Symbolic matrices. More...

#include <matrix.h>

Inheritance diagram for GiNaC::matrix:

Public Member Functions
	matrix (unsigned r, unsigned c)
	Very common ctor. More...

	matrix (unsigned r, unsigned c, const lst &l)
	Construct matrix from (flat) list of elements. More...

	matrix (std::initializer_list< std::initializer_list< ex > > l)
	Construct a matrix from an 2 dimensional initializer list. More...

size_t	nops () const override
	nops is defined to be rows x columns. More...

ex	op (size_t i) const override
	returns matrix entry at position (i/col, icol). More...

ex &	let_op (size_t i) override
	returns writable matrix entry at position (i/col, icol). More...

ex	evalm () const override
	Evaluate sums, products and integer powers of matrices. More...

ex	subs (const exmap &m, unsigned options=0) const override
	Substitute a set of objects by arbitrary expressions. More...

ex	eval_indexed (const basic &i) const override
	Automatic symbolic evaluation of an indexed matrix. More...

ex	add_indexed (const ex &self, const ex &other) const override
	Sum of two indexed matrices. More...

ex	scalar_mul_indexed (const ex &self, const numeric &other) const override
	Product of an indexed matrix with a number. More...

bool	contract_with (exvector::iterator self, exvector::iterator other, exvector &v) const override
	Contraction of an indexed matrix with something else. More...

ex	conjugate () const override
	Complex conjugate every matrix entry. More...

ex	real_part () const override

ex	imag_part () const override

void	archive (archive_node &n) const override
	Save (a.k.a. More...

void	read_archive (const archive_node &n, lst &syms) override
	Read (a.k.a. More...

unsigned	rows () const
	Get number of rows. More...

unsigned	cols () const
	Get number of columns. More...

matrix	add (const matrix &other) const
	Sum of matrices. More...

matrix	sub (const matrix &other) const
	Difference of matrices. More...

matrix	mul (const matrix &other) const
	Product of matrices. More...

matrix	mul (const numeric &other) const
	Product of matrix and scalar. More...

matrix	mul_scalar (const ex &other) const
	Product of matrix and scalar expression. More...

matrix	pow (const ex &expn) const
	Power of a matrix. More...

const ex &	operator() (unsigned ro, unsigned co) const
	operator() to access elements for reading. More...

ex &	operator() (unsigned ro, unsigned co)
	operator() to access elements for writing. More...

matrix &	set (unsigned ro, unsigned co, const ex &value)

matrix	transpose () const
	Transposed of an m x n matrix, producing a new n x m matrix object that represents the transposed. More...

ex	determinant (unsigned algo=determinant_algo::automatic) const
	Determinant of square matrix. More...

ex	trace () const
	Trace of a matrix. More...

ex	charpoly (const ex &lambda) const
	Characteristic Polynomial. More...

matrix	inverse () const
	Inverse of this matrix, with automatic algorithm selection. More...

matrix	inverse (unsigned algo) const
	Inverse of this matrix. More...

matrix	solve (const matrix &vars, const matrix &rhs, unsigned algo=solve_algo::automatic) const
	Solve a linear system consisting of a m x n matrix and a m x p right hand side by applying an elimination scheme to the augmented matrix. More...

unsigned	rank () const
	Compute the rank of this matrix. More...

unsigned	rank (unsigned solve_algo) const
	Compute the rank of this matrix using the given algorithm, which should be a member of enum solve_algo. More...

bool	is_zero_matrix () const
	Function to check that all elements of the matrix are zero. More...

Public Member Functions inherited from GiNaC::basic
virtual	~basic ()
	basic destructor, virtual because class ex will delete objects of derived classes via a basic*. More...

	basic (const basic &other)

const basic &	operator= (const basic &other)
	basic assignment operator: the other object might be of a derived class. More...

virtual basic *	duplicate () const
	Create a clone of this object on the heap. More...

virtual ex	eval () const
	Perform automatic non-interruptive term rewriting rules. More...

virtual ex	evalf () const
	Evaluate object numerically. More...

virtual ex	evalm () const
	Evaluate sums, products and integer powers of matrices. More...

virtual ex	eval_integ () const
	Evaluate integrals, if result is known. More...

virtual ex	eval_indexed (const basic &i) const
	Perform automatic symbolic evaluations on indexed expression that contains this object as the base expression. More...

virtual void	print (const print_context &c, unsigned level=0) const
	Output to stream. More...

virtual void	dbgprint () const
	Little wrapper around print to be called within a debugger. More...

virtual void	dbgprinttree () const
	Little wrapper around printtree to be called within a debugger. More...

virtual unsigned	precedence () const
	Return relative operator precedence (for parenthezing output). More...

virtual bool	info (unsigned inf) const
	Information about the object. More...

virtual size_t	nops () const
	Number of operands/members. More...

virtual ex	op (size_t i) const
	Return operand/member at position i. More...

virtual ex	operator[] (const ex &index) const

virtual ex	operator[] (size_t i) const

virtual ex &	let_op (size_t i)
	Return modifiable operand/member at position i. More...

virtual ex &	operator[] (const ex &index)

virtual ex &	operator[] (size_t i)

virtual bool	has (const ex &other, unsigned options=0) const
	Test for occurrence of a pattern. More...

virtual bool	match (const ex &pattern, exmap &repls) const
	Check whether the expression matches a given pattern. More...

virtual ex	subs (const exmap &m, unsigned options=0) const
	Substitute a set of objects by arbitrary expressions. More...

virtual ex	map (map_function &f) const
	Construct new expression by applying the specified function to all sub-expressions (one level only, not recursively). More...

virtual void	accept (GiNaC::visitor &v) const

virtual bool	is_polynomial (const ex &var) const
	Check whether this is a polynomial in the given variables. More...

virtual int	degree (const ex &s) const
	Return degree of highest power in object s. More...

virtual int	ldegree (const ex &s) const
	Return degree of lowest power in object s. More...

virtual ex	coeff (const ex &s, int n=1) const
	Return coefficient of degree n in object s. More...

virtual ex	expand (unsigned options=0) const
	Expand expression, i.e. More...

virtual ex	collect (const ex &s, bool distributed=false) const
	Sort expanded expression in terms of powers of some object(s). More...

virtual ex	series (const relational &r, int order, unsigned options=0) const
	Default implementation of ex::series(). More...

virtual ex	normal (exmap &repl, exmap &rev_lookup, lst &modifier) const
	Default implementation of ex::normal(). More...

virtual ex	to_rational (exmap &repl) const
	Default implementation of ex::to_rational(). More...

virtual ex	to_polynomial (exmap &repl) const

virtual numeric	integer_content () const

virtual ex	smod (const numeric &xi) const
	Apply symmetric modular homomorphism to an expanded multivariate polynomial. More...

virtual numeric	max_coefficient () const
	Implementation ex::max_coefficient(). More...

virtual exvector	get_free_indices () const
	Return a vector containing the free indices of an expression. More...

virtual ex	add_indexed (const ex &self, const ex &other) const
	Add two indexed expressions. More...

virtual ex	scalar_mul_indexed (const ex &self, const numeric &other) const
	Multiply an indexed expression with a scalar. More...

virtual bool	contract_with (exvector::iterator self, exvector::iterator other, exvector &v) const
	Try to contract two indexed expressions that appear in the same product. More...

virtual unsigned	return_type () const

virtual return_type_t	return_type_tinfo () const

virtual ex	conjugate () const

virtual ex	real_part () const

virtual ex	imag_part () const

template<class T >
void	print_dispatch (const print_context &c, unsigned level) const
	Like print(), but dispatch to the specified class. More...

void	print_dispatch (const registered_class_info &ri, const print_context &c, unsigned level) const
	Like print(), but dispatch to the specified class. More...

virtual void	archive (archive_node &n) const
	Save (serialize) the object into archive node. More...

virtual void	read_archive (const archive_node &n, lst &syms)
	Load (deserialize) the object from an archive node. More...

ex	subs_one_level (const exmap &m, unsigned options) const
	Helper function for subs(). More...

ex	diff (const symbol &s, unsigned nth=1) const
	Default interface of nth derivative ex::diff(s, n). More...

int	compare (const basic &other) const
	Compare objects syntactically to establish canonical ordering. More...

bool	is_equal (const basic &other) const
	Test for syntactic equality. More...

const basic &	hold () const
	Stop further evaluation. More...

unsigned	gethash () const

const basic &	setflag (unsigned f) const
	Set some status_flags. More...

const basic &	clearflag (unsigned f) const
	Clear some status_flags. More...

Public Member Functions inherited from GiNaC::refcounted
	refcounted () noexcept

unsigned int	add_reference () noexcept

unsigned int	remove_reference () noexcept

unsigned int	get_refcount () const noexcept

void	set_refcount (unsigned int r) noexcept

Protected Member Functions
	matrix (unsigned r, unsigned c, const exvector &m2)
	Ctor from representation, for internal use only. More...

	matrix (unsigned r, unsigned c, exvector &&m2)

bool	match_same_type (const basic &other) const override
	Returns true if the attributes of two objects are similar enough for a match. More...

unsigned	return_type () const override

ex	determinant_minor () const
	Recursive determinant for small matrices having at least one symbolic entry. More...

std::vector< unsigned >	echelon_form (unsigned algo, int n)

int	gauss_elimination (const bool det=false)
	Perform the steps of an ordinary Gaussian elimination to bring the m x n matrix into an upper echelon form. More...

int	division_free_elimination (const bool det=false)
	Perform the steps of division free elimination to bring the m x n matrix into an upper echelon form. More...

int	fraction_free_elimination (const bool det=false)
	Perform the steps of Bareiss' one-step fraction free elimination to bring the matrix into an upper echelon form. More...

std::vector< unsigned >	markowitz_elimination (unsigned n)

int	pivot (unsigned ro, unsigned co, bool symbolic=true)
	Partial pivoting method for matrix elimination schemes. More...

void	print_elements (const print_context &c, const char row_start, const char row_end, const char row_sep, const char col_sep) const

void	do_print (const print_context &c, unsigned level) const

void	do_print_latex (const print_latex &c, unsigned level) const

void	do_print_python_repr (const print_python_repr &c, unsigned level) const

Protected Member Functions inherited from GiNaC::basic
	basic ()

virtual ex	eval_ncmul (const exvector &v) const

virtual bool	match_same_type (const basic &other) const
	Returns true if the attributes of two objects are similar enough for a match. More...

virtual ex	derivative (const symbol &s) const
	Default implementation of ex::diff(). More...

virtual int	compare_same_type (const basic &other) const
	Returns order relation between two objects of same type. More...

virtual bool	is_equal_same_type (const basic &other) const
	Returns true if two objects of same type are equal. More...

virtual unsigned	calchash () const
	Compute the hash value of an object and if it makes sense to store it in the objects status_flags, do so. More...

void	ensure_if_modifiable () const
	Ensure the object may be modified without hurting others, throws if this is not the case. More...

void	do_print (const print_context &c, unsigned level) const
	Default output to stream. More...

void	do_print_tree (const print_tree &c, unsigned level) const
	Tree output to stream. More...

void	do_print_python_repr (const print_python_repr &c, unsigned level) const
	Python parsable output to stream. More...

Protected Attributes
unsigned	row
	number of rows More...

unsigned	col
	number of columns More...

exvector	m
	representation (cols indexed first) More...

Protected Attributes inherited from GiNaC::basic
unsigned	flags
	of type status_flags More...

unsigned	hashvalue
	hash value More...

Detailed Description

Symbolic matrices.

Definition at line 37 of file matrix.h.

Constructor & Destructor Documentation

◆ matrix() [1/5]

GiNaC::matrix::matrix	(	unsigned	r,
		unsigned	c
	)

Very common ctor.

Initializes to r x c-dimensional zero-matrix.

Parameters

r	number of rows
c	number of cols

Definition at line 71 of file matrix.cpp.

References GiNaC::status_flags::not_shareable, and GiNaC::basic::setflag().

Referenced by add(), conjugate(), determinant(), imag_part(), mul(), mul_scalar(), real_part(), sub(), subs(), and transpose().

◆ matrix() [2/5]

GiNaC::matrix::matrix	(	unsigned	r,
		unsigned	c,
		const lst &	l
	)

Construct matrix from (flat) list of elements.

If the list has fewer elements than the matrix, the remaining matrix elements are set to zero. If the list has more elements than the matrix, the excessive elements are thrown away.

Definition at line 80 of file matrix.cpp.

References c, m, GiNaC::status_flags::not_shareable, r, GiNaC::basic::setflag(), and x.

◆ matrix() [3/5]

GiNaC::matrix::matrix ( std::initializer_list< std::initializer_list< ex > > l )

Construct a matrix from an 2 dimensional initializer list.

Throws an exception if some row has a different length than all the others.

Definition at line 99 of file matrix.cpp.

References c, col, m, GiNaC::status_flags::not_shareable, r, row, and GiNaC::basic::setflag().

◆ matrix() [4/5]

GiNaC::matrix::matrix	(	unsigned	r,
		unsigned	c,
		const exvector &	m2
	)

protected

Ctor from representation, for internal use only.

Definition at line 119 of file matrix.cpp.

References GiNaC::status_flags::not_shareable, and GiNaC::basic::setflag().

◆ matrix() [5/5]

GiNaC::matrix::matrix	(	unsigned	r,
		unsigned	c,
		exvector &&	m2
	)

protected

Definition at line 124 of file matrix.cpp.

References GiNaC::status_flags::not_shareable, and GiNaC::basic::setflag().

Member Function Documentation

◆ nops()

size_t GiNaC::matrix::nops ( ) const

overridevirtual

nops is defined to be rows x columns.

Reimplemented from GiNaC::basic.

Definition at line 206 of file matrix.cpp.

References col, and row.

Referenced by let_op(), and op().

◆ op()

ex GiNaC::matrix::op ( size_t i ) const

overridevirtual

returns matrix entry at position (i/col, icol).

Reimplemented from GiNaC::basic.

Definition at line 212 of file matrix.cpp.

References GINAC_ASSERT, m, and nops().

Referenced by contract_with().

◆ let_op()

ex & GiNaC::matrix::let_op ( size_t i )

overridevirtual

returns writable matrix entry at position (i/col, icol).

Reimplemented from GiNaC::basic.

Definition at line 220 of file matrix.cpp.

References GiNaC::basic::ensure_if_modifiable(), GINAC_ASSERT, m, and nops().

◆ evalm()

ex GiNaC::matrix::evalm ( ) const

inlineoverridevirtual

Evaluate sums, products and integer powers of matrices.

Reimplemented from GiNaC::basic.

Definition at line 55 of file matrix.h.

◆ subs()

ex GiNaC::matrix::subs	(	const exmap &	m,
		unsigned	options = `0`
	)		const

overridevirtual

Substitute a set of objects by arbitrary expressions.

The ex returned will already be evaluated.

Reimplemented from GiNaC::basic.

Definition at line 228 of file matrix.cpp.

References c, col, m, matrix(), options, r, row, and subs().

Referenced by fraction_free_elimination(), and subs().

◆ eval_indexed()

ex GiNaC::matrix::eval_indexed ( const basic & i ) const

overridevirtual

Automatic symbolic evaluation of an indexed matrix.

Reimplemented from GiNaC::basic.

Definition at line 320 of file matrix.cpp.

References col, GiNaC::idx::get_dim(), GiNaC::idx::get_value(), GINAC_ASSERT, GiNaC::basic::hold(), GiNaC::is_dummy_pair(), GiNaC::ex::is_equal(), GiNaC::info_flags::nonnegint, GiNaC::basic::nops(), GiNaC::basic::op(), row, and trace().

◆ add_indexed()

ex GiNaC::matrix::add_indexed	(	const ex &	self,
		const ex &	other
	)		const

overridevirtual

Sum of two indexed matrices.

Reimplemented from GiNaC::basic.

Definition at line 397 of file matrix.cpp.

References add(), col, GINAC_ASSERT, GiNaC::ex::is_equal(), GiNaC::ex::nops(), GiNaC::ex::op(), row, and transpose().

◆ scalar_mul_indexed()

ex GiNaC::matrix::scalar_mul_indexed	(	const ex &	self,
		const numeric &	other
	)		const

overridevirtual

Product of an indexed matrix with a number.

Reimplemented from GiNaC::basic.

Definition at line 433 of file matrix.cpp.

References GINAC_ASSERT, mul(), GiNaC::ex::nops(), and GiNaC::ex::op().

◆ contract_with()

bool GiNaC::matrix::contract_with	(	exvector::iterator	self,
		exvector::iterator	other,
		exvector &	v
	)		const

overridevirtual

Contraction of an indexed matrix with something else.

Reimplemented from GiNaC::basic.

Definition at line 448 of file matrix.cpp.

References GiNaC::_ex1, col, GINAC_ASSERT, GiNaC::is_dummy_pair(), mul(), GiNaC::ex::op(), op(), row, and transpose().

◆ conjugate()

ex GiNaC::matrix::conjugate ( ) const

overridevirtual

Complex conjugate every matrix entry.

Reimplemented from GiNaC::basic.

Definition at line 239 of file matrix.cpp.

References GiNaC::are_ex_trivially_equal(), col, m, matrix(), row, and x.

◆ real_part()

ex GiNaC::matrix::real_part ( ) const

overridevirtual

Reimplemented from GiNaC::basic.

Definition at line 264 of file matrix.cpp.

References col, m, matrix(), and row.

◆ imag_part()

ex GiNaC::matrix::imag_part ( ) const

overridevirtual

Reimplemented from GiNaC::basic.

Definition at line 273 of file matrix.cpp.

References col, m, matrix(), and row.

◆ archive()

void GiNaC::matrix::archive ( archive_node & n ) const

overridevirtual

Save (a.k.a.

serialize) object into archive.

Reimplemented from GiNaC::basic.

Definition at line 152 of file matrix.cpp.

References col, m, n, and row.

◆ read_archive()

void GiNaC::matrix::read_archive	(	const archive_node &	n,
		lst &	syms
	)

overridevirtual

Read (a.k.a.

deserialize) object from archive.

Reimplemented from GiNaC::basic.

Definition at line 134 of file matrix.cpp.

References col, m, n, and row.

◆ match_same_type()

bool GiNaC::matrix::match_same_type ( const basic & other ) const

overrideprotectedvirtual

Returns true if the attributes of two objects are similar enough for a match.

This function must not match subexpressions (this is already done by basic::match()). Only attributes not accessible by op() should be compared. This is also the reason why this function doesn't take the wildcard replacement list from match() as an argument: only subexpressions are subject to wildcard matches. Also, this function only needs to be implemented for container classes because is_equal_same_type() is automatically used instead of match_same_type() if nops() == 0.

See also: basic::match

Reimplemented from GiNaC::basic.

Definition at line 309 of file matrix.cpp.

References col, cols(), GINAC_ASSERT, row, and rows().

◆ return_type()

unsigned GiNaC::matrix::return_type ( ) const

inlineoverrideprotectedvirtual

Reimplemented from GiNaC::basic.

Definition at line 71 of file matrix.h.

References GiNaC::return_types::noncommutative.

◆ rows()

unsigned GiNaC::matrix::rows ( ) const

inline

Get number of rows.

Definition at line 75 of file matrix.h.

References row.

Referenced by division_free_elimination(), fraction_free_elimination(), gauss_elimination(), GiNaC::lsolve(), match_same_type(), mul(), solve(), and transpose().

◆ cols()

unsigned GiNaC::matrix::cols ( ) const

inline

Get number of columns.

Definition at line 77 of file matrix.h.

References col.

Referenced by determinant_minor(), division_free_elimination(), fraction_free_elimination(), gauss_elimination(), GiNaC::lsolve(), match_same_type(), mul(), solve(), and transpose().

◆ add()

matrix GiNaC::matrix::add ( const matrix & other ) const

Sum of matrices.

Exceptions

logic_error (incompatible matrices)

Definition at line 555 of file matrix.cpp.

References col, m, matrix(), and row.

Referenced by add_indexed(), and GiNaC::add::evalm().

◆ sub()

matrix GiNaC::matrix::sub ( const matrix & other ) const

Difference of matrices.

Exceptions

logic_error (incompatible matrices)

Definition at line 572 of file matrix.cpp.

References col, m, matrix(), and row.

◆ mul() [1/2]

matrix GiNaC::matrix::mul ( const matrix & other ) const

Product of matrices.

Exceptions

logic_error (incompatible matrices)

Definition at line 589 of file matrix.cpp.

References c, col, cols(), GiNaC::is_zero(), m, matrix(), row, and rows().

Referenced by charpoly(), contract_with(), GiNaC::ncmul::evalm(), pow(), and scalar_mul_indexed().

◆ mul() [2/2]

matrix GiNaC::matrix::mul ( const numeric & other ) const

Product of matrix and scalar.

Definition at line 610 of file matrix.cpp.

References c, col, m, matrix(), r, and row.

◆ mul_scalar()

matrix GiNaC::matrix::mul_scalar ( const ex & other ) const

Product of matrix and scalar expression.

Definition at line 623 of file matrix.cpp.

References c, col, GiNaC::return_types::commutative, m, matrix(), r, GiNaC::ex::return_type(), and row.

◆ pow()

matrix GiNaC::matrix::pow ( const ex & expn ) const

Power of a matrix.

Currently handles integer exponents only.

Definition at line 639 of file matrix.cpp.

References GiNaC::_ex1, GiNaC::_num1_p, GiNaC::_num2_p, col, GiNaC::ex::info(), GiNaC::info_flags::integer, inverse(), GiNaC::numeric::is_odd(), GiNaC::numeric::is_zero(), mul(), GiNaC::info_flags::negative, r, and row.

◆ operator()() [1/2]

const ex & GiNaC::matrix::operator()	(	unsigned	ro,
		unsigned	co
	)		const

operator() to access elements for reading.

Parameters

ro	row of element
co	column of element

Exceptions

range_error (index out of range)

Definition at line 686 of file matrix.cpp.

References col, m, and row.

◆ operator()() [2/2]

ex & GiNaC::matrix::operator()	(	unsigned	ro,
		unsigned	co
	)

operator() to access elements for writing.

Parameters

ro	row of element
co	column of element

Exceptions

range_error (index out of range)

Definition at line 700 of file matrix.cpp.

References col, GiNaC::basic::ensure_if_modifiable(), m, and row.

◆ set()

matrix & GiNaC::matrix::set	(	unsigned	ro,
		unsigned	co,
		const ex &	value
	)

inline

Definition at line 87 of file matrix.h.

References value.

◆ transpose()

matrix GiNaC::matrix::transpose ( ) const

Transposed of an m x n matrix, producing a new n x m matrix object that represents the transposed.

Definition at line 712 of file matrix.cpp.

References c, cols(), m, matrix(), r, and rows().

Referenced by add_indexed(), and contract_with().

◆ determinant()

ex GiNaC::matrix::determinant ( unsigned algo = determinant_algo::automatic ) const

Determinant of square matrix.

This routine doesn't actually calculate the determinant, it only implements some heuristics about which algorithm to run. If all the elements of the matrix are elements of an integral domain the determinant is also in that integral domain and the result is expanded only. If one or more elements are from a quotient field the determinant is usually also in that quotient field and the result is normalized before it is returned. This implies that the determinant of the symbolic 2x2 matrix [[a/(a-b),1],[b/(a-b),1]] is returned as unity. (In this respect, it behaves like MapleV and unlike Mathematica.)

Parameters

algo	allows to chose an algorithm

Returns: the determinant as a new expression

Exceptions

logic_error (matrix not square)

See also: determinant_algo

Definition at line 737 of file matrix.cpp.

References GiNaC::_ex0, GiNaC::determinant_algo::automatic, GiNaC::determinant_algo::bareiss, c, col, GiNaC::info_flags::crational_polynomial, determinant_minor(), GiNaC::determinant_algo::divfree, division_free_elimination(), fraction_free_elimination(), GiNaC::determinant_algo::gauss, gauss_elimination(), GINAC_ASSERT, GiNaC::ex::info(), GiNaC::ex::is_zero(), GiNaC::is_zero(), GiNaC::determinant_algo::laplace, m, matrix(), GiNaC::ex::normal(), GiNaC::info_flags::numeric, GiNaC::permutation_sign(), r, GiNaC::info_flags::rational_function, and row.

Referenced by charpoly(), and GiNaC::tensepsilon::contract_with().

◆ trace()

ex GiNaC::matrix::trace ( ) const

Trace of a matrix.

The result is normalized if it is in some quotient field and expanded only otherwise. This implies that the trace of the symbolic 2x2 matrix [[a/(a-b),x],[y,b/(b-a)]] is recognized to be unity.

Returns: the sum of diagonal elements

Exceptions

logic_error (matrix not square)

Definition at line 866 of file matrix.cpp.

References col, GiNaC::info_flags::crational_polynomial, GiNaC::ex::expand(), GiNaC::ex::info(), m, GiNaC::ex::normal(), r, GiNaC::info_flags::rational_function, and row.

Referenced by charpoly(), and eval_indexed().

◆ charpoly()

ex GiNaC::matrix::charpoly ( const ex & lambda ) const

Characteristic Polynomial.

Following mathematica notation the characteristic polynomial of a matrix M is defined as the determinant of (M - lambda * 1) where 1 stands for the unit matrix of the same dimension as M. Note that some CASs define it with a sign inside the determinant which gives rise to an overall sign if the dimension is odd. This method returns the characteristic polynomial collected in powers of lambda as a new expression.

Returns: characteristic polynomial as new expression

Exceptions

logic_error (matrix not square)

See also: matrix::determinant()

Definition at line 894 of file matrix.cpp.

References c, col, GiNaC::ex::collect(), determinant(), GiNaC::basic::ex, m, mul(), GiNaC::info_flags::numeric, poly, r, row, and trace().

◆ inverse() [1/2]

matrix GiNaC::matrix::inverse ( ) const

Inverse of this matrix, with automatic algorithm selection.

Definition at line 939 of file matrix.cpp.

References GiNaC::solve_algo::automatic, and inverse().

Referenced by inverse(), and pow().

◆ inverse() [2/2]

matrix GiNaC::matrix::inverse ( unsigned algo ) const

Inverse of this matrix.

Parameters

algo	selects the algorithm (one of solve_algo)

Returns: the inverted matrix

Exceptions

logic_error	(matrix not square)
runtime_error	(singular matrix)

Definition at line 950 of file matrix.cpp.

References GiNaC::_ex1, c, col, r, row, and solve().

◆ solve()

matrix GiNaC::matrix::solve	(	const matrix &	vars,
		const matrix &	rhs,
		unsigned	algo = `solve_algo::automatic`
	)		const

Solve a linear system consisting of a m x n matrix and a m x p right hand side by applying an elimination scheme to the augmented matrix.

Parameters

vars	n x p matrix, all elements must be symbols
rhs	m x p matrix
algo	selects the solving algorithm

Returns: n x p solution matrix

Exceptions

logic_error	(incompatible matrices)
invalid_argument	(1st argument must be matrix of symbols)
runtime_error	(inconsistent linear system)

See also: solve_algo

Definition at line 995 of file matrix.cpp.

References c, cols(), echelon_form(), GiNaC::basic::info(), m, n, GiNaC::basic::normal(), r, GiNaC::rhs(), rows(), and GiNaC::info_flags::symbol.

Referenced by inverse(), GiNaC::lsolve(), and GiNaC::sqrfree_parfrac().

◆ rank() [1/2]

unsigned GiNaC::matrix::rank ( ) const

Compute the rank of this matrix.

Definition at line 1058 of file matrix.cpp.

References GiNaC::solve_algo::automatic, and rank().

Referenced by rank().

◆ rank() [2/2]

unsigned GiNaC::matrix::rank ( unsigned solve_algo ) const

Compute the rank of this matrix using the given algorithm, which should be a member of enum solve_algo.

Definition at line 1065 of file matrix.cpp.

References col, echelon_form(), GINAC_ASSERT, m, r, and row.

◆ is_zero_matrix()

bool GiNaC::matrix::is_zero_matrix ( ) const

Function to check that all elements of the matrix are zero.

Definition at line 1676 of file matrix.cpp.

References m.

◆ determinant_minor()

ex GiNaC::matrix::determinant_minor ( ) const

protected

Recursive determinant for small matrices having at least one symbolic entry.

The basic algorithm, known as Laplace-expansion, is enhanced by some bookkeeping to avoid calculation of the same submatrices ("minors") more than once. According to W.M.Gentleman and S.C.Johnson this algorithm is better than elimination schemes for matrices of sparse multivariate polynomials and also for matrices of dense univariate polynomials if the matrix' dimension is larger than 7.

Returns: the determinant as a new expression (in expanded form)

See also: matrix::determinant()

Definition at line 1096 of file matrix.cpp.

References GiNaC::_ex0, GiNaC::_ex1, c, cols(), GiNaC::ex::expand(), GiNaC::ex::is_zero(), GiNaC::is_zero(), m, n, and r.

Referenced by determinant().

◆ echelon_form()

std::vector< unsigned > GiNaC::matrix::echelon_form	(	unsigned	algo,
		int	n
	)

protected

Definition at line 1197 of file matrix.cpp.

References GiNaC::solve_algo::automatic, GiNaC::solve_algo::bareiss, c, col, GiNaC::solve_algo::divfree, division_free_elimination(), fraction_free_elimination(), GiNaC::solve_algo::gauss, gauss_elimination(), m, GiNaC::solve_algo::markowitz, markowitz_elimination(), n, GiNaC::info_flags::numeric, r, and row.

Referenced by rank(), and solve().

◆ gauss_elimination()

int GiNaC::matrix::gauss_elimination ( const bool det = false )

protected

Perform the steps of an ordinary Gaussian elimination to bring the m x n matrix into an upper echelon form.

The algorithm is ok for matrices with numeric coefficients but quite unsuited for symbolic matrices.

Parameters

det	may be set to true to save a lot of space if one is only interested in the diagonal elements (i.e. for calculating determinants). The others are set to zero in this case.

Returns: sign is 1 if an even number of rows was swapped, -1 if an odd number of rows was swapped and 0 if the matrix is singular.

Definition at line 1269 of file matrix.cpp.

References GiNaC::_ex0, c, cols(), GiNaC::basic::ensure_if_modifiable(), GINAC_ASSERT, GiNaC::basic::info(), GiNaC::is_zero(), m, n, GiNaC::ex::normal(), GiNaC::info_flags::numeric, pivot(), r, and rows().

Referenced by determinant(), and echelon_form().

◆ division_free_elimination()

int GiNaC::matrix::division_free_elimination ( const bool det = false )

protected

Perform the steps of division free elimination to bring the m x n matrix into an upper echelon form.

Parameters

det	may be set to true to save a lot of space if one is only interested in the diagonal elements (i.e. for calculating determinants). The others are set to zero in this case.

Returns: sign is 1 if an even number of rows was swapped, -1 if an odd number of rows was swapped and 0 if the matrix is singular.

Definition at line 1441 of file matrix.cpp.

References GiNaC::_ex0, c, cols(), GiNaC::basic::ensure_if_modifiable(), GINAC_ASSERT, m, n, pivot(), r, and rows().

Referenced by determinant(), and echelon_form().

◆ fraction_free_elimination()

int GiNaC::matrix::fraction_free_elimination ( const bool det = false )

protected

Perform the steps of Bareiss' one-step fraction free elimination to bring the matrix into an upper echelon form.

Fraction free elimination means that divide is used straightforwardly, without computing GCDs first. This is possible, since we know the divisor at each step.

Parameters

det	may be set to true to save a lot of space if one is only interested in the last element (i.e. for calculating determinants). The others are set to zero in this case.

Returns: sign is 1 if an even number of rows was swapped, -1 if an odd number of rows was swapped and 0 if the matrix is singular.

Definition at line 1495 of file matrix.cpp.

References GiNaC::_ex0, GiNaC::_ex1, c, cols(), GiNaC::divide(), GiNaC::basic::ensure_if_modifiable(), GiNaC::basic::expand(), GINAC_ASSERT, GiNaC::is_zero(), m, n, GiNaC::subs_options::no_pattern, GiNaC::ex::normal(), GiNaC::ex::numer_denom(), GiNaC::ex::op(), r, rows(), subs(), and GiNaC::ex::to_rational().

Referenced by determinant(), and echelon_form().

◆ markowitz_elimination()

std::vector< unsigned > GiNaC::matrix::markowitz_elimination ( unsigned n )

protected

Definition at line 1326 of file matrix.cpp.

References GiNaC::_ex0, c, col, GINAC_ASSERT, GiNaC::ex::is_zero(), GiNaC::is_zero(), k, m, n, GiNaC::basic::normal(), r, row, and std::swap().

Referenced by echelon_form().

◆ pivot()

int GiNaC::matrix::pivot	(	unsigned	ro,
		unsigned	co,
		bool	symbolic = `true`
	)

protected

Partial pivoting method for matrix elimination schemes.

Usual pivoting (symbolic==false) returns the index to the element with the largest absolute value in column ro and swaps the current row with the one where the element was found. With (symbolic==true) it does the same thing with the first non-zero element.

Parameters

ro	is the row from where to begin
co	is the column to be inspected
symbolic	signal if we want the first non-zero element to be pivoted (true) or the one with the largest absolute value (false).

Returns: 0 if no interchange occurred, -1 if all are zero (usually signaling a degeneracy) and positive integer k means that rows ro and k were swapped.

Definition at line 1636 of file matrix.cpp.

References GiNaC::abs(), c, col, GiNaC::basic::ensure_if_modifiable(), GiNaC::basic::expand(), GINAC_ASSERT, GiNaC::numeric::is_zero(), GiNaC::is_zero(), k, m, row, and GiNaC::swap().

Referenced by division_free_elimination(), and gauss_elimination().

◆ print_elements()

void GiNaC::matrix::print_elements	(	const print_context &	c,
		const char *	row_start,
		const char *	row_end,
		const char *	row_sep,
		const char *	col_sep
	)		const

protected

Definition at line 168 of file matrix.cpp.

References c, col, m, and row.

Referenced by do_print(), do_print_latex(), and do_print_python_repr().

◆ do_print()

void GiNaC::matrix::do_print	(	const print_context &	c,
		unsigned	level
	)		const

protected

Definition at line 184 of file matrix.cpp.

References c, and print_elements().

◆ do_print_latex()

void GiNaC::matrix::do_print_latex	(	const print_latex &	c,
		unsigned	level
	)		const

protected

Definition at line 191 of file matrix.cpp.

References c, col, and print_elements().

◆ do_print_python_repr()

void GiNaC::matrix::do_print_python_repr	(	const print_python_repr &	c,
		unsigned	level
	)		const

protected

Definition at line 198 of file matrix.cpp.

References c, and print_elements().

Member Data Documentation

◆ row

unsigned GiNaC::matrix::row

protected

number of rows

Definition at line 115 of file matrix.h.

Referenced by add(), add_indexed(), archive(), charpoly(), conjugate(), contract_with(), determinant(), echelon_form(), eval_indexed(), imag_part(), inverse(), markowitz_elimination(), match_same_type(), matrix(), mul(), mul_scalar(), nops(), operator()(), pivot(), pow(), print_elements(), rank(), read_archive(), real_part(), rows(), sub(), subs(), and trace().

◆ col

unsigned GiNaC::matrix::col

protected

number of columns

Definition at line 116 of file matrix.h.

Referenced by add(), add_indexed(), archive(), charpoly(), cols(), conjugate(), contract_with(), determinant(), do_print_latex(), echelon_form(), eval_indexed(), imag_part(), inverse(), markowitz_elimination(), match_same_type(), matrix(), mul(), mul_scalar(), nops(), operator()(), pivot(), pow(), print_elements(), rank(), read_archive(), real_part(), sub(), subs(), and trace().

◆ m

exvector GiNaC::matrix::m

protected

representation (cols indexed first)

Definition at line 117 of file matrix.h.

Referenced by add(), archive(), charpoly(), conjugate(), determinant(), determinant_minor(), division_free_elimination(), echelon_form(), fraction_free_elimination(), gauss_elimination(), imag_part(), is_zero_matrix(), let_op(), markowitz_elimination(), matrix(), mul(), mul_scalar(), op(), operator()(), pivot(), print_elements(), rank(), read_archive(), real_part(), solve(), sub(), subs(), trace(), and transpose().

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

Public Member Functions

Protected Member Functions

Protected Attributes

Detailed Description

Constructor & Destructor Documentation

◆ matrix() [1/5]

◆ matrix() [2/5]

◆ matrix() [3/5]

◆ matrix() [4/5]

◆ matrix() [5/5]

Member Function Documentation

◆ nops()

◆ op()

◆ let_op()

◆ evalm()

◆ subs()

◆ eval_indexed()

◆ add_indexed()

◆ scalar_mul_indexed()

◆ contract_with()

◆ conjugate()

◆ real_part()

◆ imag_part()

◆ archive()

◆ read_archive()

◆ match_same_type()

◆ return_type()

◆ rows()

◆ cols()

◆ add()

◆ sub()

◆ mul() [1/2]

◆ mul() [2/2]

◆ mul_scalar()

◆ pow()

◆ operator()() [1/2]

◆ operator()() [2/2]

◆ set()

◆ transpose()

◆ determinant()

◆ trace()

◆ charpoly()

◆ inverse() [1/2]

◆ inverse() [2/2]

◆ solve()

◆ rank() [1/2]

◆ rank() [2/2]

◆ is_zero_matrix()

◆ determinant_minor()

◆ echelon_form()

◆ gauss_elimination()

◆ division_free_elimination()

◆ fraction_free_elimination()

◆ markowitz_elimination()

◆ pivot()

◆ print_elements()

◆ do_print()

◆ do_print_latex()

◆ do_print_python_repr()

Member Data Documentation

◆ row

◆ col

◆ m