X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fmatrix.cpp;h=871c3f15d1c5f433238b6f5a3b354d1f17ef1ea2;hp=9dc46de92776ff3a8988d40e76b92e10c6effe03;hb=20159f92a9ffad034172bd36d532e5ffd24c9ff2;hpb=dfa384d36f986f5f8d7dbb5e20c0d3a63af42cd9

diff --git a/ginac/matrix.cpp b/ginac/matrix.cpp
index 9dc46de9..871c3f15 100644
--- a/ginac/matrix.cpp
+++ b/ginac/matrix.cpp
@@ -3,7 +3,7 @@
  *  Implementation of symbolic matrices */
 
 /*
- *  GiNaC Copyright (C) 1999-2003 Johannes Gutenberg University Mainz, Germany
+ *  GiNaC Copyright (C) 1999-2015 Johannes Gutenberg University Mainz, Germany
  *
  *  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
@@ -17,16 +17,9 @@
  *
  *  You should have received a copy of the GNU General Public License
  *  along with this program; if not, write to the Free Software
- *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
+ *  Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
  */
 
-#include <string>
-#include <iostream>
-#include <sstream>
-#include <algorithm>
-#include <map>
-#include <stdexcept>
-
 #include "matrix.h"
 #include "numeric.h"
 #include "lst.h"
@@ -40,12 +33,19 @@
 #include "archive.h"
 #include "utils.h"
 
+#include <algorithm>
+#include <iostream>
+#include <map>
+#include <sstream>
+#include <stdexcept>
+#include <string>
+
 namespace GiNaC {
 
 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(matrix, basic,
   print_func<print_context>(&matrix::do_print).
   print_func<print_latex>(&matrix::do_print_latex).
-  print_func<print_tree>(&basic::do_print_tree).
+  print_func<print_tree>(&matrix::do_print_tree).
   print_func<print_python_repr>(&matrix::do_print_python_repr))
 
 //////////
@@ -53,7 +53,7 @@ GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(matrix, basic,
 //////////
 
 /** Default ctor.  Initializes to 1 x 1-dimensional zero-matrix. */
-matrix::matrix() : inherited(TINFO_matrix), row(1), col(1), m(1, _ex0)
+matrix::matrix() : row(1), col(1), m(1, _ex0)
 {
 	setflag(status_flags::not_shareable);
 }
@@ -68,8 +68,7 @@ matrix::matrix() : inherited(TINFO_matrix), row(1), col(1), m(1, _ex0)
  *
  *  @param r number of rows
  *  @param c number of cols */
-matrix::matrix(unsigned r, unsigned c)
-  : inherited(TINFO_matrix), row(r), col(c), m(r*c, _ex0)
+matrix::matrix(unsigned r, unsigned c) : row(r), col(c), m(r*c, _ex0)
 {
 	setflag(status_flags::not_shareable);
 }
@@ -78,7 +77,12 @@ matrix::matrix(unsigned r, unsigned c)
 
 /** Ctor from representation, for internal use only. */
 matrix::matrix(unsigned r, unsigned c, const exvector & m2)
-  : inherited(TINFO_matrix), row(r), col(c), m(m2)
+  : row(r), col(c), m(m2)
+{
+	setflag(status_flags::not_shareable);
+}
+matrix::matrix(unsigned r, unsigned c, exvector && m2)
+  : row(r), col(c), m(std::move(m2))
 {
 	setflag(status_flags::not_shareable);
 }
@@ -88,17 +92,18 @@ matrix::matrix(unsigned r, unsigned c, const exvector & m2)
  *  If the list has more elements than the matrix, the excessive elements are
  *  thrown away. */
 matrix::matrix(unsigned r, unsigned c, const lst & l)
-  : inherited(TINFO_matrix), row(r), col(c), m(r*c, _ex0)
+  : row(r), col(c), m(r*c, _ex0)
 {
 	setflag(status_flags::not_shareable);
 
 	size_t i = 0;
-	for (lst::const_iterator it = l.begin(); it != l.end(); ++it, ++i) {
+	for (auto & it : l) {
 		size_t x = i % c;
 		size_t y = i / c;
 		if (y >= r)
 			break; // matrix smaller than list: throw away excessive elements
-		m[y*c+x] = *it;
+		m[y*c+x] = it;
+		++i;
 	}
 }
 
@@ -106,36 +111,36 @@ matrix::matrix(unsigned r, unsigned c, const lst & l)
 // archiving
 //////////
 
-matrix::matrix(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
+void matrix::read_archive(const archive_node &n, lst &sym_lst)
 {
-	setflag(status_flags::not_shareable);
+	inherited::read_archive(n, sym_lst);
 
 	if (!(n.find_unsigned("row", row)) || !(n.find_unsigned("col", col)))
 		throw (std::runtime_error("unknown matrix dimensions in archive"));
 	m.reserve(row * col);
-	for (unsigned int i=0; true; i++) {
+	// XXX: default ctor inserts a zero element, we need to erase it here.
+	m.pop_back();
+	auto first = n.find_first("m");
+	auto last = n.find_last("m");
+	++last;
+	for (auto i=first; i != last; ++i) {
 		ex e;
-		if (n.find_ex("m", e, sym_lst, i))
-			m.push_back(e);
-		else
-			break;
+		n.find_ex_by_loc(i, e, sym_lst);
+		m.push_back(e);
 	}
 }
+GINAC_BIND_UNARCHIVER(matrix);
 
 void matrix::archive(archive_node &n) const
 {
 	inherited::archive(n);
 	n.add_unsigned("row", row);
 	n.add_unsigned("col", col);
-	exvector::const_iterator i = m.begin(), iend = m.end();
-	while (i != iend) {
-		n.add_ex("m", *i);
-		++i;
+	for (auto & i : m) {
+		n.add_ex("m", i);
 	}
 }
 
-DEFAULT_UNARCHIVE(matrix)
-
 //////////
 // functions overriding virtual functions from base classes
 //////////
@@ -220,8 +225,8 @@ ex matrix::eval(int level) const
 		for (unsigned c=0; c<col; ++c)
 			m2[r*col+c] = m[r*col+c].eval(level);
 	
-	return (new matrix(row, col, m2))->setflag(status_flags::dynallocated |
-											   status_flags::evaluated);
+	return (new matrix(row, col, std::move(m2)))->setflag(status_flags::dynallocated |
+	                                                      status_flags::evaluated);
 }
 
 ex matrix::subs(const exmap & mp, unsigned options) const
@@ -231,7 +236,51 @@ ex matrix::subs(const exmap & mp, unsigned options) const
 		for (unsigned c=0; c<col; ++c)
 			m2[r*col+c] = m[r*col+c].subs(mp, options);
 
-	return matrix(row, col, m2).subs_one_level(mp, options);
+	return matrix(row, col, std::move(m2)).subs_one_level(mp, options);
+}
+
+/** Complex conjugate every matrix entry. */
+ex matrix::conjugate() const
+{
+	std::unique_ptr<exvector> ev(nullptr);
+	for (auto i=m.begin(); i!=m.end(); ++i) {
+		ex x = i->conjugate();
+		if (ev) {
+			ev->push_back(x);
+			continue;
+		}
+		if (are_ex_trivially_equal(x, *i)) {
+			continue;
+		}
+		ev.reset(new exvector);
+		ev->reserve(m.size());
+		for (auto j=m.begin(); j!=i; ++j) {
+			ev->push_back(*j);
+		}
+		ev->push_back(x);
+	}
+	if (ev) {
+		return matrix(row, col, std::move(*ev));
+	}
+	return *this;
+}
+
+ex matrix::real_part() const
+{
+	exvector v;
+	v.reserve(m.size());
+	for (auto & i : m)
+		v.push_back(i.real_part());
+	return matrix(row, col, std::move(v));
+}
+
+ex matrix::imag_part() const
+{
+	exvector v;
+	v.reserve(m.size());
+	for (auto & i : m)
+		v.push_back(i.imag_part());
+	return matrix(row, col, std::move(v));
 }
 
 // protected
@@ -513,12 +562,11 @@ matrix matrix::add(const matrix & other) const
 		throw std::logic_error("matrix::add(): incompatible matrices");
 	
 	exvector sum(this->m);
-	exvector::iterator i = sum.begin(), end = sum.end();
-	exvector::const_iterator ci = other.m.begin();
-	while (i != end)
-		*i++ += *ci++;
+	auto ci = other.m.begin();
+	for (auto & i : sum)
+		i += *ci++;
 	
-	return matrix(row,col,sum);
+	return matrix(row, col, std::move(sum));
 }
 
 
@@ -531,12 +579,11 @@ matrix matrix::sub(const matrix & other) const
 		throw std::logic_error("matrix::sub(): incompatible matrices");
 	
 	exvector dif(this->m);
-	exvector::iterator i = dif.begin(), end = dif.end();
-	exvector::const_iterator ci = other.m.begin();
-	while (i != end)
-		*i++ -= *ci++;
+	auto ci = other.m.begin();
+	for (auto & i : dif)
+		i -= *ci++;
 	
-	return matrix(row,col,dif);
+	return matrix(row, col, std::move(dif));
 }
 
 
@@ -552,13 +599,14 @@ matrix matrix::mul(const matrix & other) const
 	
 	for (unsigned r1=0; r1<this->rows(); ++r1) {
 		for (unsigned c=0; c<this->cols(); ++c) {
+			// Quick test: can we shortcut?
 			if (m[r1*col+c].is_zero())
 				continue;
 			for (unsigned r2=0; r2<other.cols(); ++r2)
-				prod[r1*other.col+r2] += (m[r1*col+c] * other.m[c*other.col+r2]).expand();
+				prod[r1*other.col+r2] += (m[r1*col+c] * other.m[c*other.col+r2]);
 		}
 	}
-	return matrix(row, other.col, prod);
+	return matrix(row, other.col, std::move(prod));
 }
 
 
@@ -571,7 +619,7 @@ matrix matrix::mul(const numeric & other) const
 		for (unsigned c=0; c<col; ++c)
 			prod[r*col+c] = m[r*col+c] * other;
 
-	return matrix(row, col, prod);
+	return matrix(row, col, std::move(prod));
 }
 
 
@@ -587,7 +635,7 @@ matrix matrix::mul_scalar(const ex & other) const
 		for (unsigned c=0; c<col; ++c)
 			prod[r*col+c] = m[r*col+c] * other;
 
-	return matrix(row, col, prod);
+	return matrix(row, col, std::move(prod));
 }
 
 
@@ -619,12 +667,12 @@ matrix matrix::pow(const ex & expn) const
 			// that this is not entirely optimal but close to optimal and
 			// "better" algorithms are much harder to implement.  (See Knuth,
 			// TAoCP2, section "Evaluation of Powers" for a good discussion.)
-			while (b!=_num1) {
+			while (b!=*_num1_p) {
 				if (b.is_odd()) {
 					C = C.mul(A);
 					--b;
 				}
-				b /= _num2;  // still integer.
+				b /= *_num2_p;  // still integer.
 				A = A.mul(A);
 			}
 			return A.mul(C);
@@ -673,7 +721,7 @@ matrix matrix::transpose() const
 		for (unsigned c=0; c<this->rows(); ++c)
 			trans[r*this->rows()+c] = m[c*this->cols()+r];
 	
-	return matrix(this->cols(),this->rows(),trans);
+	return matrix(this->cols(), this->rows(), std::move(trans));
 }
 
 /** Determinant of square matrix.  This routine doesn't actually calculate the
@@ -700,18 +748,16 @@ ex matrix::determinant(unsigned algo) const
 	bool numeric_flag = true;
 	bool normal_flag = false;
 	unsigned sparse_count = 0;  // counts non-zero elements
-	exvector::const_iterator r = m.begin(), rend = m.end();
-	while (r != rend) {
+	for (auto r : m) {
+		if (!r.info(info_flags::numeric))
+			numeric_flag = false;
 		exmap srl;  // symbol replacement list
-		ex rtest = r->to_rational(srl);
+		ex rtest = r.to_rational(srl);
 		if (!rtest.is_zero())
 			++sparse_count;
-		if (!rtest.info(info_flags::numeric))
-			numeric_flag = false;
 		if (!rtest.info(info_flags::crational_polynomial) &&
-			 rtest.info(info_flags::rational_function))
+		     rtest.info(info_flags::rational_function))
 			normal_flag = true;
-		++r;
 	}
 	
 	// Here is the heuristics in case this routine has to decide:
@@ -736,7 +782,7 @@ ex matrix::determinant(unsigned algo) const
 		else
 			return m[0].expand();
 	}
-	
+
 	// Compute the determinant
 	switch(algo) {
 		case determinant_algo::gauss: {
@@ -794,23 +840,22 @@ ex matrix::determinant(unsigned algo) const
 			}
 			std::sort(c_zeros.begin(),c_zeros.end());
 			std::vector<unsigned> pre_sort;
-			for (std::vector<uintpair>::const_iterator i=c_zeros.begin(); i!=c_zeros.end(); ++i)
-				pre_sort.push_back(i->second);
+			for (auto & i : c_zeros)
+				pre_sort.push_back(i.second);
 			std::vector<unsigned> pre_sort_test(pre_sort); // permutation_sign() modifies the vector so we make a copy here
 			int sign = permutation_sign(pre_sort_test.begin(), pre_sort_test.end());
 			exvector result(row*col);  // represents sorted matrix
 			unsigned c = 0;
-			for (std::vector<unsigned>::const_iterator i=pre_sort.begin();
-				 i!=pre_sort.end();
-				 ++i,++c) {
+			for (auto & it : pre_sort) {
 				for (unsigned r=0; r<row; ++r)
-					result[r*col+c] = m[r*col+(*i)];
+					result[r*col+c] = m[r*col+it];
+				++c;
 			}
 			
 			if (normal_flag)
-				return (sign*matrix(row,col,result).determinant_minor()).normal();
+				return (sign*matrix(row, col, std::move(result)).determinant_minor()).normal();
 			else
-				return sign*matrix(row,col,result).determinant_minor();
+				return sign*matrix(row, col, std::move(result)).determinant_minor();
 		}
 	}
 }
@@ -832,7 +877,7 @@ ex matrix::trace() const
 		tr += m[r*col+r];
 	
 	if (tr.info(info_flags::rational_function) &&
-		!tr.info(info_flags::crational_polynomial))
+	   !tr.info(info_flags::crational_polynomial))
 		return tr.normal();
 	else
 		return tr.expand();
@@ -840,7 +885,7 @@ ex matrix::trace() const
 
 
 /** Characteristic Polynomial.  Following mathematica notation the
- *  characteristic polynomial of a matrix M is defined as the determiant of
+ *  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
@@ -856,11 +901,11 @@ ex matrix::charpoly(const ex & lambda) const
 		throw (std::logic_error("matrix::charpoly(): matrix not square"));
 	
 	bool numeric_flag = true;
-	exvector::const_iterator r = m.begin(), rend = m.end();
-	while (r!=rend && numeric_flag==true) {
-		if (!r->info(info_flags::numeric))
+	for (auto & r : m) {
+		if (!r.info(info_flags::numeric)) {
 			numeric_flag = false;
-		++r;
+			break;
+		}
 	}
 	
 	// The pure numeric case is traditionally rather common.  Hence, it is
@@ -938,14 +983,15 @@ matrix matrix::inverse() const
  *
  *  @param vars n x p matrix, all elements must be symbols 
  *  @param rhs m x p matrix
+ *  @param algo selects the solving algorithm
  *  @return n x p solution matrix
  *  @exception logic_error (incompatible matrices)
  *  @exception invalid_argument (1st argument must be matrix of symbols)
  *  @exception runtime_error (inconsistent linear system)
  *  @see       solve_algo */
 matrix matrix::solve(const matrix & vars,
-					 const matrix & rhs,
-					 unsigned algo) const
+                     const matrix & rhs,
+                     unsigned algo) const
 {
 	const unsigned m = this->rows();
 	const unsigned n = this->cols();
@@ -970,11 +1016,11 @@ matrix matrix::solve(const matrix & vars,
 	
 	// Gather some statistical information about the augmented matrix:
 	bool numeric_flag = true;
-	exvector::const_iterator r = aug.m.begin(), rend = aug.m.end();
-	while (r!=rend && numeric_flag==true) {
-		if (!r->info(info_flags::numeric))
+	for (auto & r : aug.m) {
+		if (!r.info(info_flags::numeric)) {
 			numeric_flag = false;
-		++r;
+			break;
+		}
 	}
 	
 	// Here is the heuristics in case this routine has to decide:
@@ -1038,6 +1084,29 @@ matrix matrix::solve(const matrix & vars,
 }
 
 
+/** Compute the rank of this matrix. */
+unsigned matrix::rank() const
+{
+	// Method:
+	// Transform this matrix into upper echelon form and then count the
+	// number of non-zero rows.
+
+	GINAC_ASSERT(row*col==m.capacity());
+
+	// Actually, any elimination scheme will do since we are only
+	// interested in the echelon matrix' zeros.
+	matrix to_eliminate = *this;
+	to_eliminate.fraction_free_elimination();
+
+	unsigned r = row*col;  // index of last non-zero element
+	while (r--) {
+		if (!to_eliminate.m[r].is_zero())
+			return 1+r/col;
+	}
+	return 0;
+}
+
+
 // protected
 
 /** Recursive determinant for small matrices having at least one symbolic
@@ -1046,7 +1115,7 @@ matrix matrix::solve(const matrix & vars,
  *  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.
+ *  matrix' dimension is larger than 7.
  *
  *  @return the determinant as a new expression (in expanded form)
  *  @see matrix::determinant() */
@@ -1153,7 +1222,7 @@ ex matrix::determinant_minor() const
 					Pkey[j] = Pkey[j-1]+1;
 		} while(fc);
 		// next column, so change the role of A and B:
-		A = B;
+		A.swap(B);
 		B.clear();
 	}
 	
@@ -1179,8 +1248,8 @@ int matrix::gauss_elimination(const bool det)
 	int sign = 1;
 	
 	unsigned r0 = 0;
-	for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
-		int indx = pivot(r0, r1, true);
+	for (unsigned c0=0; c0<n && r0<m-1; ++c0) {
+		int indx = pivot(r0, c0, true);
 		if (indx == -1) {
 			sign = 0;
 			if (det)
@@ -1190,17 +1259,17 @@ int matrix::gauss_elimination(const bool det)
 			if (indx > 0)
 				sign = -sign;
 			for (unsigned r2=r0+1; r2<m; ++r2) {
-				if (!this->m[r2*n+r1].is_zero()) {
+				if (!this->m[r2*n+c0].is_zero()) {
 					// yes, there is something to do in this row
-					ex piv = this->m[r2*n+r1] / this->m[r0*n+r1];
-					for (unsigned c=r1+1; c<n; ++c) {
+					ex piv = this->m[r2*n+c0] / this->m[r0*n+c0];
+					for (unsigned c=c0+1; c<n; ++c) {
 						this->m[r2*n+c] -= piv * this->m[r0*n+c];
 						if (!this->m[r2*n+c].info(info_flags::numeric))
 							this->m[r2*n+c] = this->m[r2*n+c].normal();
 					}
 				}
 				// fill up left hand side with zeros
-				for (unsigned c=0; c<=r1; ++c)
+				for (unsigned c=r0; c<=c0; ++c)
 					this->m[r2*n+c] = _ex0;
 			}
 			if (det) {
@@ -1211,7 +1280,12 @@ int matrix::gauss_elimination(const bool det)
 			++r0;
 		}
 	}
-	
+	// clear remaining rows
+	for (unsigned r=r0+1; r<m; ++r) {
+		for (unsigned c=0; c<n; ++c)
+			this->m[r*n+c] = _ex0;
+	}
+
 	return sign;
 }
 
@@ -1233,8 +1307,8 @@ int matrix::division_free_elimination(const bool det)
 	int sign = 1;
 	
 	unsigned r0 = 0;
-	for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
-		int indx = pivot(r0, r1, true);
+	for (unsigned c0=0; c0<n && r0<m-1; ++c0) {
+		int indx = pivot(r0, c0, true);
 		if (indx==-1) {
 			sign = 0;
 			if (det)
@@ -1244,10 +1318,10 @@ int matrix::division_free_elimination(const bool det)
 			if (indx>0)
 				sign = -sign;
 			for (unsigned r2=r0+1; r2<m; ++r2) {
-				for (unsigned c=r1+1; c<n; ++c)
-					this->m[r2*n+c] = (this->m[r0*n+r1]*this->m[r2*n+c] - this->m[r2*n+r1]*this->m[r0*n+c]).expand();
+				for (unsigned c=c0+1; c<n; ++c)
+					this->m[r2*n+c] = (this->m[r0*n+c0]*this->m[r2*n+c] - this->m[r2*n+c0]*this->m[r0*n+c]).expand();
 				// fill up left hand side with zeros
-				for (unsigned c=0; c<=r1; ++c)
+				for (unsigned c=r0; c<=c0; ++c)
 					this->m[r2*n+c] = _ex0;
 			}
 			if (det) {
@@ -1258,7 +1332,12 @@ int matrix::division_free_elimination(const bool det)
 			++r0;
 		}
 	}
-	
+	// clear remaining rows
+	for (unsigned r=r0+1; r<m; ++r) {
+		for (unsigned c=0; c<n; ++c)
+			this->m[r*n+c] = _ex0;
+	}
+
 	return sign;
 }
 
@@ -1321,38 +1400,45 @@ int matrix::fraction_free_elimination(const bool det)
 	matrix tmp_n(*this);
 	matrix tmp_d(m,n);  // for denominators, if needed
 	exmap srl;  // symbol replacement list
-	exvector::const_iterator cit = this->m.begin(), citend = this->m.end();
-	exvector::iterator tmp_n_it = tmp_n.m.begin(), tmp_d_it = tmp_d.m.begin();
-	while (cit != citend) {
-		ex nd = cit->normal().to_rational(srl).numer_denom();
-		++cit;
+	auto tmp_n_it = tmp_n.m.begin(), tmp_d_it = tmp_d.m.begin();
+	for (auto & it : this->m) {
+		ex nd = it.normal().to_rational(srl).numer_denom();
 		*tmp_n_it++ = nd.op(0);
 		*tmp_d_it++ = nd.op(1);
 	}
 	
 	unsigned r0 = 0;
-	for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
-		int indx = tmp_n.pivot(r0, r1, true);
-		if (indx==-1) {
+	for (unsigned c0=0; c0<n && r0<m-1; ++c0) {
+		// When trying to find a pivot, we should try a bit harder than expand().
+		// Searching the first non-zero element in-place here instead of calling
+		// pivot() allows us to do no more substitutions and back-substitutions
+		// than are actually necessary.
+		unsigned indx = r0;
+		while ((indx<m) &&
+		       (tmp_n[indx*n+c0].subs(srl, subs_options::no_pattern).expand().is_zero()))
+			++indx;
+		if (indx==m) {
+			// all elements in column c0 below row r0 vanish
 			sign = 0;
 			if (det)
 				return 0;
-		}
-		if (indx>=0) {
-			if (indx>0) {
+		} else {
+			if (indx>r0) {
+				// Matrix needs pivoting, swap rows r0 and indx of tmp_n and tmp_d.
 				sign = -sign;
-				// tmp_n's rows r0 and indx were swapped, do the same in tmp_d:
-				for (unsigned c=r1; c<n; ++c)
+				for (unsigned c=c0; c<n; ++c) {
+					tmp_n.m[n*indx+c].swap(tmp_n.m[n*r0+c]);
 					tmp_d.m[n*indx+c].swap(tmp_d.m[n*r0+c]);
+				}
 			}
 			for (unsigned r2=r0+1; r2<m; ++r2) {
-				for (unsigned c=r1+1; c<n; ++c) {
-					dividend_n = (tmp_n.m[r0*n+r1]*tmp_n.m[r2*n+c]*
-					              tmp_d.m[r2*n+r1]*tmp_d.m[r0*n+c]
-					             -tmp_n.m[r2*n+r1]*tmp_n.m[r0*n+c]*
-					              tmp_d.m[r0*n+r1]*tmp_d.m[r2*n+c]).expand();
-					dividend_d = (tmp_d.m[r2*n+r1]*tmp_d.m[r0*n+c]*
-					              tmp_d.m[r0*n+r1]*tmp_d.m[r2*n+c]).expand();
+				for (unsigned c=c0+1; c<n; ++c) {
+					dividend_n = (tmp_n.m[r0*n+c0]*tmp_n.m[r2*n+c]*
+					              tmp_d.m[r2*n+c0]*tmp_d.m[r0*n+c]
+					             -tmp_n.m[r2*n+c0]*tmp_n.m[r0*n+c]*
+					              tmp_d.m[r0*n+c0]*tmp_d.m[r2*n+c]).expand();
+					dividend_d = (tmp_d.m[r2*n+c0]*tmp_d.m[r0*n+c]*
+					              tmp_d.m[r0*n+c0]*tmp_d.m[r2*n+c]).expand();
 					bool check = divide(dividend_n, divisor_n,
 					                    tmp_n.m[r2*n+c], true);
 					check &= divide(dividend_d, divisor_d,
@@ -1360,13 +1446,13 @@ int matrix::fraction_free_elimination(const bool det)
 					GINAC_ASSERT(check);
 				}
 				// fill up left hand side with zeros
-				for (unsigned c=0; c<=r1; ++c)
+				for (unsigned c=r0; c<=c0; ++c)
 					tmp_n.m[r2*n+c] = _ex0;
 			}
-			if ((r1<n-1)&&(r0<m-1)) {
+			if (c0<n && r0<m-1) {
 				// compute next iteration's divisor
-				divisor_n = tmp_n.m[r0*n+r1].expand();
-				divisor_d = tmp_d.m[r0*n+r1].expand();
+				divisor_n = tmp_n.m[r0*n+c0].expand();
+				divisor_d = tmp_d.m[r0*n+c0].expand();
 				if (det) {
 					// save space by deleting no longer needed elements
 					for (unsigned c=0; c<n; ++c) {
@@ -1378,12 +1464,17 @@ int matrix::fraction_free_elimination(const bool det)
 			++r0;
 		}
 	}
+	// clear remaining rows
+	for (unsigned r=r0+1; r<m; ++r) {
+		for (unsigned c=0; c<n; ++c)
+			tmp_n.m[r*n+c] = _ex0;
+	}
+
 	// repopulate *this matrix:
-	exvector::iterator it = this->m.begin(), itend = this->m.end();
 	tmp_n_it = tmp_n.m.begin();
 	tmp_d_it = tmp_d.m.begin();
-	while (it != itend)
-		*it++ = ((*tmp_n_it++)/(*tmp_d_it++)).subs(srl, subs_options::no_pattern);
+	for (auto & it : this->m)
+		it = ((*tmp_n_it++)/(*tmp_d_it++)).subs(srl, subs_options::no_pattern);
 	
 	return sign;
 }
@@ -1399,7 +1490,7 @@ int matrix::fraction_free_elimination(const bool det)
  *  @param co is the column to be inspected
  *  @param symbolic signal if we want the first non-zero element to be pivoted
  *  (true) or the one with the largest absolute value (false).
- *  @return 0 if no interchange occured, -1 if all are zero (usually signaling
+ *  @return 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.
  */
 int matrix::pivot(unsigned ro, unsigned co, bool symbolic)
@@ -1440,28 +1531,39 @@ int matrix::pivot(unsigned ro, unsigned co, bool symbolic)
 	return k;
 }
 
-ex lst_to_matrix(const lst & l)
+/** Function to check that all elements of the matrix are zero.
+ */
+bool matrix::is_zero_matrix() const
 {
-	lst::const_iterator itr, itc;
+	for (auto & i : m)
+		if (!i.is_zero())
+			return false;
+	return true;
+}
 
+ex lst_to_matrix(const lst & l)
+{
 	// Find number of rows and columns
 	size_t rows = l.nops(), cols = 0;
-	for (itr = l.begin(); itr != l.end(); ++itr) {
-		if (!is_a<lst>(*itr))
+	for (auto & itr : l) {
+		if (!is_a<lst>(itr))
 			throw (std::invalid_argument("lst_to_matrix: argument must be a list of lists"));
-		if (itr->nops() > cols)
-			cols = itr->nops();
+		if (itr.nops() > cols)
+			cols = itr.nops();
 	}
 
 	// Allocate and fill matrix
 	matrix &M = *new matrix(rows, cols);
 	M.setflag(status_flags::dynallocated);
 
-	unsigned i;
-	for (itr = l.begin(), i = 0; itr != l.end(); ++itr, ++i) {
-		unsigned j;
-		for (itc = ex_to<lst>(*itr).begin(), j = 0; itc != ex_to<lst>(*itr).end(); ++itc, ++j)
-			M(i, j) = *itc;
+	unsigned i = 0;
+	for (auto & itr : l) {
+		unsigned j = 0;
+		for (auto & itc : ex_to<lst>(itr)) {
+			M(i, j) = itc;
+			++j;
+		}
+		++i;
 	}
 
 	return M;
@@ -1469,16 +1571,17 @@ ex lst_to_matrix(const lst & l)
 
 ex diag_matrix(const lst & l)
 {
-	lst::const_iterator it;
 	size_t dim = l.nops();
 
 	// Allocate and fill matrix
 	matrix &M = *new matrix(dim, dim);
 	M.setflag(status_flags::dynallocated);
 
-	unsigned i;
-	for (it = l.begin(), i = 0; it != l.end(); ++it, ++i)
-		M(i, i) = *it;
+	unsigned i = 0;
+	for (auto & it : l) {
+		M(i, i) = it;
+		++i;
+	}
 
 	return M;
 }
@@ -1530,4 +1633,52 @@ ex symbolic_matrix(unsigned r, unsigned c, const std::string & base_name, const
 	return M;
 }
 
+ex reduced_matrix(const matrix& m, unsigned r, unsigned c)
+{
+	if (r+1>m.rows() || c+1>m.cols() || m.cols()<2 || m.rows()<2)
+		throw std::runtime_error("minor_matrix(): index out of bounds");
+
+	const unsigned rows = m.rows()-1;
+	const unsigned cols = m.cols()-1;
+	matrix &M = *new matrix(rows, cols);
+	M.setflag(status_flags::dynallocated | status_flags::evaluated);
+
+	unsigned ro = 0;
+	unsigned ro2 = 0;
+	while (ro2<rows) {
+		if (ro==r)
+			++ro;
+		unsigned co = 0;
+		unsigned co2 = 0;
+		while (co2<cols) {
+			if (co==c)
+				++co;
+			M(ro2,co2) = m(ro, co);
+			++co;
+			++co2;
+		}
+		++ro;
+		++ro2;
+	}
+
+	return M;
+}
+
+ex sub_matrix(const matrix&m, unsigned r, unsigned nr, unsigned c, unsigned nc)
+{
+	if (r+nr>m.rows() || c+nc>m.cols())
+		throw std::runtime_error("sub_matrix(): index out of bounds");
+
+	matrix &M = *new matrix(nr, nc);
+	M.setflag(status_flags::dynallocated | status_flags::evaluated);
+
+	for (unsigned ro=0; ro<nr; ++ro) {
+		for (unsigned co=0; co<nc; ++co) {
+			M(ro,co) = m(ro+r,co+c);
+		}
+	}
+
+	return M;
+}
+
 } // namespace GiNaC