_info_if_synchronization_allowed(
getrf, dev_info)
b = cupy.eye(m, dtype=dtype)
# solve for the inverse
getrs(cusolver_handle, 0, m, m, a.data.ptr, m, ipiv.data.ptr, b.data.ptr,
m, dev_info.data.ptr)
cupy.linalg.util._check_cusolver_dev_info_if_synchronization_allowed(
getrs, dev_info)
return b
def qr(a, mode='reduced'):
"""QR decomposition.
Decompose a given two-dimensional matrix into ``Q * R``, where ``Q``
is an orthonormal and ``R`` is an upper-triangular matrix.
Args:
a (cupy.ndarray): The input matrix.
mode (str): The mode of decomposition. Currently 'reduced',
'complete', 'r', and 'raw' modes are supported. The default mode
is 'reduced', in which matrix ``A = (M, N)`` is decomposed into
``Q``, ``R`` with dimensions ``(M, K)``, ``(K, N)``, where
``K = min(M, N)``.
Returns:
cupy.ndarray, or tuple of ndarray:
Although the type of returned object depends on the mode,
it returns a tuple of ``(Q, R)`` by default.
For details, please see the document of :func:`numpy.linalg.qr`.
.. seealso:: :func:`numpy.linalg.qr`
"""
# TODO(Saito): Current implementation only accepts two-dimensional arrays
util._assert_cupy_array(a)
util._assert_rank2(a)
if mode not in ('reduced', 'complete', 'r', 'raw'):
if mode in ('f', 'full', 'e', 'economic'):
msg = 'The deprecated mode \'{}\' is not supported'.format(mode)
raise ValueError(msg)
else:
raise ValueError('Unrecognized mode \'{}\''.format(mode))
# support float32, float64, complex64, and complex128
if a.dtype.char in 'fdFD':
dtype = a.dtype.char
else:
dtype = numpy.find_common_type((a.dtype.char, 'f'), ()).char
m, n = a.shape
x = a.transpose().astype(dtype, order='C', copy=True)
mn = min(m, n)
handle = device.get_cusolver_handle()
dev_info = cupy.empty(1, dtype=numpy.int32)
# compute working space of geqrf and orgqr, and solve R
if dtype == 'f':
geqrf_bufferSize = cusolver.sgeqrf_bufferSize
geqrf = cusolver.sgeqrf
elif dtype == 'd':
geqrf_bufferSize = cusolver.dgeqrf_bufferSize
geqrf = cusolver.dgeqrf
elif dtype == 'F':
geqrf_bufferSize = cusolver.cgeqrf_bufferSize
geqrf = cusolver.cgeqrf
elif dtype == 'D':
geqrf_bufferSize = cusolver.zgeqrf_bufferSize
geqrf = cusolver.zgeqrf
else:
msg = ('dtype must be float32, float64, complex64 or complex128'
' (actual: {})'.format(a.dtype))
raise ValueError(msg)
buffersize = geqrf_bufferSize(handle, m, n, x.data.ptr, n)
workspace = cupy.empty(buffersize, dtype=dtype)
tau = cupy.empty(mn, dtype=dtype)
geqrf(handle, m, n, x.data.ptr, m,
tau.data.ptr, workspace.data.ptr, buffersize, dev_info.data.ptr)
status = int(dev_info[0])
if status < 0:
raise linalg.LinAlgError(
'Parameter error (maybe caused by a bug in cupy.linalg?)')
if mode == 'r':
r = x[:, :mn].transpose()
return util._triu(r)
if mode == 'raw':
if a.dtype.char == 'f':
# The original numpy.linalg.qr returns float64 in raw mode,
# whereas the cusolver returns float32. We agree that the
# following code would be inappropriate, however, in this time
# we explicitly convert them to float64 for compatibility.
return x.astype(numpy.float64), tau.astype(numpy.float64)
elif a.dtype.char == 'F':
# The same applies to complex64
return x.astype(numpy.complex128), tau.astype(numpy.complex128)
return x, tau
if mode == 'complete' and m > n:
mc = m
q = cupy.empty((m, m), dtype)
else:
mc = mn
q = cupy.empty((n, m), dtype)
q[:n] = x
# solve Q
if dtype == 'f':
orgqr_bufferSize = cusolver.sorgqr_bufferSize
orgqr = cusolver.sorgqr
elif dtype == 'd':
orgqr_bufferSize = cusolver.dorgqr_bufferSize
orgqr = cusolver.dorgqr
elif dtype == 'F':
orgqr_bufferSize = cusolver.cungqr_bufferSize
orgqr = cusolver.cungqr
elif dtype == 'D':
orgqr_bufferSize = cusolver.zungqr_bufferSize
orgqr = cusolver.zungqr
buffersize = orgqr_bufferSize(handle, m, mc, mn, q.data.ptr, m,
tau.data.ptr)
workspace = cupy.empty(buffersize, dtype=dtype)
orgqr(handle, m, mc, mn, q.data.ptr, m, tau.data.ptr,
workspace.data.ptr, buffersize, dev_info.data.ptr)
q = q[:mc].transpose()
r = x[:, :mc].transpose()
return q, util._triu(r)
def batched_gesv(a, b):
"""Solves multiple linear matrix equations using cublas<t>getr[fs]Batched().
Computes the solution to system of linear equation ``ax = b``.
Args:
a (cupy.ndarray): The matrix with dimension ``(..., M, M)``.
b (cupy.ndarray): The matrix with dimension ``(..., M)`` or
``(..., M, K)``.
Returns:
cupy.ndarray:
The matrix with dimension ``(..., M)`` or ``(..., M, K)``.
"""
util._assert_cupy_array(a, b)
util._assert_nd_squareness(a)
if not ((a.ndim == b.ndim or a.ndim == b.ndim + 1)
and a.shape[:-1] == b.shape[:a.ndim - 1]):
raise ValueError(
'a must have (..., M, M) shape and b must have (..., M) '
'or (..., M, K)')
dtype = numpy.promote_types(a.dtype.char, 'f')
if dtype == 'f':
t = 's'
elif dtype == 'd':
t = 'd'
elif dtype == 'F':
t = 'c'
elif dtype == 'D':
t = 'z'
else:
raise TypeError('invalid dtype')
getrf = getattr(cublas, t + 'getrfBatched')
getrs = getattr(cublas, t + 'getrsBatched')
bs = numpy.prod(a.shape[:-2]) if a.ndim > 2 else 1
n = a.shape[-1]
nrhs = b.shape[-1] if a.ndim == b.ndim else 1
b_shape = b.shape
a_data_ptr = a.data.ptr
b_data_ptr = b.data.ptr
a = cupy.ascontiguousarray(a.reshape(bs, n, n).transpose(0, 2, 1),
dtype=dtype)
b = cupy.ascontiguousarray(b.reshape(bs, n, nrhs).transpose(0, 2, 1),
dtype=dtype)
if a.data.ptr == a_data_ptr:
a = a.copy()
if b.data.ptr == b_data_ptr:
b = b.copy()
if n > get_batched_gesv_limit():
warnings.warn('The matrix size ({}) exceeds the set limit ({})'.format(
n, get_batched_gesv_limit()))
handle = device.get_cublas_handle()
lda = n
a_step = lda * n * a.itemsize
a_array = cupy.arange(a.data.ptr,
a.data.ptr + a_step * bs,
a_step,
dtype=cupy.uintp)
ldb = n
b_step = ldb * nrhs * b.itemsize
b_array = cupy.arange(b.data.ptr,
b.data.ptr + b_step * bs,
b_step,
dtype=cupy.uintp)
pivot = cupy.empty((bs, n), dtype=numpy.int32)
dinfo = cupy.empty((bs, ), dtype=numpy.int32)
info = numpy.empty((1, ), dtype=numpy.int32)
# LU factorization (A = L * U)
getrf(handle, n, a_array.data.ptr, lda, pivot.data.ptr, dinfo.data.ptr, bs)
util._check_cublas_info_array_if_synchronization_allowed(getrf, dinfo)
# Solves Ax = b
getrs(handle, cublas.CUBLAS_OP_N, n, nrhs, a_array.data.ptr, lda,
pivot.data.ptr, b_array.data.ptr, ldb, info.ctypes.data, bs)
if info[0] != 0:
msg = 'Error reported by {} in cuBLAS. '.format(getrs.__name__)
if info[0] < 0:
msg += 'The {}-th parameter had an illegal value.'.format(-info[0])
raise linalg.LinAlgError(msg)
return b.transpose(0, 2, 1).reshape(b_shape)
def lu_solve(lu_and_piv, b, trans=0, overwrite_b=False, check_finite=True):
"""Solve an equation system, ``a * x = b``, given the LU factorization of ``a``
Args:
lu_and_piv (tuple): LU factorization of matrix ``a`` (``(M, M)``)
together with pivot indices.
b (cupy.ndarray): The matrix with dimension ``(M,)`` or
``(M, N)``.
trans ({0, 1, 2}): Type of system to solve:
======== =========
trans system
======== =========
0 a x = b
1 a^T x = b
2 a^H x = b
======== =========
overwrite_b (bool): Allow overwriting data in b (may enhance
performance)
check_finite (bool): Whether to check that the input matrices contain
only finite numbers. Disabling may give a performance gain, but may
result in problems (crashes, non-termination) if the inputs do
contain infinities or NaNs.
Returns:
cupy.ndarray:
The matrix with dimension ``(M,)`` or ``(M, N)``.
.. seealso:: :func:`scipy.linalg.lu_solve`
"""
(lu, ipiv) = lu_and_piv
util._assert_cupy_array(lu)
util._assert_rank2(lu)
util._assert_nd_squareness(lu)
m = lu.shape[0]
if m != b.shape[0]:
raise ValueError('incompatible dimensions.')
dtype = lu.dtype
if dtype.char == 'f':
getrs = cusolver.sgetrs
elif dtype.char == 'd':
getrs = cusolver.dgetrs
else:
raise NotImplementedError('Only float32 and float64 are supported.')
if trans == 0:
trans = cublas.CUBLAS_OP_N
elif trans == 1:
trans = cublas.CUBLAS_OP_T
elif trans == 2:
trans = cublas.CUBLAS_OP_C
else:
raise ValueError('unknown trans')
lu = lu.astype(dtype, order='F', copy=False)
ipiv = ipiv.astype(ipiv.dtype, order='F', copy=True)
# cuSolver uses 1-origin while SciPy uses 0-origin
ipiv += 1
b = b.astype(dtype, order='F', copy=(not overwrite_b))
if check_finite:
if lu.dtype.kind == 'f' and not cupy.isfinite(lu).all():
raise ValueError(
'array must not contain infs or NaNs.\n'
'Note that when a singular matrix is given, unlike '
'scipy.linalg.lu_factor, cupyx.scipy.linalg.lu_factor '
'returns an array containing NaN.')
if b.dtype.kind == 'f' and not cupy.isfinite(b).all():
raise ValueError('array must not contain infs or NaNs')
n = 1 if b.ndim == 1 else b.shape[1]
cusolver_handle = device.get_cusolver_handle()
dev_info = cupy.empty(1, dtype=numpy.int32)
# solve for the inverse
getrs(cusolver_handle, trans, m, n, lu.data.ptr, m, ipiv.data.ptr,
b.data.ptr, m, dev_info.data.ptr)
if dev_info[0] < 0:
raise ValueError('illegal value in %d-th argument of '
'internal getrs (lu_solve)' % -dev_info[0])
return b
def svd(a, full_matrices=True, compute_uv=True):
"""Singular Value Decomposition.
Factorizes the matrix ``a`` as ``u * np.diag(s) * v``, where ``u`` and
``v`` are unitary and ``s`` is an one-dimensional array of ``a``'s
singular values.
Args:
a (cupy.ndarray): The input matrix with dimension ``(M, N)``.
full_matrices (bool): If True, it returns u and v with dimensions
``(M, M)`` and ``(N, N)``. Otherwise, the dimensions of u and v
are respectively ``(M, K)`` and ``(K, N)``, where
``K = min(M, N)``.
compute_uv (bool): If ``False``, it only returns singular values.
Returns:
tuple of :class:`cupy.ndarray`:
A tuple of ``(u, s, v)`` such that ``a = u * np.diag(s) * v``.
.. seealso:: :func:`numpy.linalg.svd`
"""
# TODO(Saito): Current implementation only accepts two-dimensional arrays
util._assert_cupy_array(a)
util._assert_rank2(a)
# Cast to float32 or float64
a_dtype = numpy.find_common_type((a.dtype.char, 'f'), ()).char
if a_dtype == 'f':
s_dtype = 'f'
elif a_dtype == 'd':
s_dtype = 'd'
elif a_dtype == 'F':
s_dtype = 'f'
else: # a_dtype == 'D':
a_dtype = 'D'
s_dtype = 'd'
# Remark 1: gesvd only supports m >= n (WHAT?)
# Remark 2: gesvd only supports jobu = 'A' and jobvt = 'A'
# Remark 3: gesvd returns matrix U and V^H
# Remark 4: Remark 2 is removed since cuda 8.0 (new!)
n, m = a.shape
# `a` must be copied because xgesvd destroys the matrix
if m >= n:
x = a.astype(a_dtype, order='C', copy=True)
trans_flag = False
else:
m, n = a.shape
x = a.transpose().astype(a_dtype, order='C', copy=True)
trans_flag = True
mn = min(m, n)
if compute_uv:
if full_matrices:
u = cupy.empty((m, m), dtype=a_dtype)
vt = cupy.empty((n, n), dtype=a_dtype)
else:
u = cupy.empty((mn, m), dtype=a_dtype)
vt = cupy.empty((mn, n), dtype=a_dtype)
u_ptr, vt_ptr = u.data.ptr, vt.data.ptr
else:
u_ptr, vt_ptr = 0, 0 # Use nullptr
s = cupy.empty(mn, dtype=s_dtype)
handle = device.get_cusolver_handle()
dev_info = cupy.empty(1, dtype=numpy.int32)
if compute_uv:
job = ord('A') if full_matrices else ord('S')
else:
job = ord('N')
if a_dtype == 'f':
buffersize = cusolver.sgesvd_bufferSize(handle, m, n)
workspace = cupy.empty(buffersize, dtype=a_dtype)
cusolver.sgesvd(
handle, job, job, m, n, x.data.ptr, m,
s.data.ptr, u_ptr, m, vt_ptr, n,
workspace.data.ptr, buffersize, 0, dev_info.data.ptr)
elif a_dtype == 'd':
buffersize = cusolver.dgesvd_bufferSize(handle, m, n)
workspace = cupy.empty(buffersize, dtype=a_dtype)
cusolver.dgesvd(
handle, job, job, m, n, x.data.ptr, m,
s.data.ptr, u_ptr, m, vt_ptr, n,
workspace.data.ptr, buffersize, 0, dev_info.data.ptr)
elif a_dtype == 'F':
buffersize = cusolver.cgesvd_bufferSize(handle, m, n)
workspace = cupy.empty(buffersize, dtype=a_dtype)
cusolver.cgesvd(
handle, job, job, m, n, x.data.ptr, m,
s.data.ptr, u_ptr, m, vt_ptr, n,
workspace.data.ptr, buffersize, 0, dev_info.data.ptr)
else: # a_dtype == 'D':
buffersize = cusolver.zgesvd_bufferSize(handle, m, n)
workspace = cupy.empty(buffersize, dtype=a_dtype)
cusolver.zgesvd(
handle, job, job, m, n, x.data.ptr, m,
s.data.ptr, u_ptr, m, vt_ptr, n,
workspace.data.ptr, buffersize, 0, dev_info.data.ptr)
status = int(dev_info[0])
if status > 0:
raise linalg.LinAlgError(
'SVD computation does not converge')
elif status < 0:
raise linalg.LinAlgError(
'Parameter error (maybe caused by a bug in cupy.linalg?)')
# Note that the returned array may need to be transporsed
# depending on the structure of an input
if compute_uv:
if trans_flag:
return u.transpose(), s, vt.transpose()
else:
return vt, s, u
else:
return s
def qr(a, mode='reduced'):
"""QR decomposition.
Decompose a given two-dimensional matrix into ``Q * R``, where ``Q``
is an orthonormal and ``R`` is an upper-triangular matrix.
Args:
a (cupy.ndarray): The input matrix.
mode (str): The mode of decomposition. Currently 'reduced',
'complete', 'r', and 'raw' modes are supported. The default mode
is 'reduced', in which matrix ``A = (M, N)`` is decomposed into
``Q``, ``R`` with dimensions ``(M, K)``, ``(K, N)``, where
``K = min(M, N)``.
Returns:
cupy.ndarray, or tuple of ndarray:
Although the type of returned object depends on the mode,
it returns a tuple of ``(Q, R)`` by default.
For details, please see the document of :func:`numpy.linalg.qr`.
.. warning::
This function calls one or more cuSOLVER routine(s) which may yield
invalid results if input conditions are not met.
To detect these invalid results, you can set the `linalg`
configuration to a value that is not `ignore` in
:func:`cupyx.errstate` or :func:`cupyx.seterr`.
.. seealso:: :func:`numpy.linalg.qr`
"""
# TODO(Saito): Current implementation only accepts two-dimensional arrays
util._assert_cupy_array(a)
util._assert_rank2(a)
if mode not in ('reduced', 'complete', 'r', 'raw'):
if mode in ('f', 'full', 'e', 'economic'):
msg = 'The deprecated mode \'{}\' is not supported'.format(mode)
raise ValueError(msg)
else:
raise ValueError('Unrecognized mode \'{}\''.format(mode))
# support float32, float64, complex64, and complex128
if a.dtype.char in 'fdFD':
dtype = a.dtype.char
else:
dtype = numpy.promote_types(a.dtype.char, 'f').char
m, n = a.shape
mn = min(m, n)
if mn == 0:
if mode == 'reduced':
return cupy.empty((m, 0), dtype), cupy.empty((0, n), dtype)
elif mode == 'complete':
return cupy.identity(m, dtype), cupy.empty((m, n), dtype)
elif mode == 'r':
return cupy.empty((0, n), dtype)
else: # mode == 'raw'
# compatibility with numpy.linalg.qr
dtype = numpy.promote_types(dtype, 'd')
return cupy.empty((n, m), dtype), cupy.empty((0, ), dtype)
x = a.transpose().astype(dtype, order='C', copy=True)
handle = device.get_cusolver_handle()
dev_info = cupy.empty(1, dtype=numpy.int32)
if dtype == 'f':
geqrf_bufferSize = cusolver.sgeqrf_bufferSize
geqrf = cusolver.sgeqrf
elif dtype == 'd':
geqrf_bufferSize = cusolver.dgeqrf_bufferSize
geqrf = cusolver.dgeqrf
elif dtype == 'F':
geqrf_bufferSize = cusolver.cgeqrf_bufferSize
geqrf = cusolver.cgeqrf
elif dtype == 'D':
geqrf_bufferSize = cusolver.zgeqrf_bufferSize
geqrf = cusolver.zgeqrf
else:
msg = ('dtype must be float32, float64, complex64 or complex128'
' (actual: {})'.format(a.dtype))
raise ValueError(msg)
# compute working space of geqrf and solve R
buffersize = geqrf_bufferSize(handle, m, n, x.data.ptr, n)
workspace = cupy.empty(buffersize, dtype=dtype)
tau = cupy.empty(mn, dtype=dtype)
geqrf(handle, m, n, x.data.ptr, m, tau.data.ptr, workspace.data.ptr,
buffersize, dev_info.data.ptr)
cupy.linalg.util._check_cusolver_dev_info_if_synchronization_allowed(
geqrf, dev_info)
if mode == 'r':
r = x[:, :mn].transpose()
return util._triu(r)
if mode == 'raw':
if a.dtype.char == 'f':
# The original numpy.linalg.qr returns float64 in raw mode,
# whereas the cusolver returns float32. We agree that the
# following code would be inappropriate, however, in this time
# we explicitly convert them to float64 for compatibility.
return x.astype(numpy.float64), tau.astype(numpy.float64)
elif a.dtype.char == 'F':
# The same applies to complex64
return x.astype(numpy.complex128), tau.astype(numpy.complex128)
return x, tau
if mode == 'complete' and m > n:
mc = m
q = cupy.empty((m, m), dtype)
else:
mc = mn
q = cupy.empty((n, m), dtype)
q[:n] = x
# compute working space of orgqr and solve Q
if dtype == 'f':
orgqr_bufferSize = cusolver.sorgqr_bufferSize
orgqr = cusolver.sorgqr
elif dtype == 'd':
orgqr_bufferSize = cusolver.dorgqr_bufferSize
orgqr = cusolver.dorgqr
elif dtype == 'F':
orgqr_bufferSize = cusolver.cungqr_bufferSize
orgqr = cusolver.cungqr
elif dtype == 'D':
orgqr_bufferSize = cusolver.zungqr_bufferSize
orgqr = cusolver.zungqr
buffersize = orgqr_bufferSize(handle, m, mc, mn, q.data.ptr, m,
tau.data.ptr)
workspace = cupy.empty(buffersize, dtype=dtype)
orgqr(handle, m, mc, mn, q.data.ptr, m, tau.data.ptr, workspace.data.ptr,
buffersize, dev_info.data.ptr)
cupy.linalg.util._check_cusolver_dev_info_if_synchronization_allowed(
orgqr, dev_info)
q = q[:mc].transpose()
r = x[:, :mc].transpose()
return q, util._triu(r)
def solve_triangular(a,
b,
trans=0,
lower=False,
unit_diagonal=False,
overwrite_b=False,
check_finite=False):
"""Solve the equation a x = b for x, assuming a is a triangular matrix.
Args:
a (cupy.ndarray): The matrix with dimension ``(M, M)``.
b (cupy.ndarray): The matrix with dimension ``(M,)`` or
``(M, N)``.
lower (bool): Use only data contained in the lower triangle of `a`.
Default is to use upper triangle.
trans ({0, 1, 2, 'N', 'T', 'C'}): Type of system to solve:
======== =========
trans system
======== =========
0 or 'N' a x = b
1 or 'T' a^T x = b
2 or 'C' a^H x = b
======== =========
unit_diagonal (bool): If True, diagonal elements of `a` are assumed to
be 1 and will not be referenced.
overwrite_b (bool): Allow overwriting data in b (may enhance
performance)
check_finite (bool): Whether to check that the input matrices contain
only finite numbers. Disabling may give a performance gain, but may
result in problems (crashes, non-termination) if the inputs do
contain infinities or NaNs.
Returns:
cupy.ndarray:
The matrix with dimension ``(M,)`` or ``(M, N)``.
.. seealso:: :func:`scipy.linalg.solve_triangular`
"""
util._assert_cupy_array(a, b)
if len(a.shape) != 2 or a.shape[0] != a.shape[1]:
raise ValueError('expected square matrix')
if len(a) != len(b):
raise ValueError('incompatible dimensions')
# Cast to float32 or float64
if a.dtype.char in 'fd':
dtype = a.dtype
else:
dtype = numpy.find_common_type((a.dtype.char, 'f'), ())
a = cupy.array(a, dtype=dtype, order='F', copy=False)
b = cupy.array(b, dtype=dtype, order='F', copy=(not overwrite_b))
if check_finite:
if a.dtype.kind == 'f' and not cupy.isfinite(a).all():
raise ValueError('array must not contain infs or NaNs')
if b.dtype.kind == 'f' and not cupy.isfinite(b).all():
raise ValueError('array must not contain infs or NaNs')
m, n = (b.size, 1) if b.ndim == 1 else b.shape
cublas_handle = device.get_cublas_handle()
if dtype == 'f':
trsm = cublas.strsm
else: # dtype == 'd'
trsm = cublas.dtrsm
if lower:
uplo = cublas.CUBLAS_FILL_MODE_LOWER
else:
uplo = cublas.CUBLAS_FILL_MODE_UPPER
if trans == 'N':
trans = cublas.CUBLAS_OP_N
elif trans == 'T':
trans = cublas.CUBLAS_OP_T
elif trans == 'C':
trans = cublas.CUBLAS_OP_C
if unit_diagonal:
diag = cublas.CUBLAS_DIAG_UNIT
else:
diag = cublas.CUBLAS_DIAG_NON_UNIT
trsm(cublas_handle, cublas.CUBLAS_SIDE_LEFT, uplo, trans, diag, m, n, 1,
a.data.ptr, m, b.data.ptr, m)
return b
def invh(a):
"""Compute the inverse of a Hermitian matrix.
This function computes a inverse of a real symmetric or complex hermitian
positive-definite matrix using Cholesky factorization. If matrix ``a`` is
not positive definite, Cholesky factorization fails and it raises an error.
Args:
a (cupy.ndarray): Real symmetric or complex hermitian maxtix.
Returns:
cupy.ndarray: The inverse of matrix ``a``.
"""
# to prevent `a` from being overwritten
a = a.copy()
util._assert_cupy_array(a)
util._assert_rank2(a)
util._assert_nd_squareness(a)
# support float32, float64, complex64, and complex128
if a.dtype.char in 'fdFD':
dtype = a.dtype.char
else:
dtype = numpy.promote_types(a.dtype.char, 'f').char
cusolver_handle = device.get_cusolver_handle()
dev_info = cupy.empty(1, dtype=numpy.int32)
if dtype == 'f':
potrf = cusolver.spotrf
potrf_bufferSize = cusolver.spotrf_bufferSize
potrs = cusolver.spotrs
elif dtype == 'd':
potrf = cusolver.dpotrf
potrf_bufferSize = cusolver.dpotrf_bufferSize
potrs = cusolver.dpotrs
elif dtype == 'F':
potrf = cusolver.cpotrf
potrf_bufferSize = cusolver.cpotrf_bufferSize
potrs = cusolver.cpotrs
elif dtype == 'D':
potrf = cusolver.zpotrf
potrf_bufferSize = cusolver.zpotrf_bufferSize
potrs = cusolver.zpotrs
else:
msg = ('dtype must be float32, float64, complex64 or complex128'
' (actual: {})'.format(a.dtype))
raise ValueError(msg)
m = a.shape[0]
uplo = cublas.CUBLAS_FILL_MODE_LOWER
worksize = potrf_bufferSize(cusolver_handle, uplo, m, a.data.ptr, m)
workspace = cupy.empty(worksize, dtype=dtype)
# Cholesky factorization
potrf(cusolver_handle, uplo, m, a.data.ptr, m, workspace.data.ptr,
worksize, dev_info.data.ptr)
info = dev_info[0]
if info != 0:
if info < 0:
msg = '\tThe {}-th parameter is wrong'.format(-info)
else:
msg = ('\tThe leading minor of order {} is not positive definite'
.format(info))
raise RuntimeError('matrix inversion failed at potrf.\n' + msg)
b = cupy.eye(m, dtype=dtype)
# Solve: A * X = B
potrs(cusolver_handle, uplo, m, m, a.data.ptr, m, b.data.ptr, m,
dev_info.data.ptr)
info = dev_info[0]
if info > 0:
assert False, ('Unexpected output returned by potrs (actual: {})'
.format(info))
elif info < 0:
raise RuntimeError('matrix inversion failed at potrs.\n'
'\tThe {}-th parameter is wrong'.format(-info))
return b
def svd(a, full_matrices=True, compute_uv=True):
"""Singular Value Decomposition.
Factorizes the matrix ``a`` as ``u * np.diag(s) * v``, where ``u`` and
``v`` are unitary and ``s`` is an one-dimensional array of ``a``'s
singular values.
Args:
a (cupy.ndarray): The input matrix with dimension ``(M, N)``.
full_matrices (bool): If True, it returns u and v with dimensions
``(M, M)`` and ``(N, N)``. Otherwise, the dimensions of u and v
are respectively ``(M, K)`` and ``(K, N)``, where
``K = min(M, N)``.
compute_uv (bool): If ``False``, it only returns singular values.
Returns:
tuple of :class:`cupy.ndarray`:
A tuple of ``(u, s, v)`` such that ``a = u * np.diag(s) * v``.
.. warning::
This function calls one or more cuSOLVER routine(s) which may yield
invalid results if input conditions are not met.
To detect these invalid results, you can set the `linalg`
configuration to a value that is not `ignore` in
:func:`cupyx.errstate` or :func:`cupyx.seterr`.
.. seealso:: :func:`numpy.linalg.svd`
"""
# TODO(Saito): Current implementation only accepts two-dimensional arrays
util._assert_cupy_array(a)
util._assert_rank2(a)
# Cast to float32 or float64
a_dtype = numpy.promote_types(a.dtype.char, 'f').char
if a_dtype == 'f':
s_dtype = 'f'
elif a_dtype == 'd':
s_dtype = 'd'
elif a_dtype == 'F':
s_dtype = 'f'
else: # a_dtype == 'D':
a_dtype = 'D'
s_dtype = 'd'
# Remark 1: gesvd only supports m >= n (WHAT?)
# Remark 2: gesvd only supports jobu = 'A' and jobvt = 'A'
# Remark 3: gesvd returns matrix U and V^H
# Remark 4: Remark 2 is removed since cuda 8.0 (new!)
n, m = a.shape
# `a` must be copied because xgesvd destroys the matrix
if m >= n:
x = a.astype(a_dtype, order='C', copy=True)
trans_flag = False
else:
m, n = a.shape
x = a.transpose().astype(a_dtype, order='C', copy=True)
trans_flag = True
mn = min(m, n)
if compute_uv:
if full_matrices:
u = cupy.empty((m, m), dtype=a_dtype)
vt = cupy.empty((n, n), dtype=a_dtype)
else:
u = cupy.empty((mn, m), dtype=a_dtype)
vt = cupy.empty((mn, n), dtype=a_dtype)
u_ptr, vt_ptr = u.data.ptr, vt.data.ptr
else:
u_ptr, vt_ptr = 0, 0 # Use nullptr
s = cupy.empty(mn, dtype=s_dtype)
handle = device.get_cusolver_handle()
dev_info = cupy.empty(1, dtype=numpy.int32)
if compute_uv:
job = ord('A') if full_matrices else ord('S')
else:
job = ord('N')
if a_dtype == 'f':
gesvd = cusolver.sgesvd
gesvd_bufferSize = cusolver.sgesvd_bufferSize
elif a_dtype == 'd':
gesvd = cusolver.dgesvd
gesvd_bufferSize = cusolver.dgesvd_bufferSize
elif a_dtype == 'F':
gesvd = cusolver.cgesvd
gesvd_bufferSize = cusolver.cgesvd_bufferSize
else: # a_dtype == 'D':
gesvd = cusolver.zgesvd
gesvd_bufferSize = cusolver.zgesvd_bufferSize
buffersize = gesvd_bufferSize(handle, m, n)
workspace = cupy.empty(buffersize, dtype=a_dtype)
gesvd(handle, job, job, m, n, x.data.ptr, m, s.data.ptr, u_ptr, m, vt_ptr,
n, workspace.data.ptr, buffersize, 0, dev_info.data.ptr)
cupy.linalg.util._check_cusolver_dev_info_if_synchronization_allowed(
gesvd, dev_info)
# Note that the returned array may need to be transporsed
# depending on the structure of an input
if compute_uv:
if trans_flag:
return u.transpose(), s, vt.transpose()
else:
return vt, s, u
else:
return s
def _batched_inv(a):
assert (a.ndim >= 3)
util._assert_cupy_array(a)
util._assert_nd_squareness(a)
if a.dtype == cupy.float32:
getrf = cupy.cuda.cublas.sgetrfBatched
getri = cupy.cuda.cublas.sgetriBatched
elif a.dtype == cupy.float64:
getrf = cupy.cuda.cublas.dgetrfBatched
getri = cupy.cuda.cublas.dgetriBatched
elif a.dtype == cupy.complex64:
getrf = cupy.cuda.cublas.cgetrfBatched
getri = cupy.cuda.cublas.cgetriBatched
elif a.dtype == cupy.complex128:
getrf = cupy.cuda.cublas.zgetrfBatched
getri = cupy.cuda.cublas.zgetriBatched
else:
msg = ('dtype must be float32, float64, complex64 or complex128'
' (actual: {})'.format(a.dtype))
raise ValueError(msg)
if 0 in a.shape:
return cupy.empty_like(a)
a_shape = a.shape
# copy is necessary to present `a` to be overwritten.
a = a.copy().reshape(-1, a_shape[-2], a_shape[-1])
handle = device.get_cublas_handle()
batch_size = a.shape[0]
n = a.shape[1]
lda = n
step = n * lda * a.itemsize
start = a.data.ptr
stop = start + step * batch_size
a_array = cupy.arange(start, stop, step, dtype=cupy.uintp)
pivot_array = cupy.empty((batch_size, n), dtype=cupy.int32)
info_array = cupy.empty((batch_size, ), dtype=cupy.int32)
getrf(handle, n, a_array.data.ptr, lda, pivot_array.data.ptr,
info_array.data.ptr, batch_size)
err = False
err_detail = ''
for i in range(batch_size):
info = info_array[i]
if info < 0:
err = True
err_detail += ('\tmatrix[{}]: illegal value at {}-the parameter.'
'\n'.format(i, -info))
if info > 0:
err = True
err_detail += '\tmatrix[{}]: matrix is singular.\n'.format(i)
if err:
raise RuntimeError('matrix inversion failed at getrf.\n' + err_detail)
c = cupy.empty_like(a)
ldc = lda
step = n * ldc * c.itemsize
start = c.data.ptr
stop = start + step * batch_size
c_array = cupy.arange(start, stop, step, dtype=cupy.uintp)
getri(handle, n, a_array.data.ptr, lda, pivot_array.data.ptr,
c_array.data.ptr, ldc, info_array.data.ptr, batch_size)
for i in range(batch_size):
info = info_array[i]
if info > 0:
err = True
err_detail += '\tmatrix[{}]: matrix is singular.\n'.format(i)
if err:
raise RuntimeError('matrix inversion failed at getri.\n' + err_detail)
return c.reshape(a_shape)
def inv(a):
"""Computes the inverse of a matrix.
This function computes matrix ``a_inv`` from n-dimensional regular matrix
``a`` such that ``dot(a, a_inv) == eye(n)``.
Args:
a (cupy.ndarray): The regular matrix
Returns:
cupy.ndarray: The inverse of a matrix.
.. seealso:: :func:`numpy.linalg.inv`
"""
if a.ndim >= 3:
return _batched_inv(a)
# to prevent `a` to be overwritten
a = a.copy()
util._assert_cupy_array(a)
util._assert_rank2(a)
util._assert_nd_squareness(a)
# support float32, float64, complex64, and complex128
if a.dtype.char in 'fdFD':
dtype = a.dtype.char
else:
dtype = numpy.find_common_type((a.dtype.char, 'f'), ()).char
cusolver_handle = device.get_cusolver_handle()
dev_info = cupy.empty(1, dtype=numpy.int32)
ipiv = cupy.empty((a.shape[0], 1), dtype=numpy.intc)
if dtype == 'f':
getrf = cusolver.sgetrf
getrf_bufferSize = cusolver.sgetrf_bufferSize
getrs = cusolver.sgetrs
elif dtype == 'd':
getrf = cusolver.dgetrf
getrf_bufferSize = cusolver.dgetrf_bufferSize
getrs = cusolver.dgetrs
elif dtype == 'F':
getrf = cusolver.cgetrf
getrf_bufferSize = cusolver.cgetrf_bufferSize
getrs = cusolver.cgetrs
elif dtype == 'D':
getrf = cusolver.zgetrf
getrf_bufferSize = cusolver.zgetrf_bufferSize
getrs = cusolver.zgetrs
else:
msg = ('dtype must be float32, float64, complex64 or complex128'
' (actual: {})'.format(a.dtype))
raise ValueError(msg)
m = a.shape[0]
buffersize = getrf_bufferSize(cusolver_handle, m, m, a.data.ptr, m)
workspace = cupy.empty(buffersize, dtype=dtype)
# LU factorization
getrf(cusolver_handle, m, m, a.data.ptr, m, workspace.data.ptr,
ipiv.data.ptr, dev_info.data.ptr)
b = cupy.eye(m, dtype=dtype)
# solve for the inverse
getrs(cusolver_handle, 0, m, m, a.data.ptr, m, ipiv.data.ptr, b.data.ptr,
m, dev_info.data.ptr)
return b
def solve(a, b):
'''Solves a linear matrix equation.
It computes the exact solution of ``x`` in ``ax = b``,
where ``a`` is a square and full rank matrix.
Args:
a (cupy.ndarray): The matrix with dimension ``(M, M)``
b (cupy.ndarray): The vector with ``M`` elements, or
the matrix with dimension ``(M, K)``
Returns:
cupy.ndarray:
The vector with ``M`` elements, or the matrix with dimension
``(M, K)``.
.. seealso:: :func:`numpy.linalg.solve`
'''
# NOTE: Since cusolver in CUDA 8.0 does not support gesv,
# we manually solve a linear system with QR decomposition.
# For details, please see the following:
# https://docs.nvidia.com/cuda/cusolver/index.html#qr_examples
if not cuda.cusolver_enabled:
raise RuntimeError('Current cupy only supports cusolver in CUDA 8.0')
# TODO(Saito): Current implementation only accepts two-dimensional arrays
util._assert_cupy_array(a, b)
util._assert_rank2(a)
util._assert_nd_squareness(a)
if 2 < b.ndim:
raise linalg.LinAlgError('{}-dimensional array given. Array must be '
'one or two-dimensional'.format(b.ndim))
if len(a) != len(b):
raise linalg.LinAlgError('The number of rows of array a must be '
'the same as that of array b')
# Cast to float32 or float64
if a.dtype.char == 'f' or a.dtype.char == 'd':
dtype = a.dtype.char
else:
dtype = numpy.find_common_type((a.dtype.char, 'f'), ()).char
m, k = (b.size, 1) if b.ndim == 1 else b.shape
a = a.transpose().astype(dtype, order='C', copy=True)
b = b.transpose().astype(dtype, order='C', copy=True)
cusolver_handle = device.get_cusolver_handle()
cublas_handle = device.get_cublas_handle()
dev_info = cupy.empty(1, dtype=numpy.int32)
if dtype == 'f':
geqrf = cusolver.sgeqrf
geqrf_bufferSize = cusolver.sgeqrf_bufferSize
ormqr = cusolver.sormqr
trsm = cublas.strsm
else: # dtype == 'd'
geqrf = cusolver.dgeqrf
geqrf_bufferSize = cusolver.dgeqrf_bufferSize
ormqr = cusolver.dormqr
trsm = cublas.dtrsm
# 1. QR decomposition (A = Q * R)
buffersize = geqrf_bufferSize(cusolver_handle, m, m, a.data.ptr, m)
workspace = cupy.empty(buffersize, dtype=dtype)
tau = cupy.empty(m, dtype=dtype)
geqrf(cusolver_handle, m, m, a.data.ptr, m, tau.data.ptr,
workspace.data.ptr, buffersize, dev_info.data.ptr)
_check_status(dev_info)
# 2. ormqr (Q^T * B)
ormqr(cusolver_handle, cublas.CUBLAS_SIDE_LEFT, cublas.CUBLAS_OP_T, m, k,
m, a.data.ptr, m, tau.data.ptr, b.data.ptr, m, workspace.data.ptr,
buffersize, dev_info.data.ptr)
_check_status(dev_info)
# 3. trsm (X = R^{-1} * (Q^T * B))
trsm(cublas_handle, cublas.CUBLAS_SIDE_LEFT, cublas.CUBLAS_FILL_MODE_UPPER,
cublas.CUBLAS_OP_N, cublas.CUBLAS_DIAG_NON_UNIT, m, k, 1, a.data.ptr,
m, b.data.ptr, m)
return b.transpose()
