# tensor.nlinalg – Linear Algebra Ops Using Numpy¶

Note

This module is not imported by default. You need to import it to use it.

## API¶

class theano.tensor.nlinalg.AllocDiag[source]

Allocates a square matrix with the given vector as its diagonal.

make_node(_x)[source]

Create a “apply” nodes for the inputs in that order.

perform(node, inputs, outputs)[source]

Required: Calculate the function on the inputs and put the variables in the output storage. Return None.

Parameters
• node (Apply instance) – Contains the symbolic inputs and outputs.

• inputs (list) – Sequence of inputs (immutable).

• output_storage (list) – List of mutable 1-element lists (do not change the length of these lists)

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a Numpy ndarray, with the right number of dimensions, and the correct dtype. Its shape and stride pattern, can be arbitrary. It not is guaranteed that it was produced by a previous call to impl. It could be allocated by another Op impl is free to reuse it as it sees fit, or to discard it and allocate new memory.

Raises

MethodNotDefined – The subclass does not override this method.

class theano.tensor.nlinalg.Det[source]

Matrix determinant. Input should be a square matrix.

make_node(x)[source]

Create a “apply” nodes for the inputs in that order.

perform(node, inputs, outputs)[source]

Required: Calculate the function on the inputs and put the variables in the output storage. Return None.

Parameters
• node (Apply instance) – Contains the symbolic inputs and outputs.

• inputs (list) – Sequence of inputs (immutable).

• output_storage (list) – List of mutable 1-element lists (do not change the length of these lists)

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a Numpy ndarray, with the right number of dimensions, and the correct dtype. Its shape and stride pattern, can be arbitrary. It not is guaranteed that it was produced by a previous call to impl. It could be allocated by another Op impl is free to reuse it as it sees fit, or to discard it and allocate new memory.

Raises

MethodNotDefined – The subclass does not override this method.

class theano.tensor.nlinalg.Eig[source]

Compute the eigenvalues and right eigenvectors of a square array.

make_node(x)[source]

Create a “apply” nodes for the inputs in that order.

perform(node, inputs, outputs)[source]

Required: Calculate the function on the inputs and put the variables in the output storage. Return None.

Parameters
• node (Apply instance) – Contains the symbolic inputs and outputs.

• inputs (list) – Sequence of inputs (immutable).

• output_storage (list) – List of mutable 1-element lists (do not change the length of these lists)

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a Numpy ndarray, with the right number of dimensions, and the correct dtype. Its shape and stride pattern, can be arbitrary. It not is guaranteed that it was produced by a previous call to impl. It could be allocated by another Op impl is free to reuse it as it sees fit, or to discard it and allocate new memory.

Raises

MethodNotDefined – The subclass does not override this method.

class theano.tensor.nlinalg.Eigh(UPLO='L')[source]

Return the eigenvalues and eigenvectors of a Hermitian or symmetric matrix.

where [, ] corresponds to g_outputs, to inputs, and .

Analytic formulae for eigensystem gradients are well-known in perturbation theory:

make_node(x)[source]

Create a “apply” nodes for the inputs in that order.

perform(node, inputs, outputs)[source]

Required: Calculate the function on the inputs and put the variables in the output storage. Return None.

Parameters
• node (Apply instance) – Contains the symbolic inputs and outputs.

• inputs (list) – Sequence of inputs (immutable).

• output_storage (list) – List of mutable 1-element lists (do not change the length of these lists)

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a Numpy ndarray, with the right number of dimensions, and the correct dtype. Its shape and stride pattern, can be arbitrary. It not is guaranteed that it was produced by a previous call to impl. It could be allocated by another Op impl is free to reuse it as it sees fit, or to discard it and allocate new memory.

Raises

MethodNotDefined – The subclass does not override this method.

Gradient of an eigensystem of a Hermitian matrix.

make_node(x, w, v, gw, gv)[source]

Create a “apply” nodes for the inputs in that order.

perform(node, inputs, outputs)[source]

Implements the “reverse-mode” gradient for the eigensystem of a square matrix.

class theano.tensor.nlinalg.MatrixInverse[source]

Computes the inverse of a matrix .

Given a square matrix , matrix_inverse returns a square matrix such that the dot product and equals the identity matrix .

Notes

When possible, the call to this op will be optimized to the call of solve.

R_op(inputs, eval_points)[source]

where corresponds to g_outputs and to inputs. Using the matrix cookbook, one can deduce that the relation corresponds to

where corresponds to g_outputs and to inputs. Using the matrix cookbook, one can deduce that the relation corresponds to

make_node(x)[source]

Create a “apply” nodes for the inputs in that order.

perform(node, inputs, outputs)[source]

Required: Calculate the function on the inputs and put the variables in the output storage. Return None.

Parameters
• node (Apply instance) – Contains the symbolic inputs and outputs.

• inputs (list) – Sequence of inputs (immutable).

• output_storage (list) – List of mutable 1-element lists (do not change the length of these lists)

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a Numpy ndarray, with the right number of dimensions, and the correct dtype. Its shape and stride pattern, can be arbitrary. It not is guaranteed that it was produced by a previous call to impl. It could be allocated by another Op impl is free to reuse it as it sees fit, or to discard it and allocate new memory.

Raises

MethodNotDefined – The subclass does not override this method.

class theano.tensor.nlinalg.MatrixPinv[source]

Computes the pseudo-inverse of a matrix .

The pseudo-inverse of a matrix , denoted , is defined as: “the matrix that ‘solves’ [the least-squares problem] ,” i.e., if is said solution, then is that matrix such that .

Note that , so is close to the identity matrix. This method is not faster than matrix_inverse. Its strength comes from that it works for non-square matrices. If you have a square matrix though, matrix_inverse can be both more exact and faster to compute. Also this op does not get optimized into a solve op.

L_op(inputs, outputs, g_outputs)[source]

where corresponds to g_outputs and to inputs. According to Wikipedia, this corresponds to

make_node(x)[source]

Create a “apply” nodes for the inputs in that order.

perform(node, inputs, outputs)[source]

Required: Calculate the function on the inputs and put the variables in the output storage. Return None.

Parameters
• node (Apply instance) – Contains the symbolic inputs and outputs.

• inputs (list) – Sequence of inputs (immutable).

• output_storage (list) – List of mutable 1-element lists (do not change the length of these lists)

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a Numpy ndarray, with the right number of dimensions, and the correct dtype. Its shape and stride pattern, can be arbitrary. It not is guaranteed that it was produced by a previous call to impl. It could be allocated by another Op impl is free to reuse it as it sees fit, or to discard it and allocate new memory.

Raises

MethodNotDefined – The subclass does not override this method.

class theano.tensor.nlinalg.QRFull(mode)[source]

Full QR Decomposition.

Computes the QR decomposition of a matrix. Factor the matrix a as qr, where q is orthonormal and r is upper-triangular.

make_node(x)[source]

Create a “apply” nodes for the inputs in that order.

perform(node, inputs, outputs)[source]

Required: Calculate the function on the inputs and put the variables in the output storage. Return None.

Parameters
• node (Apply instance) – Contains the symbolic inputs and outputs.

• inputs (list) – Sequence of inputs (immutable).

• output_storage (list) – List of mutable 1-element lists (do not change the length of these lists)

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a Numpy ndarray, with the right number of dimensions, and the correct dtype. Its shape and stride pattern, can be arbitrary. It not is guaranteed that it was produced by a previous call to impl. It could be allocated by another Op impl is free to reuse it as it sees fit, or to discard it and allocate new memory.

Raises

MethodNotDefined – The subclass does not override this method.

class theano.tensor.nlinalg.QRIncomplete(mode)[source]

Incomplete QR Decomposition.

Computes the QR decomposition of a matrix. Factor the matrix a as qr and return a single matrix R.

make_node(x)[source]

Create a “apply” nodes for the inputs in that order.

perform(node, inputs, outputs)[source]

Required: Calculate the function on the inputs and put the variables in the output storage. Return None.

Parameters
• node (Apply instance) – Contains the symbolic inputs and outputs.

• inputs (list) – Sequence of inputs (immutable).

• output_storage (list) – List of mutable 1-element lists (do not change the length of these lists)

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a Numpy ndarray, with the right number of dimensions, and the correct dtype. Its shape and stride pattern, can be arbitrary. It not is guaranteed that it was produced by a previous call to impl. It could be allocated by another Op impl is free to reuse it as it sees fit, or to discard it and allocate new memory.

Raises

MethodNotDefined – The subclass does not override this method.

class theano.tensor.nlinalg.SVD(full_matrices=True, compute_uv=True)[source]
Parameters
• full_matrices (bool, optional) – If True (default), u and v have the shapes (M, M) and (N, N), respectively. Otherwise, the shapes are (M, K) and (K, N), respectively, where K = min(M, N).

• compute_uv (bool, optional) – Whether or not to compute u and v in addition to s. True by default.

make_node(x)[source]

Create a “apply” nodes for the inputs in that order.

perform(node, inputs, outputs)[source]

Required: Calculate the function on the inputs and put the variables in the output storage. Return None.

Parameters
• node (Apply instance) – Contains the symbolic inputs and outputs.

• inputs (list) – Sequence of inputs (immutable).

• output_storage (list) – List of mutable 1-element lists (do not change the length of these lists)

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a Numpy ndarray, with the right number of dimensions, and the correct dtype. Its shape and stride pattern, can be arbitrary. It not is guaranteed that it was produced by a previous call to impl. It could be allocated by another Op impl is free to reuse it as it sees fit, or to discard it and allocate new memory.

Raises

MethodNotDefined – The subclass does not override this method.

class theano.tensor.nlinalg.TensorInv(ind=2)[source]

Class wrapper for tensorinv() function; Theano utilization of numpy.linalg.tensorinv;

make_node(a)[source]

Create a “apply” nodes for the inputs in that order.

perform(node, inputs, outputs)[source]

Required: Calculate the function on the inputs and put the variables in the output storage. Return None.

Parameters
• node (Apply instance) – Contains the symbolic inputs and outputs.

• inputs (list) – Sequence of inputs (immutable).

• output_storage (list) – List of mutable 1-element lists (do not change the length of these lists)

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a Numpy ndarray, with the right number of dimensions, and the correct dtype. Its shape and stride pattern, can be arbitrary. It not is guaranteed that it was produced by a previous call to impl. It could be allocated by another Op impl is free to reuse it as it sees fit, or to discard it and allocate new memory.

Raises

MethodNotDefined – The subclass does not override this method.

class theano.tensor.nlinalg.TensorSolve(axes=None)[source]

Theano utilization of numpy.linalg.tensorsolve Class wrapper for tensorsolve function.

make_node(a, b)[source]

Create a “apply” nodes for the inputs in that order.

perform(node, inputs, outputs)[source]

Required: Calculate the function on the inputs and put the variables in the output storage. Return None.

Parameters
• node (Apply instance) – Contains the symbolic inputs and outputs.

• inputs (list) – Sequence of inputs (immutable).

• output_storage (list) – List of mutable 1-element lists (do not change the length of these lists)

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a Numpy ndarray, with the right number of dimensions, and the correct dtype. Its shape and stride pattern, can be arbitrary. It not is guaranteed that it was produced by a previous call to impl. It could be allocated by another Op impl is free to reuse it as it sees fit, or to discard it and allocate new memory.

Raises

MethodNotDefined – The subclass does not override this method.

theano.tensor.nlinalg.diag(x)[source]

Numpy-compatibility method If x is a matrix, return its diagonal. If x is a vector return a matrix with it as its diagonal.

• This method does not support the k argument that numpy supports.

class theano.tensor.nlinalg.lstsq[source]
make_node(x, y, rcond)[source]

Create a “apply” nodes for the inputs in that order.

perform(node, inputs, outputs)[source]

Required: Calculate the function on the inputs and put the variables in the output storage. Return None.

Parameters
• node (Apply instance) – Contains the symbolic inputs and outputs.

• inputs (list) – Sequence of inputs (immutable).

• output_storage (list) – List of mutable 1-element lists (do not change the length of these lists)

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a Numpy ndarray, with the right number of dimensions, and the correct dtype. Its shape and stride pattern, can be arbitrary. It not is guaranteed that it was produced by a previous call to impl. It could be allocated by another Op impl is free to reuse it as it sees fit, or to discard it and allocate new memory.

Raises

MethodNotDefined – The subclass does not override this method.

theano.tensor.nlinalg.matrix_dot(*args)[source]

Shorthand for product between several dots.

Given matrices , matrix_dot will generate the matrix product between all in the given order, namely .

theano.tensor.nlinalg.matrix_power(M, n)[source]

Raise a square matrix to the (integer) power n.

Parameters
• M (Tensor variable) –

• n (Python int) –

theano.tensor.nlinalg.qr(a, mode='reduced')[source]

Computes the QR decomposition of a matrix. Factor the matrix a as qr, where q is orthonormal and r is upper-triangular.

Parameters
• a (array_like, shape (M, N)) – Matrix to be factored.

• mode ({'reduced', 'complete', 'r', 'raw'}, optional) –

If K = min(M, N), then

’reduced’

returns q, r with dimensions (M, K), (K, N)

’complete’

returns q, r with dimensions (M, M), (M, N)

’r’

returns r only with dimensions (K, N)

’raw’

returns h, tau with dimensions (N, M), (K,)

Note that array h returned in ‘raw’ mode is transposed for calling Fortran.

Default mode is ‘reduced’

Returns

• q (matrix of float or complex, optional) – A matrix with orthonormal columns. When mode = ‘complete’ the result is an orthogonal/unitary matrix depending on whether or not a is real/complex. The determinant may be either +/- 1 in that case.

• r (matrix of float or complex, optional) – The upper-triangular matrix.

theano.tensor.nlinalg.svd(a, full_matrices=1, compute_uv=1)[source]

This function performs the SVD on CPU.

Parameters
• full_matrices (bool, optional) – If True (default), u and v have the shapes (M, M) and (N, N), respectively. Otherwise, the shapes are (M, K) and (K, N), respectively, where K = min(M, N).

• compute_uv (bool, optional) – Whether or not to compute u and v in addition to s. True by default.

Returns

U, V, D

Return type

matrices

theano.tensor.nlinalg.tensorinv(a, ind=2)[source]

Does not run on GPU; Theano utilization of numpy.linalg.tensorinv;

Compute the ‘inverse’ of an N-dimensional array. The result is an inverse for a relative to the tensordot operation tensordot(a, b, ind), i. e., up to floating-point accuracy, tensordot(tensorinv(a), a, ind) is the “identity” tensor for the tensordot operation.

Parameters
• a (array_like) – Tensor to ‘invert’. Its shape must be ‘square’, i. e., prod(a.shape[:ind]) == prod(a.shape[ind:]).

• ind (int, optional) – Number of first indices that are involved in the inverse sum. Must be a positive integer, default is 2.

Returns

ba’s tensordot inverse, shape a.shape[ind:] + a.shape[:ind].

Return type

ndarray

Raises

LinAlgError – If a is singular or not ‘square’ (in the above sense).

theano.tensor.nlinalg.tensorsolve(a, b, axes=None)[source]

Theano utilization of numpy.linalg.tensorsolve. Does not run on GPU!

Solve the tensor equation a x = b for x. It is assumed that all indices of x are summed over in the product, together with the rightmost indices of a, as is done in, for example, tensordot(a, x, axes=len(b.shape)).

Parameters
• a (array_like) – Coefficient tensor, of shape b.shape + Q. Q, a tuple, equals the shape of that sub-tensor of a consisting of the appropriate number of its rightmost indices, and must be such that prod(Q) == prod(b.shape) (in which sense a is said to be ‘square’).

• b (array_like) – Right-hand tensor, which can be of any shape.

• axes (tuple of ints, optional) – Axes in a to reorder to the right, before inversion. If None (default), no reordering is done.

Returns

x

Return type

ndarray, shape Q

Raises

LinAlgError – If a is singular or not ‘square’ (in the above sense).

theano.tensor.nlinalg.trace(X)[source]

Returns the sum of diagonal elements of matrix X.

Notes

Works on GPU since 0.6rc4.