a dynamic sized n-dimensional matrix with expression templates More...

#include <matrix.h>

Inheritance diagram for lass::num::Matrix< T, S >:

[legend]

Public Member Functions
	Matrix ()
	constructs an empty matrix

	Matrix (TSize rows, TSize columns)
	Construct a matrix of dimension rows x columns.

template<typename T2, typename S2>
	Matrix (const Matrix< T2, S2 > &other)
	assign storage/expression matrix to this (this should be a storage matrix).

template<typename T2, typename S2>
Matrix< T, S > &	operator= (const Matrix< T2, S2 > &other)
	assign storage/expression matrix to this (this should be a storage matrix).

const TSize	rows () const
	return number of rows in matrix.

const TSize	columns () const
	return number of columns in matrix.

const TValue	operator() (TSize row, TSize column) const
	return the component value at position (row, column)

TReference	operator() (TSize row, TSize column)
	access the component value at position (row, column)

const TValue	at (TSigned row, TSigned column) const
	return the component value at position (row, column) (row, column) will be wrapped to range [0, this->rows()) x [0, this->columns()] by using a modulus operator

TReference	at (TSigned row, TSigned column)
	access the component value at position (row, column) (row, column) will be wrapped to range [0, this->rows()) x [0, this->columns()] by using a modulus operator

ConstRow	row (TSigned index) const
	return a row of matrix.

Row	row (TSigned index)
	return a row of matrix.

ConstColumn	column (TSigned index) const
	return a column of matrix.

Column	column (TSigned index)
	return a column of matrix.

const Matrix< T, S > &	operator+ () const
	A weird way to get back the same object.

const Matrix< T, impl::MNeg< T, S > >	operator- () const
	return a vector with all components negated (-a)(i, j) == -(a(i, j)).

template<typename S2>
Matrix< T, S > &	operator+= (const Matrix< T, S2 > &other)
	componentswise addition assignment of two matrices

template<typename S2>
Matrix< T, S > &	operator-= (const Matrix< T, S2 > &other)
	componentswise subtraction assignment of two matrices This method is equivalent to this+=(-iB) .

Matrix< T, S > &	operator*= (TParam scalar)
	scalar multiplication assignment of matrix

Matrix< T, S > &	operator/= (TParam scalar)
	scalar multiplication assignment of matrix

void	setZero (TSize rows, TSize columns)
	set this to a zero matrix of rows rows and columns columns.

void	setIdentity (TSize size)
	set matrix to an identity matrix.

bool	isEmpty () const
	return true if this is a (0�0) matrix

bool	isZero () const
	test if this matrix equals a zero matrix.

bool	isIdentity () const
	test if this matrix equals a identity matrix

bool	isDiagonal () const
	test if this matrix is an diagonal matrix

bool	isSquare () const
	return true if matrix has as many rows as columns

const Matrix< T, impl::MTrans< T, S > >	transposed () const
	return transposed matrix

void	invert ()
	replace matrix by its inverse.

void	swap (Matrix< T, S > &other)
	swap two matrices

Related Symbols
(Note that these are not member symbols.)
template<typename T, typename S1, typename S2>
bool	operator== (const Matrix< T, S1 > &a, const Matrix< T, S2 > &b)

template<typename T, typename S1, typename S2>
bool	operator!= (const Matrix< T, S1 > &a, const Matrix< T, S2 > &b)

template<typename T, typename S1, typename S2>
const Matrix< T, impl::MAdd< T, S1, S2 > >	operator+ (const Matrix< T, S1 > &a, const Matrix< T, S2 > &b)
	componentwise addition

template<typename T, typename S1, typename S2>
const Matrix< T, impl::MSub< T, S1, S2 > >	operator- (const Matrix< T, S1 > &a, const Matrix< T, S2 > &b)
	componentwise addition

template<typename T, typename S1, typename S2>
const Matrix< T, impl::MProd< T, S1, S2 > >	operator* (const Matrix< T, S1 > &a, const Matrix< T, S2 > &b)
	Matrix multiplication.

template<typename T, typename S>
const Matrix< T, impl::MAdd< T, impl::MScalar< T >, S > >	operator+ (const T &a, const Matrix< T, S > &b)
	add a to all components of b

template<typename T, typename S>
const Matrix< T, impl::MSub< T, impl::MScalar< T >, S > >	operator- (const T &a, const Matrix< T, S > &b)
	add a to all negated components of b

template<typename T, typename S>
const Matrix< T, impl::MMul< T, impl::MScalar< T >, S > >	operator* (const T &a, const Matrix< T, S > &b)
	multiply a with all components of b

template<typename T, typename S>
const Matrix< T, impl::MAdd< T, S, impl::MScalar< T > > >	operator+ (const Matrix< T, S > &a, typename Matrix< T, S >::TParam b)
	add b to all components of a

template<typename T, typename S>
const Matrix< T, impl::MAdd< T, S, impl::MScalar< T > > >	operator- (const Matrix< T, S > &a, typename Matrix< T, S >::TParam b)
	subtract b from all components of a

template<typename T, typename S>
const Matrix< T, impl::MMul< T, S, impl::MScalar< T > > >	operator* (const Matrix< T, S > &a, typename Matrix< T, S >::TParam b)
	multiply all components of a with b.

template<typename T, typename S>
const Matrix< T, impl::MMul< T, S, impl::MScalar< T > > >	operator/ (const Matrix< T, S > &a, typename Matrix< T, S >::TParam b)
	divide all components of a by b.

template<typename T, typename S, typename S2>
void	solve (const Matrix< T, S > &a, Matrix< T, S2 > &bx, bool improve)
	solve equation A * X = B

Detailed Description

template<typename T, typename S = impl::MStorage<T>>
class lass::num::Matrix< T, S >

a dynamic sized n-dimensional matrix with expression templates

Author: Bram de Greve [BdG]

Definition at line 69 of file matrix.h.

Constructor & Destructor Documentation

◆ Matrix() [1/3]

template<typename T, typename S>

lass::num::Matrix< T, S >::Matrix ( )

constructs an empty matrix

Exception Safety:: strong guarentee.

Definition at line 84 of file matrix.inl.

Referenced by invert(), Matrix(), operator*=(), operator+(), operator+=(), operator-(), operator-=(), operator/=(), swap(), and transposed().

◆ Matrix() [2/3]

template<typename T, typename S>

lass::num::Matrix< T, S >::Matrix	(	TSize	rows,
		TSize	columns )

explicit

Construct a matrix of dimension rows x columns.

Can be used as default constructor for storage in containers.

Parameters

rows	the number of rows of the matrix to be created.
columns	the number of columns of the matrix to be created. By default both rows and columns are zero, what creates an empty matrix. Not very usefull though, except as place holder. So, it's safe to pass zero as arguments, but you shouldn't pass negative values.

Complexity:: O(rows * columns)

Exception Safety:: strong guarentee.

Definition at line 106 of file matrix.inl.

References columns(), and rows().

◆ Matrix() [3/3]

template<typename T, typename S>

template<typename T2, typename S2>

lass::num::Matrix< T, S >::Matrix ( const Matrix< T2, S2 > & other )

assign storage/expression matrix to this (this should be a storage matrix).

Precondition: this must be an l-value (storage matrix).

Complexity:: O(other.rows() * other.columns())

Exception Safety:: basic guarentee.

Definition at line 133 of file matrix.inl.

References columns(), LASS_META_ASSERT, Matrix(), and rows().

Member Function Documentation

◆ operator=()

template<typename T, typename S>

template<typename T2, typename S2>

Matrix< T, S > & lass::num::Matrix< T, S >::operator= ( const Matrix< T2, S2 > & other )

assign storage/expression matrix to this (this should be a storage matrix).

Precondition: this must be an l-value (storage matrix).

Complexity:: O(other.rows() * other.columns())

Exception Safety:: basic guarentee.

Definition at line 164 of file matrix.inl.

◆ rows()

template<typename T, typename S>

const Matrix< T, S >::TSize lass::num::Matrix< T, S >::rows ( ) const

inline

return number of rows in matrix.

Postcondition: should never return be a negative value.

Complexity:: O(1)

Exception safety:: no-fail

Definition at line 196 of file matrix.inl.

Referenced by at(), at(), invert(), isDiagonal(), isIdentity(), isZero(), Matrix(), Matrix(), operator()(), operator()(), operator*(), operator*(), operator*=(), operator+(), operator+(), operator+=(), operator-(), operator-(), operator-=(), operator/(), operator/=(), operator==(), row(), row(), setZero(), and solve().

◆ columns()

template<typename T, typename S>

const Matrix< T, S >::TSize lass::num::Matrix< T, S >::columns ( ) const

inline

return number of columns in matrix.

Postcondition: should never return be a negative value.

Complexity:: O(1)

Exception safety:: no-fail

Definition at line 215 of file matrix.inl.

Referenced by at(), at(), column(), column(), isDiagonal(), isIdentity(), isZero(), Matrix(), Matrix(), operator()(), operator()(), operator*(), operator*(), operator*=(), operator+=(), operator-=(), operator/(), operator/=(), operator==(), setZero(), and solve().

◆ operator()() [1/2]

template<typename T, typename S>

const Matrix< T, S >::TValue lass::num::Matrix< T, S >::operator()	(	TSize	row,
		TSize	column ) const

inline

return the component value at position (row, column)

Precondition: (row, column) shouldn't be out of the range [0, this->rows()) x [0, this->columns()], unless you're asking for trouble.

Definition at line 229 of file matrix.inl.

References column(), columns(), row(), and rows().

◆ operator()() [2/2]

template<typename T, typename S>

Matrix< T, S >::TReference lass::num::Matrix< T, S >::operator()	(	TSize	iRow,
		TSize	iColumn )

inline

access the component value at position (row, column)

Precondition: (row, column) shouldn't be out of the range [0, this->rows()) x [0, this->columns()], unless you're asking for trouble.; this must be an l-value (storage matrix).

Definition at line 246 of file matrix.inl.

References columns(), LASS_META_ASSERT, and rows().

◆ at()

template<typename T, typename S>

Matrix< T, S >::TReference lass::num::Matrix< T, S >::at	(	TSigned	row,
		TSigned	column )

inline

access the component value at position (row, column) (row, column) will be wrapped to range [0, this->rows()) x [0, this->columns()] by using a modulus operator

Precondition: this must be an l-value (storage matrix).

Definition at line 276 of file matrix.inl.

References column(), columns(), LASS_META_ASSERT, row(), and rows().

◆ row() [1/2]

template<typename T, typename S>

Matrix< T, S >::ConstRow lass::num::Matrix< T, S >::row ( TSigned index ) const

inline

return a row of matrix.

index will be wrapped to range [0, this->rows()) by using a modulus operator

Definition at line 289 of file matrix.inl.

References rows().

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

◆ row() [2/2]

template<typename T, typename S>

Matrix< T, S >::Row lass::num::Matrix< T, S >::row ( TSigned index )

inline

return a row of matrix.

index will be wrapped to range [0, this->rows()) by using a modulus operator. THIS MUST BE LVALUE (storage matrix).

Definition at line 302 of file matrix.inl.

References rows().

◆ column() [1/2]

template<typename T, typename S>

Matrix< T, S >::ConstColumn lass::num::Matrix< T, S >::column ( TSigned index ) const

inline

return a column of matrix.

index will be wrapped to range [0, this->columns()) by using a modulus operator

Definition at line 314 of file matrix.inl.

References columns().

Referenced by at(), at(), invert(), operator()(), and solve().

◆ column() [2/2]

template<typename T, typename S>

Matrix< T, S >::Column lass::num::Matrix< T, S >::column ( TSigned index )

inline

return a column of matrix.

index will be wrapped to range [0, this->columns()) by using a modulus operator THIS MUST BE LVALUE (storage matrix).

Definition at line 327 of file matrix.inl.

References columns().

◆ operator+()

template<typename T, typename S>

const Matrix< T, S > & lass::num::Matrix< T, S >::operator+ ( ) const

inline

A weird way to get back the same object.

Complexity:: O(1)

Exception safety:: no-fail

Definition at line 343 of file matrix.inl.

References Matrix().

◆ operator-()

template<typename T, typename S>

const Matrix< T, impl::MNeg< T, S > > lass::num::Matrix< T, S >::operator- ( ) const

return a vector with all components negated (-a)(i, j) == -(a(i, j)).

Complexity:: O(1)

Exception safety:: no-fail

Definition at line 361 of file matrix.inl.

References Matrix().

◆ operator+=()

template<typename T, typename S>

template<typename S2>

Matrix< T, S > & lass::num::Matrix< T, S >::operator+= ( const Matrix< T, S2 > & other )

componentswise addition assignment of two matrices

Matrix Addition: Denote the sum of two matrices
A and
B (of the same dimensions) by
C=A+B. The sum is defined by adding entries with the same indices
Cij=Aij+Bij over all
i and
j. http://mathworld.wolfram.com/MatrixAddition.html

This method implements the assignment addition what reduces to
Aij+=Bij over all
i and
j.

Precondition: this must be an l-value (storage matrix).

Complexity:: O(this->rows() * this->columns())

Exceptions

an	exception is thrown if *this and iB are not of the same dimensions

Definition at line 388 of file matrix.inl.

References columns(), LASS_META_ASSERT, Matrix(), and rows().

◆ operator-=()

template<typename T, typename S>

template<typename S2>

Matrix< T, S > & lass::num::Matrix< T, S >::operator-= ( const Matrix< T, S2 > & other )

componentswise subtraction assignment of two matrices This method is equivalent to
this+=(-iB) .

See also: Matrix<T>::operator+=

Precondition: this must be an l-value (storage matrix).

Complexity:: O(this->rows() * this->columns())

Exceptions

an	exception is thrown if *this and iB are not of the same dimensions

Definition at line 421 of file matrix.inl.

References columns(), LASS_META_ASSERT, Matrix(), and rows().

◆ operator*=()

template<typename T, typename S>

Matrix< T, S > & lass::num::Matrix< T, S >::operator*= ( TParam scalar )

scalar multiplication assignment of matrix

Precondition: this must be an l-value (storage matrix).

Complexity:: O(this->rows() * this->columns())

Definition at line 448 of file matrix.inl.

References columns(), LASS_META_ASSERT, Matrix(), and rows().

◆ operator/=()

template<typename T, typename S>

Matrix< T, S > & lass::num::Matrix< T, S >::operator/= ( TParam scalar )

scalar multiplication assignment of matrix

Precondition: this must be an l-value (storage matrix).

Definition at line 470 of file matrix.inl.

References columns(), LASS_META_ASSERT, Matrix(), and rows().

◆ setZero()

template<typename T, typename S>

void lass::num::Matrix< T, S >::setZero	(	TSize	rows,
		TSize	columns )

set this to a zero matrix of rows rows and columns columns.

See also: Matrix<T>::isZero

Precondition: this should be an lvalue

Complexity:: O(this->rows() * this->columns())

Definition at line 496 of file matrix.inl.

References columns(), LASS_META_ASSERT, and rows().

◆ setIdentity()

template<typename T, typename S>

void lass::num::Matrix< T, S >::setIdentity ( TSize iSize )

set matrix to an identity matrix.

See also: Matrix<T>::isIdentity

Precondition: this should be an lvalue

Complexity:: O(this->rows() * this->columns())

Definition at line 520 of file matrix.inl.

References LASS_META_ASSERT.

Referenced by invert().

◆ isEmpty()

template<typename T, typename S>

bool lass::num::Matrix< T, S >::isEmpty ( ) const

return true if this is a (0�0) matrix

Complexity:: O(1)

Exception safety:: no-fail

Definition at line 544 of file matrix.inl.

◆ isZero()

template<typename T, typename S>

bool lass::num::Matrix< T, S >::isZero ( ) const

test if this matrix equals a zero matrix.

A zero matrix is an
m�n matrix consisting of all 0s (MacDuffee 1943, p. 27), denoted
0. Zero matrices are sometimes also known as null matrices (Akivis and Goldberg 1972, p. 71). http://mathworld.wolfram.com/ZeroMatrix.html

an empty matrix is also a zero matrix.

Complexity:: O(this->rows() * this->columns())

Definition at line 563 of file matrix.inl.

References columns(), and rows().

◆ isIdentity()

template<typename T, typename S>

bool lass::num::Matrix< T, S >::isIdentity ( ) const

test if this matrix equals a identity matrix

The identity matrix is a the simplest nontrivial diagonal matrix, defined such that
I(X)=X for all vectors
X. An identity matrix may be denoted
1,
I, or
E (the latter being an abbreviation for the German term "Einheitsmatrix"; Courant and Hilbert 1989, p. 7). Identity matrices are sometimes also known as unit matrices (Akivis and Goldberg 1972, p. 71). The
n�n identity matrix is given explicitly by
Iij=dij for
i,
j = 1, 2, ..., n, where
dij is the Kronecker delta. http://mathworld.wolfram.com/IdentityMatrix.html

an empty matrix is also an identity matrix.

Complexity:: O(this->rows() * this->columns())

Definition at line 598 of file matrix.inl.

References columns(), isSquare(), and rows().

◆ isDiagonal()

template<typename T, typename S>

bool lass::num::Matrix< T, S >::isDiagonal ( ) const

test if this matrix is an diagonal matrix

A diagonal matrix is a square matrix
A of the form
Aij=Ci*dij where
dij is the Kronecker delta,
Ci are constants, and
i,
j = 1, 2, ..., n, with is no implied summation over indices. http://mathworld.wolfram.com/DiagonalMatrix.html

Note: both square zero matrices and identity matrices will yield true on this one.

Complexity:: O(this->rows() * this->columns())

Definition at line 634 of file matrix.inl.

References columns(), isSquare(), and rows().

◆ isSquare()

template<typename T, typename S>

bool lass::num::Matrix< T, S >::isSquare ( ) const

return true if matrix has as many rows as columns

Complexity:: O(1)

Definition at line 663 of file matrix.inl.

Referenced by invert(), isDiagonal(), isIdentity(), and solve().

◆ transposed()

template<typename T, typename S>

const Matrix< T, impl::MTrans< T, S > > lass::num::Matrix< T, S >::transposed ( ) const

return transposed matrix

Complexity:: O(1)

Definition at line 677 of file matrix.inl.

References Matrix().

◆ invert()

template<typename T, typename S>

void lass::num::Matrix< T, S >::invert ( )

replace matrix by its inverse.

If matrix has no inverse (this is singular), no exception is thrown. Instead, an empty matrix is made.

Precondition: this must be an l-value (storage matrix).

Returns: true if succeeded, false if not.

Exceptions

an	exception is thrown if this is not a square matrix

Definition at line 696 of file matrix.inl.

References column(), isSquare(), LASS_META_ASSERT, Matrix(), rows(), and setIdentity().

◆ swap()

template<typename T, typename S>

void lass::num::Matrix< T, S >::swap ( Matrix< T, S > & other )

inline

swap two matrices

Precondition: this and other must be an l-value (storage matrix).

Complexity:: O(1)

Exception safety:: no-fail

Definition at line 750 of file matrix.inl.

References LASS_META_ASSERT, and Matrix().

Friends And Related Symbol Documentation

◆ operator==()

template<typename T, typename S1, typename S2>

bool operator==	(	const Matrix< T, S1 > &	a,
		const Matrix< T, S2 > &	b )

Complexity:: O(a.rows() * a.columns())

Definition at line 774 of file matrix.inl.

◆ operator!=()

template<typename T, typename S1, typename S2>

bool operator!=	(	const Matrix< T, S1 > &	a,
		const Matrix< T, S2 > &	b )

Complexity:: O(a.rows() * a.columns())

Definition at line 804 of file matrix.inl.

◆ operator+() [1/3]

template<typename T, typename S1, typename S2>

const Matrix< T, impl::MAdd< T, S1, S2 > > operator+	(	const Matrix< T, S1 > &	a,
		const Matrix< T, S2 > &	b )

componentwise addition

Complexity:: O(1)

Definition at line 819 of file matrix.inl.

◆ operator-() [1/3]

template<typename T, typename S1, typename S2>

const Matrix< T, impl::MSub< T, S1, S2 > > operator-	(	const Matrix< T, S1 > &	a,
		const Matrix< T, S2 > &	b )

componentwise addition

Complexity:: O(1)

Definition at line 836 of file matrix.inl.

◆ operator*() [1/3]

template<typename T, typename S1, typename S2>

const Matrix< T, impl::MProd< T, S1, S2 > > operator*	(	const Matrix< T, S1 > &	a,
		const Matrix< T, S2 > &	b )

Matrix multiplication.

The product
C of two matrices
A and
B is defined by
Cik=Aij*Bjk, where
j is summed over for all possible values of
i and
k. The implied summation over repeated indices without the presence of an explicit sum sign is called Einstein summation, and is commonly used in both matrix and tensor analysis. Therefore, in order for matrix multiplication to be defined, the dimensions of the matrices must satisfy
(n�m)*(m�p)=(n�p) where
(a�b) denotes a matrix with
a rows and
b columns. http://mathworld.wolfram.com/MatrixMultiplication.html

Exceptions

an	exception is throw in a and b don't meet the requirement a.columns()==b.rows() .

Complexity:: O(1)

Definition at line 864 of file matrix.inl.

◆ operator+() [2/3]

template<typename T, typename S>

const Matrix< T, impl::MAdd< T, impl::MScalar< T >, S > > operator+	(	const T &	a,
		const Matrix< T, S > &	b )

add a to all components of b

Complexity:: O(1)

Definition at line 881 of file matrix.inl.

◆ operator-() [2/3]

template<typename T, typename S>

const Matrix< T, impl::MSub< T, impl::MScalar< T >, S > > operator-	(	const T &	a,
		const Matrix< T, S > &	b )

add a to all negated components of b

Complexity:: O(1)

Definition at line 898 of file matrix.inl.

◆ operator*() [2/3]

template<typename T, typename S>

const Matrix< T, impl::MMul< T, impl::MScalar< T >, S > > operator*	(	const T &	a,
		const Matrix< T, S > &	b )

multiply a with all components of b

Complexity:: O(1)

Definition at line 915 of file matrix.inl.

◆ operator+() [3/3]

template<typename T, typename S>

const Matrix< T, impl::MAdd< T, S, impl::MScalar< T > > > operator+	(	const Matrix< T, S > &	a,
		typename Matrix< T, S >::TParam	b )

add b to all components of a

Complexity:: O(1)

Definition at line 932 of file matrix.inl.

◆ operator-() [3/3]

template<typename T, typename S>

const Matrix< T, impl::MAdd< T, S, impl::MScalar< T > > > operator-	(	const Matrix< T, S > &	a,
		typename Matrix< T, S >::TParam	b )

subtract b from all components of a

Complexity:: O(1)

Definition at line 949 of file matrix.inl.

◆ operator*() [3/3]

template<typename T, typename S>

const Matrix< T, impl::MMul< T, S, impl::MScalar< T > > > operator*	(	const Matrix< T, S > &	a,
		typename Matrix< T, S >::TParam	b )

multiply all components of a with b.

Complexity:: O(1)

Definition at line 966 of file matrix.inl.

◆ operator/()

template<typename T, typename S>

const Matrix< T, impl::MMul< T, S, impl::MScalar< T > > > operator/	(	const Matrix< T, S > &	a,
		typename Matrix< T, S >::TParam	b )

divide all components of a by b.

Complexity:: O(1)

Definition at line 983 of file matrix.inl.

◆ solve()

template<typename T, typename S, typename S2>

void solve	(	const Matrix< T, S > &	a,
		Matrix< T, S2 > &	bx,
		bool	improve )

solve equation A * X = B

Parameters

a	[in] matrix A
bx	[in,out] serves both as the input matrix B and the output matrix X.
improve	[in] if true, it will perform an extra iteration to improve the solution, but it comes at the cost of having to take an extra copy of a.

Precondition: bx must be an l-value (storage matrix)

Exceptions

an	exception is thrown if dimensions don't match (this->isSquare() && this->columns() == bx.rows())

Definition at line 1006 of file matrix.inl.

The documentation for this class was generated from the following files:

Public Member Functions

Related Symbols

Detailed Description

Constructor & Destructor Documentation

◆ Matrix() [1/3]

◆ Matrix() [2/3]

◆ Matrix() [3/3]

Member Function Documentation

◆ operator=()

◆ rows()

◆ columns()

◆ operator()() [1/2]

◆ operator()() [2/2]

◆ at()

◆ row() [1/2]

◆ row() [2/2]

◆ column() [1/2]

◆ column() [2/2]

◆ operator+()

◆ operator-()

◆ operator+=()

◆ operator-=()

◆ operator*=()

◆ operator/=()

◆ setZero()

◆ setIdentity()

◆ isEmpty()

◆ isZero()

◆ isIdentity()

◆ isDiagonal()

◆ isSquare()

◆ transposed()

◆ invert()

◆ swap()

Friends And Related Symbol Documentation

◆ operator==()

◆ operator!=()

◆ operator+() [1/3]

◆ operator-() [1/3]

◆ operator*() [1/3]

◆ operator+() [2/3]

◆ operator-() [2/3]

◆ operator*() [2/3]

◆ operator+() [3/3]

◆ operator-() [3/3]

◆ operator*() [3/3]

◆ operator/()

◆ solve()