Make .eval() evaluate top-level only.
[ginac.git] / ginac / matrix.cpp
index f34534a..c842885 100644 (file)
@@ -3,7 +3,7 @@
  *  Implementation of symbolic matrices */
 
 /*
- *  GiNaC Copyright (C) 1999-2001 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
  *
  *  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 "archive.h"
 #include "numeric.h"
 #include "lst.h"
 #include "idx.h"
 #include "indexed.h"
-#include "utils.h"
-#include "debugmsg.h"
+#include "add.h"
 #include "power.h"
 #include "symbol.h"
+#include "operators.h"
 #include "normal.h"
+#include "archive.h"
+#include "utils.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
 //////////
 
-// public
-
 /** Default ctor.  Initializes to 1 x 1-dimensional zero-matrix. */
-matrix::matrix() : inherited(TINFO_matrix), row(1), col(1)
-{
-       debugmsg("matrix default ctor",LOGLEVEL_CONSTRUCT);
-       m.push_back(_ex0());
-}
-
-// protected
-
-/** For use by copy ctor and assignment operator. */
-void matrix::copy(const matrix & other)
-{
-       inherited::copy(other);
-       row = other.row;
-       col = other.col;
-       m = other.m;  // STL's vector copying invoked here
-}
-
-void matrix::destroy(bool call_parent)
+matrix::matrix() : row(1), col(1), m(1, _ex0)
 {
-       if (call_parent) inherited::destroy(call_parent);
+       setflag(status_flags::not_shareable);
 }
 
 //////////
-// other ctors
+// other constructors
 //////////
 
 // public
@@ -79,210 +68,225 @@ void matrix::destroy(bool call_parent)
  *
  *  @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)
+{
+       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
+ *  thrown away. */
+matrix::matrix(unsigned r, unsigned c, const lst & l)
+  : row(r), col(c), m(r*c, _ex0)
 {
-       debugmsg("matrix ctor from unsigned,unsigned",LOGLEVEL_CONSTRUCT);
-       m.resize(r*c, _ex0());
+       setflag(status_flags::not_shareable);
+
+       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] = 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)
-  : 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))
 {
-       debugmsg("matrix ctor from unsigned,unsigned,exvector",LOGLEVEL_CONSTRUCT);
+       setflag(status_flags::not_shareable);
 }
 
 //////////
 // archiving
 //////////
 
-/** Construct object from archive_node. */
-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 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);
 
-/** Unarchive the object. */
-ex matrix::unarchive(const archive_node &n, const lst &sym_lst)
-{
-       return (new matrix(n, sym_lst))->setflag(status_flags::dynallocated);
-}
-
-/** Archive the object. */
 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);
        }
 }
 
 //////////
-// functions overriding virtual functions from bases classes
+// functions overriding virtual functions from base classes
 //////////
 
 // public
 
-void matrix::print(std::ostream & os, unsigned upper_precedence) 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);
-       os << "[[ ";
-       for (unsigned r=0; r<row-1; ++r) {
-               os << "[[";
-               for (unsigned c=0; c<col-1; ++c)
-                       os << m[r*col+c] << ",";
-               os << m[col*(r+1)-1] << "]], ";
+       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;
        }
-       os << "[[";
-       for (unsigned c=0; c<col-1; ++c)
-               os << m[(row-1)*col+c] << ",";
-       os << m[row*col-1] << "]] ]]";
 }
 
-void matrix::printraw(std::ostream & os) const
+void matrix::do_print(const print_context & c, unsigned level) const
 {
-       debugmsg("matrix printraw",LOGLEVEL_PRINT);
-       os << class_name() << "(" << row << "," << col <<",";
-       for (unsigned r=0; r<row-1; ++r) {
-               os << "(";
-               for (unsigned c=0; c<col-1; ++c)
-                       os << m[r*col+c] << ",";
-               os << m[col*(r-1)-1] << "),";
-       }
-       os << "(";
-       for (unsigned c=0; c<col-1; ++c)
-               os << m[(row-1)*col+c] << ",";
-       os << m[row*col-1] << "))";
+       c.s << "[";
+       print_elements(c, "[", "]", ",", ",");
+       c.s << "]";
 }
 
-/** nops is defined to be rows x columns. */
-unsigned matrix::nops() const
+void matrix::do_print_latex(const print_latex & c, unsigned level) const
 {
-       return row*col;
+       c.s << "\\left(\\begin{array}{" << std::string(col,'c') << "}";
+       print_elements(c, "", "", "\\\\", "&");
+       c.s << "\\end{array}\\right)";
 }
 
-/** returns matrix entry at position (i/col, i%col). */
-ex matrix::op(int i) const
+void matrix::do_print_python_repr(const print_python_repr & c, unsigned level) const
 {
-       return m[i];
+       c.s << class_name() << '(';
+       print_elements(c, "[", "]", ",", ",");
+       c.s << ')';
 }
 
-/** returns matrix entry at position (i/col, i%col). */
-ex & matrix::let_op(int i)
+/** nops is defined to be rows x columns. */
+size_t matrix::nops() const
 {
-       GINAC_ASSERT(i>=0);
-       GINAC_ASSERT(i<nops());
-       
-       return m[i];
+       return static_cast<size_t>(row) * static_cast<size_t>(col);
 }
 
-/** expands the elements of a matrix entry by entry. */
-ex matrix::expand(unsigned options) const
+/** returns matrix entry at position (i/col, i%col). */
+ex matrix::op(size_t i) const
 {
-       exvector tmp(row*col);
-       for (unsigned i=0; i<row*col; ++i)
-               tmp[i] = m[i].expand(options);
+       GINAC_ASSERT(i<nops());
        
-       return matrix(row, col, tmp);
+       return m[i];
 }
 
-/** Search ocurrences.  A matrix 'has' an expression if it is the expression
- *  itself or one of the elements 'has' it. */
-bool matrix::has(const ex & other) const
+/** returns writable matrix entry at position (i/col, i%col). */
+ex & matrix::let_op(size_t i)
 {
-       GINAC_ASSERT(other.bp!=0);
-       
-       // tautology: it is the expression itself
-       if (is_equal(*other.bp)) return true;
-       
-       // search all the elements
-       for (exvector::const_iterator r=m.begin(); r!=m.end(); ++r)
-               if ((*r).has(other)) return true;
+       GINAC_ASSERT(i<nops());
        
-       return false;
+       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);
 }
 
-/** Evaluate matrix numerically entry by entry. */
-ex matrix::evalf(int level) const
+/** Complex conjugate every matrix entry. */
+ex matrix::conjugate() const
 {
-       debugmsg("matrix evalf",LOGLEVEL_MEMBER_FUNCTION);
-               
-       // check if we have to do anything at all
-       if (level==1)
-               return *this;
-       
-       // emergency break
-       if (level == -max_recursion_level) {
-               throw (std::runtime_error("matrix::evalf(): recursion limit exceeded"));
+       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);
        }
-       
-       // evalf() 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].evalf(level);
-       
-       return matrix(row, col, m2);
+       if (ev) {
+               return matrix(row, col, std::move(*ev));
+       }
+       return *this;
 }
 
-ex matrix::subs(const lst & ls, const lst & lr) const
+ex matrix::real_part() 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);
+       exvector v;
+       v.reserve(m.size());
+       for (auto & i : m)
+               v.push_back(i.real_part());
+       return matrix(row, col, std::move(v));
+}
 
-       return matrix(row, col, m2);
+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())
@@ -304,11 +308,21 @@ 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);
 
@@ -319,7 +333,7 @@ ex matrix::eval_indexed(const basic & i) const
                if (row != 1 && col != 1)
                        throw (std::runtime_error("matrix::eval_indexed(): vector must have exactly 1 index"));
 
-               const idx & i1 = ex_to_idx(i.op(1));
+               const idx & i1 = ex_to<idx>(i.op(1));
 
                if (col == 1) {
 
@@ -329,7 +343,7 @@ ex matrix::eval_indexed(const basic & i) const
 
                        // Index numeric -> return vector element
                        if (all_indices_unsigned) {
-                               unsigned n1 = ex_to_numeric(i1.get_value()).to_int();
+                               unsigned n1 = ex_to<numeric>(i1.get_value()).to_int();
                                if (n1 >= row)
                                        throw (std::runtime_error("matrix::eval_indexed(): value of index exceeds number of vector elements"));
                                return (*this)(n1, 0);
@@ -343,7 +357,7 @@ ex matrix::eval_indexed(const basic & i) const
 
                        // Index numeric -> return vector element
                        if (all_indices_unsigned) {
-                               unsigned n1 = ex_to_numeric(i1.get_value()).to_int();
+                               unsigned n1 = ex_to<numeric>(i1.get_value()).to_int();
                                if (n1 >= col)
                                        throw (std::runtime_error("matrix::eval_indexed(): value of index exceeds number of vector elements"));
                                return (*this)(0, n1);
@@ -353,8 +367,8 @@ ex matrix::eval_indexed(const basic & i) const
        } else if (i.nops() == 3) {
 
                // Two indices
-               const idx & i1 = ex_to_idx(i.op(1));
-               const idx & i2 = ex_to_idx(i.op(2));
+               const idx & i1 = ex_to<idx>(i.op(1));
+               const idx & i2 = ex_to<idx>(i.op(2));
 
                if (!i1.get_dim().is_equal(row))
                        throw (std::runtime_error("matrix::eval_indexed(): dimension of first index must match number of rows"));
@@ -367,7 +381,7 @@ ex matrix::eval_indexed(const basic & i) const
 
                // Both indices numeric -> return matrix element
                if (all_indices_unsigned) {
-                       unsigned n1 = ex_to_numeric(i1.get_value()).to_int(), n2 = ex_to_numeric(i2.get_value()).to_int();
+                       unsigned n1 = ex_to<numeric>(i1.get_value()).to_int(), n2 = ex_to<numeric>(i2.get_value()).to_int();
                        if (n1 >= row)
                                throw (std::runtime_error("matrix::eval_indexed(): value of first index exceeds number of rows"));
                        if (n2 >= col)
@@ -384,16 +398,17 @@ 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(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));
-               const matrix &other_matrix = ex_to_matrix(other.op(0));
+               const matrix &self_matrix = ex_to<matrix>(self.op(0));
+               const matrix &other_matrix = ex_to<matrix>(other.op(0));
 
                if (self.nops() == 2 && other.nops() == 2) { // vector + vector
 
@@ -416,28 +431,41 @@ ex matrix::add_indexed(const ex & self, const ex & other) const
        return self + other;
 }
 
+/** Product of an indexed matrix with a number. */
+ex matrix::scalar_mul_indexed(const ex & self, const numeric & other) const
+{
+       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));
+
+       if (self.nops() == 2)
+               return indexed(self_matrix.mul(other), self.op(1));
+       else // self.nops() == 3
+               return indexed(self_matrix.mul(other), self.op(1), self.op(2));
+}
+
 /** 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);
 
-       const matrix &self_matrix = ex_to_matrix(self->op(0));
-       const matrix &other_matrix = ex_to_matrix(other->op(0));
+       const matrix &self_matrix = ex_to<matrix>(self->op(0));
+       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) {
@@ -456,7 +484,7 @@ 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
@@ -467,7 +495,7 @@ 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;
                        }
 
@@ -477,7 +505,7 @@ 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;
                        }
                }
@@ -487,28 +515,28 @@ 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;
                }
        }
@@ -529,15 +557,14 @@ bool matrix::contract_with(exvector::iterator self, exvector::iterator other, ex
 matrix matrix::add(const matrix & other) const
 {
        if (col != other.col || row != other.row)
-               throw (std::logic_error("matrix::add(): incompatible matrices"));
+               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));
 }
 
 
@@ -547,15 +574,14 @@ matrix matrix::add(const matrix & other) const
 matrix matrix::sub(const matrix & other) const
 {
        if (col != other.col || row != other.row)
-               throw (std::logic_error("matrix::sub(): incompatible matrices"));
+               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));
 }
 
 
@@ -565,23 +591,96 @@ matrix matrix::sub(const matrix & other) const
 matrix matrix::mul(const matrix & other) const
 {
        if (this->cols() != other.rows())
-               throw (std::logic_error("matrix::mul(): incompatible matrices"));
+               throw std::logic_error("matrix::mul(): incompatible matrices");
        
        exvector prod(this->rows()*other.cols());
        
        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));
+}
+
+
+/** Product of matrix and scalar. */
+matrix matrix::mul(const numeric & other) const
+{
+       exvector prod(row * col);
+
+       for (unsigned r=0; r<row; ++r)
+               for (unsigned c=0; c<col; ++c)
+                       prod[r*col+c] = m[r*col+c] * other;
+
+       return matrix(row, col, std::move(prod));
 }
 
 
-/** operator() to access elements.
+/** Product of matrix and scalar expression. */
+matrix matrix::mul_scalar(const ex & other) const
+{
+       if (other.return_type() != return_types::commutative)
+               throw std::runtime_error("matrix::mul_scalar(): non-commutative scalar");
+
+       exvector prod(row * col);
+
+       for (unsigned r=0; r<row; ++r)
+               for (unsigned c=0; c<col; ++c)
+                       prod[r*col+c] = m[r*col+c] * other;
+
+       return matrix(row, col, std::move(prod));
+}
+
+
+/** Power of a matrix.  Currently handles integer exponents only. */
+matrix matrix::pow(const ex & expn) const
+{
+       if (col!=row)
+               throw (std::logic_error("matrix::pow(): matrix not square"));
+       
+       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 b = ex_to<numeric>(expn);
+                       matrix A(row,col);
+                       if (expn.info(info_flags::negative)) {
+                               b *= -1;
+                               A = this->inverse();
+                       } else {
+                               A = *this;
+                       }
+                       matrix C(row,col);
+                       for (unsigned r=0; r<row; ++r)
+                               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;
+                               }
+                               b /= *_num2_p;  // still integer.
+                               A = A.mul(A);
+                       }
+                       return A.mul(C);
+               }
+       }
+       throw (std::runtime_error("matrix::pow(): don't know how to handle exponent"));
+}
+
+
+/** operator() to access elements for reading.
  *
  *  @param ro row of element
  *  @param co column of element
@@ -595,23 +694,24 @@ const ex & matrix::operator() (unsigned ro, unsigned co) const
 }
 
 
-/** Set individual elements manually.
+/** operator() to access elements for writing.
  *
+ *  @param ro row of element
+ *  @param co column of element
  *  @exception range_error (index out of range) */
-matrix & matrix::set(unsigned ro, unsigned co, ex value)
+ex & matrix::operator() (unsigned ro, unsigned co)
 {
        if (ro>=row || co>=col)
-               throw (std::range_error("matrix::set(): index out of range"));
-    
+               throw (std::range_error("matrix::operator(): index out of range"));
+
        ensure_if_modifiable();
-       m[ro*col+co] = value;
-       return *this;
+       return m[ro*col+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());
        
@@ -619,10 +719,9 @@ 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
  *  determinant, it only implements some heuristics about which algorithm to
  *  run.  If all the elements of the matrix are elements of an integral domain
@@ -647,15 +746,15 @@ 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;
        }
        
@@ -681,7 +780,7 @@ ex matrix::determinant(unsigned algo) const
                else
                        return m[0].expand();
        }
-       
+
        // Compute the determinant
        switch(algo) {
                case determinant_algo::gauss: {
@@ -709,7 +808,7 @@ 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)
@@ -721,10 +820,13 @@ 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) {
@@ -734,24 +836,24 @@ 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);
-                       int sign = permutation_sign(pre_sort);
+                       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();
                }
        }
 }
@@ -763,7 +865,7 @@ 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"));
@@ -773,7 +875,7 @@ 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();
@@ -781,7 +883,7 @@ 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
@@ -791,15 +893,16 @@ 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;
                }
        }
        
@@ -807,27 +910,30 @@ 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);
+       }
 }
 
 
@@ -836,55 +942,37 @@ ex matrix::charpoly(const symbol & lambda) const
  *  @return    the inverted matrix
  *  @exception logic_error (matrix not square)
  *  @exception runtime_error (singular matrix) */
-matrix matrix::inverse(void) const
+matrix matrix::inverse() const
 {
        if (row != col)
                throw (std::logic_error("matrix::inverse(): matrix not square"));
        
-       // NOTE: the Gauss-Jordan elimination used here can in principle be
-       // replaced by two clever calls to gauss_elimination() and some to
-       // transpose().  Wouldn't be more efficient (maybe less?), just more
-       // orthogonal.
-       matrix tmp(row,col);
-       // set tmp to the unit matrix
-       for (unsigned i=0; i<col; ++i)
-               tmp.m[i*col+i] = _ex1();
+       // This routine actually doesn't do anything fancy at all.  We compute the
+       // inverse of the matrix A by solving the system A * A^{-1} == Id.
        
-       // create a copy of this matrix
-       matrix cpy(*this);
-       for (unsigned r1=0; r1<row; ++r1) {
-               int indx = cpy.pivot(r1, r1);
-               if (indx == -1) {
+       // 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;
+       
+       // Populate a dummy matrix of variables, just because of compatibility with
+       // matrix::solve() which wants this (for compatibility with under-determined
+       // systems of equations).
+       matrix vars(row,col);
+       for (unsigned r=0; r<row; ++r)
+               for (unsigned c=0; c<col; ++c)
+                       vars(r,c) = symbol();
+       
+       matrix sol(row,col);
+       try {
+               sol = this->solve(vars,identity);
+       } catch (const std::runtime_error & e) {
+           if (e.what()==std::string("matrix::solve(): inconsistent linear system"))
                        throw (std::runtime_error("matrix::inverse(): singular matrix"));
-               }
-               if (indx != 0) {  // swap rows r and indx of matrix tmp
-                       for (unsigned i=0; i<col; ++i)
-                               tmp.m[r1*col+i].swap(tmp.m[indx*col+i]);
-               }
-               ex a1 = cpy.m[r1*col+r1];
-               for (unsigned c=0; c<col; ++c) {
-                       cpy.m[r1*col+c] /= a1;
-                       tmp.m[r1*col+c] /= a1;
-               }
-               for (unsigned r2=0; r2<row; ++r2) {
-                       if (r2 != r1) {
-                               if (!cpy.m[r2*col+r1].is_zero()) {
-                                       ex a2 = cpy.m[r2*col+r1];
-                                       // yes, there is something to do in this column
-                                       for (unsigned c=0; c<col; ++c) {
-                                               cpy.m[r2*col+c] -= a2 * cpy.m[r1*col+c];
-                                               if (!cpy.m[r2*col+c].info(info_flags::numeric))
-                                                       cpy.m[r2*col+c] = cpy.m[r2*col+c].normal();
-                                               tmp.m[r2*col+c] -= a2 * tmp.m[r1*col+c];
-                                               if (!tmp.m[r2*col+c].info(info_flags::numeric))
-                                                       tmp.m[r2*col+c] = tmp.m[r2*col+c].normal();
-                                       }
-                               }
-                       }
-               }
+               else
+                       throw;
        }
-       
-       return tmp;
+       return sol;
 }
 
 
@@ -893,14 +981,15 @@ matrix matrix::inverse(void) 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();
@@ -925,9 +1014,11 @@ matrix matrix::solve(const matrix & vars,
        
        // 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))
+       for (auto & r : aug.m) {
+               if (!r.info(info_flags::numeric)) {
                        numeric_flag = false;
+                       break;
+               }
        }
        
        // Here is the heuristics in case this routine has to decide:
@@ -947,8 +1038,10 @@ matrix matrix::solve(const matrix & vars,
        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();
@@ -971,25 +1064,47 @@ matrix matrix::solve(const matrix & vars,
                                // 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.set(c,co,vars.m[c*p+co]);
+                                       sol(c,co) = vars.m[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.set(fnz-1,co,
-                                               (e/(aug.m[r*(n+p)+(fnz-1)])).normal());
+                               sol(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.set(ro,co,vars(ro,co));
+                       sol(ro,co) = vars(ro,co);
        }
        
        return sol;
 }
 
 
+/** 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
@@ -998,11 +1113,11 @@ 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();
@@ -1027,9 +1142,9 @@ ex matrix::determinant_minor(void) const
        //     for (unsigned r=0; r<minorM.rows(); ++r) {
        //         for (unsigned c=0; c<minorM.cols(); ++c) {
        //             if (r<r1)
-       //                 minorM.set(r,c,m[r*col+c+1]);
+       //                 minorM(r,c) = m[r*col+c+1];
        //             else
-       //                 minorM.set(r,c,m[(r+1)*col+c+1]);
+       //                 minorM(r,c) = m[(r+1)*col+c+1];
        //         }
        //     }
        //     // recurse down and care for sign:
@@ -1073,7 +1188,7 @@ ex matrix::determinant_minor(void) const
                        Pkey.push_back(i);
                unsigned fc = 0;  // controls logic for our strange flipper counter
                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())
@@ -1104,9 +1219,8 @@ ex matrix::determinant_minor(void) 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 = B;
-               B.clear();
+               // next column, clear B and change the role of A and B:
+               A = std::move(B);
        }
        
        return det;
@@ -1131,8 +1245,8 @@ 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)
@@ -1142,28 +1256,33 @@ 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;
 }
 
@@ -1185,8 +1304,8 @@ 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)
@@ -1196,21 +1315,26 @@ 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]).expand();
                                // 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;
 }
 
@@ -1235,7 +1359,7 @@ 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
@@ -1272,39 +1396,46 @@ 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,
@@ -1312,30 +1443,35 @@ 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;
 }
@@ -1351,7 +1487,7 @@ 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)
@@ -1363,12 +1499,12 @@ 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]));
+               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));
-                       numeric tmp = ex_to_numeric(this->m[kmax*col+co]);
+                       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;
                                k = kmax;
@@ -1392,27 +1528,168 @@ int matrix::pivot(unsigned ro, unsigned co, bool symbolic)
        return k;
 }
 
-/** Convert list of lists to matrix. */
-ex lst_to_matrix(const ex &l)
+/** 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)
 {
-       if (!is_ex_of_type(l, lst))
-               throw(std::invalid_argument("argument to lst_to_matrix() must be a lst"));
-       
        // 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);
-       for (i=0; i<rows; i++)
-               for (j=0; j<cols; j++)
-                       if (l.op(i).nops() > j)
-                               m.set(i, j, l.op(i).op(j));
-                       else
-                               m.set(i, j, ex(0));
-       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)
+{
+       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);
+
+       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;
 }
 
 } // namespace GiNaC