Detailed Description

Generic functions that wrap the BLAS and LAPACK interfaces to provide a more user-friendly approach for the most common linear algebra related tasks.

Functions
template<typename V >
auto	nda::linalg::cross_product (V const &x, V const &y)
	Compute the cross product of two 3-dimensional vectors.

template<typename... A> requires (nda::clef::is_any_lazy<A...>)
auto	nda::clef::determinant (A &&...__a)
	Lazy version of nda::determinant.

template<typename M >
auto	nda::determinant (M const &m)
	Compute the determinant of a square matrix/view.

template<typename M > requires (is_matrix_or_view_v<M>)
auto	nda::determinant_in_place (M &m)
	Compute the determinant of a square matrix/view.

template<typename X , typename Y >
auto	nda::dot (X &&x, Y &&y)
	Compute the dot product of two real arrays/views.

template<typename X , typename Y >
auto	nda::dotc (X &&x, Y &&y)
	Compute the dot product of two complex arrays/views.

template<typename M >
auto	nda::linalg::eigenelements (M const &m)
	Find the eigenvalues and eigenvectors of a symmetric (real) or hermitian (complex) matrix/view.

template<typename M >
auto	nda::linalg::eigenvalues (M const &m)
	Find the eigenvalues of a symmetric (real) or hermitian (complex) matrix/view.

template<typename M >
auto	nda::linalg::eigenvalues_in_place (M &m)
	Find the eigenvalues of a symmetric (real) or hermitian (complex) matrix/view.

template<Matrix M> requires (get_algebra<M> == 'M')
auto	nda::inverse (M const &m)
	Compute the inverse of an n-by-n matrix.

template<MemoryMatrix M> requires (get_algebra<M> == 'M' and mem::on_host<M>)
void	nda::inverse1_in_place (M &&m)
	Compute the inverse of a 1-by-1 matrix.

template<MemoryMatrix M> requires (get_algebra<M> == 'M' and mem::on_host<M>)
void	nda::inverse2_in_place (M &&m)
	Compute the inverse of a 2-by-2 matrix.

template<MemoryMatrix M> requires (get_algebra<M> == 'M' and mem::on_host<M>)
void	nda::inverse3_in_place (M &&m)
	Compute the inverse of a 3-by-3 matrix.

template<MemoryMatrix M> requires (get_algebra<M> == 'M')
void	nda::inverse_in_place (M &&m)
	Compute the inverse of an n-by-n matrix.

template<typename A >
bool	nda::is_matrix_diagonal (A const &a, bool print_error=false)
	Check if a given array/view is diagonal, i.e. if it is square (see nda::is_matrix_square) and all the the off-diagonal elements are zero.

template<typename A >
bool	nda::is_matrix_square (A const &a, bool print_error=false)
	Check if a given array/view is square, i.e. if the first dimension has the same extent as the second dimension.

template<Matrix A, Matrix B>
auto	nda::matmul (A &&a, B &&b)
	Perform a matrix-matrix multiplication.

template<Matrix A, Vector X>
auto	nda::matvecmul (A &&a, X &&x)
	Perform a matrix-vector multiplication.

template<ArrayOfRank< 1 > A>
double	nda::norm (A const &a, double p=2.0)
	Calculate the p-norm of an nda::ArrayOfRank<1> object \( \mathbf{x} \) with scalar values. The p-norm is defined as.

Function Documentation

◆ cross_product()

template<typename V >

auto nda::linalg::cross_product	(	V const &	x,
		V const &	y )

#include <nda/linalg/cross_product.hpp>

Compute the cross product of two 3-dimensional vectors.

Template Parameters

V	Vector type.

Parameters

x	Left hand side vector.
y	Right hand side vector.

Returns: nda::array of rank 1 containing the cross product of the two vectors.

Definition at line 40 of file cross_product.hpp.

◆ determinant()

template<typename M >

auto nda::determinant ( M const & m )

#include <nda/linalg/det_and_inverse.hpp>

Compute the determinant of a square matrix/view.

The given matrix/view is not modified. It first makes a copy of the given matrix/view and then calls nda::determinant_in_place with the copy.

Template Parameters

M	Type of the matrix/view.

Parameters

m	Matrix/view object.

Returns: Determinant of the matrix/view.

Definition at line 143 of file det_and_inverse.hpp.

◆ determinant_in_place()

template<typename M >
requires (is_matrix_or_view_v<M>)

auto nda::determinant_in_place ( M & m )

#include <nda/linalg/det_and_inverse.hpp>

Compute the determinant of a square matrix/view.

It uses nda::lapack::getrf to compute the LU decomposition of the matrix and then calculates the determinant by multiplying the diagonal elements of the \( \mathbf{U} \) matrix and taking into account that getrf may change the ordering of the rows/columns of the matrix.

The given matrix/view is modified in place.

Template Parameters

M	Type of the matrix/view.

Parameters

m	Matrix/view object.

Returns: Determinant of the matrix/view.

Definition at line 100 of file det_and_inverse.hpp.

◆ dot()

template<typename X , typename Y >

auto nda::dot	(	X &&	x,
		Y &&	y )

#include <nda/linalg/dot.hpp>

Compute the dot product of two real arrays/views.

It is generic in the sense that it allows the input arrays to belong to a different nda::mem::AddressSpace (as long as they are compatible).

If possible, it uses nda::blas::dot, otherwise it calls nda::blas::dot_generic.

Template Parameters

X	Type of the left hand side array/view.
Y	Type of the right hand side array/view.

Parameters

x	Left hand side array/view.
y	Right hand side array/view.

Returns: The dot product of the two arrays/views.

Definition at line 54 of file dot.hpp.

◆ dotc()

template<typename X , typename Y >

auto nda::dotc	(	X &&	x,
		Y &&	y )

#include <nda/linalg/dot.hpp>

Compute the dot product of two complex arrays/views.

It is generic in the sense that it allows the input arrays to belong to a different nda::mem::AddressSpace (as long as they are compatible).

If possible, it uses nda::blas::dotc, otherwise it calls nda::blas::dotc_generic.

Template Parameters

X	Type of the left hand side array/view.
Y	Type of the right hand side array/view.

Parameters

x	Left hand side array/view.
y	Right hand side array/view.

Returns: The dot product of the two arrays/views.

Definition at line 97 of file dot.hpp.

◆ eigenelements()

template<typename M >

auto nda::linalg::eigenelements ( M const & m )

#include <nda/linalg/eigenelements.hpp>

Find the eigenvalues and eigenvectors of a symmetric (real) or hermitian (complex) matrix/view.

For a real symmetric matrix/view, it calls the LAPACK routine syev. For a complex hermitian matrix/view, it calls the LAPACK routine heev.

The given matrix/view is copied and the original is not modified.

Template Parameters

M	Type of the matrix/view.

Parameters

m	Matrix/View to diagonalize.

Returns: std::pair consisting of the array of eigenvalues in ascending order and the matrix containing the eigenvectors in its columns.

Definition at line 98 of file eigenelements.hpp.

◆ eigenvalues()

template<typename M >

auto nda::linalg::eigenvalues ( M const & m )

#include <nda/linalg/eigenelements.hpp>

Find the eigenvalues of a symmetric (real) or hermitian (complex) matrix/view.

For a real symmetric matrix/view, it calls the LAPACK routine syev. For a complex hermitian matrix/view, it calls the LAPACK routine heev.

The given matrix/view is copied and the original is not modified.

Template Parameters

M	Type of the matrix/view.

Parameters

m	Matrix/View to diagonalize.

Returns: An nda::array of rank 1 containing the eigenvalues in ascending order.

Definition at line 117 of file eigenelements.hpp.

◆ eigenvalues_in_place()

template<typename M >

auto nda::linalg::eigenvalues_in_place ( M & m )

#include <nda/linalg/eigenelements.hpp>

Find the eigenvalues of a symmetric (real) or hermitian (complex) matrix/view.

For a real symmetric matrix/view, it calls the LAPACK routine syev. For a complex hermitian matrix/view, it calls the LAPACK routine heev.

The given matrix/view will be modified by the diagonalization process.

Template Parameters

M	Type of the matrix/view.

Parameters

m	Matrix/View to diagonalize.

Returns: An nda::array of rank 1 containing the eigenvalues in ascending order.

Definition at line 135 of file eigenelements.hpp.

◆ inverse()

template<Matrix M>
requires (get_algebra<M> == 'M')

auto nda::inverse ( M const & m )

#include <nda/linalg/det_and_inverse.hpp>

Compute the inverse of an n-by-n matrix.

The given matrix/view is not modified. It first makes copy of the given matrix/view and then calls nda::inverse_in_place with the copy.

Template Parameters

M	nda::MemoryMatrix type.

Parameters

m	nda::MemoryMatrix object to be inverted.

Returns: Inverse of the matrix.

Definition at line 297 of file det_and_inverse.hpp.

◆ inverse1_in_place()

template<MemoryMatrix M>
requires (get_algebra<M> == 'M' and mem::on_host<M>)

void nda::inverse1_in_place ( M && m )

#include <nda/linalg/det_and_inverse.hpp>

Compute the inverse of a 1-by-1 matrix.

The inversion is performed in place.

Template Parameters

M	nda::MemoryMatrix type.

Parameters

m	nda::MemoryMatrix object to be inverted.

Definition at line 169 of file det_and_inverse.hpp.

◆ inverse2_in_place()

template<MemoryMatrix M>
requires (get_algebra<M> == 'M' and mem::on_host<M>)

void nda::inverse2_in_place ( M && m )

#include <nda/linalg/det_and_inverse.hpp>

Compute the inverse of a 2-by-2 matrix.

The inversion is performed in place.

Template Parameters

M	nda::MemoryMatrix type.

Parameters

m	nda::MemoryMatrix object to be inverted.

Definition at line 184 of file det_and_inverse.hpp.

◆ inverse3_in_place()

template<MemoryMatrix M>
requires (get_algebra<M> == 'M' and mem::on_host<M>)

void nda::inverse3_in_place ( M && m )

#include <nda/linalg/det_and_inverse.hpp>

Compute the inverse of a 3-by-3 matrix.

The inversion is performed in place.

Template Parameters

M	nda::MemoryMatrix type.

Parameters

m	nda::MemoryMatrix object to be inverted.

Definition at line 210 of file det_and_inverse.hpp.

◆ inverse_in_place()

template<MemoryMatrix M>
requires (get_algebra<M> == 'M')

void nda::inverse_in_place ( M && m )

#include <nda/linalg/det_and_inverse.hpp>

Compute the inverse of an n-by-n matrix.

The inversion is performed in place.

For small matrices (1-by-1, 2-by-2, 3-by-3), we directly compute the matrix inversion using the optimized routines: nda::inverse1_in_place, nda::inverse2_in_place, nda::inverse3_in_place.

For larger matrices, it uses nda::lapack::getrf and nda::lapack::getri.

Template Parameters

M	nda::MemoryMatrix type.

Parameters

m	nda::MemoryMatrix object to be inverted.

Definition at line 254 of file det_and_inverse.hpp.

◆ is_matrix_diagonal()

template<typename A >

bool nda::is_matrix_diagonal	(	A const &	a,
		bool	print_error = false )

#include <nda/linalg/det_and_inverse.hpp>

Check if a given array/view is diagonal, i.e. if it is square (see nda::is_matrix_square) and all the the off-diagonal elements are zero.

Note: It does not check if the array/view has rank 2.

Template Parameters

A	Array/View type.

Parameters

a	Array/View object.
print_error	If true, print an error message if the matrix is not diagonal.

Returns: True if the array/view is diagonal, false otherwise.

Definition at line 80 of file det_and_inverse.hpp.

◆ is_matrix_square()

template<typename A >

bool nda::is_matrix_square	(	A const &	a,
		bool	print_error = false )

#include <nda/linalg/det_and_inverse.hpp>

Check if a given array/view is square, i.e. if the first dimension has the same extent as the second dimension.

Note: It does not check if the array/view has rank 2.

Template Parameters

A	Array/View type.

Parameters

a	Array/View object.
print_error	If true, print an error message if the matrix is not square.

Returns: True if the array/view is square, false otherwise.

Definition at line 61 of file det_and_inverse.hpp.

◆ matmul()

template<Matrix A, Matrix B>

auto nda::matmul	(	A &&	a,
		B &&	b )

#include <nda/linalg/matmul.hpp>

Perform a matrix-matrix multiplication.

It is generic in the sense that it allows the input matrices to belong to a different nda::mem::AddressSpace (as long as they are compatible).

If possible, it uses nda::blas::gemm, otherwise it calls nda::blas::gemm_generic.

Template Parameters

A	nda::Matrix type of lhs operand.
B	nda::Matrix type of rhs operand.

Parameters

a	Left hand side matrix operand.
b	Right hand side matrix operand.

Returns: Result of the matrix-matrix multiplication.

Definition at line 87 of file matmul.hpp.

◆ matvecmul()

template<Matrix A, Vector X>

auto nda::matvecmul	(	A &&	a,
		X &&	x )

#include <nda/linalg/matmul.hpp>

Perform a matrix-vector multiplication.

It is generic in the sense that it allows the input matrix and vector to belong to a different nda::mem::AddressSpace (as long as they are compatible).

If possible, it uses nda::blas::gemv, otherwise it calls nda::blas::gemv_generic.

Template Parameters

A	nda::Matrix type of lhs operand.
X	nda::Vector type of rhs operand.

Parameters

a	Left hand side matrix operand.
x	Right hand side vector operand.

Returns: Result of the matrix-vector multiplication.

Definition at line 152 of file matmul.hpp.

◆ norm()

template<ArrayOfRank< 1 > A>

double nda::norm	(	A const &	a,
		double	p = 2.0 )

#include <nda/linalg/norm.hpp>

Calculate the p-norm of an nda::ArrayOfRank<1> object \( \mathbf{x} \) with scalar values. The p-norm is defined as.

\[ || \mathbf{x} ||_p = \left( \sum_{i=0}^{N-1} |x_i|^p \right)^{1/p} \]

with the special cases (following numpy.linalg.norm convention)

\( || \mathbf{x} ||_0 = \text{number of non-zero elements} \),
\( || \mathbf{x} ||_{\infty} = \max_i |x_i| \),
\( || \mathbf{x} ||_{-\infty} = \min_i |x_i| \).

Template Parameters

A	nda::ArrayOfRank<1> type.

Parameters

a	nda::ArrayOfRank<1> object.
p	Order of the norm.

Returns: Norm of the array/view as a double.

Definition at line 58 of file norm.hpp.

Detailed Description

Functions

Function Documentation

◆ cross_product()

◆ determinant()

◆ determinant_in_place()

◆ dot()

◆ dotc()

◆ eigenelements()

◆ eigenvalues()

◆ eigenvalues_in_place()

◆ inverse()

◆ inverse1_in_place()

◆ inverse2_in_place()

◆ inverse3_in_place()

◆ inverse_in_place()

◆ is_matrix_diagonal()

◆ is_matrix_square()

◆ matmul()

◆ matvecmul()

◆ norm()