neml2::math Namespace Reference

Namespaces
namespace	linalg

Classes
struct	ConstantTensors
	A helper class to hold static data of type torch::Tensor. More...

Functions
BatchTensor	full_to_reduced (const BatchTensor &full, const torch::Tensor &rmap, const torch::Tensor &rfactors, TorchSize dim=0)
	Generic function to reduce two axes to one with some map.
BatchTensor	reduced_to_full (const BatchTensor &reduced, const torch::Tensor &rmap, const torch::Tensor &rfactors, TorchSize dim=0)
	Convert a BatchTensor from reduced notation to full notation.
BatchTensor	full_to_mandel (const BatchTensor &full, TorchSize dim=0)
	Convert a BatchTensor from full notation to Mandel notation.
BatchTensor	mandel_to_full (const BatchTensor &mandel, TorchSize dim=0)
	Convert a BatchTensor from Mandel notation to full notation.
BatchTensor	full_to_skew (const BatchTensor &full, TorchSize dim=0)
	Convert a BatchTensor from full notation to skew vector notation.
BatchTensor	skew_to_full (const BatchTensor &skew, TorchSize dim=0)
	Convert a BatchTensor from skew vector notation to full notation.
BatchTensor	jacrev (const BatchTensor &y, const BatchTensor &p)
	Use automatic differentiation (AD) to calculate the derivatives w.r.t. to the parameter.
BatchTensor	base_diag_embed (const BatchTensor &a, TorchSize offset, TorchSize d1, TorchSize d2)
SR2	skew_and_sym_to_sym (const SR2 &e, const WR2 &w)
	Product w_ik e_kj - e_ik w_kj with e SR2 and w WR2.
SSR4	d_skew_and_sym_to_sym_d_sym (const WR2 &w)
	Derivative of w_ik e_kj - e_ik w_kj wrt. e.
SWR4	d_skew_and_sym_to_sym_d_skew (const SR2 &e)
	Derivative of w_ik e_kj - e_ik w_kj wrt. w.
WR2	multiply_and_make_skew (const SR2 &a, const SR2 &b)
	Shortcut product a_ik b_kj - b_ik a_kj with both SR2.
WSR4	d_multiply_and_make_skew_d_first (const SR2 &b)
	Derivative of a_ik b_kj - b_ik a_kj wrt a.
WSR4	d_multiply_and_make_skew_d_second (const SR2 &a)
	Derivative of a_ik b_kj - b_ik a_kj wrt b.
BatchTensor	bmm (const BatchTensor &a, const BatchTensor &b)
	Batched matrix-matrix product.
BatchTensor	bmv (const BatchTensor &a, const BatchTensor &v)
	Batched matrix-vector product.
BatchTensor	bvv (const BatchTensor &a, const BatchTensor &b)
	Batched vector-vector (dot) product.
constexpr Real	mandel_factor (TorchSize i)
template<class Derived , typename = typename std::enable_if_t<std::is_base_of_v<BatchTensorBase<Derived>, Derived>>>
Derived	pow (const Derived &a, const Real &n)
template<class Derived , typename = typename std::enable_if_t<std::is_base_of_v<BatchTensorBase<Derived>, Derived>>>
Derived	pow (const Real &a, const Derived &n)
template<class Derived , typename = typename std::enable_if_t<std::is_base_of_v<BatchTensorBase<Derived>, Derived>>>
Derived	pow (const Derived &a, const Derived &n)
template<class Derived , typename = typename std::enable_if_t<std::is_base_of_v<BatchTensorBase<Derived>, Derived>>>
Derived	sign (const Derived &a)
template<class Derived , typename = typename std::enable_if_t<std::is_base_of_v<BatchTensorBase<Derived>, Derived>>>
Derived	cosh (const Derived &a)
template<class Derived , typename = typename std::enable_if_t<std::is_base_of_v<BatchTensorBase<Derived>, Derived>>>
Derived	sinh (const Derived &a)
template<class Derived , typename = typename std::enable_if_t<std::is_base_of_v<BatchTensorBase<Derived>, Derived>>>
Derived	tanh (const Derived &a)
template<class Derived , typename = typename std::enable_if_t<std::is_base_of_v<BatchTensorBase<Derived>, Derived>>>
Derived	where (const torch::Tensor &condition, const Derived &a, const Derived &b)
template<class Derived , typename = typename std::enable_if_t<std::is_base_of_v<BatchTensorBase<Derived>, Derived>>>
Derived	heaviside (const Derived &a)
template<class Derived , typename = typename std::enable_if_t<std::is_base_of_v<BatchTensorBase<Derived>, Derived>>>
Derived	macaulay (const Derived &a)
template<class Derived , typename = typename std::enable_if_t<std::is_base_of_v<BatchTensorBase<Derived>, Derived>>>
Derived	dmacaulay (const Derived &a)
template<class Derived , typename = typename std::enable_if_t<std::is_base_of_v<BatchTensorBase<Derived>, Derived>>>
Derived	sqrt (const Derived &a)
template<class Derived , typename = typename std::enable_if_t<std::is_base_of_v<BatchTensorBase<Derived>, Derived>>>
Derived	exp (const Derived &a)
template<class Derived , typename = typename std::enable_if_t<std::is_base_of_v<BatchTensorBase<Derived>, Derived>>>
Derived	abs (const Derived &a)
template<class Derived , typename = typename std::enable_if_t<std::is_base_of_v<BatchTensorBase<Derived>, Derived>>>
Derived	diff (const Derived &a, TorchSize n=1, TorchSize dim=-1)
template<class Derived , typename = typename std::enable_if_t<std::is_base_of_v<BatchTensorBase<Derived>, Derived>>>
Derived	batch_diag_embed (const Derived &a, TorchSize offset=0, TorchSize d1=-2, TorchSize d2=-1)
template<class Derived , typename = typename std::enable_if_t<std::is_base_of_v<BatchTensorBase<Derived>, Derived>>>
Derived	log (const Derived &a)
template<class Derived , typename = typename std::enable_if_t<!std::is_same_v<Derived, Scalar>>, typename = typename std::enable_if_t<std::is_base_of_v<BatchTensorBase<Derived>, Derived>>>
Derived	pow (const Derived &a, const Scalar &n)
template<class Derived , typename = typename std::enable_if_t<!std::is_same_v<Derived, Scalar>>, typename = typename std::enable_if_t<std::is_base_of_v<BatchTensorBase<Derived>, Derived>>>
Derived	pow (const Scalar &a, const Derived &n)

Variables
constexpr Real	eps = std::numeric_limits<at::scalar_value_type<Real>::type>::epsilon()
constexpr Real	sqrt2 = 1.4142135623730951
constexpr Real	invsqrt2 = 0.7071067811865475
constexpr TorchSize	mandel_reverse_index [3][3] = {{0, 5, 4}, {5, 1, 3}, {4, 3, 2}}
constexpr TorchSize	mandel_index [6][2] = {{0, 0}, {1, 1}, {2, 2}, {1, 2}, {0, 2}, {0, 1}}
constexpr TorchSize	skew_reverse_index [3][3] = {{0, 2, 1}, {2, 0, 0}, {1, 0, 0}}
constexpr Real	skew_factor [3][3] = {{0.0, -1.0, 1.0}, {1.0, 0.0, -1.0}, {-1.0, 1.0, 0.0}}

Function Documentation

abs()

template<class Derived , typename = typename std::enable_if_t<std::is_base_of_v<BatchTensorBase<Derived>, Derived>>>

Derived abs

(

const Derived &

a

)

base_diag_embed()

BatchTensor base_diag_embed	(	const BatchTensor &	a,
		TorchSize	offset,
		TorchSize	d1,
		TorchSize	d2 )

batch_diag_embed()

template<class Derived , typename = typename std::enable_if_t<std::is_base_of_v<BatchTensorBase<Derived>, Derived>>>

Derived batch_diag_embed	(	const Derived &	a,
		TorchSize	offset = 0,
		TorchSize	d1 = -2,
		TorchSize	d2 = -1 )

bmm()

BatchTensor bmm	(	const BatchTensor &	a,
		const BatchTensor &	b )

Batched matrix-matrix product.

The input matrices a and b must have exactly 2 base dimensions. The batch shapes must broadcast.

bmv()

BatchTensor bmv	(	const BatchTensor &	a,
		const BatchTensor &	v )

Batched matrix-vector product.

The input tensor a must have exactly 2 base dimensions. The input tensor v must have exactly 1 base dimension. The batch shapes must broadcast.

bvv()

BatchTensor bvv	(	const BatchTensor &	a,
		const BatchTensor &	b )

Batched vector-vector (dot) product.

The input tensor a must have exactly 1 base dimension. The input tensor vbmust have exactly 1 base dimension. The batch shapes must broadcast.

cosh()

template<class Derived , typename = typename std::enable_if_t<std::is_base_of_v<BatchTensorBase<Derived>, Derived>>>

Derived cosh

(

const Derived &

a

)

d_multiply_and_make_skew_d_first()

WSR4 d_multiply_and_make_skew_d_first

(

const SR2 &

b

)

Derivative of a_ik b_kj - b_ik a_kj wrt a.

d_multiply_and_make_skew_d_second()

WSR4 d_multiply_and_make_skew_d_second

(

const SR2 &

a

)

Derivative of a_ik b_kj - b_ik a_kj wrt b.

d_skew_and_sym_to_sym_d_skew()

SWR4 d_skew_and_sym_to_sym_d_skew

(

const SR2 &

e

)

Derivative of w_ik e_kj - e_ik w_kj wrt. w.

d_skew_and_sym_to_sym_d_sym()

SSR4 d_skew_and_sym_to_sym_d_sym

(

const WR2 &

w

)

Derivative of w_ik e_kj - e_ik w_kj wrt. e.

diff()

template<class Derived , typename = typename std::enable_if_t<std::is_base_of_v<BatchTensorBase<Derived>, Derived>>>

Derived diff	(	const Derived &	a,
		TorchSize	n = 1,
		TorchSize	dim = -1 )

dmacaulay()

template<class Derived , typename = typename std::enable_if_t<std::is_base_of_v<BatchTensorBase<Derived>, Derived>>>

Derived dmacaulay

(

const Derived &

a

)

exp()

template<class Derived , typename = typename std::enable_if_t<std::is_base_of_v<BatchTensorBase<Derived>, Derived>>>

Derived exp

(

const Derived &

a

)

full_to_mandel()

BatchTensor full_to_mandel	(	const BatchTensor &	full,
		TorchSize	dim = 0 )

Convert a BatchTensor from full notation to Mandel notation.

The tensor in full notation full can have arbitrary batch shape. The optional argument dim denotes the base dimension starting from which the conversion should take place.

For example, a full tensor has shape (2, 3, 1, 5; 2, 9, 3, 3, 2, 3) where the semicolon separates batch and base shapes. The symmetric axes have base dim 2 and 3. After converting to Mandel notation, the resulting tensor will have shape (2, 3, 1, 5; 2, 9, 6, 2, 3). Note how the shape of the symmetric dimensions (3, 3) becomes (6). In this example, the base dim (the second argument to this function) should be 2.

Parameters

full	The input tensor in full notation
dim	The base dimension where the symmetric axes start

Returns

BatchTensor The resulting tensor using Mandel notation to represent the symmetric axes.

full_to_reduced()

BatchTensor full_to_reduced	(	const BatchTensor &	full,
		const torch::Tensor &	rmap,
		const torch::Tensor &	rfactors,
		TorchSize	dim = 0 )

Generic function to reduce two axes to one with some map.

The tensor in full notation full can have arbitrary batch shape. The optional argument dim denotes the base dimension starting from which the conversion should take place.

The function will reduce the two axis at the desired location down to one, using the provided maps.

For example, a full tensor has shape (2, 3, 1, 5; 2, 9, 3, 3, 2, 3) where the semicolon separates batch and base shapes. The reduction axes have base dim 2 and 3. After applying the reduction, the resulting tensor will have shape (2, 3, 1, 5; 2, 9, X, 2, 3) where X is the reduced shape. In this example, the base dim (the second argument to this function) should be 2.

Parameters

full	The input tensor in full notation
rmap	The reduction map
rfactors	The reduction factors
dim	The base dimension where the reduced axes start

Returns

BatchTensor The reduced tensor

full_to_skew()

BatchTensor full_to_skew	(	const BatchTensor &	full,
		TorchSize	dim = 0 )

Convert a BatchTensor from full notation to skew vector notation.

The tensor in full notation full can have arbitrary batch shape. The optional argument dim denotes the base dimension starting from which the conversion should take place.

For example, a full tensor has shape (2, 3, 1, 5; 2, 9, 3, 3, 2, 3) where the semicolon separates batch and base shapes. The symmetric axes have base dim 2 and 3. After converting to skew notation, the resulting tensor will have shape (2, 3, 1, 5; 2, 9, 3, 2, 3). Note how the shape of the symmetric dimensions (3, 3) becomes (3). In this example, the base dim (the second argument to this function) should be 2.

Parameters

full	The input tensor in full notation
dim	The base dimension where the symmetric axes start

Returns

BatchTensor The resulting tensor using skew notation to represent the skew-symmetric axes.

heaviside()

template<class Derived , typename = typename std::enable_if_t<std::is_base_of_v<BatchTensorBase<Derived>, Derived>>>

Derived heaviside

(

const Derived &

a

)

This is (almost) equivalent to Torch's heaviside, except that the Torch's version is not differentiable (back-propagatable). I said "almost" because torch::heaviside allows you to set the return value in the case of input == 0. Our implementation always return 0.5 when the input == 0.

jacrev()

BatchTensor jacrev	(	const BatchTensor &	y,
		const BatchTensor &	p )

Use automatic differentiation (AD) to calculate the derivatives w.r.t. to the parameter.

Warning

Torch (and hence NEML2) AD wasn't designed to compute the full Jacobian from the very beginning. Using this method to calculate the full Jacobian is inefficient and is subjected to some restrictions on batch shapes: This method will only work when the output y and the paramter p have the same batch shape.

However, in practice, the batch shape of the output y and the batch shape of the parameter p can be different. In that case, calculating the full Jacobian is not possible, and an exception will be thrown.

One possible (inefficient) workaround is to expand and copy the parameter p batch dimensions, e.g., batch_expand_copy, before calculating the output y.

Parameters

y	The BatchTensor to to be differentiated
p	The parameter to take derivatives with respect to

Returns

BatchTensor \(\partial y/\partial p\)

log()

template<class Derived , typename = typename std::enable_if_t<std::is_base_of_v<BatchTensorBase<Derived>, Derived>>>

Derived log

(

const Derived &

a

)

macaulay()

template<class Derived , typename = typename std::enable_if_t<std::is_base_of_v<BatchTensorBase<Derived>, Derived>>>

Derived macaulay

(

const Derived &

a

)

mandel_factor()

constexpr Real mandel_factor ( TorchSize i )

inlineconstexpr

mandel_to_full()

BatchTensor mandel_to_full	(	const BatchTensor &	mandel,
		TorchSize	dim = 0 )

Convert a BatchTensor from Mandel notation to full notation.

See full_to_mandel for a detailed explanation.

Parameters

mandel	The input tensor in Mandel notation
dim	The base dimension where the symmetric axes start

Returns

BatchTensor The resulting tensor in full notation.

multiply_and_make_skew()

WR2 multiply_and_make_skew	(	const SR2 &	a,
		const SR2 &	b )

Shortcut product a_ik b_kj - b_ik a_kj with both SR2.

pow() [1/5]

template<class Derived , typename = typename std::enable_if_t<std::is_base_of_v<BatchTensorBase<Derived>, Derived>>>

Derived pow	(	const Derived &	a,
		const Derived &	n )

pow() [2/5]

template<class Derived , typename = typename std::enable_if_t<std::is_base_of_v<BatchTensorBase<Derived>, Derived>>>

Derived pow	(	const Derived &	a,
		const Real &	n )

pow() [3/5]

template<class Derived , typename = typename std::enable_if_t<!std::is_same_v<Derived, Scalar>>, typename = typename std::enable_if_t<std::is_base_of_v<BatchTensorBase<Derived>, Derived>>>

Derived pow	(	const Derived &	a,
		const Scalar &	n )

pow() [4/5]

template<class Derived , typename = typename std::enable_if_t<std::is_base_of_v<BatchTensorBase<Derived>, Derived>>>

Derived pow	(	const Real &	a,
		const Derived &	n )

pow() [5/5]

template<class Derived , typename = typename std::enable_if_t<!std::is_same_v<Derived, Scalar>>, typename = typename std::enable_if_t<std::is_base_of_v<BatchTensorBase<Derived>, Derived>>>

Derived pow	(	const Scalar &	a,
		const Derived &	n )

reduced_to_full()

BatchTensor reduced_to_full	(	const BatchTensor &	reduced,
		const torch::Tensor &	rmap,
		const torch::Tensor &	rfactors,
		TorchSize	dim = 0 )

Convert a BatchTensor from reduced notation to full notation.

See full_to_reduced for a detailed explanation.

Parameters

reduced	The input tensor in reduced notation
rmap	The unreduction map
rfactors	The unreduction factors
dim	The base dimension where the reduced axes start

Returns

BatchTensor The resulting tensor in full notation.

sign()

template<class Derived , typename = typename std::enable_if_t<std::is_base_of_v<BatchTensorBase<Derived>, Derived>>>

Derived sign

(

const Derived &

a

)

sinh()

template<class Derived , typename = typename std::enable_if_t<std::is_base_of_v<BatchTensorBase<Derived>, Derived>>>

Derived sinh

(

const Derived &

a

)

skew_and_sym_to_sym()

SR2 skew_and_sym_to_sym	(	const SR2 &	e,
		const WR2 &	w )

Product w_ik e_kj - e_ik w_kj with e SR2 and w WR2.

skew_to_full()

BatchTensor skew_to_full	(	const BatchTensor &	skew,
		TorchSize	dim = 0 )

Convert a BatchTensor from skew vector notation to full notation.

See full_to_skew for a detailed explanation.

Parameters

skew	The input tensor in skew notation
dim	The base dimension where the symmetric axes start

Returns

BatchTensor The resulting tensor in full notation.

sqrt()

template<class Derived , typename = typename std::enable_if_t<std::is_base_of_v<BatchTensorBase<Derived>, Derived>>>

Derived sqrt

(

const Derived &

a

)

tanh()

template<class Derived , typename = typename std::enable_if_t<std::is_base_of_v<BatchTensorBase<Derived>, Derived>>>

Derived tanh

(

const Derived &

a

)

where()

template<class Derived , typename = typename std::enable_if_t<std::is_base_of_v<BatchTensorBase<Derived>, Derived>>>

Derived where	(	const torch::Tensor &	condition,
		const Derived &	a,
		const Derived &	b )

Variable Documentation

eps

constexpr Real eps = std::numeric_limits<at::scalar_value_type<Real>::type>::epsilon()

constexpr

invsqrt2

constexpr Real invsqrt2 = 0.7071067811865475

constexpr

mandel_index

constexpr TorchSize mandel_index[6][2] = {{0, 0}, {1, 1}, {2, 2}, {1, 2}, {0, 2}, {0, 1}}

constexpr

mandel_reverse_index

constexpr TorchSize mandel_reverse_index[3][3] = {{0, 5, 4}, {5, 1, 3}, {4, 3, 2}}

constexpr

skew_factor

constexpr Real skew_factor[3][3] = {{0.0, -1.0, 1.0}, {1.0, 0.0, -1.0}, {-1.0, 1.0, 0.0}}

constexpr

skew_reverse_index

constexpr TorchSize skew_reverse_index[3][3] = {{0, 2, 1}, {2, 0, 0}, {1, 0, 0}}

constexpr

sqrt2

constexpr Real sqrt2 = 1.4142135623730951

constexpr