Linear algebra

Collaboration diagram for Linear algebra:

Modules
	Decompositions
	Linear equation
template<typename MatrixType >
MatrixType::value_type	mptensor::trace (const Tensor< MatrixType > &M)
	Trace of matrix (rank-2 tensor)
template<typename MatrixType >
MatrixType::value_type	mptensor::trace (const Tensor< MatrixType > &T, const Axes &axes_1, const Axes &axes_2)
	Full trace of tensor.
template<typename MatrixType >
MatrixType::value_type	mptensor::trace (const Tensor< MatrixType > &A, const Tensor< MatrixType > &B, const Axes &axes_a, const Axes &axes_b)
	Full contraction of two tensors.
template<typename MatrixType >
Tensor< MatrixType >	mptensor::contract (const Tensor< MatrixType > &T, const Axes &axes_1, const Axes &axes_2)
	Partial trace of tensor.
template<typename MatrixType >
Tensor< MatrixType >	mptensor::tensordot (const Tensor< MatrixType > &a, const Tensor< MatrixType > &b, const Axes &axes_a, const Axes &axes_b)
	Compute tensor dot product.
template<typename MatrixType >
Tensor< MatrixType >	mptensor::kron (const Tensor< MatrixType > &a, const Tensor< MatrixType > &b)
	Compute the Kronecker product.

Detailed Description

Operations for linear algebra.

Function Documentation

contract()

template<typename MatrixType >

Tensor< MatrixType > mptensor::contract	(	const Tensor< MatrixType > &	T,
		const Axes &	axes_1,
		const Axes &	axes_2
	)

Partial trace of tensor.

axes_1[k] and axes_2[k] are contracted. For example.

B = trace(A, Axes(0,2), Axes(5,4)) 

returns \( B_{ab} = \sum_{ij} A_{iajbji} \).

Parameters

T	Tensor to partially trace.
axes_1	Axes to trace.
axes_2	Axes to trace.

Returns

Contracted tensor.

kron()

template<typename MatrixType >

Tensor< MatrixType > mptensor::kron	(	const Tensor< MatrixType > &	a,
		const Tensor< MatrixType > &	b
	)

Compute the Kronecker product.

Parameters

a	Tensor to product.
b	Tensor to product.

Returns

The Kronecker product of a and b.

If A.shape() = A[r0,...,rN] and B.shape() = B[s0,...,sN], the Kronecker product has shape [r0*s0,..., rN*sN]. Each element is given as

kron(A, B)[k0,...,kN] = A[i0,...,iN] * B[j0,...,jN] 

where

kt = it + jt * st 

Note

The rank of a and b should be the same.

Warning

To be consistent with the reshape operation, the new indices of the Kronecker product are determined by the column major, unlike in NumPy. Thus, if the shapes of A and B are [r0, r1] and [s0, s1], the Kronecker product of A and B satisfies

reshape(kron(A, B), [r0, s0, r1, s1])[i, j, k, l] = A[i, k] * B[j, l] 

tensordot()

template<typename MatrixType >

Tensor< MatrixType > mptensor::tensordot	(	const Tensor< MatrixType > &	a,
		const Tensor< MatrixType > &	b,
		const Axes &	axes_a,
		const Axes &	axes_b
	)

Compute tensor dot product.

For example, \( T_{abcd} = \sum_{ij} A_{iajb} B_{cjdi} \) is

T = tensordot(A, B, Axes(0,2), Axes(3,1)); 

Parameters

a	Tensor.
b	Tensor.
axes_a	Axes of tensor a to contract.
axes_b	Axes of tensor b to contract.

Returns

Result.

trace() [1/3]

template<typename MatrixType >

MatrixType::value_type mptensor::trace	(	const Tensor< MatrixType > &	A,
		const Tensor< MatrixType > &	B,
		const Axes &	axes_a,
		const Axes &	axes_b
	)

Full contraction of two tensors.

axes_a[k] and axes_b[k] are contracted. For example,

trace(A, B, Axes(0,1,2), Axes(2,0,1)) 

returns \( \sum_{ijk} A_{ijk} B_{kij} \).

Parameters

A	Tensor to contract.
B	Tensor to contract.
axes_a	Axes order of tensor A.
axes_b	Axes order of tensor B.

Returns

Value of full contraction.

Note

This function is more effective than tensordot and MPI_Bcast.

trace() [2/3]

template<typename MatrixType >

MatrixType::value_type mptensor::trace

(

const Tensor< MatrixType > &

M

)

Trace of matrix (rank-2 tensor)

Parameters

M	Rank-2 tensor

Returns

Value of trace. \( \sum_i M_{ii} \)

trace() [3/3]

template<typename MatrixType >

MatrixType::value_type mptensor::trace	(	const Tensor< MatrixType > &	T,
		const Axes &	axes_1,
		const Axes &	axes_2
	)

Full trace of tensor.

axes_1[k] and axes_2[k] are contracted. For example,

trace(T, Axes(0,1,3), Axes(2,4,5))

returns \( \sum_{ijk} T_{ijikjk}\).

Parameters

T	Tensor to trace.
axes_1	Axes of tensor.
axes_2	Axes of tensor.

Returns

Value of tensor.