Symbolic matrices. More...

#include <matrix.h>

Inheritance diagram for GiNaC::matrix:

Public Member Functions
	matrix (unsigned r, unsigned c)
	Very common ctor.
	matrix (unsigned r, unsigned c, const exvector &m2)
	Ctor from representation, for internal use only.
	matrix (unsigned r, unsigned c, const lst &l)
	Construct matrix from (flat) list of elements.
matrix_init< ex, exvector::iterator >	operator= (const ex &x)
size_t	nops () const
	nops is defined to be rows x columns.
ex	op (size_t i) const
	returns matrix entry at position (i/col, icol).
ex &	let_op (size_t i)
	returns writable matrix entry at position (i/col, icol).
ex	eval (int level=0) const
	Evaluate matrix entry by entry.
ex	evalm () const
	Evaluate sums, products and integer powers of matrices.
ex	subs (const exmap &m, unsigned options=0) const
	Substitute a set of objects by arbitrary expressions.
ex	eval_indexed (const basic &i) const
	Automatic symbolic evaluation of an indexed matrix.
ex	add_indexed (const ex &self, const ex &other) const
	Sum of two indexed matrices.
ex	scalar_mul_indexed (const ex &self, const numeric &other) const
	Product of an indexed matrix with a number.
bool	contract_with (exvector::iterator self, exvector::iterator other, exvector &v) const
	Contraction of an indexed matrix with something else.
ex	conjugate () const
	Complex conjugate every matrix entry.
ex	real_part () const
ex	imag_part () const
void	archive (archive_node &n) const
	Save (a.k.a.
void	read_archive (const archive_node &n, lst &syms)
	Read (a.k.a.
unsigned	rows () const
	Get number of rows.
unsigned	cols () const
	Get number of columns.
matrix	add (const matrix &other) const
	Sum of matrices.
matrix	sub (const matrix &other) const
	Difference of matrices.
matrix	mul (const matrix &other) const
	Product of matrices.
matrix	mul (const numeric &other) const
	Product of matrix and scalar.
matrix	mul_scalar (const ex &other) const
	Product of matrix and scalar expression.
matrix	pow (const ex &expn) const
	Power of a matrix.
const ex &	operator() (unsigned ro, unsigned co) const
	operator() to access elements for reading.
ex &	operator() (unsigned ro, unsigned co)
	operator() to access elements for writing.
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.
ex	determinant (unsigned algo=determinant_algo::automatic) const
	Determinant of square matrix.
ex	trace () const
	Trace of a matrix.
ex	charpoly (const ex &lambda) const
	Characteristic Polynomial.
matrix	inverse () const
	Inverse of this matrix.
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.
unsigned	rank () const
	Compute the rank of this matrix.
bool	is_zero_matrix () const
	Function to check that all elements of the matrix are zero.
Protected Member Functions
bool	match_same_type (const basic &other) const
	Returns true if the attributes of two objects are similar enough for a match.
unsigned	return_type () const
ex	determinant_minor () const
	Recursive determinant for small matrices having at least one symbolic entry.
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.
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.
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.
int	pivot (unsigned ro, unsigned co, bool symbolic=true)
	Partial pivoting method for matrix elimination schemes.
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
	Default output to stream.
void	do_print_latex (const print_latex &c, unsigned level) const
void	do_print_python_repr (const print_python_repr &c, unsigned level) const
	Python parsable output to stream.
Protected Attributes
unsigned	row
	number of rows
unsigned	col
	number of columns
exvector	m
	representation (cols indexed first)

Detailed Description

Symbolic matrices.

Definition at line 94 of file matrix.h.

Constructor & Destructor Documentation

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(), eval(), imag_part(), mul(), mul_scalar(), real_part(), sub(), subs(), and transpose().

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

Ctor from representation, for internal use only.

Definition at line 79 of file matrix.cpp.

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

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 89 of file matrix.cpp.

References GiNaC::container< C >::begin(), c, GiNaC::container< C >::end(), m, GiNaC::status_flags::not_shareable, GiNaC::basic::setflag(), and x.

Member Function Documentation

matrix_init<ex, exvector::iterator> GiNaC::matrix::operator= ( const ex & x ) [inline]

Definition at line 106 of file matrix.h.

References m, and x.

size_t GiNaC::matrix::nops ( ) const [virtual]

nops is defined to be rows x columns.

Reimplemented from GiNaC::basic.

Definition at line 184 of file matrix.cpp.

References col, and row.

Referenced by add_indexed(), let_op(), GiNaC::nops(), op(), and scalar_mul_indexed().

ex GiNaC::matrix::op ( size_t i ) const [virtual]

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

Reimplemented from GiNaC::basic.

Definition at line 190 of file matrix.cpp.

References GINAC_ASSERT, m, and nops().

Referenced by add_indexed(), and scalar_mul_indexed().

ex & GiNaC::matrix::let_op ( size_t i ) [virtual]

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

Reimplemented from GiNaC::basic.

Definition at line 198 of file matrix.cpp.

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

ex GiNaC::matrix::eval ( int level = 0 ) const [virtual]

Evaluate matrix entry by entry.

Reimplemented from GiNaC::basic.

Definition at line 207 of file matrix.cpp.

References c, col, GiNaC::status_flags::dynallocated, GiNaC::status_flags::evaluated, GiNaC::basic::flags, m, matrix(), GiNaC::max_recursion_level, r, and row.

Referenced by GiNaC::eval().

ex GiNaC::matrix::evalm ( ) const [inline, virtual]

Evaluate sums, products and integer powers of matrices.

Reimplemented from GiNaC::basic.

Definition at line 118 of file matrix.h.

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

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(), r, and row.

Referenced by fraction_free_elimination().

ex GiNaC::matrix::eval_indexed ( const basic & i ) const [virtual]

Automatic symbolic evaluation of an indexed matrix.

Reimplemented from GiNaC::basic.

Definition at line 322 of file matrix.cpp.

References 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(), GiNaC::to_int(), and trace().

ex GiNaC::matrix::add_indexed	(	const ex &	self,
		const ex &	other
	)		const `[virtual]`

Sum of two indexed matrices.

Reimplemented from GiNaC::basic.

Definition at line 399 of file matrix.cpp.

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

ex GiNaC::matrix::scalar_mul_indexed	(	const ex &	self,
		const numeric &	other
	)		const `[virtual]`

Product of an indexed matrix with a number.

Reimplemented from GiNaC::basic.

Definition at line 435 of file matrix.cpp.

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

bool GiNaC::matrix::contract_with	(	exvector::iterator	self,
		exvector::iterator	other,
		exvector &	v
	)		const `[virtual]`

Contraction of an indexed matrix with something else.

Reimplemented from GiNaC::basic.

Definition at line 450 of file matrix.cpp.

References GiNaC::_ex1, GINAC_ASSERT, GiNaC::is_dummy_pair(), mul(), and transpose().

ex GiNaC::matrix::conjugate ( ) const [virtual]

Complex conjugate every matrix entry.

Reimplemented from GiNaC::basic.

Definition at line 239 of file matrix.cpp.

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

ex GiNaC::matrix::real_part ( ) const [virtual]

Reimplemented from GiNaC::basic.

Definition at line 266 of file matrix.cpp.

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

ex GiNaC::matrix::imag_part ( ) const [virtual]

Reimplemented from GiNaC::basic.

Definition at line 275 of file matrix.cpp.

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

void GiNaC::matrix::archive ( archive_node & n ) const [virtual]

Save (a.k.a.

serialize) object into archive.

Reimplemented from GiNaC::basic.

Definition at line 128 of file matrix.cpp.

References GiNaC::archive_node::add_ex(), GiNaC::archive_node::add_unsigned(), col, m, and row.

void GiNaC::matrix::read_archive	(	const archive_node &	n,
		lst &	syms
	)		`[virtual]`

Read (a.k.a.

deserialize) object from archive.

Reimplemented from GiNaC::basic.

Definition at line 108 of file matrix.cpp.

References col, GiNaC::archive_node::find_ex_by_loc(), GiNaC::archive_node::find_first(), GiNaC::archive_node::find_last(), GiNaC::archive_node::find_unsigned(), last, m, and row.

bool GiNaC::matrix::match_same_type ( const basic & other ) const [protected, virtual]

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 311 of file matrix.cpp.

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

unsigned GiNaC::matrix::return_type ( ) const [inline, protected, virtual]

Reimplemented from GiNaC::basic.

Definition at line 134 of file matrix.h.

References GiNaC::return_types::noncommutative.

unsigned GiNaC::matrix::rows ( ) const [inline]

Get number of rows.

Definition at line 138 of file matrix.h.

References row.

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

unsigned GiNaC::matrix::cols ( ) const [inline]

Get number of columns.

Definition at line 140 of file matrix.h.

References col.

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

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

Sum of matrices.

Exceptions:

logic_error (incompatible matrices)

Definition at line 557 of file matrix.cpp.

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

Referenced by GiNaC::add::evalm().

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

Difference of matrices.

Exceptions:

logic_error (incompatible matrices)

Definition at line 575 of file matrix.cpp.

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

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

Product of matrices.

Exceptions:

logic_error (incompatible matrices)

Definition at line 593 of file matrix.cpp.

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

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

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

Product of matrix and scalar.

Definition at line 614 of file matrix.cpp.

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

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

Product of matrix and scalar expression.

Definition at line 627 of file matrix.cpp.

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

Referenced by GiNaC::mul::evalm().

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

Power of a matrix.

Currently handles integer exponents only.

Definition at line 643 of file matrix.cpp.

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

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 690 of file matrix.cpp.

References m.

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 704 of file matrix.cpp.

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

matrix& GiNaC::matrix::set	(	unsigned	ro,
		unsigned	co,
		const ex &	value
	)		`[inline]`

Definition at line 150 of file matrix.h.

References value.

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 716 of file matrix.cpp.

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

Referenced by contract_with(), and GiNaC::transpose().

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 741 of file matrix.cpp.

References GiNaC::_ex0, GiNaC::determinant_algo::automatic, GiNaC::determinant_algo::bareiss, c, col, GiNaC::info_flags::crational_polynomial, 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::basic::normal(), GiNaC::info_flags::numeric, GiNaC::permutation_sign(), r, GiNaC::info_flags::rational_function, row, and GiNaC::ex::to_rational().

Referenced by charpoly(), GiNaC::determinant(), and GiNaC::resultant().

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 873 of file matrix.cpp.

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

Referenced by charpoly(), eval_indexed(), and GiNaC::trace().

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

Characteristic Polynomial.

Following mathematica notation the characteristic polynomial of a matrix M is defined as the determiant 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 901 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().

Referenced by GiNaC::charpoly().

matrix GiNaC::matrix::inverse ( ) const

Inverse of this matrix.

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().

Referenced by GiNaC::inverse(), and pow().

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 GiNaC::solve_algo::automatic, GiNaC::solve_algo::bareiss, c, col, cols(), GiNaC::solve_algo::divfree, division_free_elimination(), fraction_free_elimination(), GiNaC::solve_algo::gauss, gauss_elimination(), GiNaC::basic::info(), m, n, GiNaC::basic::normal(), GiNaC::info_flags::numeric, r, rows(), and GiNaC::info_flags::symbol.

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

unsigned GiNaC::matrix::rank ( ) const

Compute the rank of this matrix.

Definition at line 1091 of file matrix.cpp.

References col, fraction_free_elimination(), GINAC_ASSERT, and m.

Referenced by GiNaC::rank().

bool GiNaC::matrix::is_zero_matrix ( ) const

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

Definition at line 1542 of file matrix.cpp.

References m.

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' dimesion is larger than 7.

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

See also:: matrix::determinant()

Definition at line 1125 of file matrix.cpp.

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

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 1245 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 solve().

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 1304 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 solve().

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 1358 of file matrix.cpp.

References GiNaC::_ex0, GiNaC::_ex1, GiNaC::ex::begin(), 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(), rank(), and solve().

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 occured, -1 if all are zero (usually signaling a degeneracy) and positive integer k means that rows ro and k were swapped.

Definition at line 1502 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, and GiNaC::swap().

Referenced by division_free_elimination(), and gauss_elimination().

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]`