funcy::linalg Namespace Reference

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

Classes
class	LeftCauchyGreenStrainTensor
	Left Cauchy-Green strain tensor \( F^T F \) for a symmetric matrix \( F \). More...
class	RightCauchyGreenStrainTensor
	Right Cauchy-Green strain tensor \( F^T F \) for a symmetric matrix \( F \). More...
class	SecondPrincipalInvariant
struct	SquaredFrobeniusNorm
class	Transpose

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

Functions
template<Matrix M, class Index >
decltype(auto) FUNCY_ALWAYS_INLINE	at (M &&A, Index i, Index j) requires requires(std
	Access matrix entry.
template<int row, int col, ConstantSize Mat>
auto	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	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	det (const Mat &A) requires(!Function< Mat > &&!SquareMatrix< Mat >)
	Generate \(\det(A)\). More...
template<SquareMatrix Mat>
auto	det (const Mat &A)
	Generate \(\det(A)\). More...
template<Function F>
auto	det (const F &f)
	Generate \(\det\circ f\). More...
template<SquareMatrix M, int n = dim< M >()>
auto	deviator (const M &A)
template<class Matrix >
auto	deviator (const Matrix &A)
template<Function F>
auto	deviator (const F &f)
	Generate deviator \( \mathrm{dev}\circ f\).
template<Matrix M>
auto	j2 (const M &A) requires(!Function< M >)
	Second deviatoric invariant \( j_2(\sigma)=\sqrt{\bar\sigma\negthinspace:\negthinspace\bar\sigma} \) with \(\bar\sigma = \sigma - \frac{\mathrm{tr}(\sigma)}{n}I\) and \(\sigma\in\mathbb{R}^{n,n}\).
template<class Mat >
constexpr int	dim ()
	Dimension \(n\) of a fixed size matrix in \(\mathbb{R}^{n,n}\).
template<class Mat >
auto	frobenius_norm (const Mat &A)
template<Function F>
auto	frobenius_norm (const F &f)
template<Matrix Mat>
auto	i4 (const Mat &A, const Mat &M) requires(!Function< Mat >)
	Generate first mixed invariant \( \iota_4=\iota_1(AM) \) of a matrix \(A\in\mathbb{R}^{n,n}\) with respect to the structural tensor \(M\in\mathbb{R}^{n,n}\). More...
template<Function F, class Matrix >
auto	i4 (const F &f, const Matrix &M)
	Generate first mixed invariant \( \iota_4\circ f \) with \(f\mapsto\mathbb{R}^{n,n}\) and structural tensor \(M\in\mathbb{R}^{n,n}\). More...
template<Matrix Mat>
auto	i5 (const Mat &A, const Mat &M) requires(!Function< Mat >)
	Generate second mixed invariant \( \iota_5=\iota_1(A^2M) \) of a matrix \(A\in\mathbb{R}^{n,n}\) with respect to the structural tensor \(M\in\mathbb{R}^{n,n}\). More...
template<Function F, class Matrix >
auto	i5 (const F &f, const Matrix &M)
	Generate second mixed invariant \( \iota_5\circ f \) with \(f\mapsto\mathbb{R}^{n,n}\) and structural tensor \(M\in\mathbb{R}^{n,n}\). More...
template<Matrix Mat>
auto	i6 (const Mat &A, const Mat &M) requires(!Function< Mat >)
	Generate third mixed invariant \( \iota_6=\iota_1(AM^2) \) of a matrix \(A\in\mathbb{R}^{n,n}\) with respect to the structural tensor \(M\in\mathbb{R}^{n,n}\). More...
template<Function F, class Matrix >
auto	i6 (const F &f, const Matrix &M)
	Generate third mixed invariant \( \iota_6\circ f \) with \(f\mapsto\mathbb{R}^{n,n}\) and structural tensor \(M\in\mathbb{R}^{n,n}\). More...
template<class Arg , class Matrix , int n = dim< Matrix >()>
auto	mi4 (const Arg &x, const Matrix &M)
	Isochoric (volume-preserving), first modified mixed invariant \( \bar\iota_4(A)=\iota_4\iota_3^{-1/3} \), where \(\iota_4\) is the first mixed and \(\iota_3\) is the third principal invariant. More...
template<class Arg , class Matrix , int n = dim< Matrix >()>
auto	mi5 (const Arg &x, const Matrix &M)
	Isochoric (volume-preserving), second modified principal invariant \( \bar\iota_5(A)=\iota_5\iota_3^{-2/3} \), where \(\iota_5\) is the second mixed and \(\iota_3\) is the third principal invariant. More...
template<class Arg , class Matrix , int n = dim< Matrix >()>
auto	mi6 (const Arg &x, const Matrix &M)
	Isochoric (volume-preserving), second modified principal invariant \( \bar\iota_6(A)=\iota_6\iota_3^{-1/3} \), where \(\iota_6\) is the third mixed and \(\iota_3\) is the third principal invariant. More...
template<class Arg >
auto	i1 (Arg &&x)
	Generate first principal invariant. More...
template<Matrix M>
auto	i2 (M &&A)
	Convenient generation of second principal invariant \( \iota_2(A)=\mathrm{tr}(\mathrm{cof}(A)) \) for \(A\in\mathbb{R}^{n,n}\). More...
template<Function F>
auto	i2 (const F &f)
	Convenient generation of second principal invariant \( \iota_2\circ f \) for \(f:\cdot\mapsto\mathbb{R}^{n,n}\). More...
template<class Arg >
auto	i3 (const Arg &x)
	Generate third principal invariant. More...
template<class Arg , int n = dim< Arg >()>
auto	mi1 (const Arg &x)
	Isochoric (volume-preserving), first modified principal invariant \( \bar\iota_1(A)=\iota_1\iota_3^{-1/3} \), where \(\iota_1\) is the first and \(\iota_3\) is the third principal invariant. More...
template<class Arg , int n = dim< Arg >()>
auto	mi2 (const Arg &x)
	Isochoric (volume-preserving), second modified principal invariant \( \bar\iota_2(A)=\iota_2\iota_3^{-2/3} \), where \(\iota_2\) is the second and \(\iota_3\) is the third principal invariant. More...
template<class Mat >
auto	rows (const Mat &A) requires(!ConstantSize< Mat > &&static_check
	Number of rows of a dynamic size matrix.
template<ConstantSize Mat>
constexpr auto	rows ()
	Number of rows of a constant size matrix.
template<class Mat >
auto	cols (const Mat &A) requires(!ConstantSize< Mat > &&static_check
	Number of columns of a dynamic size matrix.
template<ConstantSize Mat>
constexpr auto	cols ()
	Number of columns of a constant size matrix.
template<class Mat >
auto	strain_tensor (const Mat &A)
	Generate the right Cauchy-Green strain tensor \(A*A^T\). More...
template<Function F>
auto	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	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	tensor_product (const Vector1 &v, const Vector2 &w)
	Compute tensor product \( M = v \otimes w \).
template<class Matrix , class Vector >
Matrix	tensor_product (const Vector &v)
	Compute tensor product \( M = v \otimes v \).
template<class Mat >
auto	trace (const Mat &A) requires(!Function< Mat > &&!ConstantSize< Mat >)
	Generate \(\mathrm{tr}(A)\in\mathbb{R}^{n,n}\). More...
template<ConstantSize Mat>
auto	trace (const Mat &A) requires(!Function< Mat >)
	Generate \(\mathrm{tr}(A)\in\mathbb{R}^{n,n}\). More...
template<Function F>
auto	trace (const F &f)
	Generate \(\mathrm{tr}\circ f\), where \(f:\cdot\mapsto\mathbb{R}^{n,n} \). More...
template<class Mat >
auto	transpose (const Mat &A)
	Generate \(A^T\in\mathbb{R}^{n,n}\). More...
template<Function F>
auto	transpose (const F &f)
	Generate \(f^T\), where \(f:\cdot\mapsto\mathbb{R}^{n,n} \). More...
template<ConstantSize Matrix>
Matrix	unit_matrix ()
template<class Matrix >
Matrix	unit_matrix (int rows)

Variables
auto int	col
template<class V >
concept	ConstantSize = linalg::detail::NumberOfRows< V >::value > 0
	Constant-size matrix concept.
template<class M >
concept	SquareMatrix
	Constant-size square matrix concept. More...
template<class T >
concept	Matrix
	clang-format off More...
template<class T >
concept	Vector
	Vector concept. More...

Detailed Description

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

See also

Linear Algebra