linalg Namespace Reference

Classes
struct	Cholesky
	Factors a real positive-definite matrix as A = G G^T. More...
struct	coordinate_map
	Customization point mapping space elements to finite coordinates. More...
struct	coordinate_map< CoordinateSpace< T, Dim > >
	Identity coordinate map for CoordinateSpace. More...
struct	coordinate_map< Matrix< T, Rows, Cols > >
	Identity coordinate map for one-row or one-column matrices. More...
struct	coordinate_map< MatrixSpace< T, Rows, Cols > >
	Row-major coordinate isomorphism for MatrixSpace. More...
struct	coordinate_map< SymmetricMatrix< Mat > >
struct	coordinate_map< SymmetricMatrixSpace< T, Dim > >
	Symmetric half-vectorization coordinate isomorphism. More...
struct	CoordinateSpace
	Canonical Dim-dimensional coordinate space over field T. More...
struct	EigenDecomposition
	Computes eigenvalues and eigenvectors satisfying A V = V diag(lambda). More...
struct	Endomorphism
	Endomorphism operator CRTP base. More...
struct	Fro
	compute the frobenius norm of a matrix More...
struct	Hessenberg
	Reduces A to upper Hessenberg form H = U^* A U. More...
struct	Hessian
	Hessian as a linear operator. More...
struct	HouseholderVector
	Compact representation of a real or complex Householder reflector. More...
struct	inner_product_traits
	Customization point for inner-product associated types. More...
struct	inner_product_traits< StandardInnerProduct< Vec > >
	Associated-type registration for StandardInnerProduct. More...
struct	is_symmetric_matrix_space
	symmetric matrix space type trait More...
struct	is_symmetric_matrix_space< SymmetricMatrix< Mat > >
	symmetric matrix space type trait More...
struct	L1
	forward declaration for L1 norm More...
struct	L2
	forward declaration for L2 norm More...
struct	LDL
	Factors a real symmetric matrix as A = L D L^T without pivoting. More...
struct	Linf
	forward declaration for Linf norm More...
struct	mat_traits
	Customization point for compile-time matrix metadata. More...
struct	mat_traits< Cholesky< Mat > >
	mat type traits definition for Cholesky More...
struct	mat_traits< Endomorphism< Mat, Derived > >
	mat type traits definition for linear operator More...
struct	mat_traits< Hessenberg< Mat > >
	mat type traits definition for hessenberg matrices More...
struct	mat_traits< Hessian< Mat, Type > >
	mat type traits definition for hessian matrices More...
struct	mat_traits< LDL< Mat > >
	mat type traits definition for LDL More...
struct	mat_traits< Matrix< T, Rows, Cols > >
	Traits for the Akreon Linalg's owning Matrix container. More...
struct	mat_traits< QuadraticRep< Mat, Circ > >
	mat type traits definition for P(x) More...
struct	mat_traits< SymmetricMatrix< Mat > >
	mat type traits definition for symmetric matrices More...
struct	mat_traits< SymmetricOperator< Mat, Derived > >
	mat type traits definition for symmetric operator More...
struct	Matrix
	Owning, fixed-size, row-major matrix container over a scalar field. More...
struct	MatrixInverse
	Matrix inverse via QR. More...
struct	MatrixPencil
struct	MatrixSpace
	Space of all Rows x Cols matrices over field T. More...
struct	MatrixSqrt
	Matrix sqrt (X^{1/2}) More...
struct	norm_traits
	declaring norm traits More...
struct	norm_traits< Fro< Mat > >
struct	norm_traits< L1< Vec > >
	declaring norm type traits - L1 More...
struct	norm_traits< L2< Vec > >
	declaring norm type traits - L2 More...
struct	norm_traits< Linf< Vec > >
	declaring norm type traits - Linf More...
struct	operator_traits
struct	operator_traits< Matrix< T, Rows, Cols > >
struct	operator_traits< Operator >
struct	QR
	Factors A = Q R using Householder reflections. More...
struct	QuadraticRep
	forward declaration of P(x) / P(x^{-1}) More...
struct	scalar_traits
	Customization point for scalar metadata. More...
struct	scalar_traits< std::complex< T > >
	Register std::complex<T> over a floating-point real component. More...
struct	scalar_traits< T >
	Register a standard floating-point type as a real scalar. More...
struct	space_traits
	Customization point for finite-dimensional space metadata. More...
struct	space_traits< ColVector< T, Dim > >
	Space registration for a matrix-shaped column vector. More...
struct	space_traits< CoordinateSpace< T, Dim > >
	Space metadata for CoordinateSpace. More...
struct	space_traits< Matrix< T, 1, 1 > >
	Space registration for a scalar-shaped 1 x 1 matrix. More...
struct	space_traits< Matrix< T, Rows, Cols > >
	Space registration for a general rectangular matrix container. More...
struct	space_traits< MatrixSpace< T, Rows, Cols > >
	Space metadata for the full rectangular matrix space. More...
struct	space_traits< RowVector< T, Dim > >
	Space registration for a matrix-shaped row vector. More...
struct	space_traits< SymmetricMatrix< Mat > >
	type traits definition for symmetric matrices (subspace of R^n) More...
struct	space_traits< SymmetricMatrixSpace< T, Dim > >
	Space metadata for real symmetric matrices. More...
struct	StandardInnerProduct
	Canonical inner product on a fixed-size vector over a scalar field. More...
struct	SVD
	Computes A = U Sigma V^* from the eigendecomposition of A^* A. More...
struct	SymmetricMatrix
	forward declaration for a matrix More...
struct	SymmetricMatrixInverse
	Symmetric matrix inverse via LDL^T. More...
struct	SymmetricMatrixSpace
	Space of real symmetric Dim x Dim matrices. More...
struct	SymmetricOperator
	forward declaration for symmetric operator More...
struct	Vector
	Owning, fixed-size vector container. More...
struct	vector_scalar
	Recursively obtain the non-vector scalar underlying a type. More...
struct	vector_scalar< Vector< T, Dim > >
	Recursively unwrap one Vector layer. More...
struct	vector_traits
	Customization point for fixed-size vector metadata. More...
struct	vector_traits< Cholesky< Mat > >
	vec type traits for cholesky matrices More...
struct	vector_traits< Endomorphism< Mat, Derived > >
	vec type traits for linear operator More...
struct	vector_traits< Hessenberg< Mat > >
	vec type traits for hessenberg matrices More...
struct	vector_traits< Hessian< Mat, Type > >
	vec type traits for hessian matrices More...
struct	vector_traits< LDL< Mat > >
	vec type traits for LDL matrices More...
struct	vector_traits< QuadraticRep< Mat, Circ > >
	vec type traits for quadratic rep matrices More...
struct	vector_traits< SymmetricMatrix< Mat > >
	vec type traits for symmetric matrices More...
struct	vector_traits< SymmetricOperator< Mat, Derived > >
	vec type traits for symmetric operator More...
struct	vector_traits< Vector< T, Dim > >
	Traits and constexpr construction helpers for Vector<T, Dim>. More...

Concepts
concept	MatrixContainer
	Trait-registered, fixed-shape matrix over a supported scalar field.
concept	SquareMatrix
	Matrix container whose compile-time row and column counts are equal.
concept	RealMatrix
	Matrix container with a real scalar field.
concept	ComplexMatrix
	Matrix container with a complex scalar field.
concept	MatVecCompatible
	Compile-time compatibility contract for matrix-vector multiplication.
concept	MatMulCompatible
	Compile-time compatibility contract for matrix multiplication.
concept	NormFunction
	concept for norm like
concept	InnerProduct
	Callable contract for a trait-registered inner-product operation.
concept	LinearOperator
concept	EndomorphicOperator
concept	Scalar
	Type registered with the Akreon Linalg scalar trait system.
concept	Field
	Registered scalar supporting the Akreon Linalg's field arithmetic interface.
concept	Real
	Field classified by scalar_traits as real.
concept	Complex
	Field classified by scalar_traits as complex.
concept	Space
	Trait-registered finite-dimensional mathematical space.
concept	CoordinatizedSpace
	Finite-dimensional space with explicit element-coordinate conversion.
concept	VectorSpace
	Legacy vector-space expression contract for a registered space.
concept	RnSpace
	Real, coordinatized space satisfying the vector-space expressions.
concept	VectorContainer
	Trait-registered fixed-size vector container.
concept	FieldVector
	Registered vector whose element type satisfies Field.
concept	RealVector
	Field vector over a scalar classified as real.
concept	ComplexVector
	Field vector over a scalar classified as complex.
concept	CompatibleVectors
	Pair of vectors with exactly matching element types and dimensions.

Typedefs
template<typename T , std::size_t Dim>
using	ColVector = Matrix< T, Dim, 1 >
	Matrix-shaped representation of a Dim-element column vector.
template<typename T , std::size_t Dim>
using	RowVector = Matrix< T, 1, Dim >
	Matrix-shaped representation of a Dim-element row vector.
template<class Mat >
using	matrix_scalar_t = typename mat_traits< remove_cvref_t< Mat > >::value_type
	Registered matrix scalar type after cv/ref normalization.
template<class Norm >
using	norm_traits_t = norm_traits< remove_cvref_t< Norm > >
	normalized norm traits
template<class Norm >
using	norm_domain_t = typename norm_traits_t< Norm >::domain_type
template<class Norm >
using	norm_scalar_t = typename norm_traits_t< Norm >::scalar_type
template<class Norm >
using	norm_codomain_t = typename norm_traits_t< Norm >::codomain_type
template<class Product >
using	inner_product_traits_t = inner_product_traits< remove_cvref_t< Product > >
	Normalized trait specialization for an inner-product type.
template<class Product >
using	inner_product_domain_t = typename inner_product_traits_t< Product >::domain_type
	Registered vector domain after product cv/ref normalization.
template<class Product >
using	inner_product_scalar_t = typename inner_product_traits_t< Product >::scalar_type
	Registered scalar field associated with the product domain.
template<class Product >
using	inner_product_codomain_t = typename inner_product_traits_t< Product >::codomain_type
	Registered result type of the inner product.
template<class Operator >
using	operator_traits_t = operator_traits< remove_cvref_t< Operator > >
template<class Operator >
using	operator_scalar_t = typename operator_traits_t< Operator >::scalar_type
template<class Operator >
using	operator_domain_t = typename operator_traits_t< Operator >::domain_type
template<class Operator >
using	operator_codomain_t = typename operator_traits_t< Operator >::codomain_type
template<class Operator >
using	operator_matrix_t = typename operator_traits_t< Operator >::matrix_type
template<typename T >
using	remove_cvref_t = std::remove_cvref_t< T >
	Remove top-level const, volatile, and reference qualification.
template<typename T >
using	scalar_traits_t = scalar_traits< remove_cvref_t< T > >
	Scalar traits selected after cv/ref normalization.
template<typename T >
using	scalar_type_t = typename scalar_traits_t< T >::scalar_type
	Normalized scalar representation registered for T.
template<typename T >
using	real_type_t = typename scalar_traits_t< T >::real_type
	Associated real component type registered for T.
template<class Space >
using	coordinate_map_t = coordinate_map< remove_cvref_t< Space > >
	Coordinate-map specialization selected after cv/ref normalization.
template<class Space >
using	space_type_t = typename space_traits< remove_cvref_t< Space > >::value_type
	Legacy scalar-field alias after cv/ref normalization.
template<class Space >
using	space_point_t = typename space_traits< remove_cvref_t< Space > >::Point
	Legacy point/domain alias after cv/ref normalization.
template<class Space >
using	space_scalar_t = typename space_traits< remove_cvref_t< Space > >::scalar_type
	Scalar field registered for a space.
template<class Space >
using	space_element_t = typename space_traits< remove_cvref_t< Space > >::element_type
	Canonical element representation registered for a space.
template<class Space >
using	space_coordinate_t = typename space_traits< remove_cvref_t< Space > >::coordinate_type
	Finite coordinate representation registered for a space.
template<class Vec >
using	vector_traits_t = vector_traits< remove_cvref_t< Vec > >
	Vector traits selected after cv/ref normalization.
template<class Vec >
using	vector_element_t = typename vector_traits_t< Vec >::element_type
	Registered vector coordinate type.
template<class Vec >
using	vector_scalar_t = typename vector_scalar< remove_cvref_t< Vec > >::type
	Fully unwrapped scalar type underlying a possibly nested vector.
Common scalar and matrix aliases
These aliases shorten frequently used real and complex fixed-size types. ColVector* and RowVector* remain matrices; Vector* names the dedicated vector container.
template<std::size_t Dim>
using	Vectord = Vector< double, Dim >
template<std::size_t Dim>
using	Vectorf = Vector< float, Dim >
template<std::size_t Dim>
using	Vectorcd = Vector< std::complex< double >, Dim >
template<std::size_t Dim>
using	Vectorcf = Vector< std::complex< float >, Dim >
template<std::size_t Rows, std::size_t Cols>
using	Matrixd = Matrix< double, Rows, Cols >
template<std::size_t Dim>
using	ColVectord = ColVector< double, Dim >
template<std::size_t Dim>
using	RowVectord = RowVector< double, Dim >
template<std::size_t Rows, std::size_t Cols>
using	Matrixf = Matrix< float, Rows, Cols >
template<std::size_t Dim>
using	ColVectorf = ColVector< float, Dim >
template<std::size_t Dim>
using	RowVectorf = RowVector< float, Dim >
template<std::size_t Rows, std::size_t Cols>
using	Matrixcd = Matrix< std::complex< double >, Rows, Cols >
template<std::size_t Dim>
using	ColVectorcd = ColVector< std::complex< double >, Dim >
template<std::size_t Dim>
using	RowVectorcd = RowVector< std::complex< double >, Dim >
template<std::size_t Rows, std::size_t Cols>
using	Matrixcf = Matrix< std::complex< float >, Rows, Cols >
template<std::size_t Dim>
using	ColVectorcf = ColVector< std::complex< float >, Dim >
template<std::size_t Dim>
using	RowVectorcf = RowVector< std::complex< float >, Dim >

Enumerations
enum struct	NumericalEvent : uint8_t {   NONE = 0 , RANK_DEFICIENCY , INDEFINITE_FACTORIZATION , NEAR_SINGULAR_TRIANGULAR_SOLVE ,   NUMERICAL_SAFEGUARD , DB_MAX_ITERS_REACHED , FACTORIZATION_FAILED , EVD_FAILED }
	NumericalEvent. More...
enum struct	SymmetricOperatorType {   INV = 0 , IND , PSD , PD ,   NSD , ND }
	simple enum to define symmetric matrix types INV = Invalid IND = Indefinite PSD = Positive Semidefinite PD = Positive Definite NSD = Negative Semidefinite ND = Negative Definite More...
enum struct	DifferentialType { PUSHFORWARD , PULLBACK }
	differential type More...

Functions
template<VectorContainer Vec>
constexpr auto	add (const Vec &A, const Vec &B)
	add two vectors together to get a third
template<VectorContainer Vec>
constexpr auto	operator+ (const Vec &A, const Vec &B)
	operator overload for add
template<MatrixContainer Mat>
constexpr auto	add (const Mat &A, const Mat &B)
	Add two shape-compatible matrices element by element.
template<MatrixContainer Mat>
constexpr auto	operator+ (const Mat &A, const Mat &B)
	Matrix addition operator delegating to add.
template<VectorContainer Vec, Field Alpha>
constexpr auto	scalar_multiplication (const Alpha &alpha, const Vec &A)
	multiply a vector A with a scalar alpha
template<VectorContainer Vec, Field Alpha>
constexpr auto	operator* (const Alpha &alpha, const Vec &A)
	operator overload for scalar multiplication
template<MatrixContainer Mat, Field Alpha>
constexpr auto	scalar_multiplication (const Alpha &alpha, const Mat &A)
	Multiply every matrix entry by a field scalar.
template<MatrixContainer Mat, Field Alpha>
constexpr auto	operator* (const Alpha &alpha, const Mat &A)
	Left scalar-matrix multiplication operator.
template<VectorContainer Vec>
constexpr auto	operator- (const Vec &A, const Vec &B)
	operator overload for subtraction
template<MatrixContainer Mat>
constexpr auto	operator- (const Mat &A, const Mat &B)
	Subtract two matrices as A + (-1) * B.
template<FieldVector Vec>
constexpr auto	transpose (const Vec &x)
	tranpose of a vector
template<ComplexVector Vec>
constexpr auto	conjugate (const Vec &x)
	conjugate a vector
template<FieldVector Vec> requires (vector_dim_v<Vec> == 3)
constexpr auto	cross (const Vec &x, const Vec &y)
	cross product between x x y = Z
template<FieldVector Vec> requires (vector_dim_v<Vec> > 0)
constexpr bool	all_zero (const Vec &x)
	simple function that checks if a vector contains only zeros (hence spans nullspace)
template<std::size_t idx, FieldVector Vec> requires (vector_dim_v<Vec> > 0 && idx < vector_dim_v<Vec>)
constexpr auto	splice (const Vec &x)
	splice operator
template<MatrixContainer MatA, MatrixContainer MatB> requires (MatMulCompatible<MatA, MatB>)
constexpr auto	outer_product (const MatA &A, const MatB &B)
	Multiply matrix-shaped column and row operands as an outer product.
template<MatrixContainer Mat>
constexpr auto	transpose (const Mat &A)
	transpose of a matrix
template<SquareMatrix Mat> requires (mat_rows_v<Mat> > 0)
constexpr auto	trace (const Mat &A)
	trace of a matrix
template<MatrixContainer Mat> requires (Complex<matrix_scalar_t<Mat>>)
constexpr auto	conjugate_transpose (const Mat &A)
	conjugate transpose of a matrix
template<MatrixContainer MatA, MatrixContainer MatB> requires (MatMulCompatible<MatA, MatB>)
constexpr auto	operator* (const MatA &A, const MatB &B)
	Multiply compatible matrices with the standard row-column product.
template<MatrixContainer Mat, VectorContainer Vec> requires (MatVecCompatible<Mat, Vec>)
constexpr auto	operator* (const Mat &A, const Vec &X)
	Apply a compatible matrix to a vector.
template<VectorContainer Vec> requires (vector_dim_v<Vec> > 0)
constexpr auto	min (const Vec &x)
	min element
template<VectorContainer Vec> requires (vector_dim_v<Vec> > 0)
constexpr auto	max (const Vec &x)
	max element
template<VectorContainer Vec> requires (vector_dim_v<Vec> > 0)
constexpr std::size_t	arg_min (const Vec &x)
	min element
template<VectorContainer Vec> requires (vector_dim_v<Vec> > 0)
constexpr std::size_t	arg_max (const Vec &x)
	max element
template<std::size_t Start, std::size_t End, VectorContainer Vec> requires (End - Start < vector_dim_v<Vec> && Start < vector_dim_v<Vec>)
constexpr auto	get_elements (const Vec &x)
	Copy the half-open vector slice [Start, End).
template<Field T, std::size_t N>
constexpr Matrix< T, N, N >	make_identity ()
	create an identity matrix of size N type T
template<Field T, std::size_t Rows, std::size_t Cols>
constexpr Matrix< T, Rows, Cols >	make_zeros ()
	create a matrix filled with zeros of size Rows x Cols of type T
template<VectorContainer Vec>
constexpr auto	as_col_vector (const Vec &X)
	express the vector X as a column vector (N x 1)
template<VectorContainer Vec>
constexpr auto	as_row_vector (const Vec &X)
	express the vector X as a row vector (1 x N)
template<MatrixContainer Mat>
constexpr auto	as_complex_matrix (const Mat &A)
	express matrix as complex one
template<Field T, std::size_t N>
constexpr auto	as_vector (const ColVector< T, N > &A)
	express col vector as vector
template<Field T, std::size_t N>
constexpr auto	as_vector (const RowVector< T, N > &A)
	express row vector as vector
template<MatrixContainer Mat>
constexpr auto	get_col (const Mat &A, std::size_t idx)
	get the column of matrix A based on given index
template<MatrixContainer Mat>
constexpr auto	get_row (const Mat &A, std::size_t idx)
	get the row of matrix A based on given index
template<MatrixContainer Mat>
constexpr auto	get_diag (const Mat &A)
	get the diagonal of a matrix A
template<Field T, std::size_t N>
constexpr auto	make_diag (const Vector< T, N > &X)
	make a diagonal matrix from a vector
template<std::size_t Start, std::size_t End, MatrixContainer Mat> requires (End - Start <= mat_rows_v<Mat> && Start < mat_rows_v<Mat>)
constexpr auto	get_n_rows (const Mat &A)
	Copy the half-open matrix row range [Start, End).
template<std::size_t Start, std::size_t End, MatrixContainer Mat> requires (End - Start <= mat_cols_v<Mat> && Start < mat_cols_v<Mat>)
constexpr auto	get_n_cols (const Mat &A)
	Copy the half-open matrix column range [Start, End).
template<std::size_t StartRow, std::size_t EndRow, std::size_t StartCol, std::size_t EndCol, MatrixContainer Mat> requires (EndRow - StartRow <= mat_rows_v<Mat> && StartRow < mat_rows_v<Mat> && EndCol - StartCol <= mat_cols_v<Mat> && StartCol < mat_cols_v<Mat>)
constexpr auto	get_block (const Mat &A)
	Copy a compile-time rectangular matrix block.
template<std::size_t StartRow, std::size_t EndRow, std::size_t StartCol, std::size_t EndCol, MatrixContainer Mat> requires (EndRow - StartRow <= mat_rows_v<Mat> && StartRow < mat_rows_v<Mat> && EndCol - StartCol <= mat_cols_v<Mat> && StartCol < mat_cols_v<Mat>)
constexpr auto	set_block (Mat A, const Matrix< matrix_scalar_t< Mat >, EndRow - StartRow, EndCol - StartCol > &B)
	Return a matrix copy with one compile-time block replaced.
template<MatrixContainer Mat>
constexpr auto	get_upper_right_triangular_matrix (const Mat &A)
	get upper right triangular matrix from A
template<MatrixContainer Mat>
constexpr auto	get_lower_left_triangular_matrix (const Mat &A)
	get lower left triangular matrix from A
template<VectorContainer Vec>
constexpr auto	euclidean_distance (const Vec &X, const Vec &Y)
	Compute the euclidean distance between two vectors (L2 norm of delta)
template<VectorContainer Vec>
constexpr auto	manhattan_distance (const Vec &X, const Vec &Y)
	Compute the manhattan distance between two vectors (L1 norm of delta)
template<FieldVector Vec>
constexpr bool	all_equal (const Vec &X, const Vec &Y, real_type_t< vector_element_t< Vec > > tol=1e-10)
	checks if two vectors are equivalent to a given tolerance (different than == operator on vector which must be exactly equal
template<Field T, std::size_t Rows, std::size_t Cols>
constexpr bool	all_equal (const Matrix< T, Rows, Cols > &A, const Matrix< T, Rows, Cols > &B, real_type_t< T > tol=1e-10)
	checks if two given matrix are equivalent
template<MatrixContainer MatA, MatrixContainer MatB> requires (std::same_as<matrix_scalar_t<MatA>, matrix_scalar_t<MatB>> && mat_rows_v<MatA> == mat_cols_v<MatB> && mat_cols_v<MatA> == mat_rows_v<MatB>)
constexpr bool	rows_equal_cols (const MatA &A, const MatB &B)
	checks if the row space of one matrix is equivalent to the column space of another matrix
template<std::size_t N, FieldVector Vec> requires (vector_dim_v<Vec> == N * (N + 1) / 2)
constexpr auto	make_smat (const Vec &L)
	make symmetric matrix
template<SquareMatrix Mat>
constexpr auto	get_lower_triangle_with_diag (const Mat &A)
	get lower triangle with diag
template<SquareMatrix Mat>
constexpr auto	get_svec (const Mat &A)
	get svec encoding of a symmetric matrix
template<SquareMatrix Mat>
constexpr auto	make_svec (const Mat &A)
	Compatibility spelling for get_svec(A).
template<SquareMatrix Mat>
constexpr auto	symmetric_part (const Mat &A)
	symmetric part of a square matrix
template<SquareMatrix Mat>
constexpr auto	get_lower_triangle (const Mat &A)
	get lower triangle
template<SquareMatrix Mat, VectorContainer Vec> requires (std::same_as<matrix_scalar_t<Mat>, vector_element_t<Vec>>)
constexpr auto	make_unit_lower_triangle (const Vec &x)
	Construct a unit lower-triangular matrix from strict-lower entries.
template<SquareMatrix Mat, VectorContainer Vec> requires (std::same_as<matrix_scalar_t<Mat>, vector_element_t<Vec>>)
constexpr auto	make_lower_triangle (const Vec &x)
	Construct a lower-triangular matrix including its diagonal.
template<Field T, std::size_t Cols, typename... Rows>
	Matrix (Vector< T, Cols >, Rows...) -> Matrix< T, 1+sizeof...(Rows), Cols >
	Deduce matrix scalar type and shape from one or more row vectors.
template<Field T, std::size_t Dim>
constexpr auto	as_row_matrix (const Vector< T, Dim > &vector)
	Represent a vector as a 1 x Dim matrix.
template<Field T, std::size_t Dim>
constexpr auto	as_col_matrix (const Vector< T, Dim > &vector)
	Represent a vector as a Dim x 1 matrix.
template<Field T, std::size_t Rows, std::size_t Cols> requires (Rows == 1 || Cols == 1)
constexpr auto	as_vector (const Matrix< T, Rows, Cols > &matrix)
	Convert a one-row or one-column matrix to a vector.
constexpr bool	is_fatal_event (NumericalEvent event)
	is_fatal_event
template<FieldVector Vec>
constexpr auto	inner_product (const Vec &lhs, const Vec &rhs)
	Evaluate the canonical inner product for two field vectors.
template<MatrixContainer Mat>
constexpr auto	apply (const Mat &matrix, const operator_domain_t< Mat > &vector) -> operator_codomain_t< Mat >
template<class Space >
constexpr auto	coordinates (const space_element_t< Space > &element)
	Convert a registered space element to its coordinate representation.
template<class Space >
constexpr auto	from_coordinates (const space_coordinate_t< Space > &coordinate)
	Reconstruct a registered space element from its coordinates.
template<VectorContainer Vec> requires (Complex<vector_element_t<Vec>>)
constexpr auto	make_group_permutation (const Vec &D, real_type_t< vector_element_t< Vec > > tol=1e-10)
	Group approximately equal eigenvalues into contiguous index runs.
template<std::size_t N>
constexpr auto	invert_permutation (const Vector< std::size_t, N > &p)
	Invert an index permutation.
template<MatrixContainer Mat>
constexpr auto	permute_ptjp (const Mat &J, const Vector< std::size_t, mat_rows_v< Mat > > &p)
	Reorder matrix rows and columns by the same permutation.
template<MatrixContainer Mat>
constexpr auto	unpermute_papt (const Mat &A, const Vector< std::size_t, mat_rows_v< Mat > > &p)
	Scatter matrix entries through a row-and-column permutation.
template<SquareMatrix Mat>
constexpr auto	matrix_exp (const Mat &A)
	Compute exp(A) using its eigenvector and generalized-eigenvector basis.
template<typename T , std::size_t N>
constexpr auto	array_to_tuple (const std::array< T, N > &x)
	Copy a fixed-size array into a tuple of the same length.
template<Field T>
constexpr bool	is_real (const std::complex< T > &z, real_type_t< T > tol=1e-10)
	Check whether a complex value's imaginary component is below a bound.
template<Real T>
constexpr std::optional< T >	constexpr_div (T num, T den)
	Divide two real scalars with an explicit zero-denominator result.
template<Real T>
constexpr T	constexpr_div_safe (T num, T den, T fallback=T(0))
	Divide two real scalars, returning a fallback for a zero denominator.
template<Real T>
constexpr T	constexpr_pow (T x, std::size_t power)
	Raise a real scalar to a nonnegative integral power.
template<Real T>
constexpr std::optional< T >	constexpr_sqrt (T x, T g=1e-10, T tol=1e-10)
	Approximate a nonnegative square root with Newton iteration.
template<Real T>
constexpr T	constexpr_sqrt_safe (T x, T floor=T(0), T g=1e-10, T tol=1e-10)
	Return a scalar square root with a sanitized fallback radicand.
template<Real T>
constexpr T	constexpr_abs (T x)
	Compute real absolute value or complex magnitude.
template<Complex T>
constexpr auto	constexpr_abs (const T &x)
template<Real T>
constexpr std::optional< T >	constexpr_nrt (T x, std::size_t root, T g=1e-10, T tol=1e-10)
	Approximate a real nth root with Newton iteration.
template<Real T>
constexpr T	constexpr_nrt_safe (T x, std::size_t root, T floor=T(0), T g=1e-10, T tol=1e-10)
	Return a scalar nth root with a sanitized fallback radicand.
template<Real T>
constexpr auto	constexpr_sin (T x, int terms=16)
	Approximate sine with a truncated Taylor recurrence.
template<Real T>
constexpr auto	constexpr_cos (T x, int terms=16)
	Approximate cosine with a truncated Taylor recurrence.
template<Real T>
constexpr std::optional< T >	constexpr_atanh (T x, int terms=16)
	atanh (inverse hyperbolic tangent function)
template<Real T>
constexpr T	constexpr_atanh_safe (T x, T fallback=T(0), int terms=16)
	constexpr-safe atanh with fallback
template<Real T>
constexpr auto	constexpr_expi (T theta, int terms=16)
	Approximate exp(i * theta) on the complex unit circle.
template<Real T>
constexpr auto	constexpr_exp (T x, int terms=16)
	Approximate the real exponential with a truncated power series.
template<Complex T>
constexpr auto	constexpr_exp_lambda (const T &lambda, int terms=16)
	Approximate the exponential of a complex scalar.
template<Real T>
constexpr std::optional< T >	constexpr_ln (T x_in, int terms=16)
	natural log
template<Real T>
constexpr T	constexpr_ln_safe (T x, T floor=std::numeric_limits< T >::min(), int terms=16)
	natural log with domain sanitization and fallback
template<std::size_t Start, std::size_t End, typename Func > requires (End - Start >= 0)
constexpr void	constexpr_for (Func &&f)
	constexpr for loop which recursively unrolls a dynamic for loop into compile time recursive loop
template<Complex T>
constexpr bool	is_same (const T &a, const T &b, real_type_t< T > tol=1e-10)
	Compare complex scalars by magnitude of their difference.
template<VectorContainer Vec> requires (Real<vector_element_t<Vec>>)
constexpr auto	argsort_desc (const Vec &x)
	Return coordinate indices sorted by descending real value.
template<typename T , typename... Ts>
	Vector (T, Ts...) -> Vector< std::common_type_t< T, Ts... >, 1+sizeof...(Ts)>
	Deduce coordinate type and dimension from constructor arguments.

Variables
template<Field T>
constexpr T	sqrt2_v {T(1.414213562373095048801688724209698079L)}
template<Field T>
constexpr T	inv_sqrt2_v {T(1) / sqrt2_v<T>}
template<Field T>
constexpr T	ln2_v {T(0.693147180559945309417232121458176568L)}
template<class Mat >
constexpr std::size_t	mat_rows_v = mat_traits<remove_cvref_t<Mat>>::ROWS
	Registered compile-time row count after cv/ref normalization.
template<class Mat >
constexpr std::size_t	mat_cols_v = mat_traits<remove_cvref_t<Mat>>::COLS
	Registered compile-time column count after cv/ref normalization.
template<class Norm >
constexpr std::size_t	norm_dim_v = norm_traits_t<Norm>::dim
template<class Norm >
constexpr bool	is_norm_v = norm_traits<remove_cvref_t<Norm>>{}
	helper for is norm type
template<typename Space >
constexpr bool	is_symmetric_matrix_space_v
	symmetric matrix space type trait var
template<typename T >
constexpr bool	is_real_v = scalar_traits_t<T>::is_real
	Whether normalized T is registered as a real scalar.
template<typename T >
constexpr bool	is_complex_v = scalar_traits_t<T>::is_complex
	Whether normalized T is registered as a complex scalar.
template<class Space >
constexpr std::size_t	space_dim_v
	Finite coordinate dimension after cv/ref normalization.
template<class Vec >
constexpr std::size_t	vector_dim_v = vector_traits_t<Vec>::dim
	Registered compile-time vector dimension.

Typedef Documentation

ColVector

template<typename T , std::size_t Dim>

using linalg::ColVector = typedef Matrix<T, Dim, 1>

Matrix-shaped representation of a Dim-element column vector.

Definition at line 18 of file matrix_traits.hpp.

ColVectorcd

template<std::size_t Dim>

using linalg::ColVectorcd = typedef ColVector<std::complex<double>, Dim>

Definition at line 138 of file matrix_traits.hpp.

ColVectorcf

template<std::size_t Dim>

using linalg::ColVectorcf = typedef ColVector<std::complex<float>, Dim>

Definition at line 147 of file matrix_traits.hpp.

ColVectord

template<std::size_t Dim>

using linalg::ColVectord = typedef ColVector<double, Dim>

Definition at line 120 of file matrix_traits.hpp.

ColVectorf

template<std::size_t Dim>

using linalg::ColVectorf = typedef ColVector<float, Dim>

Definition at line 129 of file matrix_traits.hpp.

coordinate_map_t

template<class Space >

using linalg::coordinate_map_t = typedef coordinate_map<remove_cvref_t<Space> >

Coordinate-map specialization selected after cv/ref normalization.

Definition at line 73 of file coordinate_map.hpp.

inner_product_codomain_t

template<class Product >

using linalg::inner_product_codomain_t = typedef typename inner_product_traits_t<Product>::codomain_type

Registered result type of the inner product.

Definition at line 93 of file inner_product.hpp.

inner_product_domain_t

template<class Product >

using linalg::inner_product_domain_t = typedef typename inner_product_traits_t<Product>::domain_type

Registered vector domain after product cv/ref normalization.

Definition at line 83 of file inner_product.hpp.

inner_product_scalar_t

template<class Product >

using linalg::inner_product_scalar_t = typedef typename inner_product_traits_t<Product>::scalar_type

Registered scalar field associated with the product domain.

Definition at line 88 of file inner_product.hpp.

inner_product_traits_t

template<class Product >

using linalg::inner_product_traits_t = typedef inner_product_traits<remove_cvref_t<Product> >

Normalized trait specialization for an inner-product type.

Definition at line 79 of file inner_product.hpp.

matrix_scalar_t

template<class Mat >

using linalg::matrix_scalar_t = typedef typename mat_traits<remove_cvref_t<Mat> >::value_type

Registered matrix scalar type after cv/ref normalization.

Definition at line 87 of file matrix_traits.hpp.

Matrixcd

template<std::size_t Rows, std::size_t Cols>

using linalg::Matrixcd = typedef Matrix<std::complex<double>, Rows, Cols>

Definition at line 135 of file matrix_traits.hpp.

Matrixcf

template<std::size_t Rows, std::size_t Cols>

using linalg::Matrixcf = typedef Matrix<std::complex<float>, Rows, Cols>

Definition at line 144 of file matrix_traits.hpp.

Matrixd

template<std::size_t Rows, std::size_t Cols>

using linalg::Matrixd = typedef Matrix<double, Rows, Cols>

Definition at line 117 of file matrix_traits.hpp.

Matrixf

template<std::size_t Rows, std::size_t Cols>

using linalg::Matrixf = typedef Matrix<float, Rows, Cols>

Definition at line 126 of file matrix_traits.hpp.

norm_codomain_t

template<class Norm >

using linalg::norm_codomain_t = typedef typename norm_traits_t<Norm>::codomain_type

Definition at line 64 of file norm_traits.hpp.

norm_domain_t

template<class Norm >

using linalg::norm_domain_t = typedef typename norm_traits_t<Norm>::domain_type

Definition at line 58 of file norm_traits.hpp.

norm_scalar_t

template<class Norm >

using linalg::norm_scalar_t = typedef typename norm_traits_t<Norm>::scalar_type

Definition at line 61 of file norm_traits.hpp.

norm_traits_t

template<class Norm >

using linalg::norm_traits_t = typedef norm_traits<remove_cvref_t<Norm> >

normalized norm traits

Definition at line 55 of file norm_traits.hpp.

operator_codomain_t

template<class Operator >

using linalg::operator_codomain_t = typedef typename operator_traits_t<Operator>::codomain_type

Definition at line 292 of file operator_traits.hpp.

operator_domain_t

template<class Operator >

using linalg::operator_domain_t = typedef typename operator_traits_t<Operator>::domain_type

Definition at line 289 of file operator_traits.hpp.

operator_matrix_t

template<class Operator >

using linalg::operator_matrix_t = typedef typename operator_traits_t<Operator>::matrix_type

Definition at line 295 of file operator_traits.hpp.

operator_scalar_t

template<class Operator >

using linalg::operator_scalar_t = typedef typename operator_traits_t<Operator>::scalar_type

Definition at line 286 of file operator_traits.hpp.

operator_traits_t

template<class Operator >

using linalg::operator_traits_t = typedef operator_traits<remove_cvref_t<Operator> >

Definition at line 283 of file operator_traits.hpp.

real_type_t

template<typename T >

using linalg::real_type_t = typedef typename scalar_traits_t<T>::real_type

Associated real component type registered for T.

For example, both double and std::complex<double> have double as their real_type_t.

Definition at line 82 of file scalar_traits.hpp.

remove_cvref_t

template<typename T >

using linalg::remove_cvref_t = typedef std::remove_cvref_t<T>

Remove top-level const, volatile, and reference qualification.

Public linalg traits normalize their lookup keys through this alias so a scalar and cv/ref-qualified views of that scalar share one specialization.

Definition at line 20 of file scalar_traits.hpp.

RowVector

template<typename T , std::size_t Dim>

using linalg::RowVector = typedef Matrix<T, 1, Dim>

Matrix-shaped representation of a Dim-element row vector.

Definition at line 22 of file matrix_traits.hpp.

RowVectorcd

template<std::size_t Dim>

using linalg::RowVectorcd = typedef RowVector<std::complex<double>, Dim>

Definition at line 141 of file matrix_traits.hpp.

RowVectorcf

template<std::size_t Dim>

using linalg::RowVectorcf = typedef RowVector<std::complex<float>, Dim>

Definition at line 150 of file matrix_traits.hpp.

RowVectord

template<std::size_t Dim>

using linalg::RowVectord = typedef RowVector<double, Dim>

Definition at line 123 of file matrix_traits.hpp.

RowVectorf

template<std::size_t Dim>

using linalg::RowVectorf = typedef RowVector<float, Dim>

Definition at line 132 of file matrix_traits.hpp.

scalar_traits_t

template<typename T >

using linalg::scalar_traits_t = typedef scalar_traits<remove_cvref_t<T> >

Scalar traits selected after cv/ref normalization.

Definition at line 69 of file scalar_traits.hpp.

scalar_type_t

template<typename T >

using linalg::scalar_type_t = typedef typename scalar_traits_t<T>::scalar_type

Normalized scalar representation registered for T.

Definition at line 73 of file scalar_traits.hpp.

space_coordinate_t

template<class Space >

using linalg::space_coordinate_t = typedef typename space_traits<remove_cvref_t<Space> >::coordinate_type

Finite coordinate representation registered for a space.

Definition at line 106 of file space_traits.hpp.

space_element_t

template<class Space >

using linalg::space_element_t = typedef typename space_traits<remove_cvref_t<Space> >::element_type

Canonical element representation registered for a space.

Definition at line 101 of file space_traits.hpp.

space_point_t

template<class Space >

using linalg::space_point_t = typedef typename space_traits<remove_cvref_t<Space> >::Point

Legacy point/domain alias after cv/ref normalization.

Definition at line 87 of file space_traits.hpp.

space_scalar_t

template<class Space >

using linalg::space_scalar_t = typedef typename space_traits<remove_cvref_t<Space> >::scalar_type

Scalar field registered for a space.

Definition at line 96 of file space_traits.hpp.

space_type_t

template<class Space >

using linalg::space_type_t = typedef typename space_traits<remove_cvref_t<Space> >::value_type

Legacy scalar-field alias after cv/ref normalization.

Definition at line 83 of file space_traits.hpp.

vector_element_t

template<class Vec >

using linalg::vector_element_t = typedef typename vector_traits_t<Vec>::element_type

Registered vector coordinate type.

Definition at line 66 of file vector_traits.hpp.

vector_scalar_t

template<class Vec >

using linalg::vector_scalar_t = typedef typename vector_scalar<remove_cvref_t<Vec> >::type

Fully unwrapped scalar type underlying a possibly nested vector.

Definition at line 90 of file vector_traits.hpp.

vector_traits_t

template<class Vec >

using linalg::vector_traits_t = typedef vector_traits<remove_cvref_t<Vec> >

Vector traits selected after cv/ref normalization.

Definition at line 62 of file vector_traits.hpp.

Vectorcd

template<std::size_t Dim>

using linalg::Vectorcd = typedef Vector<std::complex<double>, Dim>

Definition at line 111 of file matrix_traits.hpp.

Vectorcf

template<std::size_t Dim>

using linalg::Vectorcf = typedef Vector<std::complex<float>, Dim>

Definition at line 114 of file matrix_traits.hpp.

Vectord

template<std::size_t Dim>

using linalg::Vectord = typedef Vector<double, Dim>

Definition at line 105 of file matrix_traits.hpp.

Vectorf

template<std::size_t Dim>

using linalg::Vectorf = typedef Vector<float, Dim>

Definition at line 108 of file matrix_traits.hpp.

Enumeration Type Documentation

DifferentialType

enum struct linalg::DifferentialType

strong

differential type

Enumerator
PUSHFORWARD
PULLBACK

Definition at line 24 of file operator_traits.hpp.

NumericalEvent

enum struct linalg::NumericalEvent : uint8_t

strong

NumericalEvent.

Enumerator
NONE
RANK_DEFICIENCY
INDEFINITE_FACTORIZATION
NEAR_SINGULAR_TRIANGULAR_SOLVE
NUMERICAL_SAFEGUARD
DB_MAX_ITERS_REACHED
FACTORIZATION_FAILED
EVD_FAILED

Definition at line 26 of file numerical_config.hpp.

SymmetricOperatorType

enum struct linalg::SymmetricOperatorType

strong

simple enum to define symmetric matrix types INV = Invalid IND = Indefinite PSD = Positive Semidefinite PD = Positive Definite NSD = Negative Semidefinite ND = Negative Definite

Enumerator
INV
IND
PSD
PD
NSD
ND

Definition at line 21 of file operator_traits.hpp.

Function Documentation

add() [1/2]

template<MatrixContainer Mat>

constexpr auto linalg::add ( const Mat &  A, const Mat &  B  )

constexpr

Add two shape-compatible matrices element by element.

Parameters

A	Left matrix.
B	Right matrix of the same registered type and shape.

Returns

Owning Matrix containing A(row,col) + B(row,col).

Definition at line 52 of file linalg_ops.hpp.

References mat_rows_v.

add() [2/2]

template<VectorContainer Vec>

constexpr auto linalg::add ( const Vec &  A, const Vec &  B  )

constexpr

add two vectors together to get a third

A + B = C

Parameters

vector	A of type T with N elements
vector	B of type T with N elements

Returns

vector

Definition at line 25 of file linalg_ops.hpp.

References mat_rows_v.

Referenced by operator+(), and operator+().

all_equal() [1/2]

template<Field T, std::size_t Rows, std::size_t Cols>

constexpr bool linalg::all_equal ( const Matrix< T, Rows, Cols > &  A, const Matrix< T, Rows, Cols > &  B, real_type_t< T >  tol = 1e-10  )

constexpr

checks if two given matrix are equivalent

Parameters

matrix	A of type T with Rows x Cols elements
vector	B of type T with Rows x Cols elements
tolerance	set to 1e-10 by default

Returns

T/F

Definition at line 373 of file linalg_util.hpp.

References all_equal(), and mat_rows_v.

all_equal() [2/2]

template<FieldVector Vec>

constexpr bool linalg::all_equal ( const Vec &  X, const Vec &  Y, real_type_t< vector_element_t< Vec > >  tol = 1e-10  )

constexpr

checks if two vectors are equivalent to a given tolerance (different than == operator on vector which must be exactly equal

Parameters

vector	X of type T N elements
vector	Y of type T N elements

Returns

T/F

Definition at line 358 of file linalg_util.hpp.

References euclidean_distance(), and mat_rows_v.

Referenced by all_equal(), and linalg::MatrixSqrt< MatIn >::apply().

all_zero()

template<FieldVector Vec>
requires (vector_dim_v<Vec> > 0)

constexpr bool linalg::all_zero ( const Vec &  x )

constexpr

simple function that checks if a vector contains only zeros (hence spans nullspace)

Parameters

vector

of type T with N elements

Returns

T/F

Definition at line 199 of file linalg_ops.hpp.

References mat_rows_v.

apply()

template<MatrixContainer Mat>

constexpr auto linalg::apply ( const Mat &  matrix, const operator_domain_t< Mat > &  vector  ) -> operator_codomain_t<Mat>

constexpr

Definition at line 298 of file operator_traits.hpp.

References mat_rows_v.

arg_max()

template<VectorContainer Vec>
requires (vector_dim_v<Vec> > 0)

constexpr std::size_t linalg::arg_max ( const Vec &  x )

constexpr

max element

Parameters

vector

x

Returns

max element

Definition at line 422 of file linalg_ops.hpp.

References mat_rows_v.

arg_min()

template<VectorContainer Vec>
requires (vector_dim_v<Vec> > 0)

constexpr std::size_t linalg::arg_min ( const Vec &  x )

constexpr

min element

Parameters

vector

x

Returns

min element

Definition at line 403 of file linalg_ops.hpp.

References mat_rows_v.

argsort_desc()

template<VectorContainer Vec>
requires (Real<vector_element_t<Vec>>)

constexpr auto linalg::argsort_desc ( const Vec &  x )

constexpr

Return coordinate indices sorted by descending real value.

Parameters

x	Fixed-size real vector.

Returns

std::array<std::size_t, N> containing a descending permutation.

Definition at line 540 of file algebra_util.hpp.

References mat_rows_v.

array_to_tuple()

template<typename T , std::size_t N>

constexpr auto linalg::array_to_tuple ( const std::array< T, N > &  x )

constexpr

Copy a fixed-size array into a tuple of the same length.

Parameters

x	Source array in index order.

Returns

Tuple containing x[0] through x[N - 1].

Definition at line 47 of file algebra_util.hpp.

References mat_rows_v.

as_col_matrix()

template<Field T, std::size_t Dim>

constexpr auto linalg::as_col_matrix ( const Vector< T, Dim > &  vector )

constexpr

Represent a vector as a Dim x 1 matrix.

Definition at line 144 of file matrix.hpp.

References mat_rows_v.

as_col_vector()

template<VectorContainer Vec>

constexpr auto linalg::as_col_vector ( const Vec &  X )

constexpr

express the vector X as a column vector (N x 1)

Parameters

vector

X of type T and N elements

Returns

(N x 1) matrix

Definition at line 66 of file linalg_util.hpp.

References mat_rows_v.

Referenced by linalg::Hessenberg< Mat >::Hessenberg(), and linalg::Hessenberg< Mat >::U().

as_complex_matrix()

template<MatrixContainer Mat>

constexpr auto linalg::as_complex_matrix ( const Mat &  A )

constexpr

express matrix as complex one

Parameters

Matrix

A type T with Rows x Cols elements

Returns

A in C

Definition at line 98 of file linalg_util.hpp.

References mat_rows_v.

Referenced by matrix_exp().

as_row_matrix()

template<Field T, std::size_t Dim>

constexpr auto linalg::as_row_matrix ( const Vector< T, Dim > &  vector )

constexpr

Represent a vector as a 1 x Dim matrix.

Definition at line 138 of file matrix.hpp.

References mat_rows_v.

as_row_vector()

template<VectorContainer Vec>

constexpr auto linalg::as_row_vector ( const Vec &  X )

constexpr

express the vector X as a row vector (1 x N)

Parameters

vector

X of type T and N elements

Returns

(1 x N) matrix

Definition at line 82 of file linalg_util.hpp.

References mat_rows_v.

as_vector() [1/3]

template<Field T, std::size_t N>

constexpr auto linalg::as_vector ( const ColVector< T, N > &  A )

constexpr

express col vector as vector

Parameters

Matrix

A type T with N x 1 elements

Returns

vector

Definition at line 121 of file linalg_util.hpp.

References mat_rows_v.

as_vector() [2/3]

template<Field T, std::size_t Rows, std::size_t Cols>
requires (Rows == 1 || Cols == 1)

constexpr auto linalg::as_vector ( const Matrix< T, Rows, Cols > &  matrix )

constexpr

Convert a one-row or one-column matrix to a vector.

The element order follows the non-unit dimension. General rectangular matrices do not implicitly flatten to vectors.

Definition at line 158 of file matrix.hpp.

References mat_rows_v.

as_vector() [3/3]

template<Field T, std::size_t N>

constexpr auto linalg::as_vector ( const RowVector< T, N > &  A )

constexpr

express row vector as vector

Parameters

Matrix

A type T with 1 x N elements

Returns

vector

Definition at line 135 of file linalg_util.hpp.

References mat_rows_v.

conjugate()

template<ComplexVector Vec>

constexpr auto linalg::conjugate ( const Vec &  x )

constexpr

conjugate a vector

Parameters

vector

of type T with N elements

Returns

conjugated vector

Definition at line 166 of file linalg_ops.hpp.

References mat_rows_v.

conjugate_transpose()

template<MatrixContainer Mat>
requires (Complex<matrix_scalar_t<Mat>>)

constexpr auto linalg::conjugate_transpose ( const Mat &  A )

constexpr

conjugate transpose of a matrix

Like A^T, however the field is complex and the elements get conjugated

Parameters

matrix

A of type T Rows x Cols elements

Returns

A^*

Definition at line 300 of file linalg_ops.hpp.

References mat_rows_v, and transpose().

Referenced by linalg::Hessenberg< Mat >::Hessenberg(), linalg::QR< Mat >::solve(), and linalg::Hessenberg< Mat >::U().

constexpr_abs() [1/2]

template<Complex T>

constexpr auto linalg::constexpr_abs ( const T &  x )

constexpr

Definition at line 190 of file algebra_util.hpp.

References constexpr_sqrt_safe().

constexpr_abs() [2/2]

template<Real T>

constexpr T linalg::constexpr_abs ( T  x )

constexpr

Compute real absolute value or complex magnitude.

Parameters

x	Real or complex field scalar.

Returns

|x| in the associated real type.

Definition at line 181 of file algebra_util.hpp.

Referenced by linalg::SymmetricMatrixInverse< MatIn >::apply(), constexpr_atanh(), constexpr_nrt(), is_same(), linalg::L1< Vec >::operator()(), and linalg::Linf< Vec >::operator()().

constexpr_atanh()

template<Real T>

constexpr std::optional< T > linalg::constexpr_atanh ( T  x, int  terms = 16  )

constexpr

atanh (inverse hyperbolic tangent function)

atanh(x) = x + x^3 / 3 + x^5 / 5 + ... = sum_{n=0}^{infinity} frac{x^{2n+1}}{2n+1}

Parameters

x	in (-1, 1)
terms	Number of odd-power series terms.

Returns

y in R

Definition at line 333 of file algebra_util.hpp.

References constexpr_abs(), and mat_rows_v.

Referenced by constexpr_atanh_safe(), and constexpr_ln().

constexpr_atanh_safe()

template<Real T>

constexpr T linalg::constexpr_atanh_safe ( T  x, T  fallback = T(0), int  terms = 16  )

constexpr

constexpr-safe atanh with fallback

Parameters

x	input scalar
fallback	fallback value when |x| >= 1
terms	series terms

Returns

scalar atanh result or fallback

Definition at line 359 of file algebra_util.hpp.

References constexpr_atanh(), and mat_rows_v.

constexpr_cos()

template<Real T>

constexpr auto linalg::constexpr_cos ( T  x, int  terms = 16  )

constexpr

Approximate cosine with a truncated Taylor recurrence.

sum_{k=0}^{infinity}frac{(-1)^{k}x^{2k}}{(2k)!} = cos x

Parameters

x	Angle in radians.
terms	Number of series terms.

Returns

Cosine approximation.

Definition at line 308 of file algebra_util.hpp.

References mat_rows_v.

Referenced by constexpr_expi(), affine::Rotation2d< T >::operator()(), affine::Rotation3d< T >::x(), affine::Rotation3d< T >::y(), and affine::Rotation3d< T >::z().

constexpr_div()

template<Real T>

constexpr std::optional< T > linalg::constexpr_div ( T  num, T  den  )

constexpr

Divide two real scalars with an explicit zero-denominator result.

Parameters

num	numerator
den	denominator

Returns

num / den when denominator is nonzero, else nullopt

Definition at line 79 of file algebra_util.hpp.

References mat_rows_v.

Referenced by constexpr_div_safe().

constexpr_div_safe()

template<Real T>

constexpr T linalg::constexpr_div_safe ( T  num, T  den, T  fallback = T(0)  )

constexpr

Divide two real scalars, returning a fallback for a zero denominator.

Parameters

num	numerator
den	denominator
fallback	fallback value when denominator is zero

Returns

scalar division result or fallback

Definition at line 94 of file algebra_util.hpp.

References constexpr_div(), and mat_rows_v.

Referenced by constexpr_ln(), constexpr_nrt(), and constexpr_sqrt().

constexpr_exp()

template<Real T>

constexpr auto linalg::constexpr_exp ( T  x, int  terms = 16  )

constexpr

Approximate the real exponential with a truncated power series.

exp(x) = 1 + x + x^2/2! + x^3/3!... = sum_{n=0}^infinity frac{x^n}{n!}

Parameters

x	Real exponent.
terms	Number of series terms.

Returns

Real exponential approximation.

Definition at line 393 of file algebra_util.hpp.

References mat_rows_v.

Referenced by constexpr_exp_lambda().

constexpr_exp_lambda()

template<Complex T>

constexpr auto linalg::constexpr_exp_lambda ( const T &  lambda, int  terms = 16  )

constexpr

Approximate the exponential of a complex scalar.

Parameters

lambda	Complex exponent alpha + i * omega.
terms	Number of series terms used by each component approximation.

Returns

Complex exponential approximation.

Definition at line 415 of file algebra_util.hpp.

References constexpr_exp(), constexpr_expi(), and mat_rows_v.

Referenced by matrix_exp().

constexpr_expi()

template<Real T>

constexpr auto linalg::constexpr_expi ( T  theta, int  terms = 16  )

constexpr

Approximate exp(i * theta) on the complex unit circle.

Parameters

theta	Angle in radians.
terms	Number of sine and cosine series terms.

Returns

Complex unit-circle approximation.

Definition at line 372 of file algebra_util.hpp.

References constexpr_cos(), constexpr_sin(), and mat_rows_v.

Referenced by constexpr_exp_lambda().

constexpr_for()

template<std::size_t Start, std::size_t End, typename Func >
requires (End - Start >= 0)

constexpr void linalg::constexpr_for ( Func &&  f )

constexpr

constexpr for loop which recursively unrolls a dynamic for loop into compile time recursive loop

Template Parameters

Start	Inclusive compile-time start index.
End	Exclusive compile-time end index.

Parameters

f	Callable receiving each index as std::integral_constant.

Definition at line 513 of file algebra_util.hpp.

References mat_rows_v.

Referenced by linalg::Hessenberg< Mat >::Hessenberg(), and linalg::Hessenberg< Mat >::U().

constexpr_ln()

template<Real T>

constexpr std::optional< T > linalg::constexpr_ln ( T  x_in, int  terms = 16  )

constexpr

natural log

[old 1] less numerically friendly Taylor series expansion for 1 + t, t = 0 log(1 + t) = t - t^2 / 2 + t^3 / 3 - ... = sum_{k=1}^{infinity}(-1)^{k+1}frac{t^k}{k}

Restricting x in R_++, and let t = x - 1... log(1 + x - 1) = log(x) log(x) approximately (x - 1) - (x - 1)^2 / 2 + (x - 1)^3 - ...

[old 2] use atanh log(x) approximately 2 * atan(y) where y = (x - 1) / (x + 1)

[new] range reduce x = m * 2^e where m in [0.5, 2] for all x > 0 we have x = m 2^e, m in [0.5, 1] and e in Z ln(x) = ln(m) + e ln(2) ** the above makes this much more numerically stable

• perform atan(y) as before
• sum 2a with e*ln(2)

Parameters

x_in	Positive real argument.
terms	Number of inverse-hyperbolic-tangent series terms.

Returns

Logarithm approximation, or std::nullopt outside the positive domain.

Definition at line 452 of file algebra_util.hpp.

References constexpr_atanh(), constexpr_div_safe(), and mat_rows_v.

Referenced by constexpr_ln_safe().

constexpr_ln_safe()

template<Real T>

constexpr T linalg::constexpr_ln_safe ( T  x, T  floor = std::numeric_limits<T>::min(), int  terms = 16  )

constexpr

natural log with domain sanitization and fallback

Ensures no optional access is required at call sites that need a scalar. Handles NaN/invalid by clamping to a positive floor.

Parameters

x	input scalar
floor	positive floor value used when x is invalid/non-positive
terms	series terms

Returns

ln(max(x, floor)) with NaN-safe clamping semantics

Definition at line 492 of file algebra_util.hpp.

References constexpr_ln(), and mat_rows_v.

constexpr_nrt()

template<Real T>

constexpr std::optional< T > linalg::constexpr_nrt ( T  x, std::size_t  root, T  g = 1e-10, T  tol = 1e-10  )

constexpr

Approximate a real nth root with Newton iteration.

f(g) = g^n - x to df/dg = n g^{n-1} Newton formula : g_{k+1} = g_k - frac{g_k^n - x}{n g^{n-1}} equivalent to g_{k+1} = 1/n ((n - 1) g_k + x / g_k^{n-1})

Parameters

x	Real radicand. Negative values require an odd root degree.
root	Root degree; zero returns std::nullopt.
g	Positive initial guess.
tol	Absolute reconstruction tolerance.

Returns

Approximation or std::nullopt for an invalid real-domain request.

Definition at line 212 of file algebra_util.hpp.

References constexpr_abs(), constexpr_div_safe(), mat_rows_v, and MAX_STEP_EVAL.

Referenced by constexpr_nrt_safe().

constexpr_nrt_safe()

template<Real T>

constexpr T linalg::constexpr_nrt_safe ( T  x, std::size_t  root, T  floor = T(0), T  g = 1e-10, T  tol = 1e-10  )

constexpr

Return a scalar nth root with a sanitized fallback radicand.

Parameters

x	input scalar
root	root degree
floor	fallback scalar
g	initial guess
tol	convergence tolerance

Returns

scalar nth-root with optional fallback handling

Definition at line 263 of file algebra_util.hpp.

References constexpr_nrt(), and mat_rows_v.

constexpr_pow()

template<Real T>

constexpr T linalg::constexpr_pow ( T  x, std::size_t  power  )

constexpr

Raise a real scalar to a nonnegative integral power.

Parameters

x	Real base.
power	Nonnegative integral exponent; conventionally positive here.

Definition at line 106 of file algebra_util.hpp.

References mat_rows_v.

constexpr_sin()

template<Real T>

constexpr auto linalg::constexpr_sin ( T  x, int  terms = 16  )

constexpr

Approximate sine with a truncated Taylor recurrence.

sum_{k=0}^{infinity}frac{(-1)^{k}x^{2k+1}}{(2k+1)!}

Parameters

x	Angle in radians.
terms	Number of series terms.

Returns

Sine approximation.

Definition at line 284 of file algebra_util.hpp.

References mat_rows_v.

Referenced by constexpr_expi(), affine::Rotation2d< T >::operator()(), affine::Rotation3d< T >::x(), affine::Rotation3d< T >::y(), and affine::Rotation3d< T >::z().

constexpr_sqrt()

template<Real T>

constexpr std::optional< T > linalg::constexpr_sqrt ( T  x, T  g = 1e-10, T  tol = 1e-10  )

constexpr

Approximate a nonnegative square root with Newton iteration.

f(g) = g^2 - x to df/dg = 2 g Newton formula : g_{k+1} = g_k - frac{g^2 - x} / {2 g_k} equivalent to 0.5 * (g_n + x / g_n)

Parameters

x	Nonnegative radicand; a negative value returns std::nullopt.
g	Positive initial guess; nonpositive guesses are replaced with one.
tol	Absolute squared-residual tolerance.

Returns

Approximation, or std::nullopt outside the real domain. The current zero-input convention returns zero.

Definition at line 130 of file algebra_util.hpp.

References constexpr_div_safe(), mat_rows_v, and MAX_STEP_EVAL.

Referenced by constexpr_sqrt_safe().

constexpr_sqrt_safe()

template<Real T>

constexpr T linalg::constexpr_sqrt_safe ( T  x, T  floor = T(0), T  g = 1e-10, T  tol = 1e-10  )

constexpr

Return a scalar square root with a sanitized fallback radicand.

Parameters

x	input scalar
floor	minimum scalar for fallback
g	initial guess
tol	convergence tolerance

Returns

scalar sqrt with optional fallback handling

Definition at line 163 of file algebra_util.hpp.

References constexpr_sqrt(), and mat_rows_v.

Referenced by constexpr_abs(), linalg::Fro< Mat >::operator()(), linalg::L2< Vec >::operator()(), and linalg::EigenDecomposition< Mat >::spectral_radius().

coordinates()

template<class Space >

constexpr auto linalg::coordinates ( const space_element_t< Space > &  element )

constexpr

Convert a registered space element to its coordinate representation.

Parameters

element

Element in the canonical representation of Space.

Returns

Coordinate vector registered by space_traits<Space>.

Definition at line 81 of file coordinate_map.hpp.

References mat_rows_v.

cross()

template<FieldVector Vec>
requires (vector_dim_v<Vec> == 3)

constexpr auto linalg::cross ( const Vec &  x, const Vec &  y  )

constexpr

cross product between x x y = Z

Parameters

vector	of type T with 3 elements
vector	of type T with 3 elements

Returns

(3x1) vector

Definition at line 184 of file linalg_ops.hpp.

euclidean_distance()

template<VectorContainer Vec>

constexpr auto linalg::euclidean_distance ( const Vec &  X, const Vec &  Y  )

constexpr

Compute the euclidean distance between two vectors (L2 norm of delta)

Parameters

vector	X of type T with N elements
vector	Y of type T with N elements

Returns

distance between two vectors as a scalar

Definition at line 331 of file linalg_util.hpp.

References mat_rows_v.

Referenced by all_equal(), and rows_equal_cols().

from_coordinates()

template<class Space >

constexpr auto linalg::from_coordinates ( const space_coordinate_t< Space > &  coordinate )

constexpr

Reconstruct a registered space element from its coordinates.

Parameters

coordinate

Coordinate representation registered for Space.

Returns

Element in the canonical representation of Space.

Definition at line 91 of file coordinate_map.hpp.

References mat_rows_v.

get_block()

template<std::size_t StartRow, std::size_t EndRow, std::size_t StartCol, std::size_t EndCol, MatrixContainer Mat>
requires (EndRow - StartRow <= mat_rows_v<Mat> && StartRow < mat_rows_v<Mat> && EndCol - StartCol <= mat_cols_v<Mat> && StartCol < mat_cols_v<Mat>)

constexpr auto linalg::get_block ( const Mat &  A )

constexpr

Copy a compile-time rectangular matrix block.

Parameters

matrix

A of type T Rows x Cols elements

Row and column bounds are half-open: [StartRow, EndRow) x [StartCol, EndCol).

Returns

Owning matrix with the selected block shape.

Definition at line 255 of file linalg_util.hpp.

References mat_rows_v.

get_col()

template<MatrixContainer Mat>

constexpr auto linalg::get_col ( const Mat &  A, std::size_t  idx  )

constexpr

get the column of matrix A based on given index

Parameters

matrix	A of size Rows x Cols of type T
index

Returns

vector X

Definition at line 150 of file linalg_util.hpp.

References AK_ASSERT, and mat_rows_v.

Referenced by linalg::LDL< Mat >::apply(), linalg::QuadraticRep< JxJ, Circ >::apply(), linalg::Hessenberg< Mat >::Hessenberg(), and rows_equal_cols().

get_diag()

template<MatrixContainer Mat>

constexpr auto linalg::get_diag ( const Mat &  A )

constexpr

get the diagonal of a matrix A

Parameters

matrix

A of size Rows x Cols of type T

Returns

diagonal vector

Definition at line 181 of file linalg_util.hpp.

References mat_rows_v.

get_elements()

template<std::size_t Start, std::size_t End, VectorContainer Vec>
requires (End - Start < vector_dim_v<Vec> && Start < vector_dim_v<Vec>)

constexpr auto linalg::get_elements ( const Vec &  x )

constexpr

Copy the half-open vector slice [Start, End).

Parameters

vector

of type T with N elements

Returns

Vector of type T with End - Start elements.

Definition at line 26 of file linalg_util.hpp.

References mat_rows_v.

get_lower_left_triangular_matrix()

template<MatrixContainer Mat>

constexpr auto linalg::get_lower_left_triangular_matrix ( const Mat &  A )

constexpr

get lower left triangular matrix from A

Parameters

matrix

A of type T Rows x Cols elements

Returns

matrix

Definition at line 313 of file linalg_util.hpp.

References mat_rows_v.

get_lower_triangle()

template<SquareMatrix Mat>

constexpr auto linalg::get_lower_triangle ( const Mat &  A )

constexpr

get lower triangle

[ 0 0 0 0 ] [ * 0 0 0 ] [ * * 0 0 ] [ * * * 0 ]

Parameters

matrix

A

Returns

vec

Definition at line 519 of file linalg_util.hpp.

References mat_rows_v.

get_lower_triangle_with_diag()

template<SquareMatrix Mat>

constexpr auto linalg::get_lower_triangle_with_diag ( const Mat &  A )

constexpr

get lower triangle with diag

[ * 0 0 0 ] [ * * 0 0 ] [ * * * 0 ] [ * * * * ]

Parameters

matrix

A

Returns

vec

Definition at line 446 of file linalg_util.hpp.

References mat_rows_v.

get_n_cols()

template<std::size_t Start, std::size_t End, MatrixContainer Mat>
requires (End - Start <= mat_cols_v<Mat> && Start < mat_cols_v<Mat>)

constexpr auto linalg::get_n_cols ( const Mat &  A )

constexpr

Copy the half-open matrix column range [Start, End).

Parameters

matrix

A of type T Rows x Cols elements

Returns

matrix

Definition at line 232 of file linalg_util.hpp.

References mat_rows_v.

get_n_rows()

template<std::size_t Start, std::size_t End, MatrixContainer Mat>
requires (End - Start <= mat_rows_v<Mat> && Start < mat_rows_v<Mat>)

constexpr auto linalg::get_n_rows ( const Mat &  A )

constexpr

Copy the half-open matrix row range [Start, End).

Parameters

matrix

A of type T Rows x Cols elements

Returns

matrix

Definition at line 214 of file linalg_util.hpp.

References mat_rows_v.

get_row()

template<MatrixContainer Mat>

constexpr auto linalg::get_row ( const Mat &  A, std::size_t  idx  )

constexpr

get the row of matrix A based on given index

Parameters

matrix	A of size Rows x Cols of type T
index

Returns

vector X

Definition at line 168 of file linalg_util.hpp.

References AK_ASSERT, and mat_rows_v.

get_svec()

template<SquareMatrix Mat>

constexpr auto linalg::get_svec ( const Mat &  A )

constexpr

get svec encoding of a symmetric matrix

Off-diagonal entries are scaled by sqrt(2) to preserve the Frobenius inner product under vectorization.

Parameters

matrix

A

Returns

svec(A)

Definition at line 467 of file linalg_util.hpp.

References mat_rows_v.

Referenced by make_svec().

get_upper_right_triangular_matrix()

template<MatrixContainer Mat>

constexpr auto linalg::get_upper_right_triangular_matrix ( const Mat &  A )

constexpr

get upper right triangular matrix from A

Parameters

matrix

A of type T Rows x Cols elements

Returns

matrix

Definition at line 296 of file linalg_util.hpp.

References mat_rows_v.

Referenced by linalg::QR< Mat >::R().

inner_product()

template<FieldVector Vec>

constexpr auto linalg::inner_product ( const Vec &  lhs, const Vec &  rhs  )

constexpr

Evaluate the canonical inner product for two field vectors.

This is the standard call-site API and delegates to StandardInnerProduct<remove_cvref_t<Vec>>. There is deliberately no separate dot operation with different complex-number semantics.

Parameters

lhs	First vector; conjugated for a complex field.
rhs	Second vector.

Returns

Real or complex field scalar associated with Vec.

Definition at line 123 of file inner_product.hpp.

References mat_rows_v.

Referenced by linalg::L2< Vec >::operator()().

invert_permutation()

template<std::size_t N>

constexpr auto linalg::invert_permutation ( const Vector< std::size_t, N > &  p )

constexpr

Invert an index permutation.

Template Parameters

N	Number of permuted indices.

Parameters

[in]

p

Permutation of indices 0 through N-1.

Returns

Vector<std::size_t, N> satisfying p_inv(p(i)) = i.

Definition at line 53 of file matrix_exp.hpp.

References mat_rows_v.

Referenced by matrix_exp().

is_fatal_event()

constexpr bool linalg::is_fatal_event ( NumericalEvent  event )

inlineconstexpr

is_fatal_event

Parameters

event

(NumericalEvent)

Returns

T/F

Definition at line 48 of file numerical_config.hpp.

References EVD_FAILED, and FACTORIZATION_FAILED.

Referenced by linalg::SymmetricMatrixInverse< MatIn >::apply(), linalg::LDL< Mat >::apply(), and linalg::LDL< Mat >::solve().

is_real()

template<Field T>

constexpr bool linalg::is_real ( const std::complex< T > &  z, real_type_t< T >  tol = 1e-10  )

constexpr

Check whether a complex value's imaginary component is below a bound.

This function evaluates z.imag() <= tol exactly; it is primarily used where the expected imaginary residual is nonnegative. It does not take the absolute value of the imaginary component.

Parameters

z	Complex value to inspect.
tol	Upper bound for its imaginary component.

Definition at line 66 of file algebra_util.hpp.

Referenced by linalg::L2< Vec >::operator()().

is_same()

template<Complex T>

constexpr bool linalg::is_same ( const T &  a, const T &  b, real_type_t< T >  tol = 1e-10  )

constexpr

Compare complex scalars by magnitude of their difference.

Parameters

a	First complex scalar.
b	Second complex scalar.
tol	Strict magnitude tolerance.

Definition at line 528 of file algebra_util.hpp.

References constexpr_abs(), and mat_rows_v.

Referenced by make_group_permutation(), and matrix_exp().

make_diag()

template<Field T, std::size_t N>

constexpr auto linalg::make_diag ( const Vector< T, N > &  X )

constexpr

make a diagonal matrix from a vector

Parameters

vector

X of size N and type T

Returns

diagonal matrix

Definition at line 199 of file linalg_util.hpp.

References mat_rows_v.

Referenced by linalg::LDL< Mat >::D().

make_group_permutation()

template<VectorContainer Vec>
requires (Complex<vector_element_t<Vec>>)

constexpr auto linalg::make_group_permutation ( const Vec &  D, real_type_t< vector_element_t< Vec > >  tol = 1e-10  )

constexpr

Group approximately equal eigenvalues into contiguous index runs.

Template Parameters

Vec	Vector type with complex scalar entries.

Parameters

[in]	D	Complex eigenvalue vector.
[in]	tol	Absolute tolerance for comparing eigenvalues.

Returns

Vector<std::size_t, N> mapping grouped positions to original indices.

Definition at line 23 of file matrix_exp.hpp.

References is_same(), and mat_rows_v.

Referenced by matrix_exp().

make_identity()

template<Field T, std::size_t N>

constexpr Matrix< T, N, N > linalg::make_identity ( )

constexpr

create an identity matrix of size N type T

Returns

I

Definition at line 39 of file linalg_util.hpp.

References mat_rows_v.

make_lower_triangle()

template<SquareMatrix Mat, VectorContainer Vec>
requires (std::same_as<matrix_scalar_t<Mat>, vector_element_t<Vec>>)

constexpr auto linalg::make_lower_triangle ( const Vec &  x )

constexpr

Construct a lower-triangular matrix including its diagonal.

Parameters

x

(vector)

Coordinates are consumed row by row through the diagonal; entries above the diagonal remain value-initialized zero.

Returns

Lower-triangular matrix L.

Definition at line 568 of file linalg_util.hpp.

References mat_rows_v.

make_smat()

template<std::size_t N, FieldVector Vec>
requires (vector_dim_v<Vec> == N * (N + 1) / 2)

constexpr auto linalg::make_smat ( const Vec &  L )

constexpr

make symmetric matrix

Parameters

symmetric

vector (flattened)

Returns

matrix

Definition at line 412 of file linalg_util.hpp.

References mat_rows_v, and transpose().

make_svec()

template<SquareMatrix Mat>

constexpr auto linalg::make_svec ( const Mat &  A )

constexpr

Compatibility spelling for get_svec(A).

Parameters

A	Square matrix to encode in lower-triangular svec order.

Returns

Square-root-of-two-scaled symmetric coordinates.

Definition at line 487 of file linalg_util.hpp.

References get_svec(), and mat_rows_v.

make_unit_lower_triangle()

template<SquareMatrix Mat, VectorContainer Vec>
requires (std::same_as<matrix_scalar_t<Mat>, vector_element_t<Vec>>)

constexpr auto linalg::make_unit_lower_triangle ( const Vec &  x )

constexpr

Construct a unit lower-triangular matrix from strict-lower entries.

Parameters

vector

x

Coordinates are consumed row by row below the diagonal. Diagonal entries are set to one and upper-triangular entries remain value-initialized zero.

Returns

Unit lower-triangular matrix L.

Definition at line 541 of file linalg_util.hpp.

References mat_rows_v.

make_zeros()

template<Field T, std::size_t Rows, std::size_t Cols>

constexpr Matrix< T, Rows, Cols > linalg::make_zeros ( )

constexpr

create a matrix filled with zeros of size Rows x Cols of type T

Returns

Zeros

Definition at line 51 of file linalg_util.hpp.

References mat_rows_v.

manhattan_distance()

template<VectorContainer Vec>

constexpr auto linalg::manhattan_distance ( const Vec &  X, const Vec &  Y  )

constexpr

Compute the manhattan distance between two vectors (L1 norm of delta)

Parameters

vector	X of type T with N elements
vector	Y of type T with N elements

Returns

distance between two vectors as a scalar

Definition at line 344 of file linalg_util.hpp.

References mat_rows_v.

Matrix()

template<Field T, std::size_t Cols, typename... Rows>

linalg::Matrix	(	Vector< T, Cols >	,
		Rows...
	)		-> Matrix< T, 1+sizeof...(Rows), Cols >

Deduce matrix scalar type and shape from one or more row vectors.

For example, two Vector<double, 3> arguments deduce Matrix<double, 2, 3>.

matrix_exp()

template<SquareMatrix Mat>

constexpr auto linalg::matrix_exp ( const Mat &  A )

constexpr

Compute exp(A) using its eigenvector and generalized-eigenvector basis.

Algorithm flow:

1. Recover a basis V and form J = V^-1 A V. Ordinary eigenvectors give diagonal blocks; generalized eigenvectors retain Jordan-chain couplings.
2. Permute equal-eigenvalue coordinates together so each block can be evaluated independently.
3. For a block B = lambda*I + K, separate the scalar exponential from the nilpotent part: exp(B) = exp(lambda) * exp(K). For a block of size k, K^k = 0, so exp(K) = I + K + ... + K^(k-1)/(k-1)! is a finite sum.
4. Undo the coordinate grouping and return V * exp(J) * V^-1.

This path relies on the decomposition producing an invertible generalized eigenvector basis and approximately upper-triangular repeated-root blocks. Eigenvalues are grouped by the helpers' absolute comparison tolerance. Complex arithmetic is retained even for real A, whose eigenvalues may occur in complex-conjugate pairs.

Template Parameters

Mat	Square matrix type.

Parameters

[in]

A

Square input matrix.

Returns

Complex N-by-N matrix exp(A), or a zero matrix if decomposition or inversion fails.

Definition at line 121 of file matrix_exp.hpp.

References as_complex_matrix(), constexpr_exp_lambda(), invert_permutation(), is_same(), make_group_permutation(), mat_rows_v, NONE, permute_ptjp(), and unpermute_papt().

max()

template<VectorContainer Vec>
requires (vector_dim_v<Vec> > 0)

constexpr auto linalg::max ( const Vec &  x )

constexpr

max element

Parameters

vector

x

Returns

max element

Definition at line 386 of file linalg_ops.hpp.

References mat_rows_v.

min()

template<VectorContainer Vec>
requires (vector_dim_v<Vec> > 0)

constexpr auto linalg::min ( const Vec &  x )

constexpr

min element

Parameters

vector

x

Returns

min element

Definition at line 369 of file linalg_ops.hpp.

References mat_rows_v.

operator*() [1/4]

template<MatrixContainer Mat, Field Alpha>

constexpr auto linalg::operator* ( const Alpha &  alpha, const Mat &  A  )

constexpr

Left scalar-matrix multiplication operator.

Definition at line 118 of file linalg_ops.hpp.

References mat_rows_v, and scalar_multiplication().

operator*() [2/4]

template<VectorContainer Vec, Field Alpha>

constexpr auto linalg::operator* ( const Alpha &  alpha, const Vec &  A  )

constexpr

operator overload for scalar multiplication

See also

scalar_multiplication

Definition at line 94 of file linalg_ops.hpp.

References mat_rows_v, and scalar_multiplication().

operator*() [3/4]

template<MatrixContainer Mat, VectorContainer Vec>
requires (MatVecCompatible<Mat, Vec>)

constexpr auto linalg::operator* ( const Mat &  A, const Vec &  X  )

constexpr

Apply a compatible matrix to a vector.

Parameters

matrix	A of type T Rows x Cols elements
vector	X of type T Cols elements

Returns

Vector over the same field with dimension mat_rows_v<Mat>.

Definition at line 347 of file linalg_ops.hpp.

References mat_rows_v.

operator*() [4/4]

template<MatrixContainer MatA, MatrixContainer MatB>
requires (MatMulCompatible<MatA, MatB>)

constexpr auto linalg::operator* ( const MatA &  A, const MatB &  B  )

constexpr

Multiply compatible matrices with the standard row-column product.

Parameters

matrix	A of type T Rows x Cols elements
matrix	B of type T Cols x Rows elements

Returns

Matrix<T, rows(A), cols(B)>.

Definition at line 321 of file linalg_ops.hpp.

References mat_rows_v.

operator+() [1/2]

template<MatrixContainer Mat>

constexpr auto linalg::operator+ ( const Mat &  A, const Mat &  B  )

constexpr

Matrix addition operator delegating to add.

Definition at line 65 of file linalg_ops.hpp.

References add(), and mat_rows_v.

operator+() [2/2]

template<VectorContainer Vec>

constexpr auto linalg::operator+ ( const Vec &  A, const Vec &  B  )

constexpr

operator overload for add

See also

add

Definition at line 41 of file linalg_ops.hpp.

References add(), and mat_rows_v.

operator-() [1/2]

template<MatrixContainer Mat>

constexpr auto linalg::operator- ( const Mat &  A, const Mat &  B  )

constexpr

Subtract two matrices as A + (-1) * B.

Definition at line 135 of file linalg_ops.hpp.

References mat_rows_v.

operator-() [2/2]

template<VectorContainer Vec>

constexpr auto linalg::operator- ( const Vec &  A, const Vec &  B  )

constexpr

operator overload for subtraction

See also

add

scalar_multiplication

Definition at line 129 of file linalg_ops.hpp.

References mat_rows_v.

outer_product()

template<MatrixContainer MatA, MatrixContainer MatB>
requires (MatMulCompatible<MatA, MatB>)

constexpr auto linalg::outer_product ( const MatA &  A, const MatB &  B  )

constexpr

Multiply matrix-shaped column and row operands as an outer product.

A * B = C where C is a new matrix of Rows_of_A x Cols_of_B

Parameters

matrix	A of type T Rows1 x Cols1 elements
matrix	B of type T Cols1 x Cols2 elements

Returns

Matrix with mat_rows_v<MatA> rows and mat_cols_v<MatB> columns. The implementation uses A(i, 0) * B(0, j); callers should supply a one-column left operand and one-row right operand.

Definition at line 241 of file linalg_ops.hpp.

References mat_rows_v.

permute_ptjp()

template<MatrixContainer Mat>

constexpr auto linalg::permute_ptjp ( const Mat &  J, const Vector< std::size_t, mat_rows_v< Mat > > &  p  )

constexpr

Reorder matrix rows and columns by the same permutation.

Template Parameters

Mat	Square matrix type.

Parameters

[in]	J	Square matrix to reorder.
[in]	p	Permutation mapping new indices to original indices.

Returns

Matrix of type Mat with entry (i,j) = J(p(i),p(j)).

Definition at line 68 of file matrix_exp.hpp.

References mat_rows_v.

Referenced by matrix_exp().

rows_equal_cols()

template<MatrixContainer MatA, MatrixContainer MatB>
requires (std::same_as<matrix_scalar_t<MatA>, matrix_scalar_t<MatB>> && mat_rows_v<MatA> == mat_cols_v<MatB> && mat_cols_v<MatA> == mat_rows_v<MatB>)

constexpr bool linalg::rows_equal_cols ( const MatA &  A, const MatB &  B  )

constexpr

checks if the row space of one matrix is equivalent to the column space of another matrix

Parameters

matrix	A of type T with Rows x Cols elements
vector	B of type T with Cols x Rows elements

Returns

T/F

Definition at line 395 of file linalg_util.hpp.

References euclidean_distance(), get_col(), and mat_rows_v.

scalar_multiplication() [1/2]

template<MatrixContainer Mat, Field Alpha>

constexpr auto linalg::scalar_multiplication ( const Alpha &  alpha, const Mat &  A  )

constexpr

Multiply every matrix entry by a field scalar.

Parameters

alpha	Scalar multiplier.
A	Matrix operand.

Returns

Owning matrix with the same shape and registered matrix scalar type.

Definition at line 105 of file linalg_ops.hpp.

References mat_rows_v.

scalar_multiplication() [2/2]

template<VectorContainer Vec, Field Alpha>

constexpr auto linalg::scalar_multiplication ( const Alpha &  alpha, const Vec &  A  )

constexpr

multiply a vector A with a scalar alpha

alpha * A = alpha A

Parameters

alpha	of scalar type T
vector	x of type T with N elements

Returns

vector

Definition at line 80 of file linalg_ops.hpp.

References mat_rows_v.

Referenced by operator*(), and operator*().

set_block()

template<std::size_t StartRow, std::size_t EndRow, std::size_t StartCol, std::size_t EndCol, MatrixContainer Mat>
requires (EndRow - StartRow <= mat_rows_v<Mat> && StartRow < mat_rows_v<Mat> && EndCol - StartCol <= mat_cols_v<Mat> && StartCol < mat_cols_v<Mat>)

constexpr auto linalg::set_block ( Mat  A, const Matrix< matrix_scalar_t< Mat >, EndRow - StartRow, EndCol - StartCol > &  B  )

constexpr

Return a matrix copy with one compile-time block replaced.

Parameters

matrix	A of type T Rows x Cols elements
matrix	B of type T (EndRow - StartRow) x (EndCol - StartCol)

Bounds are half-open. A is accepted by value, so this operation does not mutate the caller's original matrix.

Returns

Updated matrix copy.

Definition at line 279 of file linalg_util.hpp.

References mat_rows_v.

splice()

template<std::size_t idx, FieldVector Vec>
requires (vector_dim_v<Vec> > 0 && idx < vector_dim_v<Vec>)

constexpr auto linalg::splice ( const Vec &  x )

constexpr

splice operator

Parameters

vector

type T with N elements

Returns

tuple

Definition at line 216 of file linalg_ops.hpp.

References mat_rows_v.

symmetric_part()

template<SquareMatrix Mat>

constexpr auto linalg::symmetric_part ( const Mat &  A )

constexpr

symmetric part of a square matrix

sym(A) = 0.5 * (A + A^T)

Parameters

matrix

A

Returns

symmetric matrix

Definition at line 501 of file linalg_util.hpp.

References mat_rows_v, and transpose().

trace()

template<SquareMatrix Mat>
requires (mat_rows_v<Mat> > 0)

constexpr auto linalg::trace ( const Mat &  A )

constexpr

trace of a matrix

Sum of the diagonal elements of a matrix

Parameters

matrix

A of type T Rows x Cols elements

Returns

sum

Definition at line 282 of file linalg_ops.hpp.

References mat_rows_v.

transpose() [1/2]

template<MatrixContainer Mat>

constexpr auto linalg::transpose ( const Mat &  A )

constexpr

transpose of a matrix

A^T (rows and cols flipped from A)

Parameters

matrix

A of type T Rows x Cols elements

Returns

A^T

Definition at line 261 of file linalg_ops.hpp.

References mat_rows_v.

transpose() [2/2]

template<FieldVector Vec>

constexpr auto linalg::transpose ( const Vec &  x )

constexpr

tranpose of a vector

Vector is naturally row vector in how it's stored so the transpose flips it from a row to column based matrix

Parameters

vector

of type T with N elements

Returns

(N x 1) matrix

Definition at line 150 of file linalg_ops.hpp.

References mat_rows_v.

Referenced by conjugate_transpose(), linalg::Hessenberg< Mat >::Hessenberg(), make_smat(), linalg::LDL< Mat >::solve(), linalg::QR< Mat >::solve(), symmetric_part(), and linalg::Hessenberg< Mat >::U().

unpermute_papt()

template<MatrixContainer Mat>

constexpr auto linalg::unpermute_papt ( const Mat &  A, const Vector< std::size_t, mat_rows_v< Mat > > &  p  )

constexpr

Scatter matrix entries through a row-and-column permutation.

Template Parameters

Mat	Square matrix type.

Parameters

[in]	A	Square matrix whose entries are scattered.
[in]	p	Permutation mapping source indices to destination indices.

Returns

Matrix R of type Mat satisfying R(p(i),p(j)) = A(i,j).

Definition at line 86 of file matrix_exp.hpp.

References mat_rows_v.

Referenced by matrix_exp().

Vector()

template<typename T , typename... Ts>

linalg::Vector	(	T	,
		Ts...
	)		-> Vector< std::common_type_t< T, Ts... >, 1+sizeof...(Ts)>

Deduce coordinate type and dimension from constructor arguments.

The coordinate type is their std::common_type_t; the dimension is the number of arguments. For example, Vector{1.0, 2.0} deduces Vector<double, 2>.

Variable Documentation

inv_sqrt2_v

template<Field T>

constexpr T linalg::inv_sqrt2_v {T(1) / sqrt2_v<T>}

inlineconstexpr

Definition at line 16 of file constants.hpp.

is_complex_v

template<typename T >

constexpr bool linalg::is_complex_v = scalar_traits_t<T>::is_complex

inlineconstexpr

Whether normalized T is registered as a complex scalar.

Definition at line 90 of file scalar_traits.hpp.

is_norm_v

template<class Norm >

constexpr bool linalg::is_norm_v = norm_traits<remove_cvref_t<Norm>>{}

inlineconstexpr

helper for is norm type

Definition at line 71 of file norm_traits.hpp.

is_real_v

template<typename T >

constexpr bool linalg::is_real_v = scalar_traits_t<T>::is_real

inlineconstexpr

Whether normalized T is registered as a real scalar.

Definition at line 86 of file scalar_traits.hpp.

is_symmetric_matrix_space_v

template<typename Space >

constexpr bool linalg::is_symmetric_matrix_space_v

inlineconstexpr

Initial value:

=
    is_symmetric_matrix_space<remove_cvref_t<Space>>::value

symmetric matrix space type trait var

Definition at line 256 of file operator_traits.hpp.

ln2_v

template<Field T>

constexpr T linalg::ln2_v {T(0.693147180559945309417232121458176568L)}

inlineconstexpr

Definition at line 19 of file constants.hpp.

mat_cols_v

template<class Mat >

constexpr std::size_t linalg::mat_cols_v = mat_traits<remove_cvref_t<Mat>>::COLS

inlineconstexpr

Registered compile-time column count after cv/ref normalization.

Definition at line 95 of file matrix_traits.hpp.

mat_rows_v

template<class Mat >

constexpr std::size_t linalg::mat_rows_v = mat_traits<remove_cvref_t<Mat>>::ROWS

inlineconstexpr

Registered compile-time row count after cv/ref normalization.

Definition at line 91 of file matrix_traits.hpp.

Referenced by add(), add(), all_equal(), all_equal(), all_zero(), linalg::QuadraticRep< JxJ, Circ >::apply(), linalg::MatrixInverse< MatIn >::apply(), linalg::SymmetricMatrixInverse< MatIn >::apply(), linalg::MatrixSqrt< MatIn >::apply(), apply(), linalg::LDL< Mat >::apply(), linalg::Hessian< Mat, Type >::apply(), linalg::QuadraticRep< JxJ, Circ >::apply(), linalg::Hessian< Mat, Type >::apply(), linalg::MatrixPencil< Mat, Dim >::apply_full(), linalg::MatrixPencil< Mat, Dim >::apply_sym_mat(), arg_max(), arg_min(), argsort_desc(), array_to_tuple(), as_col_matrix(), as_col_vector(), as_complex_matrix(), as_row_matrix(), as_row_vector(), as_vector(), as_vector(), as_vector(), linalg::Cholesky< Mat >::Cholesky(), linalg::Cholesky< Mat >::Cholesky(), affine::AffineMap< Mat >::compose(), linalg::SymmetricOperator< Mat, Derived >::compute_definiteness(), conjugate(), conjugate_transpose(), constexpr_atanh(), constexpr_atanh_safe(), constexpr_cos(), constexpr_div(), constexpr_div_safe(), constexpr_exp(), constexpr_exp_lambda(), constexpr_expi(), constexpr_for(), constexpr_ln(), constexpr_ln_safe(), constexpr_nrt(), constexpr_nrt_safe(), constexpr_pow(), constexpr_sin(), constexpr_sqrt(), constexpr_sqrt_safe(), coordinates(), linalg::LDL< Mat >::det(), linalg::EigenDecomposition< Mat >::EigenDecomposition(), linalg::EigenDecomposition< Mat >::EigenDecomposition(), euclidean_distance(), linalg::mat_traits< Matrix< T, Rows, Cols > >::fill(), linalg::vector_traits< Vector< T, Dim > >::fill(), linalg::coordinate_map< SymmetricMatrix< Mat > >::from_coordinates(), from_coordinates(), linalg::coordinate_map< SymmetricMatrixSpace< T, Dim > >::from_coordinates(), linalg::coordinate_map< CoordinateSpace< T, Dim > >::from_coordinates(), linalg::coordinate_map< Matrix< T, Rows, Cols > >::from_coordinates(), linalg::coordinate_map< MatrixSpace< T, Rows, Cols > >::from_coordinates(), linalg::Cholesky< Mat >::G(), get_block(), get_col(), get_diag(), get_elements(), get_lower_left_triangular_matrix(), get_lower_triangle(), get_lower_triangle_with_diag(), get_n_cols(), get_n_rows(), get_row(), get_svec(), get_upper_right_triangular_matrix(), linalg::Hessenberg< Mat >::H(), linalg::Hessenberg< Mat >::Hessenberg(), linalg::SymmetricOperator< Mat, Derived >::index(), inner_product(), affine::AffineMap< Mat >::inverse(), invert_permutation(), linalg::Hessian< Mat, Type >::is_pullback(), is_same(), linalg::LDL< Mat >::L(), linalg::LDL< Mat >::LDL(), linalg::LDL< Mat >::LDL(), make_diag(), make_group_permutation(), make_identity(), make_lower_triangle(), affine::make_map_id(), affine::make_selector_map(), make_smat(), make_svec(), make_unit_lower_triangle(), make_zeros(), manhattan_distance(), linalg::Matrix< T, Rows, Cols >::Matrix(), matrix_exp(), max(), min(), linalg::Vector< T, N >::operator!=(), linalg::StandardInnerProduct< Vec >::operator()(), linalg::Fro< Mat >::operator()(), linalg::L2< Vec >::operator()(), linalg::L1< Vec >::operator()(), linalg::Linf< Vec >::operator()(), linalg::MatrixInverse< MatIn >::operator()(), linalg::SymmetricMatrixInverse< MatIn >::operator()(), linalg::MatrixSqrt< MatIn >::operator()(), affine::Scale< T, N >::operator()(), linalg::Endomorphism< Mat, Derived >::operator()(), linalg::Endomorphism< Mat, Derived >::operator()(), linalg::Hessenberg< Mat >::operator()(), linalg::Hessian< Mat, Type >::operator()(), linalg::SymmetricOperator< Mat, Derived >::operator()(), affine::Shear< T, N >::operator()(), linalg::Vector< T, N >::operator()(), linalg::Vector< T, N >::operator()(), linalg::Matrix< T, Rows, Cols >::operator()(), linalg::Matrix< T, Rows, Cols >::operator()(), linalg::mat_traits< Matrix< T, Rows, Cols > >::operator()(), linalg::vector_traits< Vector< T, Dim > >::operator()(), operator*(), operator*(), operator*(), operator*(), operator+(), operator+(), operator-(), operator-(), linalg::Vector< T, N >::operator==(), outer_product(), permute_ptjp(), linalg::QR< Mat >::QR(), linalg::QR< Mat >::QR(), linalg::QR< Mat >::rank(), linalg::Matrix< T, Rows, Cols >::row(), rows_equal_cols(), scalar_multiplication(), scalar_multiplication(), linalg::Endomorphism< Mat, Derived >::self(), linalg::Endomorphism< Mat, Derived >::self(), linalg::SymmetricOperator< Mat, Derived >::self(), set_block(), linalg::LDL< Mat >::solve(), linalg::QR< Mat >::solve(), linalg::EigenDecomposition< Mat >::spectral_radius(), splice(), linalg::SVD< Mat >::SVD(), symmetric_part(), linalg::SymmetricMatrix< Mat >::SymmetricMatrix(), linalg::coordinate_map< SymmetricMatrix< Mat > >::to_coordinates(), linalg::coordinate_map< Matrix< T, Rows, Cols > >::to_coordinates(), linalg::coordinate_map< MatrixSpace< T, Rows, Cols > >::to_coordinates(), linalg::coordinate_map< SymmetricMatrixSpace< T, Dim > >::to_coordinates(), linalg::coordinate_map< CoordinateSpace< T, Dim > >::to_coordinates(), linalg::coordinate_map< Matrix< T, Rows, Cols > >::to_coordinates(), trace(), transpose(), transpose(), linalg::Hessenberg< Mat >::U(), unpermute_papt(), and linalg::Vector< T, N >::Vector().

norm_dim_v

template<class Norm >

constexpr std::size_t linalg::norm_dim_v = norm_traits_t<Norm>::dim

inlineconstexpr

Definition at line 67 of file norm_traits.hpp.

space_dim_v

template<class Space >

constexpr std::size_t linalg::space_dim_v

inlineconstexpr

Initial value:

=
    space_traits<remove_cvref_t<Space>>::DIM

Finite coordinate dimension after cv/ref normalization.

Definition at line 91 of file space_traits.hpp.

sqrt2_v

template<Field T>

constexpr T linalg::sqrt2_v {T(1.414213562373095048801688724209698079L)}

inlineconstexpr

Definition at line 13 of file constants.hpp.

vector_dim_v

template<class Vec >

constexpr std::size_t linalg::vector_dim_v = vector_traits_t<Vec>::dim

inlineconstexpr

Registered compile-time vector dimension.

Definition at line 70 of file vector_traits.hpp.