Linear Algebra

Functionality from linear algebra such as (modified) principal and mixed matrix invariants. More...

Collaboration diagram for Linear Algebra:

Modules
	Invariants
	Matrix Invariants (principal and mixed, modified (isochoric) invariants and deviatoric invariants).

Namespaces
	funcy::linalg
	Functionality from linear algebra such as (modified) principal and mixed matrix invariants.

Classes
struct	funcy::linalg::SquaredFrobeniusNorm< Mat >
class	funcy::linalg::RightCauchyGreenStrainTensor< Mat >
	Right Cauchy-Green strain tensor \( F^T F \) for a symmetric matrix \( F \). More...
class	funcy::linalg::LeftCauchyGreenStrainTensor< Mat >
	Left Cauchy-Green strain tensor \( F^T F \) for a symmetric matrix \( F \). More...
class	funcy::linalg::Transpose< Mat >

Typedefs
template<class Mat >
using	funcy::linalg::FrobeniusNorm = mathop::Chain< Sqrt, SquaredFrobeniusNorm< Mat > >

Functions
template<int row, int col, ConstantSize Mat>
auto	funcy::linalg::compute_cofactor (const Mat &A) requires(dim< Mat >()
	Compute the \((row,col)\)-cofactor of \( A \). Implemented for \( A\in \mathbb{R}^{n,n} \) with \( n=2,3 \). More...
template<int row, int col, ConstantSize Mat>
auto	funcy::linalg::compute_cofactor_directional_derivative (const Mat &A, const Mat &B) requires(dim< Mat >()
	Compute the first directional derivative in direction \( B \) of the \((row,col)\)-cofactor of \( A \). Implemented for \( A\in \mathbb{R}^{n,n} \) with \( n=2,3 \). More...
template<class Mat >
auto	funcy::linalg::det (const Mat &A) requires(!Function< Mat > &&!SquareMatrix< Mat >)
	Generate \(\det(A)\). More...
template<SquareMatrix Mat>
auto	funcy::linalg::det (const Mat &A)
	Generate \(\det(A)\). More...
template<Function F>
auto	funcy::linalg::det (const F &f)
	Generate \(\det\circ f\). More...
template<SquareMatrix M, int n = dim< M >()>
auto	funcy::linalg::deviator (const M &A)
template<class Matrix >
auto	funcy::linalg::deviator (const Matrix &A)
template<Function F>
auto	funcy::linalg::deviator (const F &f)
	Generate deviator \( \mathrm{dev}\circ f\).
template<class Mat >
auto	funcy::linalg::frobenius_norm (const Mat &A)
template<Function F>
auto	funcy::linalg::frobenius_norm (const F &f)
	funcy::linalg::RightCauchyGreenStrainTensor< Mat >::RightCauchyGreenStrainTensor (const Mat &F)
	Constructor. More...
void	funcy::linalg::RightCauchyGreenStrainTensor< Mat >::update (const Mat &F)
	Reset point of evaluation.
const Mat &	funcy::linalg::RightCauchyGreenStrainTensor< Mat >::d0 () const noexcept
	Function value \( F^T * F \).
Mat	funcy::linalg::RightCauchyGreenStrainTensor< Mat >::d1 (const Mat &dF1) const
	First directional derivative \( F^T dF_1 + dF_1^T F \).
Mat	funcy::linalg::RightCauchyGreenStrainTensor< Mat >::d2 (const Mat &dF1, const Mat &dF2) const
	Second directional derivative \( dF_2^T dF_1 + dF_1^T dF_2 \).
	funcy::linalg::LeftCauchyGreenStrainTensor< Mat >::LeftCauchyGreenStrainTensor (const Mat &F)
	Constructor. More...
void	funcy::linalg::LeftCauchyGreenStrainTensor< Mat >::update (const Mat &F)
	Reset point of evaluation.
const Mat &	funcy::linalg::LeftCauchyGreenStrainTensor< Mat >::d0 () const noexcept
	Function value \( F^T * F \).
Mat	funcy::linalg::LeftCauchyGreenStrainTensor< Mat >::d1 (const Mat &dF1) const
	First directional derivative \( F^T dF_1 + dF_1^T F \).
Mat	funcy::linalg::LeftCauchyGreenStrainTensor< Mat >::d2 (const Mat &dF1, const Mat &dF2) const
	Second directional derivative \( dF_2^T dF_1 + dF_1^T dF_2 \).
template<class Mat >
auto	funcy::linalg::strain_tensor (const Mat &A)
	Generate the right Cauchy-Green strain tensor \(A*A^T\). More...
template<Function F>
auto	funcy::linalg::strain_tensor (const F &f)
	Generate the right Cauchy-Green strain tensor \(f*f^T\), where \(f:\cdot\mapsto\mathbb{R}^{n,n} \). More...
template<class Mat >
auto	funcy::linalg::left_strain_tensor (const Mat &A)
	Generate the left Cauchy-Green strain tensor \(A^T*A\). More...
template<ConstantSize Matrix, class Vector1 , class Vector2 >
Matrix	funcy::linalg::tensor_product (const Vector1 &v, const Vector2 &w)
	Compute tensor product \( M = v \otimes w \).
template<class Matrix , class Vector >
Matrix	funcy::linalg::tensor_product (const Vector &v)
	Compute tensor product \( M = v \otimes v \).
template<class Mat >
auto	funcy::linalg::trace (const Mat &A) requires(!Function< Mat > &&!ConstantSize< Mat >)
	Generate \(\mathrm{tr}(A)\in\mathbb{R}^{n,n}\). More...
template<ConstantSize Mat>
auto	funcy::linalg::trace (const Mat &A) requires(!Function< Mat >)
	Generate \(\mathrm{tr}(A)\in\mathbb{R}^{n,n}\). More...
template<Function F>
auto	funcy::linalg::trace (const F &f)
	Generate \(\mathrm{tr}\circ f\), where \(f:\cdot\mapsto\mathbb{R}^{n,n} \). More...
template<class Mat >
auto	funcy::linalg::transpose (const Mat &A)
	Generate \(A^T\in\mathbb{R}^{n,n}\). More...
template<Function F>
auto	funcy::linalg::transpose (const F &f)
	Generate \(f^T\), where \(f:\cdot\mapsto\mathbb{R}^{n,n} \). More...
template<ConstantSize Matrix>
Matrix	funcy::linalg::unit_matrix ()
template<class Matrix >
Matrix	funcy::linalg::unit_matrix (int rows)

Variables
auto int	funcy::linalg::col

Detailed Description

Functionality from linear algebra such as (modified) principal and mixed matrix invariants.

Typedef Documentation

FrobeniusNorm

template<class Mat >

using funcy::linalg::FrobeniusNorm = typedef mathop::Chain< Sqrt, SquaredFrobeniusNorm< Mat > >

Frobenius norm \( \|A\| = \sqrt{A\negthinspace : \negthinspace A }= \sqrt{\mathrm{tr}(A^TA)} = \sqrt{\sum_{i,j} A_{ij}^2}. \)

Function Documentation

compute_cofactor()

template<int row, int col, ConstantSize Mat>

auto int class Mat auto funcy::linalg::compute_cofactor

(

const Mat &

A

)

Compute the \((row,col)\)-cofactor of \( A \). Implemented for \( A\in \mathbb{R}^{n,n} \) with \( n=2,3 \).

The \((i,j)\)-cofactor of a matrix \( A \) is \( (-1)^{i+j} \det(A^\#_{ij}) \), where \( A^\#_ij \) is obtained from \( A \) by deleting the \(i\)-th row and \( j \)-th column.

compute_cofactor_directional_derivative()

template<int row, int col, ConstantSize Mat>

auto int class Mat auto funcy::linalg::compute_cofactor_directional_derivative	(	const Mat &	A,
		const Mat &	B
	)

Compute the first directional derivative in direction \( B \) of the \((row,col)\)-cofactor of \( A \). Implemented for \( A\in \mathbb{R}^{n,n} \) with \( n=2,3 \).

The \((i,j)\)-cofactor of a matrix \( A \) is \( (-1)^{i+j} \det(A^\#_{ij}) \), where \( A^\#_{ij} \) is obtained from \( A \) by deleting the \(i\)-th row and \( j \)-th column. If \( A\in \mathbb{R}^{3,3} \), then the cofactors are quadratic polynomials of the entries of \( A^\#_{ij} \). In this case this function can also used to compute the second directional derivative in directions \( A \) and \( B \).

det() [1/3]

template<class Mat >

auto funcy::linalg::det

(

const Mat &

A

)

&&

Generate \(\det(A)\).

Parameters

A	square matrix

Returns

Determinant<Matrix>(A)

det() [2/3]

template<SquareMatrix Mat>

auto funcy::linalg::det

(

const Mat &

A

)

Generate \(\det(A)\).

Parameters

A	square matrix

Returns

Determinant<Matrix>(A)

det() [3/3]

template<Function F>

auto funcy::linalg::det

(

const F &

f

)

Generate \(\det\circ f\).

Parameters

f	function mapping into a space of square matrices

Returns

Determinant< std::decay_t<decltype(f())> >(f())(f)

deviator() [1/2]

template<SquareMatrix M, int n = dim< M >()>

auto funcy::linalg::deviator

(

const M &

A

)

Generate deviator \( \mathrm{dev}(A) = A - \frac{\mathrm{tr}(A)}{n}I \) of a matrix \( A\in\mathbb{R}^{n,n} \).

deviator() [2/2]

template<class Matrix >

auto funcy::linalg::deviator

(

const Matrix &

A

)

Generate deviator \( \mathrm{dev}(A) = A - \frac{\mathrm{tr}(A)}{n}I \) of a matrix \( A\in\mathbb{R}^{n,n} \).

frobenius_norm() [1/2]

template<class Mat >

auto funcy::linalg::frobenius_norm

(

const Mat &

A

)

Generate Frobenius norm \( \|A\| = \sqrt{A\negthinspace : \negthinspace A }= \sqrt{\mathrm{tr}(A^TA)} = \sqrt{\sum_{i,j} A_{ij}^2}. \)

frobenius_norm() [2/2]

template<Function F>

auto funcy::linalg::frobenius_norm

(

const F &

f

)

Generate Frobenius norm \( \|A\| = \sqrt{A\negthinspace : \negthinspace A }= \sqrt{\mathrm{tr}(A^TA)} = \sqrt{\sum_{i,j} A_{ij}^2}. \)

left_strain_tensor()

template<class Mat >

auto funcy::linalg::left_strain_tensor

(

const Mat &

A

)

Generate the left Cauchy-Green strain tensor \(A^T*A\).

Parameters

A

matrix

Returns

LeftCauchyGreenStrainTensor<Matrix>(A)

LeftCauchyGreenStrainTensor()

template<class Mat >

funcy::linalg::LeftCauchyGreenStrainTensor< Mat >::LeftCauchyGreenStrainTensor ( const Mat &  F )

inlineexplicit

Constructor.

Parameters

F	point of evaluation.

RightCauchyGreenStrainTensor()

template<class Mat >

funcy::linalg::RightCauchyGreenStrainTensor< Mat >::RightCauchyGreenStrainTensor ( const Mat &  F )

inlineexplicit

Constructor.

Parameters

F	point of evaluation.

strain_tensor() [1/2]

template<class Mat >

auto funcy::linalg::strain_tensor

(

const Mat &

A

)

Generate the right Cauchy-Green strain tensor \(A*A^T\).

Parameters

A

matrix

Returns

RightCauchyGreenStrainTensor<Matrix>(A)

strain_tensor() [2/2]

template<Function F>

auto funcy::linalg::strain_tensor

(

const F &

f

)

Generate the right Cauchy-Green strain tensor \(f*f^T\), where \(f:\cdot\mapsto\mathbb{R}^{n,n} \).

Parameters

f	function object mapping into a space of square matrices

Returns

RightCauchyGreenStrainTensor< decay_t<decltype(f())> >(f())( f )

trace() [1/3]

template<class Mat >

auto funcy::linalg::trace

(

const Mat &

A

)

&&

Generate \(\mathrm{tr}(A)\in\mathbb{R}^{n,n}\).

Parameters

A

matrix

Returns

DynamicSizeTrace<Mat>(A)

trace() [2/3]

template<ConstantSize Mat>

auto funcy::linalg::trace

(

const Mat &

A

)

Generate \(\mathrm{tr}(A)\in\mathbb{R}^{n,n}\).

Parameters

A

matrix

Returns

ConstantSizeTrace<Mat>(A)

trace() [3/3]

template<Function F>

auto funcy::linalg::trace

(

const F &

f

)

Generate \(\mathrm{tr}\circ f\), where \(f:\cdot\mapsto\mathbb{R}^{n,n} \).

Parameters

f	function object mapping into a space of square matrices

transpose() [1/2]

template<class Mat >

auto funcy::linalg::transpose

(

const Mat &

A

)

Generate \(A^T\in\mathbb{R}^{n,n}\).

Parameters

A	square matrix

Returns

Transpose<Matrix>(A)

transpose() [2/2]

template<Function F>

auto funcy::linalg::transpose

(

const F &

f

)

Generate \(f^T\), where \(f:\cdot\mapsto\mathbb{R}^{n,n} \).

Parameters

f	function object mapping into a space of square matrices

Returns

Transpose< decay_t<decltype(f())> >(f())( f )

unit_matrix() [1/2]

template<ConstantSize Matrix>

Matrix funcy::linalg::unit_matrix

(

)

Compute unit matrix for the specified constant size matrix type. This requires that a corresponding specialization of Zero is provided.

unit_matrix() [2/2]

template<class Matrix >

Matrix funcy::linalg::unit_matrix

(

int

rows

)

Compute unit matrix for the specified dynamic size matrix type. This requires that a corresponding specialization of Zero is provided.