basix::math Namespace Reference

Mathematical functions. More...

Functions
template<typename U , typename V >
std::pair< std::vector< typename U::value_type >, std::array< std::size_t, 2 > >	outer (const U &u, const V &v)
	Compute the outer product of vectors u and v.
template<typename U , typename V >
std::array< typename U::value_type, 3 >	cross (const U &u, const V &v)
	Compute the cross product u x v.
template<std::floating_point T>
std::pair< std::vector< T >, std::vector< T > >	eigh (std::span< const T > A, std::size_t n)
	Compute the eigenvalues and eigenvectors of a square Hermitian matrix A.
template<std::floating_point T>
std::vector< T >	solve (md::mdspan< const T, md::dextents< std::size_t, 2 > > A, md::mdspan< const T, md::dextents< std::size_t, 2 > > B)
	Solve A X = B.
template<std::floating_point T>
bool	is_singular (md::mdspan< const T, md::dextents< std::size_t, 2 > > A)
	Check if A is a singular matrix.
template<std::floating_point T>
std::vector< std::size_t >	transpose_lu (std::pair< std::vector< T >, std::array< std::size_t, 2 > > &A)
	Compute the LU decomposition of the transpose of a square matrix A.
template<typename U , typename V , typename W >
void	dot (const U &A, const V &B, W &&C, typename std::decay_t< U >::value_type alpha=1, typename std::decay_t< U >::value_type beta=0)
	Compute C = alpha A * B + beta C.
template<std::floating_point T>
std::vector< T >	eye (std::size_t n)
	Build an identity matrix.
template<std::floating_point T>
void	orthogonalise (md::mdspan< T, md::dextents< std::size_t, 2 > > wcoeffs, std::size_t start=0)
	Orthogonalise the rows of a matrix (in place).

Detailed Description

Mathematical functions.

Note

Functions in this namespace are designed to be called multiple times at runtime, so their performance is critical.

Function Documentation

outer()

template<typename U , typename V >

std::pair< std::vector< typename U::value_type >, std::array< std::size_t, 2 > > basix::math::outer	(	const U &	u,
		const V &	v
	)

Compute the outer product of vectors u and v.

Parameters

u	The first vector.
v	The second vector.

Returns

The outer product. The type will be the same as u.

cross()

template<typename U , typename V >

std::array< typename U::value_type, 3 > basix::math::cross	(	const U &	u,
		const V &	v
	)

Compute the cross product u x v.

Parameters

u	The first vector. It must has size 3.
v	The second vector. It must has size 3.

Returns

The cross product u x v. The type will be the same as u.

eigh()

template<std::floating_point T>

std::pair< std::vector< T >, std::vector< T > > basix::math::eigh	(	std::span< const T >	A,
		std::size_t	n
	)

Compute the eigenvalues and eigenvectors of a square Hermitian matrix A.

Parameters

[in]	A	Input matrix, row-major storage.
[in]	n	Number of rows.

Returns

Eigenvalues (0) and eigenvectors (1). The eigenvector array uses column-major storage, which each column being an eigenvector.

Precondition

The matrix A must be symmetric.

solve()

template<std::floating_point T>

std::vector< T > basix::math::solve	(	md::mdspan< const T, md::dextents< std::size_t, 2 > >	A,
		md::mdspan< const T, md::dextents< std::size_t, 2 > >	B
	)

Solve A X = B.

Parameters

[in]	A	The matrix.
[in]	B	Right-hand side matrix/vector.

Returns

A^{-1} B.

is_singular()

template<std::floating_point T>

bool basix::math::is_singular

(

md::mdspan< const T, md::dextents< std::size_t, 2 > >

A

)

Check if A is a singular matrix.

Parameters

[in]

A

The matrix.

Returns

A bool indicating if the matrix is singular.

transpose_lu()

template<std::floating_point T>

std::vector< std::size_t > basix::math::transpose_lu

(

std::pair< std::vector< T >, std::array< std::size_t, 2 > > &

A

)

Compute the LU decomposition of the transpose of a square matrix A.

Parameters

[in,out]

A

The matrix.

Returns

The LU permutation, in prepared format (see precompute::prepare_permutation).

dot()

template<typename U , typename V , typename W >

void basix::math::dot	(	const U &	A,
		const V &	B,
		W &&	C,
		typename std::decay_t< U >::value_type	alpha = 1,
		typename std::decay_t< U >::value_type	beta = 0
	)

Compute C = alpha A * B + beta C.

Parameters

[in]	A	Input matrix
[in]	B	Input matrix
[out]	C	Output matrix. Must be sized correctly before calling this function.
[in]	alpha
[in]	beta

eye()

template<std::floating_point T>

std::vector< T > basix::math::eye

(

std::size_t

n

)

Build an identity matrix.

Parameters

[in]

n

The number of rows/columns.

Returns

Identity matrix using row-major storage.

orthogonalise()

template<std::floating_point T>

void basix::math::orthogonalise	(	md::mdspan< T, md::dextents< std::size_t, 2 > >	wcoeffs,
		std::size_t	start = 0
	)

Orthogonalise the rows of a matrix (in place).

Parameters

[in]	wcoeffs	The matrix.
[in]	start	The row to start from. The rows before this should already be orthogonal.