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``. """ _util._assert_cupy_array(a) _util._assert_nd_squareness(a) # TODO: Remove this assert once cusolver supports nrhs > 1 for potrsBatched _util._assert_rank2(a) n = a.shape[-1] identity_matrix = cupy.eye(n, dtype=a.dtype) b = cupy.empty(a.shape, a.dtype) b[...] = identity_matrix return lapack.posv(a, b)
def _lu_factor(a, overwrite_a=False, check_finite=True): a = cupy.asarray(a) _util._assert_rank2(a) dtype = a.dtype if dtype.char == 'f': getrf = cusolver.sgetrf getrf_bufferSize = cusolver.sgetrf_bufferSize elif dtype.char == 'd': getrf = cusolver.dgetrf getrf_bufferSize = cusolver.dgetrf_bufferSize elif dtype.char == 'F': getrf = cusolver.cgetrf getrf_bufferSize = cusolver.cgetrf_bufferSize elif dtype.char == 'D': getrf = cusolver.zgetrf getrf_bufferSize = cusolver.zgetrf_bufferSize else: msg = 'Only float32, float64, complex64 and complex128 are supported.' raise NotImplementedError(msg) a = a.astype(dtype, order='F', copy=(not overwrite_a)) if check_finite: if a.dtype.kind == 'f' and not cupy.isfinite(a).all(): raise ValueError('array must not contain infs or NaNs') cusolver_handle = device.get_cusolver_handle() dev_info = cupy.empty(1, dtype=numpy.int32) m, n = a.shape ipiv = cupy.empty((min(m, n), ), dtype=numpy.intc) buffersize = getrf_bufferSize(cusolver_handle, m, n, a.data.ptr, m) workspace = cupy.empty(buffersize, dtype=dtype) # LU factorization getrf(cusolver_handle, m, n, a.data.ptr, m, workspace.data.ptr, ipiv.data.ptr, dev_info.data.ptr) if dev_info[0] < 0: raise ValueError('illegal value in %d-th argument of ' 'internal getrf (lu_factor)' % -dev_info[0]) elif dev_info[0] > 0: warn('Diagonal number %d is exactly zero. Singular matrix.' % dev_info[0], RuntimeWarning, stacklevel=2) # cuSolver uses 1-origin while SciPy uses 0-origin ipiv -= 1 return (a, ipiv)
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) _util._assert_cupy_array(a) _util._assert_rank2(a) _util._assert_nd_squareness(a) dtype, out_dtype = _util.linalg_common_type(a) order = 'F' if a._f_contiguous else 'C' # prevent 'a' to be overwritten a = a.astype(dtype, copy=True, order=order) b = cupy.eye(a.shape[0], dtype=dtype, order=order) if order == 'F': cupyx.lapack.gesv(a, b) else: cupyx.lapack.gesv(a.T, b.T) return b.astype(out_dtype, copy=False)
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 lstsq(a, b, rcond='warn'): """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` """ if rcond == 'warn': warnings.warn( '`rcond` parameter will change to the default of ' 'machine precision times ``max(M, N)`` where M and N ' 'are the input matrix dimensions.\n' 'To use the future default and silence this warning ' 'we advise to pass `rcond=None`, to keep using the old, ' 'explicitly pass `rcond=-1`.', FutureWarning) rcond = -1 _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, vh = cupy.linalg.svd(a, full_matrices=False) if rcond is None: rcond = numpy.finfo(s.dtype).eps * max(m, n) elif rcond <= 0 or rcond >= 1: # some doc of gelss/gelsd says "rcond < 0", but it's not true! rcond = numpy.finfo(s.dtype).eps # 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(dtype=numpy.int32) # Solve the least-squares solution # x = vh.T.conj() @ diag(s1) @ u.T.conj() @ b z = (cupy.dot(b.T, u.conj()) * s1).T x = cupy.dot(vh.T.conj(), z) # Calculate squared Euclidean 2-norm for each column in b - a*x if m <= n or rank != n: resids = cupy.empty((0,), dtype=s.dtype) else: e = b - a.dot(x) resids = cupy.atleast_1d(_nrm2_last_axis(e.T)) 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.promote_types(a.dtype.char, 'f') 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 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 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``. """ _util._assert_cupy_array(a) _util._assert_nd_squareness(a) # TODO: Remove this assert once cusolver supports nrhs > 1 for potrsBatched _util._assert_rank2(a) if a.ndim > 2: return _batched_invh(a) # to prevent `a` from being overwritten a = a.copy() # 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 if m == 0 or n == 0: s = cupy.empty((0, ), s_dtype) if compute_uv: if full_matrices: u = cupy.eye(n, dtype=a_dtype) vt = cupy.eye(m, dtype=a_dtype) else: u = cupy.empty((n, 0), dtype=a_dtype) vt = cupy.empty((0, m), dtype=a_dtype) return u, s, vt else: return s # `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 k = n # = min(m, n) where m >= n is ensured above if compute_uv: if full_matrices: u = cupy.empty((m, m), dtype=a_dtype) vt = x[:, :n] job_u = ord('A') job_vt = ord('O') else: u = x vt = cupy.empty((k, n), dtype=a_dtype) job_u = ord('O') job_vt = ord('S') u_ptr, vt_ptr = u.data.ptr, vt.data.ptr else: u_ptr, vt_ptr = 0, 0 # Use nullptr job_u = ord('N') job_vt = ord('N') s = cupy.empty(k, dtype=s_dtype) handle = device.get_cusolver_handle() dev_info = cupy.empty(1, dtype=numpy.int32) 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_u, job_vt, 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 transposed # 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 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 elif dtype.char == 'F': getrs = cusolver.cgetrs elif dtype.char == 'D': getrs = cusolver.zgetrs else: msg = 'Only float32, float64, complex64 and complex128 are supported.' raise NotImplementedError(msg) 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