chomp::homology::chaincomplex< euclidom > Class Template Reference

This is an implementation of a chain complex over an arbitrary ring. More...

#include <chains.h>

Public Member Functions
	chaincomplex (int d, int trace_generators=0, int trace_bases=0)
	The default constructor. More...
	~chaincomplex ()
	The destructor. More...
void	def_gen (int q, int_t n)
	Defines the number of generators in the given dimension. More...
void	add (int q, int_t m, int_t n, const euclidom &e)
	Adds a coefficient to the structure: D_q [m, n] += e. More...
euclidom	get (int q, int_t row, int_t col) const
	Returns an element from the boundary matrix for the given dimension. More...
const mmatrix< euclidom > &	getboundary (int i) const
	Returns a reference to the given boundary matrix. More...
int_t	getnumgen (int i) const
	Returns the number of generators at the given level. More...
int	dim () const
	Returns the dimension of the chain complex. More...
const chain< euclidom > &	gethomgen (int q, int_t i) const
	Returns the given homology generator at level q. More...
const mmatrix< euclidom > &	gethomgen (int q) const
	Returns the homology generators matrix at level q. More...
const mmatrix< euclidom > &	getchgbasis (int q) const
	Returns the base change matrix at level q. More...
void	simple_form (int which, bool quiet)
	Reduces of the given boundary matrix in the chain complex for the purpose of homology computation. More...
void	simple_form (const int *level, bool quiet)
	Runs the reduction of all the boundary matrices in the chain complex. More...
int	simple_homology (chain< euclidom > &result, int which) const
	Computes the given level of homology of the chain complex, provided it has been transformed into the simple form previously. More...
int	simple_homology (chain< euclidom > *&result, const int *level=NULL) const
	Computes the homology of the chain complex, provided it has been transformed into the simple form previously. More...
void	take_homology (const chain< euclidom > *hom_chain)
	Creates a chain complex containing exactly one generator for each homology generator. More...
outputstream &	show_homology (outputstream &out, const chain< euclidom > *hom, const int *level=NULL) const
	Writes the homology module of the chain complex to an output stream. More...
std::ostream &	show_homology (std::ostream &out, const chain< euclidom > *hom, const int *level=NULL) const
	Writes the homology module of the chain complex to an output stream. More...
outputstream &	show_generators (outputstream &out, const chain< euclidom > &list, int q) const
	Writes the homology generators of the homology module to an output stream. More...
std::ostream &	show_generators (std::ostream &out, const chain< euclidom > &list, int q) const
	Writes the homology generators of the homology module to an output stream. More...
outputstream &	compute_and_show_homology (outputstream &out, chain< euclidom > *&hom, const int *level=NULL)
	Computes the homology and shows the result. More...
std::ostream &	compute_and_show_homology (std::ostream &out, chain< euclidom > *&hom, const int *level=NULL)
	Computes the homology and shows the result. More...

Private Member Functions
	chaincomplex (const chaincomplex< euclidom > &m)
	The copy constructor is not implemented. More...
chaincomplex< euclidom > &	operator= (const chaincomplex< euclidom > &s)
	The assignment operator is not implemented. More...

Private Attributes
int	len
	The length of the chain complex (i.e., its dimension + 1). More...
mmatrix< euclidom > *	boundary
	The matrices of the boundary homomorphism. More...
mmatrix< euclidom > *	generators
	Matrices which store actual combinations of generators. More...
mmatrix< euclidom > *	chgbases
	Matrices which store the change of basis to obtain the SNF. More...
int *	generators_initialized
	Have the generator tracing matrices been initialized to the identity (of suitable size each)? More...
int *	chgbases_initialized
	Have the base change tracing matrices been initialized to the identity (of suitable size each)? More...
int_t *	numgen
	The numbers of generators in each dimension, or -1's if not defined yet. More...

Friends
class	chainmap< euclidom >
	The class "chainmap" is a friend class which has access to the internal data of the chain complex class. More...

Detailed Description

template<class euclidom>
class chomp::homology::chaincomplex< euclidom >

This is an implementation of a chain complex over an arbitrary ring.

The dimension of the chain complex must be known apriori. If there are elements not used in boundaries, use "def_gen" to set the true number of generators at each level.

Definition at line 2661 of file chains.h.

Constructor & Destructor Documentation

chaincomplex() [1/2]

template<class euclidom >

chomp::homology::chaincomplex< euclidom >::chaincomplex ( int  d, int  trace_generators = 0, int  trace_bases = 0  )

inline

The default constructor.

The dimension must be defined apriori and cannot be modified later. If requested, additional matrices are created to trace homology generators, and/or to determine the change of bases while computing the SNF.

Definition at line 2820 of file chains.h.

{
        // set the number of tables to be sufficient for 0 to d inclusive
        len = (d >= 0) ? (d + 1) : 0;
 
        // create a table of empty matrices
        boundary = len ? new mmatrix<euclidom> [len] : NULL;
 
        // create a table of matrices for tracing generators of homology
        generators = (trace_generators && len) ?
                new mmatrix<euclidom> [len] : NULL;
        if (generators)
                generators_initialized = new int [len];
        else
                generators_initialized = NULL;
 
        // create a table of matrices for tacing base changes
        chgbases = (trace_bases && len) ?
                new mmatrix<euclidom> [len] : NULL;
        if (chgbases)
                chgbases_initialized = new int [len];
        else
                chgbases_initialized = NULL;
 
        // create a table of generator numbers
        numgen = len ? new int_t [len] : NULL;
 
        // link the boundary matrices to each other,
        // as well as to the generators and to the base change matrices
        for (int i = 0; i < len; ++ i)
        {
                numgen [i] = -1;
                if (i < len - 1)
                        boundary [i]. dom_img. add (boundary [i + 1]);
                if (i > 0)
                        boundary [i]. img_dom. add (boundary [i - 1]);
                if (generators)
                {
                        boundary [i]. dom_dom. add (generators [i]);
                        if (i > 0)
                        {
                                boundary [i]. img_dom. add
                                        (generators [i - 1]);
                        }
                        generators_initialized [i] = 0;
                }
                if (chgbases)
                {
                        boundary [i]. dom_img. add (chgbases [i]); // ???
                        if (i > 0)
                        {
                                boundary [i]. img_img. add // ???
                                        (chgbases [i - 1]);
                        }
                        chgbases_initialized [i] = 0;
                }
        }
 
        return;
} /* chaincomplex<euclidom>::chaincomplex */

~chaincomplex()

template<class euclidom >

chomp::homology::chaincomplex< euclidom >::~chaincomplex

inline

The destructor.

Definition at line 2883 of file chains.h.

{
        if (boundary)
                delete [] boundary;
        if (generators)
                delete [] generators;
        if (chgbases)
                delete [] chgbases;
        if (numgen)
                delete [] numgen;
        if (generators_initialized)
                delete [] generators_initialized;
        if (chgbases_initialized)
                delete [] chgbases_initialized;
        return;
} /* chaincomplex<euclidom>::~chaincomplex */

chaincomplex() [2/2]

template<class euclidom >

chomp::homology::chaincomplex< euclidom >::chaincomplex ( const chaincomplex< euclidom > &  m )

inlineprivate

The copy constructor is not implemented.

Definition at line 2779 of file chains.h.

                {throw "Copy constructor not implemented "
                "for a chain complex.";}

Member Function Documentation

add()

template<class euclidom >

void chomp::homology::chaincomplex< euclidom >::add ( int  q, int_t  m, int_t  n, const euclidom &  e  )

inline

Adds a coefficient to the structure: D_q [m, n] += e.

In other words, boundary of element n += e * element m.

Definition at line 2920 of file chains.h.

{
        if ((q <= 0) || (q >= len))
                throw "Trying to add a boundary for dimension out of range";
 
        if (numgen [q] <= n)
                numgen [q] = n + 1;
        if (numgen [q - 1] <= m)
                numgen [q - 1] = m + 1;
 
        boundary [q]. add (m, n, e);
        return;
} /* chaincomplex<euclidom>::add */

compute_and_show_homology() [1/2]

template<class euclidom >

outputstream & chomp::homology::chaincomplex< euclidom >::compute_and_show_homology ( outputstream &  out, chain< euclidom > *&  hom, const int *  level = NULL  )

inline

Computes the homology and shows the result.

Definition at line 3189 of file chains.h.

{
        simple_form (level, false);
        simple_homology (hom, level);
        show_homology (out, hom, level);
        return out;
} /* chaincomplex<euclidom>::compute_and_show_homology */

References chomp::homology::show_homology().

compute_and_show_homology() [2/2]

template<class euclidom >

std::ostream & chomp::homology::chaincomplex< euclidom >::compute_and_show_homology ( std::ostream &  out, chain< euclidom > *&  hom, const int *  level = NULL  )

inline

Computes the homology and shows the result.

Definition at line 3199 of file chains.h.

{
        outputstream tout (out);
        compute_and_show_homology (tout, hom, level);
        return out;
} /* chaincomplex<euclidom>::compute_and_show_homology */

def_gen()

template<class euclidom >

void chomp::homology::chaincomplex< euclidom >::def_gen ( int  q, int_t  n  )

inline

Defines the number of generators in the given dimension.

This number is automatically increased while boundary formulas are added. However, it must be used if some generators do not appear in the boundaries or have zero boundaries.

Definition at line 2901 of file chains.h.

{
        if ((q < 0) || (q >= len))
                return;
 
        if (numgen [q] < n)
                numgen [q] = n;
 
        if (q == 0)
                boundary [0]. define (0, numgen [q]);
        if ((q > 0) && (numgen [q - 1] >= 0))
                boundary [q]. define (numgen [q - 1], numgen [q]);
        if ((q < dim ()) && (numgen [q + 1] >= 0))
                boundary [q + 1]. define (numgen [q], numgen [q + 1]);
 
        return;
} /* chaincomplex<euclidom>::def_gen */

dim()

template<class euclidom >

int chomp::homology::chaincomplex< euclidom >::dim

inline

Returns the dimension of the chain complex.

Definition at line 2968 of file chains.h.

{
        return len - 1;
} /* chaincomplex<euclidom>::dim */

get()

template<class euclidom >

euclidom chomp::homology::chaincomplex< euclidom >::get ( int  q, int_t  row, int_t  col  ) const

inline

Returns an element from the boundary matrix for the given dimension.

Definition at line 2936 of file chains.h.

{
        if ((q < 0) || (q >= len))
                throw "Boundary coefficient out of range from chain cplx.";
 
        return boundary [q]. get (row, col);
} /* chaincomplex<euclidom>::get */

getboundary()

template<class euclidom >

const mmatrix< euclidom > & chomp::homology::chaincomplex< euclidom >::getboundary ( int  i ) const

inline

Returns a reference to the given boundary matrix.

Definition at line 2945 of file chains.h.

{
        if ((i < 0) || (i >= len))
                throw "Boundary matrix out of range from chain complex.";
 
        return boundary [i];
} /* chaincomplex<euclidom>::getboundary */

getchgbasis()

template<class euclidom >

const mmatrix< euclidom > & chomp::homology::chaincomplex< euclidom >::getchgbasis ( int  q ) const

inline

Returns the base change matrix at level q.

Definition at line 2996 of file chains.h.

{
        if ((q < 0) || (q >= len))
                throw "Level for basis change matrix out of range.";
        if (!chgbases || !chgbases_initialized [q])
                throw "Trying to get non-existent basis change matrix.";
        return chgbases [q];
} /* chaincomplex<euclidom>::getchgbasis */

gethomgen() [1/2]

template<class euclidom >

const mmatrix< euclidom > & chomp::homology::chaincomplex< euclidom >::gethomgen ( int  q ) const

inline

Returns the homology generators matrix at level q.

Definition at line 2985 of file chains.h.

{
        if ((q < 0) || (q >= len))
                throw "Level for homology generators out of range.";
        if (!generators || !generators_initialized [q])
                throw "Trying to get non-existent homology generators.";
        return generators [q];
} /* chaincomplex<euclidom>::gethomgen */

gethomgen() [2/2]

template<class euclidom >

const chain< euclidom > & chomp::homology::chaincomplex< euclidom >::gethomgen ( int  q, int_t  i  ) const

inline

Returns the given homology generator at level q.

Note: 'i' must be the number of chain generator.

Definition at line 2974 of file chains.h.

{
        if ((q < 0) || (q >= len))
                throw "Level for homology generators out of range.";
        if (!generators || !generators_initialized [q])
                throw "Trying to get non-existent homology generators.";
        return generators [q]. getcol (i);
} /* chaincomplex<euclidom>::gethomgen */

getnumgen()

template<class euclidom >

int_t chomp::homology::chaincomplex< euclidom >::getnumgen ( int  i ) const

inline

Returns the number of generators at the given level.

Definition at line 2955 of file chains.h.

{
        if ((i < 0) || (i >= len))
        //      throw "Level for the number of generators out of range.";
                return 0;
 
        if (numgen [i] < 0)
                return 0;
        else
                return numgen [i];
} /* chaincomplex<euclidom>::getnumgen */

operator=()

template<class euclidom >

chaincomplex< euclidom > & chomp::homology::chaincomplex< euclidom >::operator= ( const chaincomplex< euclidom > &  s )

inlineprivate

The assignment operator is not implemented.

Definition at line 2784 of file chains.h.

                {throw "Operator = not implemented "
                "for a chain complex."; return *this;}

show_generators() [1/2]

template<class euclidom >

outputstream & chomp::homology::chaincomplex< euclidom >::show_generators ( outputstream &  out, const chain< euclidom > &  list, int  q  ) const

inline

Writes the homology generators of the homology module to an output stream.

Each generator as a combination of the original ones is shown on a separate line.

Definition at line 3169 of file chains.h.

{
        if (!generators || (q < 0) || (q >= len))
                return out;
        for (int_t i = 0; i < list. size (); ++ i)
                out << gethomgen (q, list. num (i)) << '\n';
        return out;
} /* chaincomplex<euclidom>::show_generators */

show_generators() [2/2]

template<class euclidom >

std::ostream & chomp::homology::chaincomplex< euclidom >::show_generators ( std::ostream &  out, const chain< euclidom > &  list, int  q  ) const

inline

Writes the homology generators of the homology module to an output stream.

Each generator as a combination of the original ones is shown on a separate line.

Definition at line 3180 of file chains.h.

{
        outputstream tout (out);
        show_generators (tout, list, q);
        return out;
} /* chaincomplex<euclidom>::show_generators */

show_homology() [1/2]

template<class euclidom >

outputstream & chomp::homology::chaincomplex< euclidom >::show_homology ( outputstream &  out, const chain< euclidom > *  hom, const int *  level = NULL  ) const

inline

Writes the homology module of the chain complex to an output stream.

If a table of levels is given, shows only these levels for which the table's entry is nonzero.

Definition at line 3139 of file chains.h.

{
        int max_level = len - 1;
        while ((max_level >= 0) && !hom [max_level]. size ())
                -- max_level;
        ++ max_level;
        for (int q = 0; q < max_level; ++ q)
        {
                if (!level || level [q])
                {
                        out << "H_" << q << " = ";
                        chomp::homology::show_homology (out, hom [q]);
                        out << '\n';
                }
        }
        return out;
} /* chaincomplex<euclidom>::show_homology */

References chomp::homology::show_homology().

show_homology() [2/2]

template<class euclidom >

std::ostream & chomp::homology::chaincomplex< euclidom >::show_homology ( std::ostream &  out, const chain< euclidom > *  hom, const int *  level = NULL  ) const

inline

Writes the homology module of the chain complex to an output stream.

If a table of levels is given, shows only these levels for which the table's entry is nonzero.

Definition at line 3160 of file chains.h.

{
        outputstream tout (out);
        show_homology (tout, hom, level);
        return out;
} /* chaincomplex<euclidom>::show_homology */

References chomp::homology::show_homology().

simple_form() [1/2]

template<class euclidom >

void chomp::homology::chaincomplex< euclidom >::simple_form ( const int *  level, bool  quiet  )

inline

Runs the reduction of all the boundary matrices in the chain complex.

If the array of levels is given, computes only simple forms necessary for homology levels for which the entries in the array are nonzero.

Definition at line 3067 of file chains.h.

{
        for (int i = len - 1; i >= 0; -- i)
        {
                if (!level || level [i] || (i && level [i - 1]))
                        simple_form (i, quiet);
        }
        return;
} /* chaincomplex<euclidom>::simple_form */

simple_form() [2/2]

template<class euclidom >

void chomp::homology::chaincomplex< euclidom >::simple_form ( int  which, bool  quiet  )

inline

Reduces of the given boundary matrix in the chain complex for the purpose of homology computation.

Definition at line 3007 of file chains.h.

{
//      if ((which < 0) || (which >= len))
//              throw "Trying to find the simple form of a wrong matrix.";
 
        // if the generator tracing matrices are not initialized, do it now
        if (generators)
        {
                if (!generators_initialized [which])
                        generators [which]. identity (numgen [which]);
                generators_initialized [which] = 1;
                if ((which > 0) && (!generators_initialized [which - 1]))
                {
                        generators [which - 1]. identity
                                (numgen [which - 1]);
                        generators_initialized [which - 1] = 1;
                }
        }
 
        // if the base change tracing matrices are not initialized, do it now
        if (chgbases)
        {
                if (!chgbases_initialized [which])
                        chgbases [which]. identity (numgen [which]);
                chgbases_initialized [which] = 1;
                if ((which > 0) && (!chgbases_initialized [which - 1]))
                {
                        chgbases [which - 1]. identity
                                (numgen [which - 1]);
                        chgbases_initialized [which - 1] = 1;
                }
        }
 
        // verify that the adjacent matrices have sufficient size
        if (which > 0)
        {
                int_t n = boundary [which]. getnrows ();
                mmatrix<euclidom> &other = boundary [which - 1];
                if (other. getncols () < n)
                        other. define (other. getnrows (), n);
        }
        if (which < len - 1)
        {
                int_t n = boundary [which]. getncols ();
                mmatrix<euclidom> &other = boundary [which + 1];
                if (other. getnrows () < n)
                        other. define (n, other. getncols ());
        }
 
        // compute simple form of the desired boundary matrix
        if (!quiet && which)
                sout << "Reducing D_" << which << ": ";
        boundary [which]. simple_form (quiet);
        if (!quiet && which)
                sout << '\n';
 
        return;
} /* chaincomplex<euclidom>::simple_form */

References chomp::homology::sout.

simple_homology() [1/2]

template<class euclidom >

int chomp::homology::chaincomplex< euclidom >::simple_homology ( chain< euclidom > &  result, int  which  ) const

inline

Computes the given level of homology of the chain complex, provided it has been transformed into the simple form previously.

Encodes this homology group as a chain which is a combination generator numbers together with their torsion coefficients (or with 1's if none).

Definition at line 3079 of file chains.h.

{
        int_t g = boundary [which]. findcol (-1, 0);
        while (g >= 0)
        {
                euclidom e;
                if (which == dim ())
                        e = 0;
                else
                        e = boundary [which + 1]. get (g, -1);
                if (e == 0)
                {
                        e = 1;
                        result. add (g, e);
                }
                else if (e. delta () > 1)
                        result. add (g, e. normalized ());
                g = boundary [which]. findcol (-1, g + 1);
        }
 
        return which;
} /* chaincomplex<euclidom>::simple_homology */

simple_homology() [2/2]

template<class euclidom >

int chomp::homology::chaincomplex< euclidom >::simple_homology ( chain< euclidom > *&  result, const int *  level = NULL  ) const

inline

Computes the homology of the chain complex, provided it has been transformed into the simple form previously.

Encodes the homology as a table of chains (one chain for each dimension) which are combinations of generator numbers together with their torsion coefficients (or with 1's if none). Returns the dimension of the chain complex. If a table of levels is given, computes only these levels of homology for which the table's entry is nonzero.

Definition at line 3104 of file chains.h.

{
        // if the chain complex is empty, then just set the result to NULL
        if (!len)
        {
                result = NULL;
                return dim ();
        }
 
        result = new chain<euclidom> [len];
        if (!result)
                throw "Not enough memory to create homology chains.";
 
        for (int q = 0; q < len; ++ q)
        {
                if (!level || level [q])
                        simple_homology (result [q], q);
        }
 
        return dim ();
} /* chaincomplex<euclidom>::simple_homology */

take_homology()

template<class euclidom >

void chomp::homology::chaincomplex< euclidom >::take_homology ( const chain< euclidom > *  hom_chain )

inline

Creates a chain complex containing exactly one generator for each homology generator.

This function is used for extracting the map induced in homology by a chain map.

Definition at line 3128 of file chains.h.

{
        if (!hom_chain)
                return;
        for (int q = 0; q < len; ++ q)
                def_gen (q, hom_chain [q]. size ());
        return;
} /* chaincomplex<euclidom>::take_homology */

Friends And Related Function Documentation

chainmap< euclidom >

template<class euclidom >

friend class chainmap< euclidom >

friend

The class "chainmap" is a friend class which has access to the internal data of the chain complex class.

Definition at line 2770 of file chains.h.

Member Data Documentation

boundary

template<class euclidom >

mmatrix<euclidom>* chomp::homology::chaincomplex< euclidom >::boundary

private

The matrices of the boundary homomorphism.

"D_q" is stored in "boundary [q]".

Definition at line 2794 of file chains.h.

chgbases

template<class euclidom >

mmatrix<euclidom>* chomp::homology::chaincomplex< euclidom >::chgbases

private

Matrices which store the change of basis to obtain the SNF.

Definition at line 2801 of file chains.h.

chgbases_initialized

template<class euclidom >

int* chomp::homology::chaincomplex< euclidom >::chgbases_initialized

private

Have the base change tracing matrices been initialized to the identity (of suitable size each)?

Definition at line 2809 of file chains.h.

generators

template<class euclidom >

mmatrix<euclidom>* chomp::homology::chaincomplex< euclidom >::generators

private

Matrices which store actual combinations of generators.

Used for the extraction of generators of homology.

Definition at line 2798 of file chains.h.

generators_initialized

template<class euclidom >

int* chomp::homology::chaincomplex< euclidom >::generators_initialized

private

Have the generator tracing matrices been initialized to the identity (of suitable size each)?

Definition at line 2805 of file chains.h.

len

template<class euclidom >

int chomp::homology::chaincomplex< euclidom >::len

private

The length of the chain complex (i.e., its dimension + 1).

Definition at line 2790 of file chains.h.

numgen

template<class euclidom >

int_t* chomp::homology::chaincomplex< euclidom >::numgen

private

The numbers of generators in each dimension, or -1's if not defined yet.

Definition at line 2813 of file chains.h.

---

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

• chomp/homology/chains.h