X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fmatrix.cpp;h=97abbc9fd325f9620c55c0cb4e9057f34456e9f5;hp=9f9f67a82c83b62e76431de79ea4bf4054a05047;hb=79f30c335f1ddbd3c76dfee5d76128b992b6b19c;hpb=199b64938ab86af572d0816c15d7838730567b2d

diff --git a/ginac/matrix.cpp b/ginac/matrix.cpp
index 9f9f67a8..97abbc9f 100644
--- a/ginac/matrix.cpp
+++ b/ginac/matrix.cpp
@@ -3,7 +3,7 @@
  *  Implementation of symbolic matrices */
 
 /*
- *  GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
+ *  GiNaC Copyright (C) 1999-2021 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
@@ -17,53 +17,49 @@
  *
  *  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 <algorithm>
-#include <map>
-#include <stdexcept>
-
 #include "matrix.h"
 #include "numeric.h"
 #include "lst.h"
 #include "idx.h"
 #include "indexed.h"
+#include "add.h"
 #include "power.h"
 #include "symbol.h"
+#include "operators.h"
 #include "normal.h"
-#include "print.h"
 #include "archive.h"
 #include "utils.h"
-#include "debugmsg.h"
+
+#include <algorithm>
+#include <iostream>
+#include <map>
+#include <sstream>
+#include <stdexcept>
+#include <string>
 
 namespace GiNaC {
 
-GINAC_IMPLEMENT_REGISTERED_CLASS(matrix, basic)
+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>(&matrix::do_print_tree).
+  print_func<print_python_repr>(&matrix::do_print_python_repr))
 
 //////////
-// default ctor, dtor, copy ctor, assignment operator and helpers:
+// default constructor
 //////////
 
 /** Default ctor.  Initializes to 1 x 1-dimensional zero-matrix. */
-matrix::matrix() : inherited(TINFO_matrix), row(1), col(1)
+matrix::matrix() : row(1), col(1), m(1, _ex0)
 {
-	debugmsg("matrix default ctor",LOGLEVEL_CONSTRUCT);
-	m.push_back(_ex0());
+	setflag(status_flags::not_shareable);
 }
 
-void matrix::copy(const matrix & other)
-{
-	inherited::copy(other);
-	row = other.row;
-	col = other.col;
-	m = other.m;  // STL's vector copying invoked here
-}
-
-DEFAULT_DESTROY(matrix)
-
 //////////
-// other ctors
+// other constructors
 //////////
 
 // public
@@ -72,20 +68,9 @@ DEFAULT_DESTROY(matrix)
  *
  *  @param r number of rows
  *  @param c number of cols */
-matrix::matrix(unsigned r, unsigned c)
-  : inherited(TINFO_matrix), row(r), col(c)
+matrix::matrix(unsigned r, unsigned c) : row(r), col(c), m(r*c, _ex0)
 {
-	debugmsg("matrix ctor from unsigned,unsigned",LOGLEVEL_CONSTRUCT);
-	m.resize(r*c, _ex0());
-}
-
-// protected
-
-/** 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)
-{
-	debugmsg("matrix ctor from unsigned,unsigned,exvector",LOGLEVEL_CONSTRUCT);
+	setflag(status_flags::not_shareable);
 }
 
 /** Construct matrix from (flat) list of elements. If the list has fewer
@@ -93,151 +78,213 @@ 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)
+  : row(r), col(c), m(r*c, _ex0)
 {
-	debugmsg("matrix ctor from unsigned,unsigned,lst",LOGLEVEL_CONSTRUCT);
-	m.resize(r*c, _ex0());
+	setflag(status_flags::not_shareable);
 
-	for (unsigned i=0; i<l.nops(); i++) {
-		unsigned x = i % c;
-		unsigned y = i / c;
+	size_t i = 0;
+	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] = l.op(i);
+		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
 //////////
 
-matrix::matrix(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
+void matrix::read_archive(const archive_node &n, lst &sym_lst)
 {
-	debugmsg("matrix ctor from archive_node", LOGLEVEL_CONSTRUCT);
+	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 range = n.find_property_range("m", "m");
+	for (auto i=range.begin; i != range.end; ++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.emplace_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 bases classes
+// functions overriding virtual functions from base classes
 //////////
 
 // public
 
-void matrix::print(const print_context & c, unsigned level) const
+void matrix::print_elements(const print_context & c, const char *row_start, const char *row_end, const char *row_sep, const char *col_sep) const
 {
-	debugmsg("matrix print", LOGLEVEL_PRINT);
-
-	if (is_of_type(c, print_tree)) {
-
-		inherited::print(c, level);
+	for (unsigned ro=0; ro<row; ++ro) {
+		c.s << row_start;
+		for (unsigned co=0; co<col; ++co) {
+			m[ro*col+co].print(c);
+			if (co < col-1)
+				c.s << col_sep;
+			else
+				c.s << row_end;
+		}
+		if (ro < row-1)
+			c.s << row_sep;
+	}
+}
 
-	} else {
+void matrix::do_print(const print_context & c, unsigned level) const
+{
+	c.s << "[";
+	print_elements(c, "[", "]", ",", ",");
+	c.s << "]";
+}
 
-		c.s << "[";
-		for (unsigned y=0; y<row-1; ++y) {
-			c.s << "[";
-			for (unsigned x=0; x<col-1; ++x) {
-				m[y*col+x].print(c);
-				c.s << ",";
-			}
-			m[col*(y+1)-1].print(c);
-			c.s << "],";
-		}
-		c.s << "[";
-		for (unsigned x=0; x<col-1; ++x) {
-			m[(row-1)*col+x].print(c);
-			c.s << ",";
-		}
-		m[row*col-1].print(c);
-		c.s << "]]";
+void matrix::do_print_latex(const print_latex & c, unsigned level) const
+{
+	c.s << "\\left(\\begin{array}{" << std::string(col,'c') << "}";
+	print_elements(c, "", "", "\\\\", "&");
+	c.s << "\\end{array}\\right)";
+}
 
-	}
+void matrix::do_print_python_repr(const print_python_repr & c, unsigned level) const
+{
+	c.s << class_name() << '(';
+	print_elements(c, "[", "]", ",", ",");
+	c.s << ')';
 }
 
 /** nops is defined to be rows x columns. */
-unsigned matrix::nops() const
+size_t matrix::nops() const
 {
-	return row*col;
+	return static_cast<size_t>(row) * static_cast<size_t>(col);
 }
 
 /** returns matrix entry at position (i/col, i%col). */
-ex matrix::op(int i) const
+ex matrix::op(size_t i) const
 {
+	GINAC_ASSERT(i<nops());
+	
 	return m[i];
 }
 
-/** returns matrix entry at position (i/col, i%col). */
-ex & matrix::let_op(int i)
+/** returns writable matrix entry at position (i/col, i%col). */
+ex & matrix::let_op(size_t i)
 {
-	GINAC_ASSERT(i>=0);
 	GINAC_ASSERT(i<nops());
 	
+	ensure_if_modifiable();
 	return m[i];
 }
 
-/** Evaluate matrix entry by entry. */
-ex matrix::eval(int level) const
+ex matrix::subs(const exmap & mp, unsigned options) const
 {
-	debugmsg("matrix eval",LOGLEVEL_MEMBER_FUNCTION);
-	
-	// 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;
+	exvector m2(row * col);
 	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 );
+			m2[r*col+c] = m[r*col+c].subs(mp, options);
+
+	return matrix(row, col, std::move(m2)).subs_one_level(mp, options);
 }
 
-ex matrix::subs(const lst & ls, const lst & lr, bool no_pattern) const
+/** Complex conjugate every matrix entry. */
+ex matrix::conjugate() const
 {
-	exvector m2(row * col);
-	for (unsigned r=0; r<row; ++r)
-		for (unsigned c=0; c<col; ++c)
-			m2[r*col+c] = m[r*col+c].subs(ls, lr, no_pattern);
+	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));
+}
 
-	return ex(matrix(row, col, m2)).bp->basic::subs(ls, lr, no_pattern);
+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
 
 int matrix::compare_same_type(const basic & other) const
 {
-	GINAC_ASSERT(is_exactly_of_type(other, matrix));
-	const matrix & o = static_cast<matrix &>(const_cast<basic &>(other));
+	GINAC_ASSERT(is_exactly_a<matrix>(other));
+	const matrix &o = static_cast<const matrix &>(other);
 	
 	// compare number of rows
 	if (row != o.rows())
@@ -259,11 +306,21 @@ int matrix::compare_same_type(const basic & other) const
 	return 0;
 }
 
+bool matrix::match_same_type(const basic & other) const
+{
+	GINAC_ASSERT(is_exactly_a<matrix>(other));
+	const matrix & o = static_cast<const matrix &>(other);
+	
+	// The number of rows and columns must be the same. This is necessary to
+	// prevent a 2x3 matrix from matching a 3x2 one.
+	return row == o.rows() && col == o.cols();
+}
+
 /** Automatic symbolic evaluation of an indexed matrix. */
 ex matrix::eval_indexed(const basic & i) const
 {
-	GINAC_ASSERT(is_of_type(i, indexed));
-	GINAC_ASSERT(is_ex_of_type(i.op(0), matrix));
+	GINAC_ASSERT(is_a<indexed>(i));
+	GINAC_ASSERT(is_a<matrix>(i.op(0)));
 
 	bool all_indices_unsigned = static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint);
 
@@ -339,13 +396,13 @@ ex matrix::eval_indexed(const basic & i) const
 /** Sum of two indexed matrices. */
 ex matrix::add_indexed(const ex & self, const ex & other) const
 {
-	GINAC_ASSERT(is_ex_of_type(self, indexed));
-	GINAC_ASSERT(is_ex_of_type(self.op(0), matrix));
-	GINAC_ASSERT(is_ex_of_type(other, indexed));
+	GINAC_ASSERT(is_a<indexed>(self));
+	GINAC_ASSERT(is_a<matrix>(self.op(0)));
+	GINAC_ASSERT(is_a<indexed>(other));
 	GINAC_ASSERT(self.nops() == 2 || self.nops() == 3);
 
 	// Only add two matrices
-	if (is_ex_of_type(other.op(0), matrix)) {
+	if (is_a<matrix>(other.op(0))) {
 		GINAC_ASSERT(other.nops() == 2 || other.nops() == 3);
 
 		const matrix &self_matrix = ex_to<matrix>(self.op(0));
@@ -375,8 +432,8 @@ ex matrix::add_indexed(const ex & self, const ex & other) const
 /** Product of an indexed matrix with a number. */
 ex matrix::scalar_mul_indexed(const ex & self, const numeric & other) const
 {
-	GINAC_ASSERT(is_ex_of_type(self, indexed));
-	GINAC_ASSERT(is_ex_of_type(self.op(0), matrix));
+	GINAC_ASSERT(is_a<indexed>(self));
+	GINAC_ASSERT(is_a<matrix>(self.op(0)));
 	GINAC_ASSERT(self.nops() == 2 || self.nops() == 3);
 
 	const matrix &self_matrix = ex_to<matrix>(self.op(0));
@@ -390,13 +447,13 @@ ex matrix::scalar_mul_indexed(const ex & self, const numeric & other) const
 /** Contraction of an indexed matrix with something else. */
 bool matrix::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 {
-	GINAC_ASSERT(is_ex_of_type(*self, indexed));
-	GINAC_ASSERT(is_ex_of_type(*other, indexed));
+	GINAC_ASSERT(is_a<indexed>(*self));
+	GINAC_ASSERT(is_a<indexed>(*other));
 	GINAC_ASSERT(self->nops() == 2 || self->nops() == 3);
-	GINAC_ASSERT(is_ex_of_type(self->op(0), matrix));
+	GINAC_ASSERT(is_a<matrix>(self->op(0)));
 
 	// Only contract with other matrices
-	if (!is_ex_of_type(other->op(0), matrix))
+	if (!is_a<matrix>(other->op(0)))
 		return false;
 
 	GINAC_ASSERT(other->nops() == 2 || other->nops() == 3);
@@ -405,10 +462,8 @@ bool matrix::contract_with(exvector::iterator self, exvector::iterator other, ex
 	const matrix &other_matrix = ex_to<matrix>(other->op(0));
 
 	if (self->nops() == 2) {
-		unsigned self_dim = (self_matrix.col == 1) ? self_matrix.row : self_matrix.col;
 
 		if (other->nops() == 2) { // vector * vector (scalar product)
-			unsigned other_dim = (other_matrix.col == 1) ? other_matrix.row : other_matrix.col;
 
 			if (self_matrix.col == 1) {
 				if (other_matrix.col == 1) {
@@ -427,7 +482,7 @@ bool matrix::contract_with(exvector::iterator self, exvector::iterator other, ex
 					*self = self_matrix.mul(other_matrix.transpose())(0, 0);
 				}
 			}
-			*other = _ex1();
+			*other = _ex1;
 			return true;
 
 		} else { // vector * matrix
@@ -438,7 +493,7 @@ bool matrix::contract_with(exvector::iterator self, exvector::iterator other, ex
 					*self = indexed(self_matrix.mul(other_matrix), other->op(2));
 				else
 					*self = indexed(self_matrix.transpose().mul(other_matrix), other->op(2));
-				*other = _ex1();
+				*other = _ex1;
 				return true;
 			}
 
@@ -448,7 +503,7 @@ bool matrix::contract_with(exvector::iterator self, exvector::iterator other, ex
 					*self = indexed(other_matrix.mul(self_matrix), other->op(1));
 				else
 					*self = indexed(other_matrix.mul(self_matrix.transpose()), other->op(1));
-				*other = _ex1();
+				*other = _ex1;
 				return true;
 			}
 		}
@@ -458,28 +513,28 @@ bool matrix::contract_with(exvector::iterator self, exvector::iterator other, ex
 		// A_ij * B_jk = (A*B)_ik
 		if (is_dummy_pair(self->op(2), other->op(1))) {
 			*self = indexed(self_matrix.mul(other_matrix), self->op(1), other->op(2));
-			*other = _ex1();
+			*other = _ex1;
 			return true;
 		}
 
 		// A_ij * B_kj = (A*Btrans)_ik
 		if (is_dummy_pair(self->op(2), other->op(2))) {
 			*self = indexed(self_matrix.mul(other_matrix.transpose()), self->op(1), other->op(1));
-			*other = _ex1();
+			*other = _ex1;
 			return true;
 		}
 
 		// A_ji * B_jk = (Atrans*B)_ik
 		if (is_dummy_pair(self->op(1), other->op(1))) {
 			*self = indexed(self_matrix.transpose().mul(other_matrix), self->op(2), other->op(2));
-			*other = _ex1();
+			*other = _ex1;
 			return true;
 		}
 
 		// A_ji * B_kj = (B*A)_ki
 		if (is_dummy_pair(self->op(1), other->op(2))) {
 			*self = indexed(other_matrix.mul(self_matrix), other->op(1), self->op(2));
-			*other = _ex1();
+			*other = _ex1;
 			return true;
 		}
 	}
@@ -503,12 +558,11 @@ matrix matrix::add(const matrix & other) const
 		throw std::logic_error("matrix::add(): incompatible matrices");
 	
 	exvector sum(this->m);
-	exvector::iterator i;
-	exvector::const_iterator ci;
-	for (i=sum.begin(), ci=other.m.begin(); i!=sum.end(); ++i, ++ci)
-		(*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));
 }
 
 
@@ -521,12 +575,11 @@ matrix matrix::sub(const matrix & other) const
 		throw std::logic_error("matrix::sub(): incompatible matrices");
 	
 	exvector dif(this->m);
-	exvector::iterator i;
-	exvector::const_iterator ci;
-	for (i=dif.begin(), ci=other.m.begin(); i!=dif.end(); ++i, ++ci)
-		(*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));
 }
 
 
@@ -542,13 +595,14 @@ 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));
 }
 
 
@@ -561,7 +615,7 @@ 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));
 }
 
 
@@ -577,7 +631,7 @@ 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));
 }
 
 
@@ -587,36 +641,37 @@ matrix matrix::pow(const ex & expn) const
 	if (col!=row)
 		throw (std::logic_error("matrix::pow(): matrix not square"));
 	
-	if (is_ex_exactly_of_type(expn, numeric)) {
+	if (is_exactly_a<numeric>(expn)) {
 		// Integer cases are computed by successive multiplication, using the
 		// obvious shortcut of storing temporaries, like A^4 == (A*A)*(A*A).
 		if (expn.info(info_flags::integer)) {
-			numeric k;
-			matrix prod(row,col);
+			numeric b = ex_to<numeric>(expn);
+			matrix A(row,col);
 			if (expn.info(info_flags::negative)) {
-				k = -ex_to<numeric>(expn);
-				prod = this->inverse();
+				b *= -1;
+				A = this->inverse();
 			} else {
-				k = ex_to<numeric>(expn);
-				prod = *this;
+				A = *this;
 			}
-			matrix result(row,col);
+			matrix C(row,col);
 			for (unsigned r=0; r<row; ++r)
-				result(r,r) = _ex1();
-			numeric b(1);
-			// this loop computes the representation of k in base 2 and
-			// multiplies the factors whenever needed:
-			while (b.compare(k)<=0) {
-				b *= numeric(2);
-				numeric r(mod(k,b));
-				if (!r.is_zero()) {
-					k -= r;
-					result = result.mul(prod);
+				C(r,r) = _ex1;
+			if (b.is_zero())
+				return C;
+			// This loop computes the representation of b in base 2 from right
+			// to left and multiplies the factors whenever needed.  Note
+			// 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_p) {
+				if (b.is_odd()) {
+					C = C.mul(A);
+					--b;
 				}
-				if (b.compare(k)<=0)
-					prod = prod.mul(prod);
+				b /= *_num2_p;  // still integer.
+				A = A.mul(A);
 			}
-			return result;
+			return A.mul(C);
 		}
 	}
 	throw (std::runtime_error("matrix::pow(): don't know how to handle exponent"));
@@ -654,7 +709,7 @@ ex & matrix::operator() (unsigned ro, unsigned co)
 
 /** Transposed of an m x n matrix, producing a new n x m matrix object that
  *  represents the transposed. */
-matrix matrix::transpose(void) const
+matrix matrix::transpose() const
 {
 	exvector trans(this->cols()*this->rows());
 	
@@ -662,7 +717,7 @@ matrix matrix::transpose(void) 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
@@ -689,15 +744,15 @@ ex matrix::determinant(unsigned algo) const
 	bool numeric_flag = true;
 	bool normal_flag = false;
 	unsigned sparse_count = 0;  // counts non-zero elements
-	for (exvector::const_iterator r=m.begin(); r!=m.end(); ++r) {
-		lst srl;  // symbol replacement list
-		ex rtest = (*r).to_rational(srl);
+	for (auto r : m) {
+		if (!r.info(info_flags::numeric))
+			numeric_flag = false;
+		exmap srl;  // symbol replacement list
+		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;
 	}
 	
@@ -723,7 +778,7 @@ ex matrix::determinant(unsigned algo) const
 		else
 			return m[0].expand();
 	}
-	
+
 	// Compute the determinant
 	switch(algo) {
 		case determinant_algo::gauss: {
@@ -751,7 +806,7 @@ ex matrix::determinant(unsigned algo) const
 			int sign;
 			sign = tmp.division_free_elimination(true);
 			if (sign==0)
-				return _ex0();
+				return _ex0;
 			ex det = tmp.m[row*col-1];
 			// factor out accumulated bogus slag
 			for (unsigned d=0; d<row-2; ++d)
@@ -763,10 +818,13 @@ ex matrix::determinant(unsigned algo) const
 		default: {
 			// This is the minor expansion scheme.  We always develop such
 			// that the smallest minors (i.e, the trivial 1x1 ones) are on the
-			// rightmost column.  For this to be efficient it turns out that
-			// the emptiest columns (i.e. the ones with most zeros) should be
-			// the ones on the right hand side.  Therefore we presort the
-			// columns of the matrix:
+			// rightmost column.  For this to be efficient, empirical tests
+			// have shown that the emptiest columns (i.e. the ones with most
+			// zeros) should be the ones on the right hand side -- although
+			// this might seem counter-intuitive (and in contradiction to some
+			// literature like the FORM manual).  Please go ahead and test it
+			// if you don't believe me!  Therefore we presort the columns of
+			// the matrix:
 			typedef std::pair<unsigned,unsigned> uintpair;
 			std::vector<uintpair> c_zeros;  // number of zeros in column
 			for (unsigned c=0; c<col; ++c) {
@@ -776,25 +834,24 @@ ex matrix::determinant(unsigned algo) const
 						++acc;
 				c_zeros.push_back(uintpair(acc,c));
 			}
-			sort(c_zeros.begin(),c_zeros.end());
+			std::sort(c_zeros.begin(),c_zeros.end());
 			std::vector<unsigned> pre_sort;
-			for (std::vector<uintpair>::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>::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();
 		}
 	}
 }
@@ -806,7 +863,7 @@ ex matrix::determinant(unsigned algo) const
  *
  *  @return    the sum of diagonal elements
  *  @exception logic_error (matrix not square) */
-ex matrix::trace(void) const
+ex matrix::trace() const
 {
 	if (row != col)
 		throw (std::logic_error("matrix::trace(): matrix not square"));
@@ -816,7 +873,7 @@ ex matrix::trace(void) 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();
@@ -824,7 +881,7 @@ ex matrix::trace(void) 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
@@ -834,15 +891,16 @@ ex matrix::trace(void) const
  *  @return    characteristic polynomial as new expression
  *  @exception logic_error (matrix not square)
  *  @see       matrix::determinant() */
-ex matrix::charpoly(const symbol & lambda) const
+ex matrix::charpoly(const ex & lambda) const
 {
 	if (row != col)
 		throw (std::logic_error("matrix::charpoly(): matrix not square"));
 	
 	bool numeric_flag = true;
-	for (exvector::const_iterator r=m.begin(); r!=m.end(); ++r) {
-		if (!(*r).info(info_flags::numeric)) {
+	for (auto & r : m) {
+		if (!r.info(info_flags::numeric)) {
 			numeric_flag = false;
+			break;
 		}
 	}
 	
@@ -850,36 +908,46 @@ ex matrix::charpoly(const symbol & lambda) const
 	// trapped and we use Leverrier's algorithm which goes as row^3 for
 	// every coefficient.  The expensive part is the matrix multiplication.
 	if (numeric_flag) {
+
 		matrix B(*this);
 		ex c = B.trace();
-		ex poly = power(lambda,row)-c*power(lambda,row-1);
+		ex poly = power(lambda, row) - c*power(lambda, row-1);
 		for (unsigned i=1; i<row; ++i) {
 			for (unsigned j=0; j<row; ++j)
 				B.m[j*col+j] -= c;
 			B = this->mul(B);
-			c = B.trace()/ex(i+1);
-			poly -= c*power(lambda,row-i-1);
+			c = B.trace() / ex(i+1);
+			poly -= c*power(lambda, row-i-1);
 		}
 		if (row%2)
 			return -poly;
 		else
 			return poly;
-	}
+
+	} else {
 	
-	matrix M(*this);
-	for (unsigned r=0; r<col; ++r)
-		M.m[r*col+r] -= lambda;
+		matrix M(*this);
+		for (unsigned r=0; r<col; ++r)
+			M.m[r*col+r] -= lambda;
 	
-	return M.determinant().collect(lambda);
+		return M.determinant().collect(lambda);
+	}
 }
 
 
+/** 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(void) const
+matrix matrix::inverse(unsigned algo) const
 {
 	if (row != col)
 		throw (std::logic_error("matrix::inverse(): matrix not square"));
@@ -890,7 +958,7 @@ matrix matrix::inverse(void) const
 	// First populate the identity matrix supposed to become the right hand side.
 	matrix identity(row,col);
 	for (unsigned i=0; i<row; ++i)
-		identity(i,i) = _ex1();
+		identity(i,i) = _ex1;
 	
 	// Populate a dummy matrix of variables, just because of compatibility with
 	// matrix::solve() which wants this (for compatibility with under-determined
@@ -902,7 +970,7 @@ matrix matrix::inverse(void) 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"));
@@ -916,23 +984,24 @@ matrix matrix::inverse(void) 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
  *  @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();
 	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)
@@ -947,39 +1016,9 @@ 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;
-	for (exvector::const_iterator r=aug.m.begin(); r!=aug.m.end(); ++r) {
-		if (!(*r).info(info_flags::numeric))
-			numeric_flag = false;
-	}
-	
-	// 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);
@@ -987,34 +1026,60 @@ 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());
+
+	matrix to_eliminate = *this;
+	to_eliminate.echelon_form(solve_algo, col);
+
+	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
 
@@ -1024,23 +1089,14 @@ 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() */
-ex matrix::determinant_minor(void) const
+ex matrix::determinant_minor() const
 {
-	// for small matrices the algorithm does not make any sense:
 	const unsigned n = this->cols();
-	if (n==1)
-		return m[0].expand();
-	if (n==2)
-		return (m[0]*m[3]-m[2]*m[1]).expand();
-	if (n==3)
-		return (m[0]*m[4]*m[8]-m[0]*m[5]*m[7]-
-		        m[1]*m[3]*m[8]+m[2]*m[3]*m[7]+
-		        m[1]*m[5]*m[6]-m[2]*m[4]*m[6]).expand();
-	
+
 	// This algorithm can best be understood by looking at a naive
 	// implementation of Laplace-expansion, like this one:
 	// ex det;
@@ -1073,71 +1129,133 @@ ex matrix::determinant_minor(void) const
 	// calculated in step c-1.  We therefore only have to store at most 
 	// 2*binomial(n,n/2) minors.
 	
-	// Unique flipper counter for partitioning into minors
-	std::vector<unsigned> Pkey;
-	Pkey.reserve(n);
-	// key for minor determinant (a subpartition of Pkey)
-	std::vector<unsigned> Mkey;
+	// we store the minors in maps, keyed by the rows they arise from
+	typedef std::vector<unsigned> keyseq;
+	typedef std::map<keyseq, ex> Rmap;
+
+	Rmap M, N;  // minors used in current and next column, respectively
+	// populate M with dummy unit, to be used as factor in rightmost column
+	M[keyseq{}] = _ex1;
+
+	// keys to identify minor of M and N (Mkey is a subsequence of Nkey)
+	keyseq Mkey, Nkey;
 	Mkey.reserve(n-1);
-	// we store our subminors in maps, keys being the rows they arise from
-	typedef std::map<std::vector<unsigned>,class ex> Rmap;
-	typedef std::map<std::vector<unsigned>,class ex>::value_type Rmap_value;
-	Rmap A;
-	Rmap B;
+	Nkey.reserve(n);
+
 	ex det;
-	// initialize A with last column:
-	for (unsigned r=0; r<n; ++r) {
-		Pkey.erase(Pkey.begin(),Pkey.end());
-		Pkey.push_back(r);
-		A.insert(Rmap_value(Pkey,m[n*(r+1)-1]));
-	}
 	// proceed from right to left through matrix
-	for (int c=n-2; c>=0; --c) {
-		Pkey.erase(Pkey.begin(),Pkey.end());  // don't change capacity
-		Mkey.erase(Mkey.begin(),Mkey.end());
+	for (int c=n-1; c>=0; --c) {
+		Nkey.clear();
+		Mkey.clear();
 		for (unsigned i=0; i<n-c; ++i)
-			Pkey.push_back(i);
-		unsigned fc = 0;  // controls logic for our strange flipper counter
+			Nkey.push_back(i);
+		unsigned fc = 0;  // controls logic for minor key generator
 		do {
-			det = _ex0();
+			det = _ex0;
 			for (unsigned r=0; r<n-c; ++r) {
 				// maybe there is nothing to do?
-				if (m[Pkey[r]*n+c].is_zero())
+				if (m[Nkey[r]*n+c].is_zero())
 					continue;
-				// create the sorted key for all possible minors
-				Mkey.erase(Mkey.begin(),Mkey.end());
-				for (unsigned i=0; i<n-c; ++i)
-					if (i!=r)
-						Mkey.push_back(Pkey[i]);
-				// Fetch the minors and compute the new determinant
+				// Mkey is same as Nkey, but with element r removed
+				Mkey.clear();
+				Mkey.insert(Mkey.begin(), Nkey.begin(), Nkey.begin() + r);
+				Mkey.insert(Mkey.end(), Nkey.begin() + r + 1, Nkey.end());
+				// add product of matrix element and minor M to determinant
 				if (r%2)
-					det -= m[Pkey[r]*n+c]*A[Mkey];
+					det -= m[Nkey[r]*n+c]*M[Mkey];
 				else
-					det += m[Pkey[r]*n+c]*A[Mkey];
+					det += m[Nkey[r]*n+c]*M[Mkey];
 			}
-			// prevent build-up of deep nesting of expressions saves time:
+			// prevent nested expressions to save time
 			det = det.expand();
-			// store the new determinant at its place in B:
+			// if the next computed minor is zero, don't store it in N:
+			// (if key is not found, operator[] will just return a zero ex)
 			if (!det.is_zero())
-				B.insert(Rmap_value(Pkey,det));
-			// increment our strange flipper counter
+				N[Nkey] = det;
+			// compute next minor key
 			for (fc=n-c; fc>0; --fc) {
-				++Pkey[fc-1];
-				if (Pkey[fc-1]<fc+c)
+				++Nkey[fc-1];
+				if (Nkey[fc-1]<fc+c)
 					break;
 			}
 			if (fc<n-c && fc>0)
 				for (unsigned j=fc; j<n-c; ++j)
-					Pkey[j] = Pkey[j-1]+1;
+					Nkey[j] = Nkey[j-1]+1;
 		} while(fc);
-		// next column, so change the role of A and B:
-		A = B;
-		B.clear();
+		// if N contains no minors, then they all vanished
+		if (N.empty())
+			return _ex0;
+
+		// proceed to next column: switch roles of M and N, clear N
+		M = std::move(N);
 	}
 	
 	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
@@ -1157,8 +1275,8 @@ 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)
@@ -1168,31 +1286,149 @@ 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)
-					this->m[r2*n+c] = _ex0();
+				for (unsigned c=r0; c<=c0; ++c)
+					this->m[r2*n+c] = _ex0;
 			}
 			if (det) {
 				// save space by deleting no longer needed elements
 				for (unsigned c=r0+1; c<n; ++c)
-					this->m[r0*n+c] = _ex0();
+					this->m[r0*n+c] = _ex0;
 			}
 			++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;
 }
 
+/* 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.
@@ -1211,8 +1447,8 @@ 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)
@@ -1222,21 +1458,26 @@ 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]).normal();
 				// fill up left hand side with zeros
-				for (unsigned c=0; c<=r1; ++c)
-					this->m[r2*n+c] = _ex0();
+				for (unsigned c=r0; c<=c0; ++c)
+					this->m[r2*n+c] = _ex0;
 			}
 			if (det) {
 				// save space by deleting no longer needed elements
 				for (unsigned c=r0+1; c<n; ++c)
-					this->m[r0*n+c] = _ex0();
+					this->m[r0*n+c] = _ex0;
 			}
 			++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;
 }
 
@@ -1261,7 +1502,7 @@ int matrix::fraction_free_elimination(const bool det)
 	//
 	// Bareiss (fraction-free) elimination in addition divides that element
 	// by m[k-1](k-1,k-1) for k>1, where it can be shown by means of the
-	// Sylvester determinant that this really divides m[k+1](r,c).
+	// Sylvester identity that this really divides m[k+1](r,c).
 	//
 	// We also allow rational functions where the original prove still holds.
 	// However, we must care for numerator and denominator separately and
@@ -1298,39 +1539,46 @@ int matrix::fraction_free_elimination(const bool det)
 	// makes things more complicated than they need to be.
 	matrix tmp_n(*this);
 	matrix tmp_d(m,n);  // for denominators, if needed
-	lst srl;  // symbol replacement list
-	exvector::iterator it = this->m.begin();
-	exvector::iterator tmp_n_it = tmp_n.m.begin();
-	exvector::iterator tmp_d_it = tmp_d.m.begin();
-	for (; it!= this->m.end(); ++it, ++tmp_n_it, ++tmp_d_it) {
-		(*tmp_n_it) = (*it).normal().to_rational(srl);
-		(*tmp_d_it) = (*tmp_n_it).denom();
-		(*tmp_n_it) = (*tmp_n_it).numer();
+	exmap srl;  // symbol replacement list
+	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,
@@ -1338,30 +1586,35 @@ 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)
-					tmp_n.m[r2*n+c] = _ex0();
+				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) {
-						tmp_n.m[r0*n+c] = _ex0();
-						tmp_d.m[r0*n+c] = _ex1();
+						tmp_n.m[r0*n+c] = _ex0;
+						tmp_d.m[r0*n+c] = _ex1;
 					}
 				}
 			}
 			++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:
-	it = this->m.begin();
 	tmp_n_it = tmp_n.m.begin();
 	tmp_d_it = tmp_d.m.begin();
-	for (; it!= this->m.end(); ++it, ++tmp_n_it, ++tmp_d_it)
-		(*it) = ((*tmp_n_it)/(*tmp_d_it)).subs(srl);
+	for (auto & it : this->m)
+		it = ((*tmp_n_it++)/(*tmp_d_it++)).subs(srl, subs_options::no_pattern);
 	
 	return sign;
 }
@@ -1377,7 +1630,7 @@ 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)
@@ -1389,11 +1642,11 @@ int matrix::pivot(unsigned ro, unsigned co, bool symbolic)
 			++k;
 	} else {
 		// search largest element in column co beginning at row ro
-		GINAC_ASSERT(is_ex_of_type(this->m[k*col+co],numeric));
+		GINAC_ASSERT(is_exactly_a<numeric>(this->m[k*col+co]));
 		unsigned kmax = k+1;
 		numeric mmax = abs(ex_to<numeric>(m[kmax*col+co]));
 		while (kmax<row) {
-			GINAC_ASSERT(is_ex_of_type(this->m[kmax*col+co],numeric));
+			GINAC_ASSERT(is_exactly_a<numeric>(this->m[kmax*col+co]));
 			numeric tmp = ex_to<numeric>(this->m[kmax*col+co]);
 			if (abs(tmp) > mmax) {
 				mmax = tmp;
@@ -1418,36 +1671,168 @@ int matrix::pivot(unsigned ro, unsigned co, bool symbolic)
 	return k;
 }
 
+/** Function to check that all elements of the matrix are zero.
+ */
+bool matrix::is_zero_matrix() const
+{
+	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
-	unsigned rows = l.nops(), cols = 0, i, j;
-	for (i=0; i<rows; i++)
-		if (l.op(i).nops() > cols)
-			cols = l.op(i).nops();
+	size_t rows = l.nops(), cols = 0;
+	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();
+	}
 
 	// Allocate and fill matrix
-	matrix &m = *new matrix(rows, cols);
-	m.setflag(status_flags::dynallocated);
-	for (i=0; i<rows; i++)
-		for (j=0; j<cols; j++)
-			if (l.op(i).nops() > j)
-				m(i, j) = l.op(i).op(j);
-			else
-				m(i, j) = _ex0();
-	return m;
+	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;
 }
 
 ex diag_matrix(const lst & l)
 {
-	unsigned dim = l.nops();
+	size_t dim = l.nops();
+
+	// 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 diag_matrix(std::initializer_list<ex> l)
+{
+	size_t dim = l.size();
+
+	// Allocate and fill matrix
+	matrix & M = dynallocate<matrix>(dim, dim);
 
-	matrix &m = *new matrix(dim, dim);
-	m.setflag(status_flags::dynallocated);
-	for (unsigned i=0; i<dim; i++)
-		m(i, i) = l.op(i);
+	unsigned i = 0;
+	for (auto & it : l) {
+		M(i, i) = it;
+		++i;
+	}
+
+	return M;
+}
+
+ex unit_matrix(unsigned r, unsigned c)
+{
+	matrix & Id = dynallocate<matrix>(r, c);
+	Id.setflag(status_flags::evaluated);
+	for (unsigned i=0; i<r && i<c; i++)
+		Id(i,i) = _ex1;
+
+	return Id;
+}
+
+ex symbolic_matrix(unsigned r, unsigned c, const std::string & base_name, const std::string & tex_base_name)
+{
+	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);
+
+	for (unsigned i=0; i<r; i++) {
+		for (unsigned j=0; j<c; j++) {
+			std::ostringstream s1, s2;
+			s1 << base_name;
+			s2 << tex_base_name << "_{";
+			if (single_row) {
+				if (c == 1) {
+					s1 << i;
+					s2 << i << '}';
+				} else {
+					s1 << j;
+					s2 << j << '}';
+				}
+			} else {
+				if (long_format) {
+					s1 << '_' << i << '_' << j;
+					s2 << i << ';' << j << "}";
+				} else {
+					s1 << i << j;
+					s2 << i << j << '}';
+				}
+			}
+			M(i, j) = symbol(s1.str(), s2.str());
+		}
+	}
+
+	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 = dynallocate<matrix>(rows, cols);
+	M.setflag(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 = dynallocate<matrix>(nr, nc);
+	M.setflag(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;
+	return M;
 }
 
 } // namespace GiNaC