Decompositions

template<template< typename > class Matrix, typename C >
int	mptensor::svd (const Tensor< Matrix, C > &a, std::vector< double > &s)
	Singular value decomposition for rank-2 tensor (matrix) More...
template<template< typename > class Matrix, typename C >
int	mptensor::svd (const Tensor< Matrix, C > &a, Tensor< Matrix, C > &u, std::vector< double > &s, Tensor< Matrix, C > &vt)
	Singular value decomposition for rank-2 tensor (matrix) More...
template<template< typename > class Matrix, typename C >
int	mptensor::svd (const Tensor< Matrix, C > &a, const Axes &axes_row, const Axes &axes_col, std::vector< double > &s)
	Singular value decomposition for tensor. More...
template<template< typename > class Matrix, typename C >
int	mptensor::svd (const Tensor< Matrix, C > &a, const Axes &axes_row, const Axes &axes_col, Tensor< Matrix, C > &u, std::vector< double > &s, Tensor< Matrix, C > &vt)
	Singular value decomposition for tensor. More...
template<template< typename > class Matrix, typename C >
int	mptensor::psvd (const Tensor< Matrix, C > &a, std::vector< double > &s, const size_t target_rank)
	Partial SVD for rank-2 tensor (matrix) More...
template<template< typename > class Matrix, typename C >
int	mptensor::psvd (const Tensor< Matrix, C > &a, Tensor< Matrix, C > &u, std::vector< double > &s, Tensor< Matrix, C > &vt, const size_t target_rank)
	Partial SVD for rank-2 tensor (matrix) More...
template<template< typename > class Matrix, typename C >
int	mptensor::psvd (const Tensor< Matrix, C > &a, const Axes &axes_row, const Axes &axes_col, std::vector< double > &s, const size_t target_rank)
	Partial SVD for tensor. More...
template<template< typename > class Matrix, typename C >
int	mptensor::psvd (const Tensor< Matrix, C > &a, const Axes &axes_row, const Axes &axes_col, Tensor< Matrix, C > &u, std::vector< double > &s, Tensor< Matrix, C > &vt, const size_t target_rank)
	Partial SVD for tensor. More...
template<template< typename > class Matrix, typename C >
int	mptensor::qr (const Tensor< Matrix, C > &a, Tensor< Matrix, C > &q, Tensor< Matrix, C > &r)
	QR decomposition of matrix (rank-2 tensor) More...
template<template< typename > class Matrix, typename C >
int	mptensor::qr (const Tensor< Matrix, C > &a, const Axes &axes_row, const Axes &axes_col, Tensor< Matrix, C > &q, Tensor< Matrix, C > &r)
	QR decomposition of tensor. More...
template<template< typename > class Matrix, typename C >
int	mptensor::eigh (const Tensor< Matrix, C > &a, std::vector< double > &eigval, Tensor< Matrix, C > &eigvec)
template<template< typename > class Matrix, typename C >
int	mptensor::eigh (const Tensor< Matrix, C > &a, std::vector< double > &eigval)
template<template< typename > class Matrix, typename C >
int	mptensor::eigh (const Tensor< Matrix, C > &a, const Axes &axes_row, const Axes &axes_col, std::vector< double > &eigval, Tensor< Matrix, C > &eigvec)
template<template< typename > class Matrix, typename C >
int	mptensor::eigh (const Tensor< Matrix, C > &a, const Axes &axes_row, const Axes &axes_col, std::vector< double > &w)
	Compute the eigenvalues of a complex Hermitian or real symmetric tensor. More...
template<template< typename > class Matrix, typename C >
int	mptensor::eigh (const Tensor< Matrix, C > &a, const Axes &axes_row_a, const Axes &axes_col_a, const Tensor< Matrix, C > &b, const Axes &axes_row_b, const Axes &axes_col_b, std::vector< double > &eigval, Tensor< Matrix, C > &eigvec)
template<template< typename > class Matrix, typename C >
int	mptensor::eig (const Tensor< Matrix, C > &a, std::vector< complex > &w, Tensor< Matrix, complex > &z)
	Compute the eigenvalues and eigenvectors of a general matrix (rank-2 tensor) More...
template<template< typename > class Matrix, typename C >
int	mptensor::eig (const Tensor< Matrix, C > &a, std::vector< complex > &w)
	Compute only the eigenvalues of a general matrix (rank-2 tensor) More...
template<template< typename > class Matrix, typename C >
int	mptensor::eig (const Tensor< Matrix, C > &a, const Axes &axes_row, const Axes &axes_col, std::vector< complex > &w, Tensor< Matrix, complex > &z)
	Compute the eigenvalues and eigenvectors of a general tensor. More...
template<template< typename > class Matrix, typename C >
int	mptensor::eig (const Tensor< Matrix, C > &a, const Axes &axes_row, const Axes &axes_col, std::vector< complex > &w)
	Compute the eigenvalues of a general tensor. More...

Detailed Description

Functions to decompose a tensor into some tensors.

Function Documentation

eig() [1/4]

template<template< typename > class Matrix, typename C >

int mptensor::eig	(	const Tensor< Matrix, C > &	a,
		const Axes &	axes_row,
		const Axes &	axes_col,
		std::vector< complex > &	w
	)

Compute the eigenvalues of a general tensor.

Parameters

[in]	a	A tensor
[in]	axes_row	Axes for left singular vectors.
[in]	axes_col	Axes for right singular vectors.
[out]	w	Eigenvalues

Returns

Information from linear-algebra library.

Warning

This function may require huge memory because it uses LAPACK routines.

eig() [2/4]

template<template< typename > class Matrix, typename C >

int mptensor::eig	(	const Tensor< Matrix, C > &	a,
		const Axes &	axes_row,
		const Axes &	axes_col,
		std::vector< complex > &	w,
		Tensor< Matrix, complex > &	z
	)

Compute the eigenvalues and eigenvectors of a general tensor.

For example,

eig(A, Axes(0,3), Axes(1,2), w, U)

calculates the following decomposition.

\[ A_{abcd} = A_{(ad)(bc)} = \sum_i U_{adi} w_i (U^\dagger)_{ibc}. \]

Parameters

[in]	a	A tensor
[in]	axes_row	Axes for left singular vectors.
[in]	axes_col	Axes for right singular vectors.
[out]	w	Eigenvalues
[out]	z	Tensor corresponds to Eigenvectors

Returns

Information from linear-algebra library.

Warning

This function may require huge memory because it uses LAPACK routines.

eig() [3/4]

template<template< typename > class Matrix, typename C >

int mptensor::eig	(	const Tensor< Matrix, C > &	a,
		std::vector< complex > &	w
	)

Compute only the eigenvalues of a general matrix (rank-2 tensor)

Parameters

[in]	a	Rank-2 tensor
[out]	w	Eigenvalues

Returns

Information from linear-algebra library.

Warning

This function may require huge memory because it uses LAPACK routines.

eig() [4/4]

template<template< typename > class Matrix, typename C >

int mptensor::eig	(	const Tensor< Matrix, C > &	a,
		std::vector< complex > &	w,
		Tensor< Matrix, complex > &	z
	)

Compute the eigenvalues and eigenvectors of a general matrix (rank-2 tensor)

Parameters

[in]	a	Rank-2 tensor
[out]	w	Eigenvalues
[out]	z	Eigenvectors

Returns

Information from linear-algebra library.

Warning

This function may require huge memory because it uses LAPACK routines.

eigh() [1/5]

template<template< typename > class Matrix, typename C >

int mptensor::eigh	(	const Tensor< Matrix, C > &	a,
		const Axes &	axes_row,
		const Axes &	axes_col,
		std::vector< double > &	w,
		Tensor< Matrix, C > &	z
	)

Compute the eigenvalues and eigenvectors of a complex Hermitian or real symmetric tensor

For example,

eigh(A, Axes(0,3), Axes(1,2), w, U)

calculates the following decomposition.

\[ A_{abcd} = A_{(ad)(bc)} = \sum_i U_{adi} w_i (U^\dagger)_{ibc}. \]

Parameters

[in]	a	A tensor
[in]	axes_row	Axes for left singular vectors.
[in]	axes_col	Axes for right singular vectors.
[out]	w	Eigenvalues
[out]	z	Tensor corresponds to Eigenvectors

Returns

Information from linear-algebra library.

Warning

Input tensor should be Hermite.

eigh() [2/5]

template<template< typename > class Matrix, typename C >

int mptensor::eigh	(	const Tensor< Matrix, C > &	a,
		const Axes &	axes_row,
		const Axes &	axes_col,
		std::vector< double > &	w
	)

Compute the eigenvalues of a complex Hermitian or real symmetric tensor.

Parameters

[in]	a	A tensor
[in]	axes_row	Axes for left singular vectors.
[in]	axes_col	Axes for right singular vectors.
[out]	w	Eigenvalues

Returns

Information from linear-algebra library.

Warning

Input tensor should be Hermite.

eigh() [3/5]

template<template< typename > class Matrix, typename C >

int mptensor::eigh	(	const Tensor< Matrix, C > &	a,
		const Axes &	axes_row_a,
		const Axes &	axes_col_a,
		const Tensor< Matrix, C > &	b,
		const Axes &	axes_row_b,
		const Axes &	axes_col_b,
		std::vector< double > &	w,
		Tensor< Matrix, C > &	z
	)

Solve a generalized eigenvalue problem \( A \vec{v} = \lambda B \vec{v} \) for a complex Hermitian or real symmetric tensor

Parameters

[in]	a	A complex Hermitian or real symmetric tensor.
[in]	axes_row_a	Axes of tensor a to be row after matricization.
[in]	axes_col_a	Axes of tensor a to be column after matricization.
[in]	b	A complex Hermitian or real symmetric definitive positive tensor.
[in]	axes_row_b	Axes of tensor b to be row after matricization.
[in]	axes_col_b	Axes of tensor b to be column after matricization.
[out]	w	Eigenvalues
[out]	z	Tensor corresponds to Eigenvectors

Returns

Information from linear-algebra library.

For example,

eigh(A, Axes(0,3), Axes(1,2), B, Axes(1, 3), Axes(2, 0), w, U)

solve a generalized eigenvalue problem

\[ \sum_{bc} A_{abcd} U_{bck} = \sum_{bc} B_{cabd} U_{bck} w_k. \]

In the matricized representation, it is written as

\[ \sum_{bc} A'_{(ad)(bc)} U'_{(bc)(k)} = \sum_{bc} B_{(ad)(bc)} U'_{(bc)(k)} w_k, \]

where \( A \) and \( B \) are matricized as \( A_{abcd} = A'_{(ad)(bc)} \) and \( B_{cabd} = B'_{(ad)(bc)} \).

Warning

The matricies obtained by a and b should be square and have the same size.

Note

The shape of z is determined by axes_col_a.

eigh() [4/5]

template<template< typename > class Matrix, typename C >

int mptensor::eigh	(	const Tensor< Matrix, C > &	a,
		std::vector< double > &	w
	)

Compute only the eigenvalues of a complex Hermitian or real symmetric matrix (rank-2 tensor)

Parameters

[in]	a	Rank-2 tensor
[out]	w	Eigenvalues

Returns

Information from linear-algebra library.

eigh() [5/5]

template<template< typename > class Matrix, typename C >

int mptensor::eigh	(	const Tensor< Matrix, C > &	a,
		std::vector< double > &	w,
		Tensor< Matrix, C > &	z
	)

Compute the eigenvalues and eigenvectors of a complex Hermitian or real symmetric matrix (rank-2 tensor)

Parameters

[in]	a	Rank-2 tensor
[out]	w	Eigenvalues
[out]	z	Eigenvectors

Returns

Information from linear-algebra library.

psvd() [1/4]

template<template< typename > class Matrix, typename C >

int mptensor::psvd	(	const Tensor< Matrix, C > &	a,
		const Axes &	axes_row,
		const Axes &	axes_col,
		std::vector< double > &	s,
		const size_t	target_rank
	)

Partial SVD for tensor.

This function only calculates singular values.

Parameters

[in]	a	Tensor to be decomposed.
[in]	axes_row	Axes for left singular vectors.
[in]	axes_col	Axes for right singular vectors.
[out]	s	Singular values.
[in]	target_rank	The number of singular values to be calculated.

Returns

Information from linear-algebra library.

Attention

Computational cost is the same as full SVD.

psvd() [2/4]

template<template< typename > class Matrix, typename C >

int mptensor::psvd	(	const Tensor< Matrix, C > &	a,
		const Axes &	axes_row,
		const Axes &	axes_col,
		Tensor< Matrix, C > &	u,
		std::vector< double > &	s,
		Tensor< Matrix, C > &	vt,
		const size_t	target_rank
	)

Partial SVD for tensor.

This may be useful for creation of an isometry. For example,

psvd(A, Axes(0,3), Axes(1,2), U, S, VT, n)

calculates the following decomposition.

\[ A_{abcd} \simeq \sum_{i=0}^{n-1} U_{adi} S_i (V^\dagger)_{ibc}. \]

Parameters

[in]	a	A tensor to be decomposed.
[in]	axes_row	Axes for left singular vectors.
[in]	axes_col	Axes for right singular vectors.
[out]	u	Tensor \( U \) corresponds to left singular vectors.
[out]	s	Singular values.
[out]	vt	Tensor \( V^\dagger \) corresponds to right singular vectors.
[in]	target_rank	The number of singular values to be calculated.

Returns

Information from linear-algebra library.

Attention

Computational cost is the same as full SVD.

psvd() [3/4]

template<template< typename > class Matrix, typename C >

int mptensor::psvd	(	const Tensor< Matrix, C > &	a,
		std::vector< double > &	s,
		const size_t	target_rank
	)

Partial SVD for rank-2 tensor (matrix)

This function only caluculates singular values.

Parameters

[in]	a	Rank-2 tensor to be decomposed.
[out]	s	Singular values.
[in]	target_rank	The number of singular values to be calculated.

Returns

Information from linear-algebra library.

Attention

Computational cost is the same as full SVD.

psvd() [4/4]

template<template< typename > class Matrix, typename C >

int mptensor::psvd	(	const Tensor< Matrix, C > &	a,
		Tensor< Matrix, C > &	u,
		std::vector< double > &	s,
		Tensor< Matrix, C > &	vt,
		const size_t	target_rank
	)

Partial SVD for rank-2 tensor (matrix)

\[ a_{ij} = \sum_k u_{ik} s_k (v^{\dagger})_{kj} \]

Parameters

[in]	a	Rank-2 tensor to be decomposed.
[out]	u	Tensor correspond to left singular vectors.
[out]	s	Singular values.
[out]	vt	Tensor correspond to right singular vectors.
[in]	target_rank	The number of singular values to be calculated.

Returns

Information from linear-algebra library.

Attention

Computational cost is the same as full SVD.

qr() [1/2]

template<template< typename > class Matrix, typename C >

int mptensor::qr	(	const Tensor< Matrix, C > &	a,
		const Axes &	axes_row,
		const Axes &	axes_col,
		Tensor< Matrix, C > &	q,
		Tensor< Matrix, C > &	r
	)

QR decomposition of tensor.

This may be useful for creation of an isometry. For example,

qr(A, Axes(0,3), Axes(1,2), Q, R)

calculates the following decomposition,

\[ A_{abcd} = \sum_i Q_{adi} R_{ibc}, \]

where the tensor Q satisfies

\[ \sum_{ab} Q_{abi} Q_{abj}^{*} = \delta_{ij}, \]

and the tensor R corresponds to the upper triangular matrix.

Parameters

[in]	a	A tensor to be decomposed.
[in]	axes_row	Axes for the orthogonal matrix.
[in]	axes_col	Axes for the upper triangular matrix.
[out]	q	Decomposed tensor corresponds to orthogonal matrix
[out]	r	Decomposed tensor corresponds to upper triangular matrix

Returns

Information from linear-algebra library.

qr() [2/2]

template<template< typename > class Matrix, typename C >

int mptensor::qr	(	const Tensor< Matrix, C > &	a,
		Tensor< Matrix, C > &	q,
		Tensor< Matrix, C > &	r
	)

QR decomposition of matrix (rank-2 tensor)

The sizes of q and r are reduced. When a is \( m\times n \) matrix and \( k=\min\{m,n\} \), the dimensions of q and r is \( m\times k \) and \( k\times n \).

Parameters

[in]	a	A matrix (rank-2 tensor) to be decomposed
[out]	q	Orthogonal matrix
[out]	r	Upper triangular matrix

Returns

Information from linear-algebra library.

svd() [1/4]

template<template< typename > class Matrix, typename C >

int mptensor::svd	(	const Tensor< Matrix, C > &	a,
		const Axes &	axes_row,
		const Axes &	axes_col,
		std::vector< double > &	s
	)

Singular value decomposition for tensor.

This function only calculates singular values.

Parameters

[in]	a	Tensor to be decomposed.
[in]	axes_row	Axes for left singular vectors.
[in]	axes_col	Axes for right singular vectors.
[out]	s	Singular values.

Returns

Information from linear-algebra library.

svd() [2/4]

template<template< typename > class Matrix, typename C >

int mptensor::svd	(	const Tensor< Matrix, C > &	a,
		const Axes &	axes_row,
		const Axes &	axes_col,
		Tensor< Matrix, C > &	u,
		std::vector< double > &	s,
		Tensor< Matrix, C > &	vt
	)

Singular value decomposition for tensor.

This may be useful for creation of an isometry. For example,

svd(A, Axes(0,3), Axes(1,2), U, S, VT)

calculates the following decomposition.

\[ A_{abcd} = \sum_i U_{adi} S_i (V^\dagger)_{ibc}. \]

Parameters

[in]	a	A tensor to be decomposed.
[in]	axes_row	Axes for left singular vectors.
[in]	axes_col	Axes for right singular vectors.
[out]	u	Tensor \( U \) corresponds to left singular vectors.
[out]	s	Singular values.
[out]	vt	Tensor \( V^\dagger \) corresponds to right singular vectors.

Returns

Information from linear-algebra library.

svd() [3/4]

template<template< typename > class Matrix, typename C >

int mptensor::svd	(	const Tensor< Matrix, C > &	a,
		std::vector< double > &	s
	)

Singular value decomposition for rank-2 tensor (matrix)

This function only caluculates singular values.

Parameters

[in]	a	Rank-2 tensor to be decomposed.
[out]	s	Singular values.

Returns

Information from linear-algebra library.

svd() [4/4]

template<template< typename > class Matrix, typename C >

int mptensor::svd	(	const Tensor< Matrix, C > &	a,
		Tensor< Matrix, C > &	u,
		std::vector< double > &	s,
		Tensor< Matrix, C > &	vt
	)

Singular value decomposition for rank-2 tensor (matrix)

\[ a_{ij} = \sum_k u_{ik} s_k (v^{\dagger})_{kj} \]

Parameters

[in]	a	Rank-2 tensor to be decomposed.
[out]	u	Tensor correspond to left singular vectors.
[out]	s	Singular values.
[out]	vt	Tensor correspond to right singular vectors.

Returns

Information from linear-algebra library.