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* Normaliz
* Copyright (C) 2007-2014 Winfried Bruns, Bogdan Ichim, Christof Soeger
* This program is free software: you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation, either version 3 of the License, or
* (at your option) any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program. If not, see <http://www.gnu.org/licenses/>.
*
* As an exception, when this program is distributed through (i) the App Store
* by Apple Inc.; (ii) the Mac App Store by Apple Inc.; or (iii) Google Play
* by Google Inc., then that store may impose any digital rights management,
* device limits and/or redistribution restrictions that are required by its
* terms of service.
*/
//---------------------------------------------------------------------------
#ifndef MATRIX_HPP
#define MATRIX_HPP
//---------------------------------------------------------------------------
#include <vector>
#include <list>
#include <iostream>
#include <string>
#include <libnormaliz/libnormaliz.h>
#include <libnormaliz/integer.h>
#include <libnormaliz/convert.h>
//---------------------------------------------------------------------------
namespace libnormaliz {
using std::list;
using std::vector;
using std::string;
template<typename Integer> class Matrix {
template<typename> friend class Matrix;
template<typename> friend class Lineare_Transformation;
template<typename> friend class Sublattice_Representation;
// public:
size_t nr;
size_t nc;
vector< vector<Integer> > elem;
//---------------------------------------------------------------------------
// Private routines, used in the public routines
//---------------------------------------------------------------------------
//---------------------------------------------------------------------------
// Rows and columns exchange
//---------------------------------------------------------------------------
void exchange_rows(const size_t& row1, const size_t& row2); //row1 is exchanged with row2
void exchange_columns(const size_t& col1, const size_t& col2); // col1 is exchanged with col2
//---------------------------------------------------------------------------
// Row and column reduction
//---------------------------------------------------------------------------
// return value false undicates failure because of overflow
// for all the routines below
// reduction via integer division and elemntary transformations
bool reduce_row(size_t corner); //reduction by the corner-th row
bool reduce_row (size_t row, size_t col); // corner at position (row,col)
// replaces two columns by linear combinations of them
bool linear_comb_columns(const size_t& col,const size_t& j,
const Integer& u,const Integer& w,const Integer& v,const Integer& z);
// column reduction with gcd method
bool gcd_reduce_column (size_t corner, Matrix<Integer>& Right);
//---------------------------------------------------------------------------
// Work horses
//---------------------------------------------------------------------------
// takes |product of the diagonal elem|
Integer compute_vol(bool& success);
// Solve system with coefficients and right hand side in one matrix, using elementary transformations
// solution replaces right hand side
bool solve_destructive_inner(bool ZZinvertible, Integer& denom);
// asembles the matrix of the system (left side the submatrix of mother given by key
// right side from column vectors pointed to by RS
// both in a single matrix
void solve_system_submatrix_outer(const Matrix<Integer>& mother, const vector<key_t>& key, const vector<vector<Integer>* >& RS,
Integer& denom, bool ZZ_invertible, bool transpose, size_t red_col, size_t sign_col,
bool compute_denom=true, bool make_sol_prime=false);
size_t row_echelon_inner_elem(bool& success); // does the work and checks for overflows
// size_t row_echelon_inner_bareiss(bool& success, Integer& det);
// NOTE: Bareiss cannot be used if z-invertible transformations are needed
size_t row_echelon(bool& success); // transforms this into row echelon form and returns rank
size_t row_echelon(bool& success, Integer& det); // computes also |det|
size_t row_echelon(bool& success, bool do_compute_vol, Integer& det); // chooses elem (or bareiss in former time)
// reduces the rows a matrix in row echelon form upwards, from left to right
bool reduce_rows_upwards();
size_t row_echelon_reduce(bool& success); // combines row_echelon and reduce_rows_upwards
// computes rank and index simultaneously, returns rank
Integer full_rank_index(bool& success);
vector<key_t> max_rank_submatrix_lex_inner(bool& success) const;
// A version of invert that circumvents protection and leaves it to the calling routine
Matrix invert_unprotected(Integer& denom, bool& sucess) const;
bool SmithNormalForm_inner(size_t& rk, Matrix<Integer>& Right);
//---------------------------------------------------------------------------
// Pivots for rows/columns operations
//---------------------------------------------------------------------------
vector<long> pivot(size_t corner); //Find the position of an element x with
//0<abs(x)<=abs(y) for all y!=0 in the right-lower submatrix of this
//described by an int corner
long pivot_column(size_t col); //Find the position of an element x with
//0<abs(x)<=abs(y) for all y!=0 in the lower half of the column of this
//described by an int col
long pivot_column(size_t row,size_t col); //in column col starting from row
//---------------------------------------------------------------------------
// Helpers for linear systems
//---------------------------------------------------------------------------
Matrix bundle_matrices(const Matrix<Integer>& Right_side)const;
Matrix extract_solution() const;
vector<vector<Integer>* > row_pointers();
void customize_solution(size_t dim, Integer& denom, size_t red_col,
size_t sign_col, bool make_sol_prime);
public:
size_t row_echelon_inner_bareiss(bool& success, Integer& det);
vector<vector<Integer>* > submatrix_pointers(const vector<key_t>& key);
//---------------------------------------------------------------------------
//---------------------------------------------------------------------------
// Construction and destruction
//---------------------------------------------------------------------------
Matrix();
Matrix(size_t dim); //constructor of unit matrix
Matrix(size_t row, size_t col); //main constructor, all entries 0
Matrix(size_t row, size_t col, Integer value); //constructor, all entries set to value
Matrix(const vector< vector<Integer> >& elem); //constuctor, elem=elem
Matrix(const list< vector<Integer> >& elems);
//---------------------------------------------------------------------------
// Data access
//---------------------------------------------------------------------------
void write(std::istream& in = std::cin); // to be modified, just for tests
void write(size_t row, const vector<Integer>& data); //write a row
void write(size_t row, const vector<int>& data); //write a row
void write_column(size_t col, const vector<Integer>& data); //write a column
void write(size_t row, size_t col, Integer data); // write data at (row,col)
void print(const string& name, const string& suffix) const; // writes matrix into name.suffix
void print_append(const string& name,const string& suffix) const; // the same, but appends matrix
void print(std::ostream& out) const; // writes matrix to the stream
void pretty_print(std::ostream& out, bool with_row_nr=false) const; // writes matrix in a nice format to the stream
void read() const; // to be modified, just for tests
vector<Integer> read(size_t row) const; // read a row
Integer read (size_t row, size_t col) const; // read data at (row,col)
size_t nr_of_rows() const; // returns nr
size_t nr_of_columns() const; // returns nc
/* generates a pseudo random matrix for tests, entries form 0 to mod-1 */
void random(int mod=3);
void set_zero(); // sets all entries to 0
/* returns a submatrix with rows corresponding to indices given by
* the entries of rows, Numbering from 0 to n-1 ! */
Matrix submatrix(const vector<key_t>& rows) const;
Matrix submatrix(const vector<int>& rows) const;
Matrix submatrix(const vector<bool>& rows) const;
void swap (Matrix<Integer>& x);
// returns the permutation created by sorting the rows with a grading function
// or by 1-norm if computed is false
vector<key_t> perm_sort_by_degree(const vector<key_t>& key, const vector<Integer>& grading, bool computed) const;
vector<key_t> perm_by_weights(const Matrix<Integer>& Weights, vector<bool> absolute);
void select_submatrix(const Matrix<Integer>& mother, const vector<key_t>& rows);
void select_submatrix_trans(const Matrix<Integer>& mother, const vector<key_t>& rows);
Matrix& remove_zero_rows(); // remove zero rows, modifies this
// resizes the matrix to the given number of rows/columns
// if the size shrinks it will keep all its allocated memory
// useful when the size varies
void resize(size_t nr_rows);
void resize(size_t nr_rows, size_t nr_cols);
void resize_columns(size_t nr_cols);
void Shrink_nr_rows(size_t new_nr_rows);
vector<Integer> diagonal() const; //returns the diagonale of this
//this should be a quadratic matrix
size_t maximal_decimal_length() const; //return the maximal number of decimals
//needed to write an entry
vector<size_t> maximal_decimal_length_columnwise() const; // the same per column
void append(const Matrix& M); // appends the rows of M to this
void append(const vector<vector<Integer> >& M); // the same, but for another type of matrix
void append(const vector<Integer>& v); // append the row v to this
void append_column(const vector<Integer>& v); // append the column v to this
void remove_row(const vector<Integer>& row); // removes all appearances of this row, not very efficient!
void remove_duplicate_and_zero_rows();
inline const Integer& get_elem(size_t row, size_t col) const {
return elem[row][col];
}
inline const vector< vector<Integer> >& get_elements() const {
return elem;
}
inline vector<Integer> const& operator[] (size_t row) const {
return elem[row];
}
inline vector<Integer>& operator[] (size_t row) {
return elem[row];
}
void set_nc(size_t col){
nc=col;
}
void set_nr(size_t rows){
nc=rows;
}
//---------------------------------------------------------------------------
// Basic matrices operations
//---------------------------------------------------------------------------
Matrix add(const Matrix& A) const; // returns this+A
Matrix multiplication(const Matrix& A) const; // returns this*A
Matrix multiplication(const Matrix& A, long m) const;// returns this*A (mod m)
Matrix<Integer> multiplication_cut(const Matrix<Integer>& A, const size_t& c) const; // returns
// this*(first c columns of A)
bool equal(const Matrix& A) const; // returns this==A
bool equal(const Matrix& A, long m) const; // returns this==A (mod m)
Matrix transpose() const; // returns the transpose of this
bool is_diagonal() const;
//---------------------------------------------------------------------------
// Scalar operations
//---------------------------------------------------------------------------
void scalar_multiplication(const Integer& scalar); //this=this*scalar
void scalar_division(const Integer& scalar);
//this=this div scalar, all the elem of this must be divisible with the scalar
void reduction_modulo(const Integer& modulo); //this=this mod scalar
Integer matrix_gcd() const; //returns the gcd of all elem
vector<Integer> make_prime(); //each row of this is reduced by its gcd,
// vector of gcds returned
void make_cols_prime(size_t from_col, size_t to_col);
// the columns of this in the specified range are reduced by their gcd
Matrix multiply_rows(const vector<Integer>& m) const; //returns matrix were row i is multiplied by m[i]
//---------------------------------------------------------------------------
// Vector operations
//---------------------------------------------------------------------------
void MxV(vector<Integer>& result, const vector<Integer>& v) const;//result = this*V
vector<Integer> MxV(const vector<Integer>& v) const;//returns this*V
vector<Integer> VxM(const vector<Integer>& v) const;//returns V*this
vector<Integer> VxM_div(const vector<Integer>& v, const Integer& divisor,bool& success) const; // additionally divides by divisor
//---------------------------------------------------------------------------
// Matrix operations
// --- these are more complicated algorithms ---
//---------------------------------------------------------------------------
// Normal forms
// converts this to row echelon form over ZZ and returns rank, GMP protected, uses only elementary transformations over ZZ
size_t row_echelon();
// public version of row_echelon_reduce, GMP protected, uses only elementary transformations over ZZ
size_t row_echelon_reduce();
// transforms matrix into lower triangular form via column transformations
// assumes that rk is the rank and that the matrix is zero after the first rk rows
// Right = Right*(column transformation of this call)
bool column_trigonalize(size_t rk, Matrix<Integer>& Right);
// combines row_echelon_reduce and column_trigonalize
// returns column transformation matrix
Matrix<Integer> row_column_trigonalize(size_t& rk, bool& success);
// rank and determinant
size_t rank() const; //returns rank
Integer full_rank_index() const; // returns index of full rank sublattice
size_t rank_submatrix(const vector<key_t>& key) const; //returns rank of submarix defined by key
// returns rank of submatrix of mother. "this" is used as work space
size_t rank_submatrix(const Matrix<Integer>& mother, const vector<key_t>& key);
// vol stands for |det|
Integer vol() const;
Integer vol_submatrix(const vector<key_t>& key) const;
Integer vol_submatrix(const Matrix<Integer>& mother, const vector<key_t>& key);
// find linearly indepenpendent submatrix of maximal rank
vector<key_t> max_rank_submatrix_lex() const; //returns a vector with entries
//the indices of the first rows in lexicographic order of this forming
//a submatrix of maximal rank.
// Solution of linear systems with square matrix
// In the following routines, denom is the absolute value of the determinant of the
// left side matrix.
// If the diagonal is asked for, ZZ-invertible transformations are used.
// Otherwise ther is no restriction on the used algorithm
//The diagonal of left hand side after transformation into an upper triangular matrix
//is saved in diagonal, denom is |determinant|.
// System with "this" as left side
Matrix solve(const Matrix& Right_side, Integer& denom) const;
Matrix solve(const Matrix& Right_side, vector< Integer >& diagonal, Integer& denom) const;
// solve the system this*Solution=denom*Right_side.
// system is defined by submatrix of mother given by key (left side) and column vectors pointed to by RS (right side)
// NOTE: this is used as the matrix for the work
void solve_system_submatrix(const Matrix& mother, const vector<key_t>& key, const vector<vector<Integer>* >& RS,
vector< Integer >& diagonal, Integer& denom, size_t red_col, size_t sign_col);
void solve_system_submatrix(const Matrix& mother, const vector<key_t>& key, const vector<vector<Integer>* >& RS,
Integer& denom, size_t red_col, size_t sign_col, bool compute_denom=true, bool make_sol_prime=false);
// the left side gets transposed
void solve_system_submatrix_trans(const Matrix& mother, const vector<key_t>& key, const vector<vector<Integer>* >& RS,
Integer& denom, size_t red_col, size_t sign_col);
// For non-square matrices
// The next two solve routines do not require the matrix to be square.
// However, we want rank = number of columns, ensuring unique solvability
vector<Integer> solve_rectangular(const vector<Integer>& v, Integer& denom) const;
// computes solution vector for right side v, solution over the rationals
// matrix needs not be quadratic, but must have rank = number of columns
// with denominator denom.
// gcd of denom and solution is extracted !!!!!
vector<Integer> solve_ZZ(const vector<Integer>& v) const;
// computes solution vector for right side v
// insists on integrality of the solution
// homogenous linear systems
Matrix<Integer> kernel () const;
// computes a ZZ-basis of the solutions of (*this)x=0
// the basis is formed by the ROWS of the returned matrix
// inverse matrix
//this*Solution=denom*I. "this" should be a quadratic matrix with nonzero determinant.
Matrix invert(Integer& denom) const;
void invert_submatrix(const vector<key_t>& key, Integer& denom, Matrix<Integer>& Inv,
bool compute_denom=true, bool make_sol_prime=false) const;
// find linear form that is constant on the rows
vector<Integer> find_linear_form () const;
// Tries to find a linear form which gives the same value an all rows of this
// this should be a m x n matrix (m>=n) of maxinal rank
// returns an empty vector if there does not exist such a linear form
vector<Integer> find_linear_form_low_dim () const;
//same as find_linear_form but also works with not maximal rank
//uses a linear transformation to get a full rank matrix
// normal forms
Matrix AlmostHermite(size_t& rk);
// Converts "this" into lower trigonal column Hermite normal form, returns column
// transformation matrix
// Almost: elements left of diagonal are not reduced mod diagonal
// Computes Smith normal form and returns column transformation matrix
Matrix SmithNormalForm(size_t& rk);
//for simplicial subcones
// computes support hyperplanes and volume
void simplex_data(const vector<key_t>& key, Matrix<Integer>& Supp, Integer& vol, bool compute_vol) const;
// Sorting of rows
Matrix& sort_by_weights(const Matrix<Integer>& Weights, vector<bool> absolute);
Matrix& sort_lex();
void order_rows_by_perm(const vector<key_t>& perm);
// solve homogeneous congruences
Matrix<Integer> solve_congruences(bool& zero_modulus) const;
// saturate sublattice
void saturate();
// find the indices of the rows in which the linear form L takes its max and min values
vector<key_t> max_and_min(const vector<Integer>& L, const vector<Integer>& norm) const;
// try to sort the rows in such a way that the extreme points of the polytope spanned by the rows come first
size_t extreme_points_first(const vector<Integer> norm=vector<Integer>(0));
// find an inner point in the cone spanned by the rows of the matrix
vector<Integer> find_inner_point();
};
//class end *****************************************************************
template<typename Integer> class order_helper {
public:
vector<Integer> weight;
key_t index;
vector<Integer>* v;
};
template<typename Integer>
vector<vector<Integer> > to_matrix(const vector<Integer>& v){
vector<vector<Integer> > mat(1);
mat[0]=v;
return mat;
}
//---------------------------------------------------------------------------
// Conversion between integer types
//---------------------------------------------------------------------------
template<typename ToType, typename FromType>
void convert(Matrix<ToType>& to_mat, const Matrix<FromType>& from_mat);
template<typename Integer>
void mat_to_mpz(const Matrix<Integer>& mat, Matrix<mpz_class>& mpz_mat);
template<typename Integer>
void mat_to_Int(const Matrix<mpz_class>& mpz_mat, Matrix<Integer>& mat);
template<typename Integer>
void mpz_submatrix(Matrix<mpz_class>& sub, const Matrix<Integer>& mother, const vector<key_t>& selection);
template<typename Integer>
void mpz_submatrix_trans(Matrix<mpz_class>& sub, const Matrix<Integer>& mother, const vector<key_t>& selection);
} // namespace
//---------------------------------------------------------------------------
#endif
//---------------------------------------------------------------------------
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