X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fmatrix.cpp;h=299cacfc98fa58ce8a6d78322ca49be9928794ce;hp=66f3999a4e37574d142f57d4dc68d486c572864c;hb=8cffcdf13d817a47f217f1a1043317d95969e070;hpb=67edef78ce992a8f6ad704bfac228b8dec6eacd2

diff --git a/ginac/matrix.cpp b/ginac/matrix.cpp
index 66f3999a..299cacfc 100644
--- a/ginac/matrix.cpp
+++ b/ginac/matrix.cpp
@@ -3,7 +3,7 @@
  *  Implementation of symbolic matrices */
 
 /*
- *  GiNaC Copyright (C) 1999-2010 Johannes Gutenberg University Mainz, Germany
+ *  GiNaC Copyright (C) 1999-2019 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
@@ -73,15 +73,6 @@ matrix::matrix(unsigned r, unsigned c) : row(r), col(c), m(r*c, _ex0)
 	setflag(status_flags::not_shareable);
 }
 
-// protected
-
-/** Ctor from representation, for internal use only. */
-matrix::matrix(unsigned r, unsigned c, const exvector & m2)
-  : row(r), col(c), m(m2)
-{
-	setflag(status_flags::not_shareable);
-}
-
 /** 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
@@ -92,15 +83,50 @@ matrix::matrix(unsigned r, unsigned c, const lst & l)
 	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;
+	}
+}
+
+/** Construct a matrix from an 2 dimensional initializer list.
+ *  Throws an exception if some row has a different length than all the others.
+ */
+matrix::matrix(std::initializer_list<std::initializer_list<ex>> l)
+  : row(l.size()), col(l.begin()->size())
+{
+	setflag(status_flags::not_shareable);
+
+	m.reserve(row*col);
+	for (const auto & r : l) {
+		unsigned c = 0;
+		for (const auto & e : r) {
+			m.push_back(e);
+			++c;
+		}
+		if (c != col)
+			throw std::invalid_argument("matrix::matrix{{}}: wrong dimension");
 	}
 }
 
+// protected
+
+/** Ctor from representation, for internal use only. */
+matrix::matrix(unsigned r, unsigned c, const exvector & 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);
+}
+
 //////////
 // archiving
 //////////
@@ -114,10 +140,10 @@ void matrix::read_archive(const archive_node &n, lst &sym_lst)
 	m.reserve(row * col);
 	// XXX: default ctor inserts a zero element, we need to erase it here.
 	m.pop_back();
-	archive_node::archive_node_cit first = n.find_first("m");
-	archive_node::archive_node_cit last = n.find_last("m");
+	auto first = n.find_first("m");
+	auto last = n.find_last("m");
 	++last;
-	for (archive_node::archive_node_cit i=first; i != last; ++i) {
+	for (auto i=first; i != last; ++i) {
 		ex e;
 		n.find_ex_by_loc(i, e, sym_lst);
 		m.push_back(e);
@@ -130,10 +156,8 @@ 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);
 	}
 }
 
@@ -203,28 +227,6 @@ ex & matrix::let_op(size_t i)
 	return m[i];
 }
 
-/** Evaluate matrix entry by entry. */
-ex matrix::eval(int level) const
-{
-	// check if we have to do anything at all
-	if ((level==1)&&(flags & status_flags::evaluated))
-		return *this;
-	
-	// emergency break
-	if (level == -max_recursion_level)
-		throw (std::runtime_error("matrix::eval(): recursion limit exceeded"));
-	
-	// eval() entry by entry
-	exvector m2(row*col);
-	--level;
-	for (unsigned r=0; r<row; ++r)
-		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);
-}
-
 ex matrix::subs(const exmap & mp, unsigned options) const
 {
 	exvector m2(row * col);
@@ -232,14 +234,14 @@ 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
 {
-	exvector * ev = 0;
-	for (exvector::const_iterator i=m.begin(); i!=m.end(); ++i) {
+	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);
@@ -248,17 +250,15 @@ ex matrix::conjugate() const
 		if (are_ex_trivially_equal(x, *i)) {
 			continue;
 		}
-		ev = new exvector;
+		ev.reset(new exvector);
 		ev->reserve(m.size());
-		for (exvector::const_iterator j=m.begin(); j!=i; ++j) {
+		for (auto j=m.begin(); j!=i; ++j) {
 			ev->push_back(*j);
 		}
 		ev->push_back(x);
 	}
 	if (ev) {
-		ex result = matrix(row, col, *ev);
-		delete ev;
-		return result;
+		return matrix(row, col, std::move(*ev));
 	}
 	return *this;
 }
@@ -267,18 +267,18 @@ ex matrix::real_part() const
 {
 	exvector v;
 	v.reserve(m.size());
-	for (exvector::const_iterator i=m.begin(); i!=m.end(); ++i)
-		v.push_back(i->real_part());
-	return matrix(row, col, v);
+	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 (exvector::const_iterator i=m.begin(); i!=m.end(); ++i)
-		v.push_back(i->imag_part());
-	return matrix(row, col, v);
+	for (auto & i : m)
+		v.push_back(i.imag_part());
+	return matrix(row, col, std::move(v));
 }
 
 // protected
@@ -560,12 +560,11 @@ 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));
 }
 
 
@@ -578,12 +577,11 @@ 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));
 }
 
 
@@ -606,7 +604,7 @@ matrix matrix::mul(const matrix & other) const
 				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));
 }
 
 
@@ -619,7 +617,7 @@ 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));
 }
 
 
@@ -635,7 +633,7 @@ 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));
 }
 
 
@@ -721,7 +719,7 @@ 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
@@ -748,18 +746,16 @@ 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) {
-		if (!r->info(info_flags::numeric))
+	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::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:
@@ -842,23 +838,22 @@ 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();
 		}
 	}
 }
@@ -888,7 +883,7 @@ 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
@@ -904,11 +899,11 @@ 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
@@ -942,12 +937,19 @@ ex matrix::charpoly(const ex & lambda) const
 }
 
 
+/** Inverse of this matrix, with automatic algorithm selection. */
+matrix matrix::inverse() const
+{
+	return inverse(solve_algo::automatic);
+}
+
 /** Inverse of this matrix.
  *
+ *  @param algo selects the algorithm (one of solve_algo)
  *  @return    the inverted matrix
  *  @exception logic_error (matrix not square)
  *  @exception runtime_error (singular matrix) */
-matrix matrix::inverse() const
+matrix matrix::inverse(unsigned algo) const
 {
 	if (row != col)
 		throw (std::logic_error("matrix::inverse(): matrix not square"));
@@ -970,7 +972,7 @@ matrix matrix::inverse() const
 	
 	matrix sol(row,col);
 	try {
-		sol = this->solve(vars,identity);
+		sol = this->solve(vars, identity, algo);
 	} catch (const std::runtime_error & e) {
 	    if (e.what()==std::string("matrix::solve(): inconsistent linear system"))
 			throw (std::runtime_error("matrix::inverse(): singular matrix"));
@@ -984,7 +986,7 @@ matrix matrix::inverse() 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.
  *
- *  @param vars n x p matrix, all elements must be symbols 
+ *  @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
@@ -1000,8 +1002,8 @@ matrix matrix::solve(const matrix & vars,
 	const unsigned n = this->cols();
 	const unsigned p = rhs.cols();
 	
-	// syntax checks    
-	if ((rhs.rows() != m) || (vars.rows() != n) || (vars.col != p))
+	// syntax checks
+	if ((rhs.rows() != m) || (vars.rows() != n) || (vars.cols() != p))
 		throw (std::logic_error("matrix::solve(): incompatible matrices"));
 	for (unsigned ro=0; ro<n; ++ro)
 		for (unsigned co=0; co<p; ++co)
@@ -1016,41 +1018,9 @@ matrix matrix::solve(const matrix & vars,
 		for (unsigned c=0; c<p; ++c)
 			aug.m[r*(n+p)+c+n] = rhs.m[r*p+c];
 	}
-	
-	// 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))
-			numeric_flag = false;
-		++r;
-	}
-	
-	// Here is the heuristics in case this routine has to decide:
-	if (algo == solve_algo::automatic) {
-		// Bareiss (fraction-free) elimination is generally a good guess:
-		algo = solve_algo::bareiss;
-		// For m<3, Bareiss elimination is equivalent to division free
-		// elimination but has more logistic overhead
-		if (m<3)
-			algo = solve_algo::divfree;
-		// This overrides any prior decisions.
-		if (numeric_flag)
-			algo = solve_algo::gauss;
-	}
-	
+
 	// Eliminate the augmented matrix:
-	switch(algo) {
-		case solve_algo::gauss:
-			aug.gauss_elimination();
-			break;
-		case solve_algo::divfree:
-			aug.division_free_elimination();
-			break;
-		case solve_algo::bareiss:
-		default:
-			aug.fraction_free_elimination();
-	}
+	auto colid = aug.echelon_form(algo, n);
 	
 	// assemble the solution matrix:
 	matrix sol(n,p);
@@ -1058,48 +1028,51 @@ matrix matrix::solve(const matrix & vars,
 		unsigned last_assigned_sol = n+1;
 		for (int r=m-1; r>=0; --r) {
 			unsigned fnz = 1;    // first non-zero in row
-			while ((fnz<=n) && (aug.m[r*(n+p)+(fnz-1)].is_zero()))
+			while ((fnz<=n) && (aug.m[r*(n+p)+(fnz-1)].normal().is_zero()))
 				++fnz;
 			if (fnz>n) {
 				// row consists only of zeros, corresponding rhs must be 0, too
-				if (!aug.m[r*(n+p)+n+co].is_zero()) {
+				if (!aug.m[r*(n+p)+n+co].normal().is_zero()) {
 					throw (std::runtime_error("matrix::solve(): inconsistent linear system"));
 				}
 			} else {
 				// assign solutions for vars between fnz+1 and
 				// last_assigned_sol-1: free parameters
 				for (unsigned c=fnz; c<last_assigned_sol-1; ++c)
-					sol(c,co) = vars.m[c*p+co];
+					sol(colid[c],co) = vars.m[colid[c]*p+co];
 				ex e = aug.m[r*(n+p)+n+co];
 				for (unsigned c=fnz; c<n; ++c)
-					e -= aug.m[r*(n+p)+c]*sol.m[c*p+co];
-				sol(fnz-1,co) = (e/(aug.m[r*(n+p)+(fnz-1)])).normal();
+					e -= aug.m[r*(n+p)+c]*sol.m[colid[c]*p+co];
+				sol(colid[fnz-1],co) = (e/(aug.m[r*(n+p)+fnz-1])).normal();
 				last_assigned_sol = fnz;
 			}
 		}
 		// assign solutions for vars between 1 and
 		// last_assigned_sol-1: free parameters
 		for (unsigned ro=0; ro<last_assigned_sol-1; ++ro)
-			sol(ro,co) = vars(ro,co);
+			sol(colid[ro],co) = vars(colid[ro],co);
 	}
 	
 	return sol;
 }
 
-
 /** Compute the rank of this matrix. */
 unsigned matrix::rank() const
+{
+	return rank(solve_algo::automatic);
+}
+
+/** Compute the rank of this matrix using the given algorithm,
+ *  which should be a member of enum solve_algo. */
+unsigned matrix::rank(unsigned solve_algo) 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();
+	to_eliminate.echelon_form(solve_algo, col);
 
 	unsigned r = row*col;  // index of last non-zero element
 	while (r--) {
@@ -1118,7 +1091,7 @@ unsigned matrix::rank() const
  *  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() */
@@ -1224,14 +1197,76 @@ ex matrix::determinant_minor() const
 				for (unsigned j=fc; j<n-c; ++j)
 					Pkey[j] = Pkey[j-1]+1;
 		} while(fc);
-		// next column, so change the role of A and B:
-		A.swap(B);
-		B.clear();
+		// next column, clear B and change the role of A and B:
+		A = std::move(B);
 	}
 	
 	return det;
 }
 
+std::vector<unsigned>
+matrix::echelon_form(unsigned algo, int n)
+{
+	// Here is the heuristics in case this routine has to decide:
+	if (algo == solve_algo::automatic) {
+		// Gather some statistical information about the augmented matrix:
+		bool numeric_flag = true;
+		for (const auto & r : m) {
+			if (!r.info(info_flags::numeric)) {
+				numeric_flag = false;
+				break;
+			}
+		}
+		unsigned density = 0;
+		for (const auto & r : m) {
+			density += !r.is_zero();
+		}
+		unsigned ncells = col*row;
+		if (numeric_flag) {
+			// For numerical matrices Gauss is good, but Markowitz becomes
+			// better for large sparse matrices.
+			if ((ncells > 200) && (density < ncells/2)) {
+				algo = solve_algo::markowitz;
+			} else {
+				algo = solve_algo::gauss;
+			}
+		} else {
+			// For symbolic matrices Markowitz is good, but Bareiss/Divfree
+			// is better for small and dense matrices.
+			if ((ncells < 120) && (density*5 > ncells*3)) {
+				if (ncells <= 12) {
+					algo = solve_algo::divfree;
+				} else {
+					algo = solve_algo::bareiss;
+				}
+			} else {
+				algo = solve_algo::markowitz;
+			}
+		}
+	}
+	// Eliminate the augmented matrix:
+	std::vector<unsigned> colid(col);
+	for (unsigned c = 0; c < col; c++) {
+		colid[c] = c;
+	}
+	switch(algo) {
+		case solve_algo::gauss:
+			gauss_elimination();
+			break;
+		case solve_algo::divfree:
+			division_free_elimination();
+			break;
+		case solve_algo::bareiss:
+			fraction_free_elimination();
+			break;
+		case solve_algo::markowitz:
+			colid = markowitz_elimination(n);
+			break;
+		default:
+			throw std::invalid_argument("matrix::echelon_form(): 'algo' is not one of the solve_algo enum");
+	}
+	return colid;
+}
 
 /** 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
@@ -1292,6 +1327,119 @@ int matrix::gauss_elimination(const bool det)
 	return sign;
 }
 
+/* Perform Markowitz-ordered Gaussian elimination (with full
+ * pivoting) on a matrix, constraining the choice of pivots to
+ * the first n columns (this simplifies handling of augmented
+ * matrices). Return the column id vector v, such that v[column]
+ * is the original number of the column before shuffling (v[i]==i
+ * for i >= n). */
+std::vector<unsigned>
+matrix::markowitz_elimination(unsigned n)
+{
+	GINAC_ASSERT(n <= col);
+	std::vector<int> rowcnt(row, 0);
+	std::vector<int> colcnt(col, 0);
+	// Normalize everything before start. We'll keep all the
+	// cells normalized throughout the algorithm to properly
+	// handle unnormal zeros.
+	for (unsigned r = 0; r < row; r++) {
+		for (unsigned c = 0; c < col; c++) {
+			if (!m[r*col + c].is_zero()) {
+				m[r*col + c] = m[r*col + c].normal();
+				rowcnt[r]++;
+				colcnt[c]++;
+			}
+		}
+	}
+	std::vector<unsigned> colid(col);
+	for (unsigned c = 0; c < col; c++) {
+		colid[c] = c;
+	}
+	exvector ab(row);
+	for (unsigned k = 0; (k < col) && (k < row - 1); k++) {
+		// Find the pivot that minimizes (rowcnt[r]-1)*(colcnt[c]-1).
+		unsigned pivot_r = row + 1;
+		unsigned pivot_c = col + 1;
+		int pivot_m = row*col;
+		for (unsigned r = k; r < row; r++) {
+			for (unsigned c = k; c < n; c++) {
+				const ex &mrc = m[r*col + c];
+				if (mrc.is_zero())
+					continue;
+				GINAC_ASSERT(rowcnt[r] > 0);
+				GINAC_ASSERT(colcnt[c] > 0);
+				int measure = (rowcnt[r] - 1)*(colcnt[c] - 1);
+				if (measure < pivot_m) {
+					pivot_m = measure;
+					pivot_r = r;
+					pivot_c = c;
+				}
+			}
+		}
+		if (pivot_m == row*col) {
+			// The rest of the matrix is zero.
+			break;
+		}
+		GINAC_ASSERT(k <= pivot_r && pivot_r < row);
+		GINAC_ASSERT(k <= pivot_c && pivot_c < col);
+		// Swap the pivot into (k, k).
+		if (pivot_c != k) {
+			for (unsigned r = 0; r < row; r++) {
+				m[r*col + pivot_c].swap(m[r*col + k]);
+			}
+			std::swap(colid[pivot_c], colid[k]);
+			std::swap(colcnt[pivot_c], colcnt[k]);
+		}
+		if (pivot_r != k) {
+			for (unsigned c = k; c < col; c++) {
+				m[pivot_r*col + c].swap(m[k*col + c]);
+			}
+			std::swap(rowcnt[pivot_r], rowcnt[k]);
+		}
+		// No normalization before is_zero() here, because
+		// we maintain the matrix normalized throughout the
+		// algorithm.
+		ex a = m[k*col + k];
+		GINAC_ASSERT(!a.is_zero());
+		// Subtract the pivot row KJI-style (so: loop by pivot, then
+		// column, then row) to maximally exploit pivot row zeros (at
+		// the expense of the pivot column zeros). The speedup compared
+		// to the usual KIJ order is not really significant though...
+		for (unsigned r = k + 1; r < row; r++) {
+			const ex &b = m[r*col + k];
+			if (!b.is_zero()) {
+				ab[r] = b/a;
+				rowcnt[r]--;
+			}
+		}
+		colcnt[k] = rowcnt[k] = 0;
+		for (unsigned c = k + 1; c < col; c++) {
+			const ex &mr0c = m[k*col + c];
+			if (mr0c.is_zero())
+				continue;
+			colcnt[c]--;
+			for (unsigned r = k + 1; r < row; r++) {
+				if (ab[r].is_zero())
+					continue;
+				bool waszero = m[r*col + c].is_zero();
+				m[r*col + c] = (m[r*col + c] - ab[r]*mr0c).normal();
+				bool iszero = m[r*col + c].is_zero();
+				if (waszero && !iszero) {
+					rowcnt[r]++;
+					colcnt[c]++;
+				}
+				if (!waszero && iszero) {
+					rowcnt[r]--;
+					colcnt[c]--;
+				}
+			}
+		}
+		for (unsigned r = k + 1; r < row; r++) {
+			ab[r] = m[r*col + k] = _ex0;
+		}
+	}
+	return colid;
+}
 
 /** Perform the steps of division free elimination to bring the m x n matrix
  *  into an upper echelon form.
@@ -1322,7 +1470,7 @@ int matrix::division_free_elimination(const bool det)
 				sign = -sign;
 			for (unsigned r2=r0+1; r2<m; ++r2) {
 				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();
+					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]).normal();
 				// fill up left hand side with zeros
 				for (unsigned c=r0; c<=c0; ++c)
 					this->m[r2*n+c] = _ex0;
@@ -1403,11 +1551,9 @@ 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);
 	}
@@ -1476,11 +1622,10 @@ int matrix::fraction_free_elimination(const bool det)
 	}
 
 	// 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;
 }
@@ -1496,7 +1641,7 @@ 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)
@@ -1541,34 +1686,34 @@ int matrix::pivot(unsigned ro, unsigned co, bool symbolic)
  */
 bool matrix::is_zero_matrix() const
 {
-	for (exvector::const_iterator i=m.begin(); i!=m.end(); ++i) 
-		if(!(i->is_zero()))
+	for (auto & i : m)
+		if (!i.is_zero())
 			return false;
 	return true;
 }
 
 ex lst_to_matrix(const lst & l)
 {
-	lst::const_iterator itr, itc;
-
 	// 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;
+	matrix & M = dynallocate<matrix>(rows, cols);
+
+	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;
@@ -1576,24 +1721,40 @@ 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);
+	matrix & M = dynallocate<matrix>(dim, dim);
 
-	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;
+}
+
+ex diag_matrix(std::initializer_list<ex> l)
+{
+	size_t dim = l.size();
+
+	// Allocate and fill matrix
+	matrix & M = dynallocate<matrix>(dim, dim);
+
+	unsigned i = 0;
+	for (auto & it : l) {
+		M(i, i) = it;
+		++i;
+	}
 
 	return M;
 }
 
 ex unit_matrix(unsigned r, unsigned c)
 {
-	matrix &Id = *new matrix(r, c);
-	Id.setflag(status_flags::dynallocated);
+	matrix & Id = dynallocate<matrix>(r, c);
+	Id.setflag(status_flags::evaluated);
 	for (unsigned i=0; i<r && i<c; i++)
 		Id(i,i) = _ex1;
 
@@ -1602,8 +1763,8 @@ ex unit_matrix(unsigned r, unsigned c)
 
 ex symbolic_matrix(unsigned r, unsigned c, const std::string & base_name, const std::string & tex_base_name)
 {
-	matrix &M = *new matrix(r, c);
-	M.setflag(status_flags::dynallocated | status_flags::evaluated);
+	matrix & M = dynallocate<matrix>(r, c);
+	M.setflag(status_flags::evaluated);
 
 	bool long_format = (r > 10 || c > 10);
 	bool single_row = (r == 1 || c == 1);
@@ -1644,8 +1805,8 @@ ex reduced_matrix(const matrix& m, unsigned r, unsigned c)
 
 	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);
+	matrix & M = dynallocate<matrix>(rows, cols);
+	M.setflag(status_flags::evaluated);
 
 	unsigned ro = 0;
 	unsigned ro2 = 0;
@@ -1673,8 +1834,8 @@ 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);
+	matrix & M = dynallocate<matrix>(nr, nc);
+	M.setflag(status_flags::evaluated);
 
 	for (unsigned ro=0; ro<nr; ++ro) {
 		for (unsigned co=0; co<nc; ++co) {