An n x n matrix (i.e., a vector of length n*n), stored row-wise: that is, A(i,j) = A[ij], where ij = i*n + j. More...

#include <elementary/eagle/QuadraticMatrix.hpp>

Inheritance diagram for Eagle::Quadratic< N, Value >:

Public Types
typedef FixedArray< int, N >	PermutationVector_t
	Type of the permutation vector that is required for the functions GaussDecompose() and GaussSolve().

Public Types inherited from Eagle::Matrix< N, N, Value >
using	Matrix_t = Matrix
	Matrix type.

typedef Operator<' *'>	Mult
	Computational constructor argument for multiplicative operations.

typedef Operator<'~'>	TransposeOperation
	Computational constructor argument for transpose operations.

Public Types inherited from Eagle::Vector< Value, R *C >
enum
	The number of elements in this array.

Public Types inherited from Eagle::FixedArray< value, N >
typedef FlattenedArray_t::value_type	Element_t
	The maximally reduced type of elements stored here.

typedef GetFixedArrayType< value >::result_t	ElementAsFixedArray_t
	Element type as fixed array, if it is convertible to a fixed array.

typedef FixedArray	FixedArray_t
	Exporting the type definition of this array cass type.

typedef FixedArrayFlattener< FixedArrayOfElements_t >	FixedArrayFlattener_t
	Internal helper class.

typedef FixedArray< ElementAsFixedArray_t, N >	FixedArrayOfElements_t
	If the value is derived from a fixed array type itself, then the FixedArrayOfElements_t is an FixedArray<FixedArray<>> type.

typedef FixedArrayFlattener_t::result_t	FlattenedArray_t
	Reduction of this array type, if it is nested, to an flattened type which contains elementary types.

typedef value	value_type
	Export the type of the respective element value type.

Public Member Functions
bool	GaussDecompose (FixedArray< int, N > &perm, double EPSWEIGHT=1E-3)
	Perform gauss decomposition of the matrix, which can and should be used instead of computing the inverse of a matrix.

void	GaussSolve (const PermutationVector_t &perm, const Column< N, value > &b, Column< N, value > &x) const
	Instead of inverting a matrix, solve a linear system of equations by forward and backward elimination.

value	maxoffd () const
	Return maximum off-diagonal element of n by n square matrix A.

	Quadratic ()
	Default constructor.

	Quadratic (const LowerTriangular< N, value > &)
	Construct a quadratic matrix from a lower triangular matrix.

template<dimension_t C, class ValueA , class ValueB >
	Quadratic (const Matrix< N, C, ValueA > &A, const Matrix< C, N, ValueB > &B)
	Construct quadratic matrix from multiplication for two fitting non-quadratic matrices.

template<class OtherValue >
	Quadratic (const Matrix< N, N, OtherValue > &A)
	Construct quadratic matrix from base class.

Public Member Functions inherited from Eagle::Matrix< N, N, Value >
constexpr const Matrix_t &	matrix () const
	yield matrix here, useful for derived classes.

constexpr	Matrix ()
	Default constructor.

constexpr	Matrix (const Matrix &M)
	Copy constructor.

constexpr	Matrix (const Base_t &V)
	Construct matrix from linear vector.

constexpr	Matrix (const std::array< T, 16 > &Init)
	Construct matrix from initializer list.

constexpr	Matrix (const Matrix &A, const Value &V, const Operator<' *'> &M)
	Computational constructor for multiplying matrix by scalar.

constexpr	Matrix (const Value &V, const Matrix &A, const Operator<' *'> &M)
	Computational constructor for multiplying scalar by matrix.

constexpr	Matrix (const Matrix &A, const Value &V, const Operator<'/'> &Op)
	Computational constructor for dividing matrix by scalar.

constexpr	Matrix (const Matrix &A, const Matrix &B, const Operator<'+'> &Op)
	Computational constructor for matrix addition.

constexpr	Matrix (const Matrix &A, const Matrix &B, const Operator<'-'> &Op)
	Computational constructor for matrix subtraction.

constexpr	Matrix (const FixedArray< AType, R C > &l, const FixedArray< BType, R C > &r, const Operator< CompType > &Op)
	Generic computational constructor for two arrays of size compatible with the matrix.

constexpr	Matrix (const FixedArray< Type, R *C > &Data)
	Explicit constructor from arbitrary compatible type.

constexpr	Matrix (const FixedArray< ValueType, R > &l, const Operator< CompType > &Op, const Value &scalar)
	Generic computational constructor for vector/scalar operations.

constexpr	Matrix (const Operator< CompType > &Op, const Value &scalar, const FixedArray< ValueType, R > &l)
	Generic computational constructor for scalar/vector operations.

constexpr	Matrix (const Matrix &A, const Operator<'-'> &Op)
	Computational constructor for unary minus.

constexpr	Matrix (const Matrix< R, M, ValueA > &A, const Matrix< M, C, ValueB > &B, const Operator<' *'> &)
	Matrix multiplication .

constexpr	Matrix (const Matrix< R, M, ValueA > &A, const Matrix< C, M, ValueB > &B, const Operator<' *','~'> &)
	Transposed matrix multiplication $A B^{\tau}$ .

constexpr	Matrix (const Matrix< R, M, Value > &A, const Matrix< C, M, Value > &B, const Mult &, const TransposeOperation &)
	Matrix multiplication with transpose of matrix B.

constexpr	Matrix (const Matrix< M, R, Value > &A, const Matrix< M, C, Value > &B, const TransposeOperation &, const Mult &)
	Matrix multiplication with transpose of matrix A.

constexpr	Matrix (const Matrix< R, C, Value > &A, const LowerTriangular< C, Value > &B, const Mult &)
	Matrix multiplication where B is a symmetric matrix stored in lower triangular form.

constexpr	Matrix (const Matrix< C, R, Value > &A, const LowerTriangular< C, Value > &B, const TransposeOperation &, const Mult &)
	Matrix multiplication where B is a symmetric matrix stored in lower triangular form.

constexpr const Column< R, Value > &	column (int i) const
	Return the ith column of this matrix.

constexpr Column< R, Value > &	column (int i)
	Return the ith column of this matrix, writeable.

constexpr Value &	operator() (dimension_t row, dimension_t column)
	Access element for writing.

constexpr const Value &	operator() (dimension_t row, dimension_t column) const
	Element access operation for reading.

constexpr Row_t	row (dimension_t rowNr)
	Return writable reference to the ith row of this matrix.

constexpr const Row_t	row (dimension_t rowNr) const
	Return read-only reference to the ith row of this matrix.

constexpr const auto	operator[] (dimension_t r) const
	Allow nested access to matrix elements via bracket operator.

constexpr auto &	operator[] (dimension_t r)
	Allow nested access to matrix elements via bracket operator.

Public Member Functions inherited from Eagle::Vector< Value, R *C >
Base_t &	fixed_array ()
	Provide access to the base class, writable.

const Base_t &	fixed_array () const
	Provide access to the base class, read-only.

Vector &	*operator=** (const Value &value)
	Self-multiplication.

Vector &	operator+= (const Vector &v)
	Self-addition.

Vector	operator- () const
	Unary minus.

Vector &	operator-= (const Vector &v)
	Self-subtraction.

Vector &	operator/= (const Value &value)
	Self-division.

Assignment< Vector, 0 >	operator= (const scalar &x)
	Assign first element with scalar and use consecutive comma expressions for assignment of further elements As such a Vector may be assigned as.

Vector &	operator= (const Vector &src)
	Assignment operator.

void	set (const V &value)
	Assign all values to the same scalar.

constexpr	Vector ()
	Default constructor.

	Vector (const FixedArray< AType, n > &l, const FixedArray< BType, n > &r, const Operator< CompType > &Op)
	Computational constructor for binary operations on two fixed arrays, calls the Operator's ternary() member function for each element.

	Vector (const FixedArray< AType, n > &l, const Operator< CompType > &, const ScalarType &scalar)
	Computational constructor from vector with scalar, calls the Operator's ternary() member function for each element and the scalar as its 3rd argument.

	Vector (const FixedArray< InputType, SIZE > &V)
	Type conversion constructor.

	Vector (const Indicable &I, const AnyType &)
	Construct from indicable object, i.e.

	Vector (const Operator< CompType > &, const ScalarType &scalar, const FixedArray< AType, n > &v)
	Computational constructor from scalar with vector calls the Operator's ternary() member function for each element and the scalar as its 2nd argument.

constexpr	Vector (const std::initializer_list< InputType > &Init)
	Construct from a initialization array.

	Vector (const Value &a)
	Convenience Constructor: initialize the first element.

	Vector (const Value &a, const Value &b)
	Convenience Constructor: initialize the first two elements.

	Vector (const Value &a, const Value &b, const Value &c)
	Convenience Constructor: initialize the first three elements.

	Vector (const Value &a, const Value &b, const Value &c, const Value &d)
	Convenience Constructor: initialize the first four elements.

	Vector (const Vector &l, const Vector &r, const Operator< CompType > &)
	Computational constructor for binary operations on two vectors calls the Operator's ternary() member function for each element.

	Vector (const Vector &NV, const Sub &, const Sub &)
	Computational constructor for unary minus.

constexpr	Vector (const Vector &v)
	Copy constructor.

	Vector (const Vector &v, const Operator<'-'> &)
	Unitary minus computational constructor Applies the unitary minus to each element.

	Vector (const Vector< InputType, SIZE > &V)
	Type conversion constructor.

Public Member Functions inherited from Eagle::FixedArray< value, N >
void	expandMinMax (FixedArray &MinValues, FixedArray &MaxValues) const
	Expand a range given by MinValues,MaxValues to include the values of this array, done per component.

constexpr	FixedArray () noexcept
	Default constructor.

template<class AType , class BType , OperatorID_t CompType>
constexpr	FixedArray (const FixedArray< AType, N > &l, const FixedArray< BType, N > &r, const Operator< CompType > &)
	Computational constructor for binary operations on two fixed arrays, calls the Operator's ternary() member function for each element.

template<typename T , dimension_t M>
constexpr	FixedArray (const FixedArray< T, M > &F, const T &DefaultValue)
	Construct array from another array type of variable length.

template<typename T >
constexpr	FixedArray (const FixedArray< T, N > &F)
	Automatic type conversion function.

template<typename T , typename ConversionFunctor >
constexpr	FixedArray (const FixedArray< T, N > &F, const ConversionFunctor &CF, const FA_CONVERT &)
	Construct array from another array type of same length.

template<class InitType >
constexpr	FixedArray (const std::initializer_list< InitType > &Init)
	Construct from a initialization array.

constexpr	FixedArray (const value &a)
	Convenience Constructor: initialize the first element.

constexpr	FixedArray (const value &a, const value &b)
	Convenience Constructor: initialize the first two elements.

	FixedArray (const value &a, const value &b, const value &c)
	Convenience Constructor: initialize the first three elements.

constexpr	FixedArray (const value &a, const value &b, const value &c, const value &d)
	Convenience Constructor: initialize the first four elements.

constexpr	FixedArray (const value &a, const value &b, const value &c, const value &d, const value &e)
	Convenience Constructor: initialize the first five elements.

constexpr value &	get (int i)
	writable element access

constexpr const value &	get (int i) const
	read-only element access

template<int subsize, int offset>
constexpr const FixedArray< value, subsize > *	get_subarray () const
	Helper class for retrieve a subset of the current array.

constexpr const FixedArray< value, SIZE-1 > &	get_subdim () const
	Mainly introduced because of Interpolate::operator[].

const FixedArray_t &	getFixedArray () const noexcept
	convenience function for base classes

FixedArray_t &	getFixedArray () noexcept
	convenience function for base classes

constexpr value	getMax () const
	Compute the maximum element value of this array.

constexpr value	getMin () const
	Compute the maximum element value of this array.

template<typename T >
constexpr bool	operator!= (const FixedArray< T, N > &F) const
	Compare two arrays.

template<typename T >
constexpr bool	operator== (const FixedArray< T, N > &F) const
	Compare two arrays.

constexpr value &	operator[] (int i)
	writeable element access operator

constexpr const value &	operator[] (int i) const
	read-only element access operator

constexpr value *	ptr (int i=0) noexcept
	Return a pointer to the ith element, no range checking.

template<class Value >
constexpr void	set (const Value &x)
	Set all array elements to the same value.

void	setFromArray (const value *x, int length=SIZE, int offset=0)
	Set all array elements to an array's element value.

void	sort (FixedArray< int, N > &I) const
	Sort the elements of an array into the index array.

Additional Inherited Members
Static Public Member Functions inherited from Eagle::FixedArray< value, N >
template<typename T , typename ConversionFunctor >
static constexpr FixedArray	convert (const FixedArray< T, N > &F, const ConversionFunctor &CF)
	Perform a type conversion on a given fixed array into another basis type:

static constexpr dimension_t	size ()
	The number of elements here.

Protected Attributes inherited from Eagle::FixedArray< value, N >
value	data [SIZE]
	The actual data values, intentional no-initialization for data member, which increases performance on vector creation.

Related Symbols inherited from Eagle::Matrix< N, N, Value >
void	UnsortedEigenVectors (LowerTriangular< N, value > &A, Quadratic< N, value > &RRmatrix, Row< N, value > &E, double precision=1E-10)
	Eigenvalues and eigenvectors of a real symmetric matrix.

Detailed Description

template<dimension_t N, class Value>
class Eagle::Quadratic< N, Value >

An n x n matrix (i.e., a vector of length n*n), stored row-wise: that is, A(i,j) = A[ij], where ij = i*n + j.

See also: Matrix

Member Function Documentation

◆ GaussDecompose()

template<dimension_t N, class Value >

bool Eagle::Quadratic< N, Value >::GaussDecompose	(	FixedArray< int, N > &	perm,
		double	EPSWEIGHT = `1E-3`
	)

inline

Perform gauss decomposition of the matrix, which can and should be used instead of computing the inverse of a matrix.

The permutation operations during the composition is stored in the perm parameter, which then needs to be provided to the GaussSolve() routine.

Author: Marcus Weber, ZIB, based on "Formelsammlung zur Numerischen Mathematik mit C-Programmen" von G.Engeln-Maellges und F.Reutter"

Note: The original matrix is destroyed during this operation.

Parameters

EPSWEIGHT square root of the absolut precision of the computation

References Eagle::Matrix< N, N, Value >::matrix(), Eagle::norm(), and Eagle::Matrix< R, C, Value >::row().

◆ GaussSolve()

template<dimension_t N, class Value >

void Eagle::Quadratic< N, Value >::GaussSolve	(	const PermutationVector_t &	perm,
		const Column< N, value > &	b,
		Column< N, value > &	x
	)		const

inline

Instead of inverting a matrix, solve a linear system of equations $ Ax = b $ by forward and backward elimination.

Parameters

perm	A permutation vector as computed by GaussDecompose()
b	The right-hand side of the equation
x	On return, will contain the solution.

Eagle::Quadratic<3, double> A;
Eagle::Column<3, double> b;
Eagle::Column<3, double> x;
Eagle::FixedArray<int, 3> perm;
 
A = 2.5,  3.0, 1.1,
    0.2,  0.1, 2.1,
    0.12, 1.0, 0.3;
 
b = 39.0, 1.0, 34.33;
 
A.GaussDecompose(perm);
A.GaussSolve(perm, b, x);

The result in x then is < 16.809817, -24.730900, 16.013638 >.

References Eagle::Matrix< N, N, Value >::matrix().

Public Types

Public Member Functions

Additional Inherited Members

Detailed Description

Member Function Documentation

◆ GaussDecompose()

◆ GaussSolve()