# Copyright (c) 2020 PaddlePaddle Authors. All Rights Reserved. # # Licensed under the Apache License, Version 2.0 (the "License"); # you may not use this file except in compliance with the License. # You may obtain a copy of the License at # # http://www.apache.org/licenses/LICENSE-2.0 # # Unless required by applicable law or agreed to in writing, software # distributed under the License is distributed on an "AS IS" BASIS, # WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. # See the License for the specific language governing permissions and # limitations under the License. import numpy as np from ..fluid.layer_helper import LayerHelper from ..fluid.framework import in_dygraph_mode, _varbase_creator, Variable, _dygraph_tracer from ..fluid.data_feeder import check_variable_and_dtype, check_type, check_dtype from ..fluid.layers import transpose, cast # noqa: F401 from ..fluid import layers import paddle from paddle.common_ops_import import core from paddle.common_ops_import import VarDesc from paddle import _C_ops import paddle __all__ = [] def matmul(x, y, transpose_x=False, transpose_y=False, name=None): """ Applies matrix multiplication to two tensors. `matmul` follows the complete broadcast rules, and its behavior is consistent with `np.matmul`. Currently, the input tensors' number of dimensions can be any, `matmul` can be used to achieve the `dot`, `matmul` and `batchmatmul`. The actual behavior depends on the shapes of :math:`x`, :math:`y` and the flag values of :attr:`transpose_x`, :attr:`transpose_y`. Specifically: - If a transpose flag is specified, the last two dimensions of the tensor are transposed. If the tensor is ndim-1 of shape, the transpose is invalid. If the tensor is ndim-1 of shape :math:`[D]`, then for :math:`x` it is treated as :math:`[1, D]`, whereas for :math:`y` it is the opposite: It is treated as :math:`[D, 1]`. The multiplication behavior depends on the dimensions of `x` and `y`. Specifically: - If both tensors are 1-dimensional, the dot product result is obtained. - If both tensors are 2-dimensional, the matrix-matrix product is obtained. - If the `x` is 1-dimensional and the `y` is 2-dimensional, a `1` is prepended to its dimension in order to conduct the matrix multiply. After the matrix multiply, the prepended dimension is removed. - If the `x` is 2-dimensional and `y` is 1-dimensional, the matrix-vector product is obtained. - If both arguments are at least 1-dimensional and at least one argument is N-dimensional (where N > 2), then a batched matrix multiply is obtained. If the first argument is 1-dimensional, a 1 is prepended to its dimension in order to conduct the batched matrix multiply and removed after. If the second argument is 1-dimensional, a 1 is appended to its dimension for the purpose of the batched matrix multiple and removed after. The non-matrix (exclude the last two dimensions) dimensions are broadcasted according the broadcast rule. For example, if input is a (j, 1, n, m) tensor and the other is a (k, m, p) tensor, out will be a (j, k, n, p) tensor. Args: x (Tensor): The input tensor which is a Tensor. y (Tensor): The input tensor which is a Tensor. transpose_x (bool): Whether to transpose :math:`x` before multiplication. transpose_y (bool): Whether to transpose :math:`y` before multiplication. name(str|None): A name for this layer(optional). If set None, the layer will be named automatically. Returns: Tensor: The output Tensor. Examples: .. code-block:: python import paddle import numpy as np # vector * vector x_data = np.random.random([10]).astype(np.float32) y_data = np.random.random([10]).astype(np.float32) x = paddle.to_tensor(x_data) y = paddle.to_tensor(y_data) z = paddle.matmul(x, y) print(z.numpy().shape) # [1] # matrix * vector x_data = np.random.random([10, 5]).astype(np.float32) y_data = np.random.random([5]).astype(np.float32) x = paddle.to_tensor(x_data) y = paddle.to_tensor(y_data) z = paddle.matmul(x, y) print(z.numpy().shape) # [10] # batched matrix * broadcasted vector x_data = np.random.random([10, 5, 2]).astype(np.float32) y_data = np.random.random([2]).astype(np.float32) x = paddle.to_tensor(x_data) y = paddle.to_tensor(y_data) z = paddle.matmul(x, y) print(z.numpy().shape) # [10, 5] # batched matrix * batched matrix x_data = np.random.random([10, 5, 2]).astype(np.float32) y_data = np.random.random([10, 2, 5]).astype(np.float32) x = paddle.to_tensor(x_data) y = paddle.to_tensor(y_data) z = paddle.matmul(x, y) print(z.numpy().shape) # [10, 5, 5] # batched matrix * broadcasted matrix x_data = np.random.random([10, 1, 5, 2]).astype(np.float32) y_data = np.random.random([1, 3, 2, 5]).astype(np.float32) x = paddle.to_tensor(x_data) y = paddle.to_tensor(y_data) z = paddle.matmul(x, y) print(z.numpy().shape) # [10, 3, 5, 5] """ op_type = 'matmul_v2' if in_dygraph_mode(): op = getattr(_C_ops, op_type) return op(x, y, 'trans_x', transpose_x, 'trans_y', transpose_y) attrs = { 'trans_x': transpose_x, 'trans_y': transpose_y, } def __check_input(x, y): var_names = {'x': x, 'y': y} for name, val in var_names.items(): check_variable_and_dtype( val, name, ['float16', 'float32', 'float64'], 'matmul') __check_input(x, y) helper = LayerHelper('matmul_v2', **locals()) out = helper.create_variable_for_type_inference(dtype=x.dtype) helper.append_op( type='matmul_v2', inputs={'X': x, 'Y': y}, outputs={'Out': out}, attrs=attrs) return out def norm(x, p='fro', axis=None, keepdim=False, name=None): """ Returns the matrix norm (Frobenius) or vector norm (the 1-norm, the Euclidean or 2-norm, and in general the p-norm for p > 0) of a given tensor. .. note:: This norm API is different from `numpy.linalg.norm`. This api supports high-order input tensors (rank >= 3), and certain axis need to be pointed out to calculate the norm. But `numpy.linalg.norm` only supports 1-D vector or 2-D matrix as input tensor. For p-order matrix norm, this api actually treats matrix as a flattened vector to calculate the vector norm, NOT REAL MATRIX NORM. Args: x (Tensor): The input tensor could be N-D tensor, and the input data type could be float32 or float64. p (float|string, optional): Order of the norm. Supported values are `fro`, `0`, `1`, `2`, `inf`, `-inf` and any positive real number yielding the corresponding p-norm. Not supported: ord < 0 and nuclear norm. Default value is `fro`. axis (int|list|tuple, optional): The axis on which to apply norm operation. If axis is int or list(int)/tuple(int) with only one element, the vector norm is computed over the axis. If `axis < 0`, the dimension to norm operation is rank(input) + axis. If axis is a list(int)/tuple(int) with two elements, the matrix norm is computed over the axis. Defalut value is `None`. keepdim (bool, optional): Whether to reserve the reduced dimension in the output Tensor. The result tensor will have fewer dimension than the :attr:`input` unless :attr:`keepdim` is true, default value is False. name (str, optional): The default value is None. Normally there is no need for user to set this property. For more information, please refer to :ref:`api_guide_Name`. Returns: Tensor: results of norm operation on the specified axis of input tensor, it's data type is the same as input's Tensor. Examples: .. code-block:: python import paddle import numpy as np shape=[2, 3, 4] np_input = np.arange(24).astype('float32') - 12 np_input = np_input.reshape(shape) x = paddle.to_tensor(np_input) #[[[-12. -11. -10. -9.] [ -8. -7. -6. -5.] [ -4. -3. -2. -1.]] # [[ 0. 1. 2. 3.] [ 4. 5. 6. 7.] [ 8. 9. 10. 11.]]] # compute frobenius norm along last two dimensions. out_fro = paddle.linalg.norm(x, p='fro', axis=[0,1]) # out_fro.numpy() [17.435596 16.911535 16.7332 16.911535] # compute 2-order vector norm along last dimension. out_pnorm = paddle.linalg.norm(x, p=2, axis=-1) #out_pnorm.numpy(): [[21.118711 13.190906 5.477226] # [ 3.7416575 11.224972 19.131126]] # compute 2-order norm along [0,1] dimension. out_pnorm = paddle.linalg.norm(x, p=2, axis=[0,1]) #out_pnorm.numpy(): [17.435596 16.911535 16.7332 16.911535] # compute inf-order norm out_pnorm = paddle.linalg.norm(x, p=np.inf) #out_pnorm.numpy() = [12.] out_pnorm = paddle.linalg.norm(x, p=np.inf, axis=0) #out_pnorm.numpy(): [[12. 11. 10. 9.] [8. 7. 6. 7.] [8. 9. 10. 11.]] # compute -inf-order norm out_pnorm = paddle.linalg.norm(x, p=-np.inf) #out_pnorm.numpy(): [0.] out_pnorm = paddle.linalg.norm(x, p=-np.inf, axis=0) #out_pnorm.numpy(): [[0. 1. 2. 3.] [4. 5. 6. 5.] [4. 3. 2. 1.]] """ def frobenius_norm(input, dim=None, keepdim=False, name=None): """ The frobenius norm OP is to calculate the frobenius norm of certain two dimensions of Tensor `input`. Args: input (Variable): Tensor, data type float32, float64. dim (list, optional): None for last two dimensions. keepdim (bool, optional): Whether keep the dimensions as the `input`, Default False. """ if dim is not None and not (isinstance(dim, list) and len(dim) == 2): raise ValueError( "The dim of frobenius norm op should be None or two elements list!" ) if in_dygraph_mode(): if dim is None: return _C_ops.frobenius_norm(input, 'keep_dim', keepdim, 'reduce_all', True) return _C_ops.frobenius_norm(input, 'dim', dim, 'keep_dim', keepdim, 'reduce_all', False) attrs = {'dim': dim, 'keep_dim': keepdim, 'reduce_all': False} if dim is None: attrs['reduce_all'] = True check_variable_and_dtype(input, 'input', ['float32', 'float64'], 'frobenius_norm') helper = LayerHelper('frobenius_norm', **locals()) out = helper.create_variable_for_type_inference( dtype=helper.input_dtype()) helper.append_op( type='frobenius_norm', inputs={'X': input}, outputs={'Out': out}, attrs=attrs) return out def vector_norm(input, porder=None, axis=None, keepdim=False, asvector=False, name=None): """ Calculate the p-order vector norm for certain dimension of Tensor `input`. Args: input (Variable): Tensor, data type float32, float64. porder (float, optional): None for porder=2.0. axis (int, optional): None for last dimension. keepdim (bool, optional): Whether keep the dimensions as the `input`, Default False. """ if in_dygraph_mode(): if axis is None: axis = -1 return _C_ops.p_norm(input, 'porder', porder, 'axis', axis, 'keepdim', keepdim, 'asvector', asvector) if porder is not None: check_type(porder, 'porder', (float, int), 'p_norm') if axis is not None: check_type(axis, 'axis', (int), 'p_norm') check_variable_and_dtype(input, 'input', ['float32', 'float64'], 'p_norm') attrs = { 'axis': axis if axis is not None else -1, 'porder': float(porder) if porder is not None else 2.0, 'keepdim': keepdim, 'asvector': asvector, 'epsilon': 1e-12, } helper = LayerHelper('p_norm', **locals()) out = helper.create_variable_for_type_inference( dtype=helper.input_dtype()) helper.append_op( type='p_norm', inputs={'X': input}, outputs={'Out': out}, attrs=attrs) return out def inf_norm(input, porder=None, axis=axis, keepdim=False, asvector=False, name=None): helper = LayerHelper('frobenius_norm', **locals()) out = helper.create_variable_for_type_inference( dtype=helper.input_dtype()) helper.append_op(type='abs', inputs={'X': input}, outputs={'Out': out}) reduce_out = helper.create_variable_for_type_inference( dtype=helper.input_dtype()) reduce_all = True if axis == None or axis == [] or asvector == True else False axis = axis if axis != None and axis != [] else [0] reduce_type = 'reduce_max' if porder == np.float( 'inf') else 'reduce_min' helper.append_op( type=reduce_type, inputs={'X': out}, outputs={'Out': reduce_out}, attrs={'dim': axis, 'keep_dim': keepdim, 'reduce_all': reduce_all}) return reduce_out def p_matrix_norm(input, porder=1., axis=axis, keepdim=False, name=None): """ NOTE: This function actually treats the matrix as flattened vector to calculate vector norm instead of matrix norm. """ block = LayerHelper('norm', **locals()) out = block.create_variable_for_type_inference( dtype=block.input_dtype()) abs_out = block.create_variable_for_type_inference( dtype=block.input_dtype()) block.append_op( type='abs', inputs={'X': input}, outputs={'Out': abs_out}) pow_out = block.create_variable_for_type_inference( dtype=block.input_dtype()) block.append_op( type='pow', inputs={'X': abs_out}, outputs={'Out': pow_out}, attrs={'factor': porder}) sum_out = block.create_variable_for_type_inference( dtype=block.input_dtype()) block.append_op( type='reduce_sum', inputs={'X': pow_out}, outputs={'Out': sum_out}, attrs={ 'dim': axis, 'keep_dim': keepdim, 'reduce_all': True if axis is None else False }) porder block.append_op( type='pow', inputs={'X': sum_out}, outputs={'Out': out}, attrs={'factor': float(1. / porder)}) return out if axis is None and p is not None: if isinstance(p, str): if p == "fro": return frobenius_norm(x, dim=axis, keepdim=keepdim, name=name) else: raise ValueError( "only valid string values are 'fro', found {}".format(p)) elif isinstance(p, (int, float)): return vector_norm( x, porder=p, axis=axis, keepdim=keepdim, asvector=True, name=name) else: raise ValueError("only valid p type is string or float, found {}". format(type(p))) if isinstance(axis, tuple): axis = list(axis) if isinstance(axis, list) and len(axis) == 1: axis = axis[0] #calculate vector norm, where axis is int or list with only one integer if isinstance(axis, int): if isinstance(p, str): if p == "fro": return vector_norm( x, porder=2, axis=axis, keepdim=keepdim, asvector=False, name=name) else: raise ValueError( "only valid string values are 'fro', found {}".format(p)) elif isinstance(p, (int, float)): return vector_norm( x, axis=axis, porder=p, keepdim=keepdim, asvector=False, name=name) else: raise ValueError( "unspport p for p-order vector norm. except float, found {}". format(p)) #calculate matrix norm, where axis is list with two integers elif isinstance(axis, list) and len(axis) == 2: if p == "fro": return frobenius_norm(x, dim=axis, keepdim=keepdim, name=name) elif p == np.inf or p == -np.inf: return inf_norm(x, porder=p, axis=axis, keepdim=keepdim, name=name) elif p == 0: raise ValueError( "just suport axis type int or list (length of list <=1) if p = 0, found {}". format(axis)) else: return p_matrix_norm( x, porder=p, axis=axis, keepdim=keepdim, name=name) else: raise ValueError( "except axis type int or list (length of list <=2), found {}". format(axis)) def dist(x, y, p=2, name=None): r""" This OP returns the p-norm of (x - y). It is not a norm in a strict sense, only as a measure of distance. The shapes of x and y must be broadcastable. The definition is as follows, for details, please refer to the `numpy's broadcasting `_: - Each input has at least one dimension. - Match the two input dimensions from back to front, the dimension sizes must either be equal, one of them is 1, or one of them does not exist. Where, z = x - y, the shapes of x and y are broadcastable, then the shape of z can be obtained as follows: 1. If the number of dimensions of x and y are not equal, prepend 1 to the dimensions of the tensor with fewer dimensions. For example, The shape of x is [8, 1, 6, 1], the shape of y is [7, 1, 5], prepend 1 to the dimension of y. x (4-D Tensor): 8 x 1 x 6 x 1 y (4-D Tensor): 1 x 7 x 1 x 5 2. Determine the size of each dimension of the output z: choose the maximum value from the two input dimensions. z (4-D Tensor): 8 x 7 x 6 x 5 If the number of dimensions of the two inputs are the same, the size of the output can be directly determined in step 2. When p takes different values, the norm formula is as follows: When p = 0, defining $0^0=0$, the zero-norm of z is simply the number of non-zero elements of z. .. math:: ||z||_{0}=\lim_{p \\rightarrow 0}\sum_{i=1}^{m}|z_i|^{p} When p = inf, the inf-norm of z is the maximum element of z. .. math:: ||z||_\infty=\max_i |z_i| When p = -inf, the negative-inf-norm of z is the minimum element of z. .. math:: ||z||_{-\infty}=\min_i |z_i| Otherwise, the p-norm of z follows the formula, .. math:: ||z||_{p}=(\sum_{i=1}^{m}|z_i|^p)^{\\frac{1}{p}} Args: x (Tensor): 1-D to 6-D Tensor, its data type is float32 or float64. y (Tensor): 1-D to 6-D Tensor, its data type is float32 or float64. p (float, optional): The norm to be computed, its data type is float32 or float64. Default: 2. Returns: Tensor: Tensor that is the p-norm of (x - y). Examples: .. code-block:: python import paddle import numpy as np x = paddle.to_tensor(np.array([[3, 3],[3, 3]]), "float32") y = paddle.to_tensor(np.array([[3, 3],[3, 1]]), "float32") out = paddle.dist(x, y, 0) print(out) # out = [1.] out = paddle.dist(x, y, 2) print(out) # out = [2.] out = paddle.dist(x, y, float("inf")) print(out) # out = [2.] out = paddle.dist(x, y, float("-inf")) print(out) # out = [0.] """ check_variable_and_dtype(x, 'dtype', ['float32', 'float64'], 'dist') check_variable_and_dtype(y, 'dtype', ['float32', 'float64'], 'dist') check_type(p, 'p', (float, int), 'dist') helper = LayerHelper("dist", **locals()) out = helper.create_variable_for_type_inference(x.dtype) inputs = {"X": [x], "Y": [y]} outputs = {'Out': [out]} attrs = {"p": float(p)} helper.append_op( type='dist', inputs=inputs, outputs={'Out': out}, attrs=attrs) return out def cond(x, p=None, name=None): """ Computes the condition number of a matrix or batches of matrices with respect to a matrix norm ``p``. Args: x (Tensor): The input tensor could be tensor of shape ``(*, m, n)`` where ``*`` is zero or more batch dimensions for ``p`` in ``(2, -2)``, or of shape ``(*, n, n)`` where every matrix is invertible for any supported ``p``. And the input data type could be ``float32`` or ``float64``. p (float|string, optional): Order of the norm. Supported values are `fro`, `nuc`, `1`, `-1`, `2`, `-2`, `inf`, `-inf`. Default value is `None`, meaning that the order of the norm is `2`. name (str, optional): The default value is `None`. Normally there is no need for user to set this property. For more information, please refer to :ref:`api_guide_Name`. Returns: Tensor: computing results of condition number, its data type is the same as input Tensor ``x``. Examples: .. code-block:: python import paddle import numpy as np x = paddle.to_tensor([[1., 0, -1], [0, 1, 0], [1, 0, 1]]) # compute conditional number when p is None out = paddle.linalg.cond(x) # out.numpy() [1.4142135] # compute conditional number when order of the norm is 'fro' out_fro = paddle.linalg.cond(x, p='fro') # out_fro.numpy() [3.1622777] # compute conditional number when order of the norm is 'nuc' out_nuc = paddle.linalg.cond(x, p='nuc') # out_nuc.numpy() [9.2426405] # compute conditional number when order of the norm is 1 out_1 = paddle.linalg.cond(x, p=1) # out_1.numpy() [2.] # compute conditional number when order of the norm is -1 out_minus_1 = paddle.linalg.cond(x, p=-1) # out_minus_1.numpy() [1.] # compute conditional number when order of the norm is 2 out_2 = paddle.linalg.cond(x, p=2) # out_2.numpy() [1.4142135] # compute conditional number when order of the norm is -1 out_minus_2 = paddle.linalg.cond(x, p=-2) # out_minus_2.numpy() [0.70710677] # compute conditional number when order of the norm is inf out_inf = paddle.linalg.cond(x, p=np.inf) # out_inf.numpy() [2.] # compute conditional number when order of the norm is -inf out_minus_inf = paddle.linalg.cond(x, p=-np.inf) # out_minus_inf.numpy() [1.] a = paddle.to_tensor(np.random.randn(2, 4, 4).astype('float32')) # a.numpy() # [[[ 0.14063153 -0.996288 0.7996131 -0.02571543] # [-0.16303636 1.5534962 -0.49919784 -0.04402903] # [-1.1341571 -0.6022629 0.5445269 0.29154757] # [-0.16816919 -0.30972657 1.7521842 -0.5402487 ]] # [[-0.58081484 0.12402827 0.7229862 -0.55046535] # [-0.15178485 -1.1604939 0.75810957 0.30971205] # [-0.9669573 1.0940945 -0.27363303 -0.35416734] # [-1.216529 2.0018666 -0.7773689 -0.17556527]]] a_cond_fro = paddle.linalg.cond(a, p='fro') # a_cond_fro.numpy() [31.572273 28.120834] b = paddle.to_tensor(np.random.randn(2, 3, 4).astype('float64')) # b.numpy() # [[[ 1.61707487 0.46829144 0.38130416 0.82546736] # [-1.72710298 0.08866375 -0.62518804 0.16128892] # [-0.02822879 -1.67764516 0.11141444 0.3220113 ]] # [[ 0.22524372 0.62474921 -0.85503233 -1.03960523] # [-0.76620689 0.56673047 0.85064753 -0.45158196] # [ 1.47595418 2.23646462 1.5701758 0.10497519]]] b_cond_2 = paddle.linalg.cond(b, p=2) # b_cond_2.numpy() [3.30064451 2.51976252] """ def mat_norm(input, porder=1., axis=None): """ NOTE: Calculate the matrix norm of a square matrix or batches of square matrices, when porder is in (1, -1, inf, -inf) """ reduce_all = True if axis is None or axis == [] else False axis = axis if axis != None and axis != [] else [0] keepdim = False if in_dygraph_mode(): abs_out = _C_ops.abs(input) sum_out = _C_ops.reduce_sum(abs_out, 'dim', axis, 'keepdim', keepdim, 'reduce_all', reduce_all) if porder == 1 or porder == np.inf: return _C_ops.reduce_max(sum_out, 'dim', [-1], 'keepdim', keepdim, 'reduce_all', reduce_all) if porder == -1 or porder == -np.inf: return _C_ops.reduce_min(sum_out, 'dim', [-1], 'keepdim', keepdim, 'reduce_all', reduce_all) block = LayerHelper('norm', **locals()) abs_out = block.create_variable_for_type_inference( dtype=block.input_dtype()) sum_out = block.create_variable_for_type_inference( dtype=block.input_dtype()) out = block.create_variable_for_type_inference( dtype=block.input_dtype()) block.append_op( type='abs', inputs={'X': input}, outputs={'Out': abs_out}) block.append_op( type='reduce_sum', inputs={'X': abs_out}, outputs={'Out': sum_out}, attrs={'dim': axis, 'keep_dim': keepdim, 'reduce_all': reduce_all}) if porder == 1 or porder == np.inf: block.append_op( type='reduce_max', inputs={'X': sum_out}, outputs={'Out': out}, attrs={ 'dim': [-1], 'keep_dim': keepdim, 'reduce_all': reduce_all }) if porder == -1 or porder == -np.inf: block.append_op( type='reduce_min', inputs={'X': sum_out}, outputs={'Out': out}, attrs={ 'dim': [-1], 'keep_dim': keepdim, 'reduce_all': reduce_all }) return out def fro_norm(input, porder=2, axis=[-1]): """ NOTE: Calculate the frobenius norm of a square matrix or batches of square matrices. """ reduce_all = True if axis is None or axis == [] else False keepdim = False if in_dygraph_mode(): pow_out = _C_ops.pow(input, 'factor', porder) sum_out_1 = _C_ops.reduce_sum(pow_out, 'dim', axis, 'keepdim', keepdim, 'reduce_all', reduce_all) sum_out_2 = _C_ops.reduce_sum(sum_out_1, 'dim', axis, 'keepdim', keepdim, 'reduce_all', reduce_all) return _C_ops.pow(sum_out_2, 'factor', float(1. / porder)) block = LayerHelper('norm', **locals()) pow_out = block.create_variable_for_type_inference( dtype=block.input_dtype()) sum_out_1 = block.create_variable_for_type_inference( dtype=block.input_dtype()) sum_out_2 = block.create_variable_for_type_inference( dtype=block.input_dtype()) out = block.create_variable_for_type_inference( dtype=block.input_dtype()) block.append_op( type='pow', inputs={'X': input}, outputs={'Out': pow_out}, attrs={'factor': porder}) block.append_op( type='reduce_sum', inputs={'X': pow_out}, outputs={'Out': sum_out_1}, attrs={'dim': axis, 'keep_dim': keepdim, 'reduce_all': reduce_all}) block.append_op( type='reduce_sum', inputs={'X': sum_out_1}, outputs={'Out': sum_out_2}, attrs={'dim': axis, 'keep_dim': keepdim, 'reduce_all': reduce_all}) block.append_op( type='pow', inputs={'X': sum_out_2}, outputs={'Out': out}, attrs={'factor': float(1. / porder)}) return out def svd_norm(input, porder, axis=[-1]): """ NOTE: Calculate the matrix norm, which is related to singular values, of a matrix or batches of matrices, including nuclear norm, 2-norm and (-2)-norm. """ reduce_all = True if axis is None or axis == [] else False keepdim = False u, s, vh = svd(input, full_matrices=False) if in_dygraph_mode(): if porder == "nuc": return _C_ops.reduce_sum(s, 'dim', axis, 'keepdim', keepdim, 'reduce_all', reduce_all) max_out = _C_ops.reduce_max(s, 'dim', axis, 'keepdim', keepdim, 'reduce_all', reduce_all) min_out = _C_ops.reduce_min(s, 'dim', axis, 'keepdim', keepdim, 'reduce_all', reduce_all) if porder == 2: return _C_ops.elementwise_div(max_out, min_out, 'aixs', axis, 'use_mkldnn', False) if porder == -2: return _C_ops.elementwise_div(min_out, max_out, 'aixs', axis, 'use_mkldnn', False) block = LayerHelper('norm', **locals()) out = block.create_variable_for_type_inference( dtype=block.input_dtype()) if porder == "nuc": block.append_op( type='reduce_sum', inputs={'X': s}, outputs={'Out': out}, attrs={ 'dim': axis, 'keep_dim': keepdim, 'reduce_all': reduce_all }) return out max_out = block.create_variable_for_type_inference( dtype=block.input_dtype()) min_out = block.create_variable_for_type_inference( dtype=block.input_dtype()) block.append_op( type='reduce_max', inputs={'X': s}, outputs={'Out': max_out}, attrs={'dim': axis, 'keep_dim': keepdim, 'reduce_all': reduce_all}) block.append_op( type='reduce_min', inputs={'X': s}, outputs={'Out': min_out}, attrs={'dim': axis, 'keep_dim': keepdim, 'reduce_all': reduce_all}) if porder == 2: block.append_op( type='elementwise_div', inputs={'X': max_out, 'Y': min_out}, outputs={'Out': out}, attrs={'aixs': axis, 'use_mkldnn': False}) return out if porder == -2: block.append_op( type='elementwise_div', inputs={'X': min_out, 'Y': max_out}, outputs={'Out': out}, attrs={'aixs': axis, 'use_mkldnn': False}) return out def empty_tensor(input, shape): if in_dygraph_mode(): return input.reshape(shape) raise ValueError("only support x is nonempty tensor in static mode") x_shape = list(x.shape) if not len(x_shape) >= 2: raise ValueError("input should be a matrix or batches of matrices, " + "but the dimention of received input is {}".format( len(x_shape))) if p == None: p = 2 x_size = 0 if (0 in x_shape) else 1 if p in ("fro", "nuc", 1, -1, np.inf, -np.inf): if x_shape[len(x_shape) - 1] == x_shape[len(x_shape) - 2]: if x_size == 0: return empty_tensor(x, x_shape[:-2]) x_inv = x.inverse() if p == "fro": return fro_norm(x) * fro_norm(x_inv) if p == "nuc": return svd_norm(x, p) * svd_norm(x_inv, p) if p in (1, -1): return mat_norm( x, porder=p, axis=[-2]) * mat_norm( x_inv, porder=p, axis=[-2]) if p in (np.inf, -np.inf): return mat_norm( x, porder=p, axis=[-1]) * mat_norm( x_inv, porder=p, axis=[-1]) else: raise ValueError("only support p is {} when input is a ".format(p) + "square matrix or batches of square matrices") elif p in (2, -2): if x_size == 0: return empty_tensor(x, x_shape[:-2]) return svd_norm(x, porder=p) else: raise ValueError( "unsupported {} for p, only supporting ('fro', 'nuc', ".format( p) + "1, -1, 2, -2, inf, -inf) or none") def dot(x, y, name=None): """ This operator calculates inner product for vectors. .. note:: Support 1-d and 2-d Tensor. When it is 2d, the first dimension of this matrix is the batch dimension, which means that the vectors of multiple batches are dotted. Parameters: x(Tensor): 1-D or 2-D ``Tensor``. Its dtype should be ``float32``, ``float64``, ``int32``, ``int64`` y(Tensor): 1-D or 2-D ``Tensor``. Its dtype soulde be ``float32``, ``float64``, ``int32``, ``int64`` name(str, optional): Name of the output. Default is None. It's used to print debug info for developers. Details: :ref:`api_guide_Name` Returns: Tensor: the calculated result Tensor. Examples: .. code-block:: python import paddle import numpy as np x_data = np.random.uniform(0.1, 1, [10]).astype(np.float32) y_data = np.random.uniform(1, 3, [10]).astype(np.float32) x = paddle.to_tensor(x_data) y = paddle.to_tensor(y_data) z = paddle.dot(x, y) print(z) """ op_type = 'dot' # skip var type check in dygraph mode to improve efficiency if in_dygraph_mode(): op = getattr(_C_ops, op_type) return op(x, y) assert x is not None, 'x cannot be None in {}'.format(op_type) assert y is not None, 'y cannot be None in {}'.format(op_type) check_variable_and_dtype(x, 'x', ['float32', 'float64', 'int32', 'int64'], op_type) check_variable_and_dtype(y, 'y', ['float32', 'float64', 'int32', 'int64'], op_type) helper = LayerHelper(op_type, **locals()) if name is None: out = helper.create_variable_for_type_inference(dtype=x.dtype) else: out = helper.create_variable( name=name, dtype=x.dtype, persistable=False) helper.append_op( type="dot", inputs={'X': x, 'Y': y}, attrs={}, outputs={"Out": out}) return out def t(input, name=None): """ Transpose <=2-D tensor. 0-D and 1-D tensors are returned as it is and 2-D tensor is equal to the paddle.transpose function which perm dimensions set 0 and 1. Args: input (Tensor): The input Tensor. It is a N-D (N<=2) Tensor of data types float16, float32, float64, int32. name(str, optional): The default value is None. Normally there is no need for user to set this property. For more information, please refer to :ref:`api_guide_Name` Returns: Tensor: A transposed n-D Tensor, with data type being float16, float32, float64, int32, int64. For Example: .. code-block:: text # Example 1 (0-D tensor) x = tensor([0.79]) paddle.t(x) = tensor([0.79]) # Example 2 (1-D tensor) x = tensor([0.79, 0.84, 0.32]) paddle.t(x) = tensor([0.79, 0.84, 0.32]) # Example 3 (2-D tensor) x = tensor([0.79, 0.84, 0.32], [0.64, 0.14, 0.57]) paddle.t(x) = tensor([0.79, 0.64], [0.84, 0.14], [0.32, 0.57]) Examples: .. code-block:: python import paddle x = paddle.ones(shape=[2, 3], dtype='int32') x_transposed = paddle.t(x) print(x_transposed.shape) # [3, 2] """ if len(input.shape) > 2: raise ValueError( "Input(input) only support N-D (N<=2) tensor, but received " "length of Input(input) is %s. Perhaps you can use paddle." "tensor.transpose() instead." % len(input.shape)) if in_dygraph_mode(): if len(input.shape) == 1: return input # 2-D tensor perm = [1, 0] out, _ = _C_ops.transpose2(input, 'axis', perm) return out check_variable_and_dtype( input, 'input', ['float16', 'float32', 'float64', 'int32', 'int64'], 'transpose') helper = LayerHelper('t', **locals()) out = helper.create_variable_for_type_inference(input.dtype) input_shape = helper.create_variable_for_type_inference(input.dtype) if len(input.shape) == 1: out = input else: helper.append_op( type='transpose2', inputs={'X': [input]}, outputs={'Out': [out], 'XShape': [input_shape]}, attrs={'axis': [1, 0]}) return out def cross(x, y, axis=None, name=None): """ Computes the cross product between two tensors along an axis. Inputs must have the same shape, and the length of their axes should be equal to 3. If `axis` is not given, it defaults to the first axis found with the length 3. Args: x (Tensor): The first input tensor. y (Tensor): The second input tensor. axis (int, optional): The axis along which to compute the cross product. It defaults to the first axis found with the length 3. name (str, optional): Name for the operation (optional, default is None). For more information, please refer to :ref:`api_guide_Name`. Returns: Tensor. A Tensor with same data type as `x`. Examples: .. code-block:: python import paddle x = paddle.to_tensor([[1.0, 1.0, 1.0], [2.0, 2.0, 2.0], [3.0, 3.0, 3.0]]) y = paddle.to_tensor([[1.0, 1.0, 1.0], [1.0, 1.0, 1.0], [1.0, 1.0, 1.0]]) z1 = paddle.cross(x, y) # [[-1. -1. -1.] # [ 2. 2. 2.] # [-1. -1. -1.]] z2 = paddle.cross(x, y, axis=1) # [[0. 0. 0.] # [0. 0. 0.] # [0. 0. 0.]] """ if in_dygraph_mode(): if axis is not None: return _C_ops.cross(x, y, 'dim', axis) else: return _C_ops.cross(x, y) helper = LayerHelper("cross", **locals()) out = helper.create_variable_for_type_inference(x.dtype) attrs = dict() attrs['dim'] = axis helper.append_op( type='cross', inputs={'X': x, 'Y': y}, outputs={'Out': out}, attrs=attrs) return out def cholesky(x, upper=False, name=None): r""" Computes the Cholesky decomposition of one symmetric positive-definite matrix or batches of symmetric positive-definite matrice. If `upper` is `True`, the decomposition has the form :math:`A = U^{T}U` , and the returned matrix :math:`U` is upper-triangular. Otherwise, the decomposition has the form :math:`A = LL^{T}` , and the returned matrix :math:`L` is lower-triangular. Args: x (Tensor): The input tensor. Its shape should be `[*, M, M]`, where * is zero or more batch dimensions, and matrices on the inner-most 2 dimensions all should be symmetric positive-definite. Its data type should be float32 or float64. upper (bool): The flag indicating whether to return upper or lower triangular matrices. Default: False. Returns: Tensor: A Tensor with same shape and data type as `x`. It represents \ triangular matrices generated by Cholesky decomposition. Examples: .. code-block:: python import paddle import numpy as np a = np.random.rand(3, 3) a_t = np.transpose(a, [1, 0]) x_data = np.matmul(a, a_t) + 1e-03 x = paddle.to_tensor(x_data) out = paddle.linalg.cholesky(x, upper=False) print(out) # [[1.190523 0. 0. ] # [0.9906703 0.27676893 0. ] # [1.25450498 0.05600871 0.06400121]] """ if in_dygraph_mode(): return _C_ops.cholesky(x, "upper", upper) check_variable_and_dtype(x, 'dtype', ['float32', 'float64'], 'cholesky') check_type(upper, 'upper', bool, 'cholesky') helper = LayerHelper('cholesky', **locals()) out = helper.create_variable_for_type_inference(dtype=x.dtype) helper.append_op( type='cholesky', inputs={'X': [x]}, outputs={'Out': out}, attrs={'upper': upper}) return out def matrix_rank(x, tol=None, hermitian=False, name=None): r""" Computes the rank of a matrix. The rank of a matrix is the number of singular values that are greater than the specified `tol` threshold when hermitian=False, or the number of eigenvalues in absolute value that are greater than the specified `tol` threshold when hermitian=True. Args: x (Tensor): The input tensor. Its shape should be `[..., m, n]`, where `...` is zero or more batch dimensions. If `x` is a batch of matrices then the output has the same batch dimensions. The data type of `x` should be float32 or float64. tol (float,Tensor,optional): the tolerance value. Default: None. If `tol` is not specified, and `sigma` is the largest singular value (or eigenvalues in absolute value), and `eps` is the epsilon value for the dtype of `x`, then `tol` is computed with formula `tol=sigma * max(m,n) * eps`. Note that if `x` is a batch of matrices, `tol` is computed this way for every batch. hermitian (bool,optional): indicates whether `x` is Hermitian. Default: False. When hermitian=True, `x` is assumed to be Hermitian, enabling a more efficient method for finding eigenvalues, but `x` is not checked inside the function. Instead, We just use the lower triangular of the matrix to compute. name (str, optional): Name for the operation (optional, default is None). For more information, please refer to :ref:`api_guide_Name`. Returns: Tensor: Rank of tensor x. Examples: .. code-block:: python import paddle a = paddle.eye(10) b = paddle.linalg.matrix_rank(a) print(b) # b = [10] c = paddle.ones(shape=[3, 4, 5, 5]) d = paddle.linalg.matrix_rank(c, tol=0.01, hermitian=True) print(d) # d = [[1, 1, 1, 1], # [1, 1, 1, 1], # [1, 1, 1, 1]] """ if in_dygraph_mode(): if tol is None: tol_tensor = None tol_attr = 0.0 use_default_tol = True elif isinstance(tol, Variable): if tol.dtype != x.dtype: tol_tensor = cast(tol, x.dtype) else: tol_tensor = tol tol_attr = 0.0 use_default_tol = False else: tol_tensor = None tol_attr = float(tol) use_default_tol = False return _C_ops.matrix_rank(x, tol_tensor, "tol", tol_attr, 'hermitian', hermitian, 'use_default_tol', use_default_tol) inputs = {} attrs = {} check_variable_and_dtype(x, 'x', ['float32', 'float64'], 'matrix_rank') inputs['X'] = x if tol is None: attrs['use_default_tol'] = True elif isinstance(tol, Variable): check_variable_and_dtype(tol, 'tol', ['float32'], 'matrix_rank') attrs['use_default_tol'] = False if tol.dtype != x.dtype: inputs['TolTensor'] = cast(tol, x.dtype) else: inputs['TolTensor'] = tol else: check_type(tol, 'tol', float, 'matrix_rank') attrs['use_default_tol'] = False attrs['tol'] = tol check_type(hermitian, 'hermitian', bool, 'matrix_rank') attrs['hermitian'] = hermitian helper = LayerHelper('matrix_rank', **locals()) out = helper.create_variable_for_type_inference(dtype='int32') helper.append_op( type='matrix_rank', inputs=inputs, outputs={'Out': out}, attrs=attrs) return out def bmm(x, y, name=None): """ Applies batched matrix multiplication to two tensors. Both of the two input tensors must be three-dementional and share the same batch size. if x is a (b, m, k) tensor, y is a (b, k, n) tensor, the output will be a (b, m, n) tensor. Args: x (Tensor): The input Tensor. y (Tensor): The input Tensor. name(str|None): A name for this layer(optional). If set None, the layer will be named automatically. Returns: Tensor: The product Tensor. Examples: .. code-block:: python import paddle # In imperative mode: # size x: (2, 2, 3) and y: (2, 3, 2) x = paddle.to_tensor([[[1.0, 1.0, 1.0], [2.0, 2.0, 2.0]], [[3.0, 3.0, 3.0], [4.0, 4.0, 4.0]]]) y = paddle.to_tensor([[[1.0, 1.0],[2.0, 2.0],[3.0, 3.0]], [[4.0, 4.0],[5.0, 5.0],[6.0, 6.0]]]) out = paddle.bmm(x, y) #output size: (2, 2, 2) #output value: #[[[6.0, 6.0],[12.0, 12.0]],[[45.0, 45.0],[60.0, 60.0]]] out_np = out.numpy() """ x_shape = x.shape y_shape = y.shape if not len(x_shape) == len(y_shape) == 3: raise ValueError( "x and y should be 3-dimensional. But received x's dimention: {}, y's dimention: {}". format(x_shape, y_shape)) if x_shape[2] != y_shape[1]: raise ValueError( "x's width must be equal with y's height. But received x's shape: {}, y's shape: {}". format(x_shape, y_shape)) if x_shape[0] != y_shape[0]: raise ValueError( "x's batch (shape[0]) must be equal with y's batch (shape[0]). But received x's shape: {}, y's shape: {}". format(x_shape, y_shape)) if in_dygraph_mode(): return _C_ops.bmm(x, y) helper = LayerHelper('bmm', **locals()) out = helper.create_variable_for_type_inference(dtype=x.dtype) helper.append_op(type='bmm', inputs={'X': x, 'Y': y}, outputs={'Out': out}) return out def histogram(input, bins=100, min=0, max=0, name=None): """ Computes the histogram of a tensor. The elements are sorted into equal width bins between min and max. If min and max are both zero, the minimum and maximum values of the data are used. Args: input (Tensor): A Tensor(or LoDTensor) with shape :math:`[N_1, N_2,..., N_k]` . The data type of the input Tensor should be float32, float64, int32, int64. bins (int): number of histogram bins min (int): lower end of the range (inclusive) max (int): upper end of the range (inclusive) Returns: Tensor: data type is int64, shape is (nbins,). Examples: .. code-block:: python import paddle inputs = paddle.to_tensor([1, 2, 1]) result = paddle.histogram(inputs, bins=4, min=0, max=3) print(result) # [0, 2, 1, 0] """ if in_dygraph_mode(): return _C_ops.histogram(input, "bins", bins, "min", min, "max", max) helper = LayerHelper('histogram', **locals()) check_variable_and_dtype( input, 'X', ['int32', 'int64', 'float32', 'float64'], 'histogram') out = helper.create_variable_for_type_inference(VarDesc.VarType.INT64) helper.append_op( type='histogram', inputs={'X': input}, outputs={'Out': out}, attrs={'bins': bins, 'min': min, 'max': max}) return out def bincount(x, weights=None, minlength=0, name=None): """ Computes frequency of each value in the input tensor. Args: x (Tensor): A Tensor with non-negative integer. Should be 1-D tensor. weights (Tensor, optional): Weight for each value in the input tensor. Should have the same shape as input. Default is None. minlength (int, optional): Minimum number of bins. Should be non-negative integer. Default is 0. name(str, optional): The default value is None. Normally there is no need for user to set this property. For more information, please refer to :ref:`api_guide_Name`. Returns: Tensor: The tensor of frequency. Examples: .. code-block:: python import paddle x = paddle.to_tensor([1, 2, 1, 4, 5]) result1 = paddle.bincount(x) print(result1) # [0, 2, 1, 0, 1, 1] w = paddle.to_tensor([2.1, 0.4, 0.1, 0.5, 0.5]) result2 = paddle.bincount(x, weights=w) print(result2) # [0., 2.19999981, 0.40000001, 0., 0.50000000, 0.50000000] """ if x.dtype not in [paddle.int32, paddle.int64]: raise TypeError("Elements in Input(x) should all be integers") if in_dygraph_mode(): return _C_ops.bincount(x, weights, "minlength", minlength) helper = LayerHelper('bincount', **locals()) check_variable_and_dtype(x, 'X', ['int32', 'int64'], 'bincount') if weights is not None: check_variable_and_dtype(weights, 'Weights', ['int32', 'int64', 'float32', 'float64'], 'bincount') out = helper.create_variable_for_type_inference(dtype=weights.dtype) else: out = helper.create_variable_for_type_inference(dtype=x.dtype) helper.append_op( type='bincount', inputs={'X': x, 'Weights': weights}, outputs={'Out': out}, attrs={'minlength': minlength}) return out def mv(x, vec, name=None): """ Performs a matrix-vector product of the matrix x and the vector vec. Args: x (Tensor): A tensor with shape :math:`[M, N]` , The data type of the input Tensor x should be one of float32, float64. vec (Tensor): A tensor with shape :math:`[N]` , The data type of the input Tensor x should be one of float32, float64. name(str, optional): The default value is None. Normally there is no need for user to set this property. For more information, please refer to :ref:`api_guide_Name`. Returns: Tensor: The tensor which is producted by x and vec. Examples: .. code-block:: python # x: [M, N], vec: [N] # paddle.mv(x, vec) # out: [M] import numpy as np import paddle x_data = np.array([[2, 1, 3], [3, 0, 1]]).astype("float64") x = paddle.to_tensor(x_data) vec_data = np.array([3, 5, 1]) vec = paddle.to_tensor(vec_data).astype("float64") out = paddle.mv(x, vec) """ if in_dygraph_mode(): out = _C_ops.mv(x, vec) return out def __check_input(x, vec): var_names = {'x': x, 'vec': vec} for name, val in var_names.items(): check_variable_and_dtype(val, name, ['float32', 'float64'], 'mv') x_shape = list(x.shape) vec_shape = list(vec.shape) if len(x_shape) != 2: raise ValueError( "x should be 2-dimensional. But received x's dimention: {}". format(x_shape)) if len(vec_shape) != 1: raise ValueError( "vec should be 1-dimensional. But received vec's dimention: {}". format(vec_shape)) __check_input(x, vec) helper = LayerHelper('mv', **locals()) out = helper.create_variable_for_type_inference(dtype=x.dtype) helper.append_op( type='mv', inputs={'X': x, 'Vec': vec}, outputs={'Out': out}) return out def det(x, name=None): """ Calculates determinant value of a square matrix or batches of square matrices. Args: x (Tensor): input (Tensor): the input matrix of size `(n, n)` or the batch of matrices of size `(*, n, n)` where `*` is one or more batch dimensions. Returns: y (Tensor):the determinant value of a square matrix or batches of square matrices. Examples: .. code-block:: python import paddle x = paddle.randn([3,3,3]) A = paddle.linalg.det(x) print(A) # [ 0.02547996, 2.52317095, -6.15900707]) """ if in_dygraph_mode(): return _C_ops.determinant(x) check_dtype(x.dtype, 'Input', ['float32', 'float64'], 'det') input_shape = list(x.shape) assert len(input_shape) >= 2, \ "The x must be at least 2-dimensional, " \ "but received Input x's dimensional: %s.\n" % \ len(input_shape) assert (input_shape[-1] == input_shape[-2]), \ "Expect squared input," \ "but received %s by %s matrix.\n" \ %(input_shape[-2], input_shape[-1]) \ helper = LayerHelper('determinant', **locals()) out = helper.create_variable_for_type_inference(dtype=x.dtype) helper.append_op( type='determinant', inputs={'Input': [x]}, outputs={'Out': [out]}) return out def slogdet(x, name=None): """ Calculates the sign and natural logarithm of the absolute value of a square matrix's or batches square matrices' determinant. The determinant can be computed with ``sign * exp(logabsdet) Supports input of float, double Note that for matrices that have zero determinant, this returns ``(0, -inf)`` Args: x (Tensor): the batch of matrices of size :math:`(*, n, n)` where math:`*` is one or more batch dimensions. Returns: y (Tensor): A tensor containing the sign of the determinant and the natural logarithm of the absolute value of determinant, respectively. Examples: .. code-block:: python import paddle x = paddle.randn([3,3,3]) A = paddle.linalg.slogdet(x) print(A) # [[ 1. , 1. , -1. ], # [-0.98610914, -0.43010661, -0.10872950]]) """ if in_dygraph_mode(): return _C_ops.slogdeterminant(x) check_dtype(x.dtype, 'Input', ['float32', 'float64'], 'slogdet') input_shape = list(x.shape) assert len(input_shape) >= 2, \ "The x must be at least 2-dimensional, " \ "but received Input x's dimensional: %s.\n" % \ len(input_shape) assert (input_shape[-1] == input_shape[-2]), \ "Expect squared input," \ "but received %s by %s matrix.\n" \ %(input_shape[-2], input_shape[-1]) \ helper = LayerHelper('slogdeterminant', **locals()) out = helper.create_variable_for_type_inference(dtype=x.dtype) helper.append_op( type='slogdeterminant', inputs={'Input': [x]}, outputs={'Out': [out]}) return out def svd(x, full_matrices=False, name=None): r""" Computes the singular value decomposition of one matrix or a batch of regular matrices. Let :math:`X` be the input matrix or a batch of input matrices, the output should satisfies: .. math:: X = U * diag(S) * VT Args: x (Tensor): The input tensor. Its shape should be `[..., N, M]`, where `...` is zero or more batch dimensions. N and M can be arbitraty positive number. Note that if x is sigular matrices, the grad is numerical instable. The data type of x should be float32 or float64. full_matrices (bool): A flag to control the behavor of svd. If full_matrices = True, svd op will compute full U and V matrics, which means shape of U is `[..., N, N]`, shape of V is `[..., M, M]`. K = min(M, N). If full_matrices = False, svd op will use a economic method to store U and V. which means shape of U is `[..., N, K]`, shape of V is `[..., M, K]`. K = min(M, N). name (str, optional): Name for the operation (optional, default is None). For more information, please refer to :ref:`api_guide_Name`. Returns: Tuple of 3 tensors: (U, S, VH). VH is the conjugate transpose of V. S is the singlar value vectors of matrics with shape `[..., K]` Examples: .. code-block:: python import paddle x = paddle.to_tensor([[1.0, 2.0], [1.0, 3.0], [4.0, 6.0]]).astype('float64') x = x.reshape([3, 2]) u, s, vh = paddle.linalg.svd(x) print (u) #U = [[ 0.27364809, -0.21695147 ], # [ 0.37892198, -0.87112408 ], # [ 0.8840446 , 0.44053933 ]] print (s) #S = [8.14753743, 0.78589688] print (vh) #VT= [[ 0.51411221, 0.85772294], # [ 0.85772294, -0.51411221]] # one can verify : U * S * VT == X # U * UH == I # V * VH == I """ if in_dygraph_mode(): return _C_ops.svd(x, 'full_matrices', full_matrices) check_variable_and_dtype(x, 'dtype', ['float32', 'float64'], 'svd') check_type(full_matrices, 'full_matrices', bool, 'svd') helper = LayerHelper('svd', **locals()) u = helper.create_variable_for_type_inference(dtype=x.dtype) vh = helper.create_variable_for_type_inference(dtype=x.dtype) s = helper.create_variable_for_type_inference(dtype=x.dtype) attrs = dict() attrs['full_matrices'] = full_matrices helper.append_op( type='svd', inputs={'X': [x]}, outputs={'U': u, 'VH': vh, 'S': s}, attrs=attrs, ) return u, s, vh def matrix_power(x, n, name=None): r""" Computes the n-th power of a square matrix or a batch of square matrices. Let :math:`X` be a sqaure matrix or a batch of square matrices, :math:`n` be an exponent, the equation should be: .. math:: Out = X ^ {n} Specifically, - If `n > 0`, it returns the matrix or a batch of matrices raised to the power of `n`. - If `n = 0`, it returns the identity matrix or a batch of identity matrices. - If `n < 0`, it returns the inverse of each matrix (if invertible) raised to the power of `abs(n)`. Args: x (Tensor): A square matrix or a batch of square matrices to be raised to power `n`. Its shape should be `[*, M, M]`, where `*` is zero or more batch dimensions. Its data type should be float32 or float64. n (int): The exponent. It can be any positive, negative integer or zero. name (str, optional): Name for the operation (optional, default is None). For more information, please refer to :ref:`api_guide_Name`. Returns: Tensor: The n-th power of the matrix (or the batch of matrices) `x`. Its data type should be the same as that of `x`. Examples: .. code-block:: python import paddle x = paddle.to_tensor([[1, 2, 3], [1, 4, 9], [1, 8, 27]], dtype='float64') print(paddle.linalg.matrix_power(x, 2)) # [[6. , 34. , 102.], # [14. , 90. , 282.], # [36. , 250., 804.]] print(paddle.linalg.matrix_power(x, 0)) # [[1., 0., 0.], # [0., 1., 0.], # [0., 0., 1.]] print(paddle.linalg.matrix_power(x, -2)) # [[ 12.91666667, -12.75000000, 2.83333333 ], # [-7.66666667 , 8. , -1.83333333 ], # [ 1.80555556 , -1.91666667 , 0.44444444 ]] """ if in_dygraph_mode(): return _C_ops.matrix_power(x, "n", n) check_variable_and_dtype(x, 'dtype', ['float32', 'float64'], 'matrix_power') check_type(n, 'n', int, 'matrix_power') helper = LayerHelper('matrix_power', **locals()) out = helper.create_variable_for_type_inference(dtype=x.dtype) helper.append_op( type='matrix_power', inputs={'X': x}, outputs={'Out': out}, attrs={'n': n}) return out def qr(x, mode="reduced", name=None): r""" Computes the QR decomposition of one matrix or batches of matrice (backward is unsupported now). Args: x (Tensor): The input tensor. Its shape should be `[..., M, N]`, where ... is zero or more batch dimensions. M and N can be arbitrary positive number. The data type of x should be float32 or float64. mode (str, optional): A flag to control the behavior of qr, the default is "reduced". Suppose x's shape is `[..., M, N]` and denoting `K = min(M, N)`: If mode = "reduced", qr op will return reduced Q and R matrices, which means Q's shape is `[..., M, K]` and R's shape is `[..., K, N]`. If mode = "complete", qr op will return complete Q and R matrices, which means Q's shape is `[..., M, M]` and R's shape is `[..., M, N]`. If mode = "r", qr op will only return reduced R matrix, which means R's shape is `[..., K, N]`. name (str, optional): Name for the operation (optional, default is None). For more information, please refer to :ref:`api_guide_Name`. Returns: If mode = "reduced" or mode = "complete", qr will return a two tensor-tuple, which represents Q and R. If mode = "r", qr will return a tensor which represents R. Examples: .. code-block:: python import paddle x = paddle.to_tensor([[1.0, 2.0], [3.0, 4.0], [5.0, 6.0]]).astype('float64') q, r = paddle.linalg.qr(x) print (q) print (r) # Q = [[-0.16903085, 0.89708523], # [-0.50709255, 0.27602622], # [-0.84515425, -0.34503278]]) # R = [[-5.91607978, -7.43735744], # [ 0. , 0.82807867]]) # one can verify : X = Q * R ; """ if in_dygraph_mode(): q, r = _C_ops.qr(x, 'mode', mode) if mode == "r": return r else: return q, r check_variable_and_dtype(x, 'dtype', ['float32', 'float64'], 'qr') check_type(mode, 'mode', str, 'qr') helper = LayerHelper('qr', **locals()) q = helper.create_variable_for_type_inference(dtype=x.dtype) r = helper.create_variable_for_type_inference(dtype=x.dtype) attrs = dict() attrs['mode'] = mode helper.append_op( type='qr', inputs={'X': [x]}, outputs={'Q': q, 'R': r}, attrs=attrs) if mode == "r": return r else: return q, r def eig(x, name=None): """ This API performs the eigenvalue decomposition of a square matrix or a batch of square matrices. .. note:: If the matrix is a Hermitian or a real symmetric matrix, please use :ref:`paddle.linalg.eigh` instead, which is much faster. If only eigenvalues is needed, please use :ref:`paddle.linalg.eigvals` instead. If the matrix is of any shape, please use :ref:`paddle.linalg.svd`. This API is only supported on CPU device. The output datatype is always complex for both real and complex input. Args: x (Tensor): A tensor with shape math:`[*, N, N]`, The data type of the x should be one of ``float32``, ``float64``, ``compplex64`` or ``complex128``. name (str, optional): The default value is `None`. Normally there is no need for user to set this property. For more information, please refer to :ref:`api_guide_Name`. Returns: Eigenvalues(Tensors): A tensor with shape math:`[*, N]` refers to the eigen values. Eigenvectors(Tensors): A tensor with shape math:`[*, N, N]` refers to the eigen vectors. Examples: .. code-block:: python import paddle import numpy as np paddle.device.set_device("cpu") x_data = np.array([[1.6707249, 7.2249975, 6.5045543], [9.956216, 8.749598, 6.066444 ], [4.4251957, 1.7983172, 0.370647 ]]).astype("float32") x = paddle.to_tensor(x_data) w, v = paddle.linalg.eig(x) print(w) # Tensor(shape=[3, 3], dtype=complex128, place=CPUPlace, stop_gradient=False, # [[(-0.5061363550800655+0j) , (-0.7971760990842826+0j) , # (0.18518077798279986+0j)], # [(-0.8308237755993192+0j) , (0.3463813401919749+0j) , # (-0.6837005269141947+0j) ], # [(-0.23142567697893396+0j), (0.4944999840400175+0j) , # (0.7058765252952796+0j) ]]) print(v) # Tensor(shape=[3], dtype=complex128, place=CPUPlace, stop_gradient=False, # [ (16.50471283351188+0j) , (-5.5034820550763515+0j) , # (-0.21026087843552282+0j)]) """ if in_dygraph_mode(): w, v = _C_ops.eig(x) return w, v check_variable_and_dtype( x, 'X', ['float32', 'float64', 'complex64', 'complex128'], 'eig') helper = LayerHelper('eig', **locals()) w = helper.create_variable_for_type_inference(x.dtype) v = helper.create_variable_for_type_inference(x.dtype) inputs = {'X': x} outputs = {'Eigenvalues': w, 'Eigenvectors': v} helper.append_op(type='eig', inputs=inputs, outputs=outputs) return w, v def eigvals(x, name=None): """ Compute the eigenvalues of one or more general matrices. Warning: The gradient kernel of this operator does not yet developed. If you need back propagation through this operator, please replace it with paddle.linalg.eig. Args: x (Tensor): A square matrix or a batch of square matrices whose eigenvalues will be computed. Its shape should be `[*, M, M]`, where `*` is zero or more batch dimensions. Its data type should be float32, float64, complex64, or complex128. name (str, optional): Name for the operation (optional, default is None). For more information, please refer to :ref:`api_guide_Name`. Returns: Tensor: A tensor containing the unsorted eigenvalues which has the same batch dimensions with `x`. The eigenvalues are complex-valued even when `x` is real. Examples: .. code-block:: python import paddle paddle.set_device("cpu") paddle.seed(1234) x = paddle.rand(shape=[3, 3], dtype='float64') # [[0.02773777, 0.93004224, 0.06911496], # [0.24831591, 0.45733623, 0.07717843], # [0.48016702, 0.14235102, 0.42620817]]) print(paddle.linalg.eigvals(x)) # [(-0.27078833542132674+0j), (0.29962280156230725+0j), (0.8824477020120244+0j)] #complex128 """ check_variable_and_dtype(x, 'dtype', ['float32', 'float64', 'complex64', 'complex128'], 'eigvals') x_shape = list(x.shape) if len(x_shape) < 2: raise ValueError( "The dimension of Input(x) should be at least 2, but received x's dimention = {}, x's shape = {}". format(len(x_shape), x_shape)) if x_shape[-1] != x_shape[-2]: raise ValueError( "The last two dimensions of Input(x) should be equal, but received x's shape = {}". format(x_shape)) if in_dygraph_mode(): return _C_ops.eigvals(x) helper = LayerHelper('eigvals', **locals()) out = helper.create_variable_for_type_inference(dtype=x.dtype) helper.append_op(type='eigvals', inputs={'X': x}, outputs={'Out': out}) return out def multi_dot(x, name=None): """ Multi_dot is an operator that calculates multiple matrix multiplications. Supports inputs of float16(only GPU support), float32 and float64 dtypes. This function does not support batched inputs. The input tensor in [x] must be 2-D except for the first and last can be 1-D. If the first tensor is a 1-D vector of shape(n, ) it is treated as row vector of shape(1, n), similarly if the last tensor is a 1D vector of shape(n, ), it is treated as a column vector of shape(n, 1). If the first and last tensor are 2-D matrix, then the output is also 2-D matrix, otherwise the output is a 1-D vector. Multi_dot will select the lowest cost multiplication order for calculation. The cost of multiplying two matrices with shapes (a, b) and (b, c) is a * b * c. Given matrices A, B, C with shapes (20, 5), (5, 100), (100, 10) respectively, we can calculate the cost of different multiplication orders as follows: - Cost((AB)C) = 20x5x100 + 20x100x10 = 30000 - Cost(A(BC)) = 5x100x10 + 20x5x10 = 6000 In this case, multiplying B and C first, then multiply A, which is 5 times faster than sequential calculation. Args: x ([Tensor]): The input tensors which is a list Tensor. name(str|None): A name for this layer(optional). If set None, the layer will be named automatically. Returns: Tensor: The output Tensor. Examples: .. code-block:: python import paddle import numpy as np # A * B A_data = np.random.random([3, 4]).astype(np.float32) B_data = np.random.random([4, 5]).astype(np.float32) A = paddle.to_tensor(A_data) B = paddle.to_tensor(B_data) out = paddle.linalg.multi_dot([A, B]) print(out.numpy().shape) # [3, 5] # A * B * C A_data = np.random.random([10, 5]).astype(np.float32) B_data = np.random.random([5, 8]).astype(np.float32) C_data = np.random.random([8, 7]).astype(np.float32) A = paddle.to_tensor(A_data) B = paddle.to_tensor(B_data) C = paddle.to_tensor(C_data) out = paddle.linalg.multi_dot([A, B, C]) print(out.numpy().shape) # [10, 7] """ if in_dygraph_mode(): return _C_ops.multi_dot(x) check_type(x, 'x', (list, tuple), 'multi_dot') for id, item in enumerate(x): check_variable_and_dtype(item, 'x[' + str(id) + ']', ['float16', 'float32', 'float64'], 'multi_dot') if item.dtype != x[0].dtype: raise TypeError( "All the Tensors in the input must have the same data type.") helper = LayerHelper('multi_dot', **locals()) dtype = helper.input_dtype(input_param_name='x') out = helper.create_variable_for_type_inference(dtype) helper.append_op(type='multi_dot', inputs={"X": x}, outputs={"Out": out}) return out def eigh(x, UPLO='L', name=None): """ Compute the eigenvalues and eigenvectors of a complex Hermitian (conjugate symmetric) or a real symmetric matrix. Args: x (Tensor): A tensor with shape :math:`[*, N, N]` , The data type of the input Tensor x should be one of float32, float64, complex64, complex128. UPLO(str, optional): (string, default 'L'), 'L' represents the lower triangular matrix, "'U' represents the upper triangular matrix.". name(str, optional): The default value is None. Normally there is no need for user to set this property. For more information, please refer to :ref:`api_guide_Name`. Returns: out_value(Tensor): A Tensor with shape [*, N] and data type of float32 and float64. The eigenvalues of eigh op. out_vector(Tensor): A Tensor with shape [*, N, N] and data type of float32,float64,complex64 and complex128. The eigenvectors of eigh op. Examples: .. code-block:: python import numpy as np import paddle x_data = np.array([[1, -2j], [2j, 5]]) x = paddle.to_tensor(x_data) out_value, out_vector = paddle.linalg.eigh(x, UPLO='L') print(out_value) #[0.17157288, 5.82842712] print(out_vector) #[(-0.9238795325112867+0j), (-0.3826834323650898+0j)], #[ 0.3826834323650898j , -0.9238795325112867j ]] """ if in_dygraph_mode(): return _C_ops.eigh(x, 'UPLO', UPLO) def __check_input(x, UPLO): x_shape = list(x.shape) if len(x.shape) < 2: raise ValueError( "Input(input) only support >=2 tensor, but received " "length of Input(input) is %s." % len(x.shape)) if x_shape[-1] != x_shape[-2]: raise ValueError( "The input matrix must be batches of square matrices. But received x's dimention: {}". format(x_shape)) if UPLO != 'L' and UPLO != 'U': raise ValueError( "UPLO must be L or U. But received UPLO is: {}".format(UPLO)) __check_input(x, UPLO) helper = LayerHelper('eigh', **locals()) check_variable_and_dtype( x, 'dtype', ['float32', 'float64', 'complex64', 'complex128'], 'eigh') out_value = helper.create_variable_for_type_inference(dtype=x.dtype) out_vector = helper.create_variable_for_type_inference(dtype=x.dtype) helper.append_op( type='eigh', inputs={'X': x}, outputs={'Eigenvalues': out_value, 'Eigenvectors': out_vector}, attrs={'UPLO': UPLO}) return out_value, out_vector def pinv(x, rcond=1e-15, hermitian=False, name=None): r""" Calculate pseudo inverse via SVD(singular value decomposition) of one matrix or batches of regular matrix. .. math:: if hermitian == False: x = u * s * vt (SVD) out = v * 1/s * ut else: x = u * s * ut (eigh) out = u * 1/s * u.conj().transpose(-2,-1) If x is hermitian or symmetric matrix, svd will be replaced with eigh. Args: x(Tensor): The input tensor. Its shape should be (*, m, n) where * is zero or more batch dimensions. m and n can be arbitraty positive number. The data type of x should be float32 or float64 or complex64 or complex128. When data type is complex64 or cpmplex128, hermitian should be set True. rcond(Tensor, optional): the tolerance value to determine when is a singular value zero. Defalut:1e-15. hermitian(bool, optional): indicates whether x is Hermitian if complex or symmetric if real. Default: False. name(str|None): A name for this layer(optional). If set None, the layer will be named automatically. Returns: Tensor: The tensor with same data type with x. it represents pseudo inverse of x. Its shape should be (*, n, m). Examples: .. code-block:: python import paddle x = paddle.arange(15).reshape((3, 5)).astype('float64') input = paddle.to_tensor(x) out = paddle.linalg.pinv(input) print(input) print(out) # input: # [[0. , 1. , 2. , 3. , 4. ], # [5. , 6. , 7. , 8. , 9. ], # [10., 11., 12., 13., 14.]] # out: # [[-0.22666667, -0.06666667, 0.09333333], # [-0.12333333, -0.03333333, 0.05666667], # [-0.02000000, 0.00000000, 0.02000000], # [ 0.08333333, 0.03333333, -0.01666667], # [ 0.18666667, 0.06666667, -0.05333333]] # one can verify : x * out * x = x ; # or out * x * out = x ; """ if in_dygraph_mode(): if not hermitian: # combine svd and matmul op u, s, vt = _C_ops.svd(x, 'full_matrices', False) max_singular_val = _C_ops.reduce_max(s, 'dim', [-1], 'keep_dim', True, \ 'reduce_all', False) rcond = paddle.to_tensor(rcond, dtype=x.dtype) cutoff = rcond * max_singular_val y = float('inf') y = paddle.to_tensor(y, dtype=x.dtype) condition = s > cutoff cond_int = layers.cast(condition, s.dtype) cond_not_int = layers.cast(layers.logical_not(condition), s.dtype) out1 = layers.elementwise_mul(1 / s, cond_int) out2 = layers.elementwise_mul(1 / y, cond_not_int) singular = layers.elementwise_add(out1, out2) st, _ = _C_ops.unsqueeze2(singular, 'axes', [-2]) dims = list(range(len(vt.shape))) perm = dims[:-2] + [dims[-1]] + [dims[-2]] v, _ = _C_ops.transpose2(vt, 'axis', perm) out_1 = v * st out_2 = _C_ops.matmul_v2(out_1, u, 'trans_x', False, 'trans_y', True) return out_2 else: # combine eigh and matmul op s, u = _C_ops.eigh(x, 'UPLO', 'L') s_abs = paddle.abs(s) max_singular_val = _C_ops.reduce_max(s_abs, 'dim', [-1], 'keep_dim', True, \ 'reduce_all', False) rcond = paddle.to_tensor(rcond, dtype=s.dtype) cutoff = rcond * max_singular_val y = float('inf') y = paddle.to_tensor(y, dtype=s.dtype) condition = s_abs > cutoff cond_int = layers.cast(condition, s.dtype) cond_not_int = layers.cast(layers.logical_not(condition), s.dtype) out1 = layers.elementwise_mul(1 / s, cond_int) out2 = layers.elementwise_mul(1 / y, cond_not_int) singular = layers.elementwise_add(out1, out2) st, _ = _C_ops.unsqueeze2(singular, 'axes', [-2]) out_1 = u * st u_conj = _C_ops.conj(u) out_2 = _C_ops.matmul_v2(out_1, u_conj, 'trans_x', False, 'trans_y', True) return out_2 else: if not hermitian: helper = LayerHelper('pinv', **locals()) dtype = x.dtype check_variable_and_dtype(x, 'x', ['float32', 'float64'], 'pinv') u = helper.create_variable_for_type_inference(dtype) s = helper.create_variable_for_type_inference(dtype) vt = helper.create_variable_for_type_inference(dtype) helper.append_op( type='svd', inputs={'X': [x]}, outputs={'U': u, 'VH': vt, 'S': s}, attrs={'full_matrices': False}, ) max_singular_val = helper.create_variable_for_type_inference(dtype) helper.append_op( type='reduce_max', inputs={'X': s}, outputs={'Out': max_singular_val}, attrs={'dim': [-1], 'keep_dim': True, 'reduce_all': False}) rcond = layers.fill_constant(shape=[1], value=rcond, dtype=dtype) cutoff = rcond * max_singular_val y = float('inf') y = layers.fill_constant(shape=[1], value=y, dtype=dtype) condition = s > cutoff cond_int = layers.cast(condition, dtype) cond_not_int = layers.cast(layers.logical_not(condition), dtype) out1 = layers.elementwise_mul(1 / s, cond_int) out2 = layers.elementwise_mul(1 / y, cond_not_int) singular = layers.elementwise_add(out1, out2) st = helper.create_variable_for_type_inference(dtype=dtype) st_shape = helper.create_variable_for_type_inference(dtype=dtype) helper.append_op( type='unsqueeze2', inputs={'X': singular}, attrs={'axes': [-2]}, outputs={'Out': st, 'XShape': st_shape}) dims = list(range(len(vt.shape))) perm = dims[:-2] + [dims[-1]] + [dims[-2]] v = helper.create_variable_for_type_inference(dtype) v_shape = helper.create_variable_for_type_inference(dtype) helper.append_op( type='transpose2', inputs={'X': [vt]}, outputs={'Out': [v], 'XShape': [v_shape]}, attrs={'axis': perm}) out_1 = helper.create_variable_for_type_inference(dtype) helper.append_op( type='elementwise_mul', inputs={'X': v, 'Y': st}, outputs={'Out': out_1}, attrs={'axis': -1, 'use_mkldnn': False}) out_1 = helper.append_activation(out_1) out_2 = helper.create_variable_for_type_inference(dtype) helper.append_op( type='matmul_v2', inputs={'X': out_1, 'Y': u}, outputs={'Out': out_2}, attrs={'trans_x': False, 'trans_y': True}, ) return out_2 else: helper = LayerHelper('pinv', **locals()) dtype = x.dtype check_variable_and_dtype( x, 'dtype', ['float32', 'float64', 'complex64', 'complex128'], 'pinv') if dtype == paddle.complex128: s_type = 'float64' elif dtype == paddle.complex64: s_type = 'float32' else: s_type = dtype u = helper.create_variable_for_type_inference(dtype) s = helper.create_variable_for_type_inference(s_type) helper.append_op( type='eigh', inputs={'X': x}, outputs={'Eigenvalues': s, 'Eigenvectors': u}, attrs={'UPLO': 'L'}) s_abs = helper.create_variable_for_type_inference(s_type) helper.append_op( type='abs', inputs={'X': s}, outputs={'Out': s_abs}) max_singular_val = helper.create_variable_for_type_inference(s_type) helper.append_op( type='reduce_max', inputs={'X': s_abs}, outputs={'Out': max_singular_val}, attrs={'dim': [-1], 'keep_dim': True, 'reduce_all': False}) rcond = layers.fill_constant(shape=[1], value=rcond, dtype=s_type) cutoff = rcond * max_singular_val y = float('inf') y = layers.fill_constant(shape=[1], value=y, dtype=s_type) condition = s_abs > cutoff cond_int = layers.cast(condition, s_type) cond_not_int = layers.cast(layers.logical_not(condition), s_type) out1 = layers.elementwise_mul(1 / s, cond_int) out2 = layers.elementwise_mul(1 / y, cond_not_int) singular = layers.elementwise_add(out1, out2) st = helper.create_variable_for_type_inference(dtype=s_type) st_shape = helper.create_variable_for_type_inference(dtype=s_type) helper.append_op( type='unsqueeze2', inputs={'X': singular}, attrs={'axes': [-2]}, outputs={'Out': st, 'XShape': st_shape}) out_1 = helper.create_variable_for_type_inference(dtype) helper.append_op( type='elementwise_mul', inputs={'X': u, 'Y': st}, outputs={'Out': out_1}, attrs={'axis': -1, 'use_mkldnn': False}) out_1 = helper.append_activation(out_1) u_conj = helper.create_variable_for_type_inference(dtype) helper.append_op( type='conj', inputs={'X': u}, outputs={'Out': [u_conj]}) out_2 = helper.create_variable_for_type_inference(dtype) helper.append_op( type='matmul_v2', inputs={'X': out_1, 'Y': u_conj}, outputs={'Out': out_2}, attrs={'trans_x': False, 'trans_y': True}, ) return out_2 def solve(x, y, name=None): r""" Computes the solution of a square system of linear equations with a unique solution for input 'X' and 'Y'. Let :math: `X` be a sqaure matrix or a batch of square matrices, :math:`Y` be a vector/matrix or a batch of vectors/matrices, the equation should be: .. math:: Out = X^-1 * Y Specifically, - This system of linear equations has one solution if and only if input 'X' is invertible. Args: x (Tensor): A square matrix or a batch of square matrices. Its shape should be `[*, M, M]`, where `*` is zero or more batch dimensions. Its data type should be float32 or float64. y (Tensor): A vector/matrix or a batch of vectors/matrices. Its shape should be `[*, M, K]`, where `*` is zero or more batch dimensions. Its data type should be float32 or float64. name(str, optional): Name for the operation (optional, default is None). For more information, please refer to :ref:`api_guide_Name`. Returns: Tensor: The solution of a square system of linear equations with a unique solution for input 'x' and 'y'. Its data type should be the same as that of `x`. Examples: .. code-block:: python # a square system of linear equations: # 2*X0 + X1 = 9 # X0 + 2*X1 = 8 import paddle import numpy as np np_x = np.array([[3, 1],[1, 2]]) np_y = np.array([9, 8]) x = paddle.to_tensor(np_x, dtype="float64") y = paddle.to_tensor(np_y, dtype="float64") out = paddle.linalg.solve(x, y) print(out) # [2., 3.]) """ if in_dygraph_mode(): return _C_ops.solve(x, y) inputs = {"X": [x], "Y": [y]} helper = LayerHelper("solve", **locals()) check_variable_and_dtype(x, 'x', ['float32', 'float64'], 'solve') check_variable_and_dtype(y, 'y', ['float32', 'float64'], 'solve') out = helper.create_variable_for_type_inference(dtype=x.dtype) helper.append_op( type="solve", inputs={"X": x, "Y": y}, outputs={"Out": out}) return out def triangular_solve(x, y, upper=True, transpose=False, unitriangular=False, name=None): r""" Computes the solution of a system of equations with a triangular coefficient matrix `x` and multiple right-hand sides `y` . Input `x` and `y` is 2D matrices or batches of 2D matrices. If the inputs are batches, the outputs is also batches. Args: x (Tensor): The input triangular coefficient matrix. Its shape should be `[*, M, M]`, where `*` is zero or more batch dimensions. Its data type should be float32 or float64. y (Tensor): Multiple right-hand sides of system of equations. Its shape should be `[*, M, K]`, where `*` is zero or more batch dimensions. Its data type should be float32 or float64. upper (bool, optional): Whether to solve the upper-triangular system of equations (default) or the lower-triangular system of equations. Default: True. transpose (bool, optional): whether `x` should be transposed before calculation. Default: False. unitriangular (bool, optional): whether `x` is unit triangular. If True, the diagonal elements of `x` are assumed to be 1 and not referenced from `x` . Default: False. name(str, optional): Name for the operation (optional, default is None). For more information, please refer to :ref:`api_guide_Name`. Returns: Tensor: The solution of the system of equations. Its data type should be the same as that of `x`. Examples: .. code-block:: python # a square system of linear equations: # x1 + x2 + x3 = 0 # 2*x2 + x3 = -9 # -x3 = 5 import paddle import numpy as np x = paddle.to_tensor([[1, 1, 1], [0, 2, 1], [0, 0,-1]], dtype="float64") y = paddle.to_tensor([[0], [-9], [5]], dtype="float64") out = paddle.linalg.triangular_solve(x, y, upper=True) print(out) # [7, -2, -5] """ if in_dygraph_mode(): return _C_ops.triangular_solve(x, y, 'upper', upper, 'transpose', transpose, 'unitriangular', unitriangular) inputs = {"X": [x], "Y": [y]} helper = LayerHelper("triangular_solve", **locals()) check_variable_and_dtype(x, 'x', ['float32', 'float64'], 'triangular_solve') check_variable_and_dtype(y, 'y', ['float32', 'float64'], 'triangular_solve') out = helper.create_variable_for_type_inference(dtype=x.dtype) helper.append_op( type='triangular_solve', inputs={'X': x, 'Y': y}, outputs={'Out': out}, attrs={ 'upper': upper, 'transpose': transpose, 'unitriangular': unitriangular }) return out def cholesky_solve(x, y, upper=False, name=None): r""" Solves a linear system of equations A @ X = B, given A's Cholesky factor matrix u and matrix B. Input `x` and `y` is 2D matrices or batches of 2D matrices. If the inputs are batches, the outputs is also batches. Args: x (Tensor): The input matrix which is upper or lower triangular Cholesky factor of square matrix A. Its shape should be `[*, M, M]`, where `*` is zero or more batch dimensions. Its data type should be float32 or float64. y (Tensor): Multiple right-hand sides of system of equations. Its shape should be `[*, M, K]`, where `*` is zero or more batch dimensions. Its data type should be float32 or float64. upper (bool, optional): whether to consider the Cholesky factor as a lower or upper triangular matrix. Default: False. name(str, optional): Name for the operation (optional, default is None). For more information, please refer to :ref:`api_guide_Name`. Returns: Tensor: The solution of the system of equations. Its data type is the same as that of `x`. Examples: .. code-block:: python import paddle u = paddle.to_tensor([[1, 1, 1], [0, 2, 1], [0, 0,-1]], dtype="float64") b = paddle.to_tensor([[0], [-9], [5]], dtype="float64") out = paddle.linalg.cholesky_solve(b, u, upper=True) print(out) # [-2.5, -7, 9.5] """ if in_dygraph_mode(): return _C_ops.cholesky_solve(x, y, 'upper', upper) helper = LayerHelper("cholesky_solve", **locals()) check_variable_and_dtype(x, 'x', ['float32', 'float64'], 'cholesky_solve') check_variable_and_dtype(y, 'y', ['float32', 'float64'], 'cholesky_solve') out = helper.create_variable_for_type_inference(dtype=x.dtype) helper.append_op( type='cholesky_solve', inputs={'X': x, 'Y': y}, outputs={'Out': out}, attrs={'upper': upper}) return out def eigvalsh(x, UPLO='L', name=None): """ Computes the eigenvalues of a complex Hermitian (conjugate symmetric) or a real symmetric matrix. Args: x (Tensor): A tensor with shape :math:`[_, M, M]` , The data type of the input Tensor x should be one of float32, float64, complex64, complex128. UPLO(str, optional): Lower triangular part of a (‘L’, default) or the upper triangular part (‘U’). name(str, optional): The default value is None. Normally there is no need for user to set this property. For more information, please refer to :ref:`api_guide_Name`. Returns: Tensor: The tensor eigenvalues in ascending order. Examples: .. code-block:: python import numpy as np import paddle x_data = np.array([[1, -2j], [2j, 5]]) x = paddle.to_tensor(x_data) out_value = paddle.eigvalsh(x, UPLO='L') print(out_value) #[0.17157288, 5.82842712] """ if in_dygraph_mode(): is_test = x.stop_gradient values, _ = _C_ops.eigvalsh(x, 'UPLO', UPLO, 'is_test', is_test) return values def __check_input(x, UPLO): x_shape = list(x.shape) if len(x.shape) < 2: raise ValueError( "Input(input) only support >=2 tensor, but received " "length of Input(input) is %s." % len(x.shape)) if x_shape[-1] != x_shape[-2]: raise ValueError( "The input matrix must be batches of square matrices. But received x's dimention: {}". format(x_shape)) if UPLO is not 'L' and UPLO is not 'U': raise ValueError( "UPLO must be L or U. But received UPLO is: {}".format(UPLO)) __check_input(x, UPLO) helper = LayerHelper('eigvalsh', **locals()) check_variable_and_dtype(x, 'dtype', ['float32', 'float64', 'complex64', 'complex128'], 'eigvalsh') out_value = helper.create_variable_for_type_inference(dtype=x.dtype) out_vector = helper.create_variable_for_type_inference(dtype=x.dtype) is_test = x.stop_gradient helper.append_op( type='eigvalsh', inputs={'X': x}, outputs={'Eigenvalues': out_value, 'Eigenvectors': out_vector}, attrs={'UPLO': UPLO, 'is_test': is_test}) return out_value