X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fmatrix.cpp;h=6240ce20e9e36a14557e38a9918841ecf63923c7;hp=34d5f2f93c008a0deeac8aeaf0e4aff64992ed18;hb=db5765dc91202851739e196ba11bfccb0b3fe7bc;hpb=487e5659efe401683eee0381b0d23f967ffffc3c

diff --git a/ginac/matrix.cpp b/ginac/matrix.cpp
index 34d5f2f9..6240ce20 100644
--- a/ginac/matrix.cpp
+++ b/ginac/matrix.cpp
@@ -3,7 +3,7 @@
  *  Implementation of symbolic matrices */
 
 /*
- *  GiNaC Copyright (C) 1999 Johannes Gutenberg University Mainz, Germany
+ *  GiNaC Copyright (C) 1999-2000 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
@@ -21,9 +21,24 @@
  */
 
 #include <algorithm>
+#include <map>
 #include <stdexcept>
 
 #include "matrix.h"
+#include "archive.h"
+#include "numeric.h"
+#include "lst.h"
+#include "utils.h"
+#include "debugmsg.h"
+#include "power.h"
+#include "symbol.h"
+#include "normal.h"
+
+#ifndef NO_NAMESPACE_GINAC
+namespace GiNaC {
+#endif // ndef NO_NAMESPACE_GINAC
+
+GINAC_IMPLEMENT_REGISTERED_CLASS(matrix, basic)
 
 //////////
 // default constructor, destructor, copy constructor, assignment operator
@@ -34,10 +49,10 @@
 
 /** Default ctor.  Initializes to 1 x 1-dimensional zero-matrix. */
 matrix::matrix()
-    : basic(TINFO_matrix), row(1), col(1)
+    : inherited(TINFO_matrix), row(1), col(1)
 {
     debugmsg("matrix default constructor",LOGLEVEL_CONSTRUCT);
-    m.push_back(exZERO());
+    m.push_back(_ex0());
 }
 
 matrix::~matrix()
@@ -45,13 +60,13 @@ matrix::~matrix()
     debugmsg("matrix destructor",LOGLEVEL_DESTRUCT);
 }
 
-matrix::matrix(matrix const & other)
+matrix::matrix(const matrix & other)
 {
     debugmsg("matrix copy constructor",LOGLEVEL_CONSTRUCT);
     copy(other);
 }
 
-matrix const & matrix::operator=(matrix const & other)
+const matrix & matrix::operator=(const matrix & other)
 {
     debugmsg("matrix operator=",LOGLEVEL_ASSIGNMENT);
     if (this != &other) {
@@ -63,17 +78,17 @@ matrix const & matrix::operator=(matrix const & other)
 
 // protected
 
-void matrix::copy(matrix const & other)
+void matrix::copy(const matrix & other)
 {
-    basic::copy(other);
-    row=other.row;
-    col=other.col;
-    m=other.m;  // use STL's vector copying
+    inherited::copy(other);
+    row = other.row;
+    col = other.col;
+    m = other.m;  // STL's vector copying invoked here
 }
 
 void matrix::destroy(bool call_parent)
 {
-    if (call_parent) basic::destroy(call_parent);
+    if (call_parent) inherited::destroy(call_parent);
 }
 
 //////////
@@ -86,20 +101,59 @@ void matrix::destroy(bool call_parent)
  *
  *  @param r number of rows
  *  @param c number of cols */
-matrix::matrix(int r, int c)
-    : basic(TINFO_matrix), row(r), col(c)
+matrix::matrix(unsigned r, unsigned c)
+    : inherited(TINFO_matrix), row(r), col(c)
 {
-    debugmsg("matrix constructor from int,int",LOGLEVEL_CONSTRUCT);
-    m.resize(r*c, exZERO());
+    debugmsg("matrix constructor from unsigned,unsigned",LOGLEVEL_CONSTRUCT);
+    m.resize(r*c, _ex0());
 }
 
 // protected
 
 /** Ctor from representation, for internal use only. */
-matrix::matrix(int r, int c, vector<ex> const & m2)
-    : basic(TINFO_matrix), row(r), col(c), m(m2)
+matrix::matrix(unsigned r, unsigned c, const exvector & m2)
+    : inherited(TINFO_matrix), row(r), col(c), m(m2)
 {
-    debugmsg("matrix constructor from int,int,vector<ex>",LOGLEVEL_CONSTRUCT);
+    debugmsg("matrix constructor from unsigned,unsigned,exvector",LOGLEVEL_CONSTRUCT);
+}
+
+//////////
+// archiving
+//////////
+
+/** Construct object from archive_node. */
+matrix::matrix(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
+{
+    debugmsg("matrix constructor from archive_node", LOGLEVEL_CONSTRUCT);
+    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++) {
+        ex e;
+        if (n.find_ex("m", e, sym_lst, i))
+            m.push_back(e);
+        else
+            break;
+    }
+}
+
+/** 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;
+    }
 }
 
 //////////
@@ -114,41 +168,82 @@ basic * matrix::duplicate() const
     return new matrix(*this);
 }
 
+void matrix::print(std::ostream & os, unsigned upper_precedence) 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] << "]], ";
+    }
+    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
+{
+    debugmsg("matrix printraw",LOGLEVEL_PRINT);
+    os << "matrix(" << 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] << "))";
+}
+
 /** nops is defined to be rows x columns. */
-int matrix::nops() const
+unsigned matrix::nops() const
 {
     return row*col;
 }
 
 /** returns matrix entry at position (i/col, i%col). */
-ex & matrix::let_op(int const i)
+ex matrix::op(int i) const
 {
     return m[i];
 }
 
+/** returns matrix entry at position (i/col, i%col). */
+ex & matrix::let_op(int i)
+{
+    GINAC_ASSERT(i>=0);
+    GINAC_ASSERT(i<nops());
+    
+    return m[i];
+}
+
 /** expands the elements of a matrix entry by entry. */
 ex matrix::expand(unsigned options) const
 {
-    vector<ex> tmp(row*col);
-    for (int i=0; i<row*col; ++i) {
-        tmp[i]=m[i].expand(options);
-    }
+    exvector tmp(row*col);
+    for (unsigned i=0; i<row*col; ++i)
+        tmp[i] = m[i].expand(options);
+    
     return matrix(row, col, tmp);
 }
 
 /** Search ocurrences.  A matrix 'has' an expression if it is the expression
  *  itself or one of the elements 'has' it. */
-bool matrix::has(ex const & other) const
+bool matrix::has(const ex & other) const
 {
-    ASSERT(other.bp!=0);
+    GINAC_ASSERT(other.bp!=0);
     
     // tautology: it is the expression itself
     if (is_equal(*other.bp)) return true;
     
     // search all the elements
-    for (vector<ex>::const_iterator r=m.begin(); r!=m.end(); ++r) {
+    for (exvector::const_iterator r=m.begin(); r!=m.end(); ++r)
         if ((*r).has(other)) return true;
-    }
+    
     return false;
 }
 
@@ -158,23 +253,19 @@ ex matrix::eval(int level) const
     debugmsg("matrix eval",LOGLEVEL_MEMBER_FUNCTION);
     
     // check if we have to do anything at all
-    if ((level==1)&&(flags & status_flags::evaluated)) {
+    if ((level==1)&&(flags & status_flags::evaluated))
         return *this;
-    }
     
     // emergency break
-    if (level == -max_recursion_level) {
+    if (level == -max_recursion_level)
         throw (std::runtime_error("matrix::eval(): recursion limit exceeded"));
-    }
     
     // eval() entry by entry
-    vector<ex> m2(row*col);
-    --level;    
-    for (int r=0; r<row; ++r) {
-        for (int c=0; c<col; ++c) {
+    exvector m2(row*col);
+    --level;
+    for (unsigned r=0; r<row; ++r)
+        for (unsigned c=0; c<col; ++c)
             m2[r*col+c] = m[r*col+c].eval(level);
-        }
-    }
     
     return (new matrix(row, col, m2))->setflag(status_flags::dynallocated |
                                                status_flags::evaluated );
@@ -186,9 +277,8 @@ ex matrix::evalf(int level) const
     debugmsg("matrix evalf",LOGLEVEL_MEMBER_FUNCTION);
         
     // check if we have to do anything at all
-    if (level==1) {
+    if (level==1)
         return *this;
-    }
     
     // emergency break
     if (level == -max_recursion_level) {
@@ -196,38 +286,35 @@ ex matrix::evalf(int level) const
     }
     
     // evalf() entry by entry
-    vector<ex> m2(row*col);
+    exvector m2(row*col);
     --level;
-    for (int r=0; r<row; ++r) {
-        for (int c=0; c<col; ++c) {
+    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);
 }
 
 // protected
 
-int matrix::compare_same_type(basic const & other) const
+int matrix::compare_same_type(const basic & other) const
 {
-    ASSERT(is_exactly_of_type(other, matrix));
-    matrix const & o=static_cast<matrix &>(const_cast<basic &>(other));
+    GINAC_ASSERT(is_exactly_of_type(other, matrix));
+    const matrix & o = static_cast<matrix &>(const_cast<basic &>(other));
     
     // compare number of rows
-    if (row != o.rows()) {
+    if (row != o.rows())
         return row < o.rows() ? -1 : 1;
-    }
     
     // compare number of columns
-    if (col != o.cols()) {
+    if (col != o.cols())
         return col < o.cols() ? -1 : 1;
-    }
     
     // equal number of rows and columns, compare individual elements
     int cmpval;
-    for (int r=0; r<row; ++r) {
-        for (int c=0; c<col; ++c) {
-            cmpval=((*this)(r,c)).compare(o(r,c));
+    for (unsigned r=0; r<row; ++r) {
+        for (unsigned c=0; c<col; ++c) {
+            cmpval = ((*this)(r,c)).compare(o(r,c));
             if (cmpval!=0) return cmpval;
         }
     }
@@ -244,316 +331,311 @@ int matrix::compare_same_type(basic const & other) const
 /** Sum of matrices.
  *
  *  @exception logic_error (incompatible matrices) */
-matrix matrix::add(matrix const & other) const
+matrix matrix::add(const matrix & other) const
 {
-    if (col != other.col || row != other.row) {
+    if (col != other.col || row != other.row)
         throw (std::logic_error("matrix::add(): incompatible matrices"));
-    }
     
-    vector<ex> sum(this->m);
-    vector<ex>::iterator i;
-    vector<ex>::const_iterator ci;
-    for (i=sum.begin(), ci=other.m.begin();
-         i!=sum.end();
-         ++i, ++ci) {
+    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);
-    }
+    
     return matrix(row,col,sum);
 }
 
+
 /** Difference of matrices.
  *
  *  @exception logic_error (incompatible matrices) */
-matrix matrix::sub(matrix const & other) const
+matrix matrix::sub(const matrix & other) const
 {
-    if (col != other.col || row != other.row) {
+    if (col != other.col || row != other.row)
         throw (std::logic_error("matrix::sub(): incompatible matrices"));
-    }
     
-    vector<ex> dif(this->m);
-    vector<ex>::iterator i;
-    vector<ex>::const_iterator ci;
-    for (i=dif.begin(), ci=other.m.begin();
-         i!=dif.end();
-         ++i, ++ci) {
+    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);
-    }
+    
     return matrix(row,col,dif);
 }
 
+
 /** Product of matrices.
  *
  *  @exception logic_error (incompatible matrices) */
-matrix matrix::mul(matrix const & other) const
+matrix matrix::mul(const matrix & other) const
 {
-    if (col != other.row) {
+    if (this->cols() != other.rows())
         throw (std::logic_error("matrix::mul(): incompatible matrices"));
-    }
     
-    vector<ex> prod(row*other.col);
-    for (int i=0; i<row; ++i) {
-        for (int j=0; j<other.col; ++j) {
-            for (int l=0; l<col; ++l) {
-                prod[i*other.col+j] += m[i*col+l] * other.m[l*other.col+j];
-            }
+    exvector prod(this->rows()*other.cols());
+    
+    for (unsigned r1=0; r1<this->rows(); ++r1) {
+        for (unsigned c=0; c<this->cols(); ++c) {
+            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();
         }
     }
     return matrix(row, other.col, prod);
 }
 
+
 /** operator() to access elements.
  *
  *  @param ro row of element
- *  @param co column of element 
+ *  @param co column of element
  *  @exception range_error (index out of range) */
-ex const & matrix::operator() (int ro, int co) const
+const ex & matrix::operator() (unsigned ro, unsigned co) const
 {
-    if (ro<0 || ro>=row || co<0 || co>=col) {
+    if (ro<0 || ro>=row || co<0 || co>=col)
         throw (std::range_error("matrix::operator(): index out of range"));
-    }
-    
+
     return m[ro*col+co];
 }
 
+
 /** Set individual elements manually.
  *
  *  @exception range_error (index out of range) */
-matrix & matrix::set(int ro, int co, ex value)
+matrix & matrix::set(unsigned ro, unsigned co, ex value)
 {
-    if (ro<0 || ro>=row || co<0 || co>=col) {
+    if (ro<0 || ro>=row || co<0 || co>=col)
         throw (std::range_error("matrix::set(): index out of range"));
-    }
     
     ensure_if_modifiable();
-    m[ro*col+co]=value;
+    m[ro*col+co] = value;
     return *this;
 }
 
+
 /** Transposed of an m x n matrix, producing a new n x m matrix object that
  *  represents the transposed. */
 matrix matrix::transpose(void) const
 {
-    vector<ex> trans(col*row);
+    exvector trans(this->cols()*this->rows());
     
-    for (int r=0; r<col; ++r) {
-        for (int c=0; c<row; ++c) {
-            trans[r*row+c] = m[c*col+r];
-        }
-    }
-    return matrix(col,row,trans);
-}
-
-/* Determiant of purely numeric matrix, using pivoting. This routine is only
- * called internally by matrix::determinant(). */
-ex determinant_numeric(const matrix & M)
-{
-    ASSERT(M.rows()==M.cols());  // cannot happen, just in case...
-    matrix tmp(M);
-    ex det=exONE();
-    ex piv;
+    for (unsigned r=0; r<this->cols(); ++r)
+        for (unsigned c=0; c<this->rows(); ++c)
+            trans[r*this->rows()+c] = m[c*this->cols()+r];
     
-    for (int r1=0; r1<M.rows(); ++r1) {
-        int indx = tmp.pivot(r1);
-        if (indx == -1) {
-            return exZERO();
-        }
-        if (indx != 0) {
-            det *= exMINUSONE();
-        }
-        det = det * tmp.m[r1*M.cols()+r1];
-        for (int r2=r1+1; r2<M.rows(); ++r2) {
-            piv = tmp.m[r2*M.cols()+r1] / tmp.m[r1*M.cols()+r1];
-            for (int c=r1+1; c<M.cols(); c++) {
-                tmp.m[r2*M.cols()+c] -= piv * tmp.m[r1*M.cols()+c];
-            }
-        }
-    }
-    return det;
+    return matrix(this->cols(),this->rows(),trans);
 }
 
-// Compute the sign of a permutation of a vector of things, used internally
-// by determinant_symbolic_perm() where it is instantiated for int.
-template <class T>
-int permutation_sign(vector<T> s)
-{
-    if (s.size() < 2)
-        return 0;
-    int sigma=1;
-    for (typename vector<T>::iterator i=s.begin(); i!=s.end()-1; ++i) {
-        for (typename vector<T>::iterator j=i+1; j!=s.end(); ++j) {
-            if (*i == *j)
-                return 0;
-            if (*i > *j) {
-                iter_swap(i,j);
-                sigma = -sigma;
-            }
-        }
-    }
-    return sigma;
-}
 
-/** Determinant built by application of the full permutation group. This
- *  routine is only called internally by matrix::determinant(). */
-ex determinant_symbolic_perm(const matrix & M)
+/** 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
+ *  the determinant is also in that integral domain and the result is expanded
+ *  only.  If one or more elements are from a quotient field the determinant is
+ *  usually also in that quotient field and the result is normalized before it
+ *  is returned.  This implies that the determinant of the symbolic 2x2 matrix
+ *  [[a/(a-b),1],[b/(a-b),1]] is returned as unity.  (In this respect, it
+ *  behaves like MapleV and unlike Mathematica.)
+ *
+ *  @param     algo allows to chose an algorithm
+ *  @return    the determinant as a new expression
+ *  @exception logic_error (matrix not square)
+ *  @see       determinant_algo */
+ex matrix::determinant(unsigned algo) const
 {
-    ASSERT(M.rows()==M.cols());  // cannot happen, just in case...
+    if (row!=col)
+        throw (std::logic_error("matrix::determinant(): matrix not square"));
+    GINAC_ASSERT(row*col==m.capacity());
     
-    if (M.rows()==1) {  // speed things up
-        return M(0,0);
+    // Gather some statistical information about this matrix:
+    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);
+        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))
+            normal_flag = true;
     }
     
-    ex det;
-    ex term;
-    vector<int> sigma(M.cols());
-    for (int i=0; i<M.cols(); ++i) sigma[i]=i;
-    
-    do {
-        term = M(sigma[0],0);
-        for (int i=1; i<M.cols(); ++i) term *= M(sigma[i],i);
-        det += permutation_sign(sigma)*term;
-    } while (next_permutation(sigma.begin(), sigma.end()));
-    
-    return det;
-}
-
-/** Recursive determiant for small matrices having at least one symbolic entry.
- *  This algorithm is also known as Laplace-expansion. This routine is only
- *  called internally by matrix::determinant(). */
-ex determinant_symbolic_minor(const matrix & M)
-{
-    ASSERT(M.rows()==M.cols());  // cannot happen, just in case...
-    
-    if (M.rows()==1) {  // end of recursion
-        return M(0,0);
+    // Here is the heuristics in case this routine has to decide:
+    if (algo == determinant_algo::automatic) {
+        // Minor expansion is generally a good guess:
+        algo = determinant_algo::laplace;
+        // Does anybody know when a matrix is really sparse?
+        // Maybe <~row/2.236 nonzero elements average in a row?
+        if (row>3 && 5*sparse_count<=row*col)
+            algo = determinant_algo::bareiss;
+        // Purely numeric matrix can be handled by Gauss elimination.
+        // This overrides any prior decisions.
+        if (numeric_flag)
+            algo = determinant_algo::gauss;
     }
-    if (M.rows()==2) {  // speed things up
-        return (M(0,0)*M(1,1)-
-                M(1,0)*M(0,1));
-    }
-    if (M.rows()==3) {  // speed things up even a little more
-        return ((M(2,1)*M(0,2)-M(2,2)*M(0,1))*M(1,0)+
-                (M(1,2)*M(0,1)-M(1,1)*M(0,2))*M(2,0)+
-                (M(2,2)*M(1,1)-M(2,1)*M(1,2))*M(0,0));
+    
+    // Trap the trivial case here, since some algorithms don't like it
+    if (this->row==1) {
+        // for consistency with non-trivial determinants...
+        if (normal_flag)
+            return m[0].normal();
+        else
+            return m[0].expand();
     }
     
-    ex det;
-    matrix minorM(M.rows()-1,M.cols()-1);
-    for (int r1=0; r1<M.rows(); ++r1) {
-        // assemble the minor matrix
-        for (int r=0; r<minorM.rows(); ++r) {
-            for (int c=0; c<minorM.cols(); ++c) {
-                if (r<r1) {
-                    minorM.set(r,c,M(r,c+1));
-                } else {
-                    minorM.set(r,c,M(r+1,c+1));
-                }
-            }
+    // Compute the determinant
+    switch(algo) {
+        case determinant_algo::gauss: {
+            ex det = 1;
+            matrix tmp(*this);
+            int sign = tmp.gauss_elimination(true);
+            for (unsigned d=0; d<row; ++d)
+                det *= tmp.m[d*col+d];
+            if (normal_flag)
+                return (sign*det).normal();
+            else
+                return (sign*det).normal().expand();
         }
-        // recurse down
-        if (r1%2) {
-            det -= M(r1,0) * determinant_symbolic_minor(minorM);
-        } else {
-            det += M(r1,0) * determinant_symbolic_minor(minorM);
+        case determinant_algo::bareiss: {
+            matrix tmp(*this);
+            int sign;
+            sign = tmp.fraction_free_elimination(true);
+            if (normal_flag)
+                return (sign*tmp.m[row*col-1]).normal();
+            else
+                return (sign*tmp.m[row*col-1]).expand();
         }
-    }
-    return det;
-}
-
-/*  Leverrier algorithm for large matrices having at least one symbolic entry.
- *  This routine is only called internally by matrix::determinant(). The
- *  algorithm is deemed bad for symbolic matrices since it returns expressions
- *  that are very hard to canonicalize. */
-/*ex determinant_symbolic_leverrier(const matrix & M)
- *{
- *    ASSERT(M.rows()==M.cols());  // cannot happen, just in case...
- *    
- *    matrix B(M);
- *    matrix I(M.row, M.col);
- *    ex c=B.trace();
- *    for (int i=1; i<M.row; ++i) {
- *        for (int j=0; j<M.row; ++j)
- *            I.m[j*M.col+j] = c;
- *        B = M.mul(B.sub(I));
- *        c = B.trace()/ex(i+1);
- *    }
- *    if (M.row%2) {
- *        return c;
- *    } else {
- *        return -c;
- *    }
- *}*/
-
-/** Determinant of square matrix.  This routine doesn't actually calculate the
- *  determinant, it only implements some heuristics about which algorithm to
- *  call.  When the parameter for normalization is explicitly turned off this
- *  method does not normalize its result at the end, which might imply that
- *  the symbolic 2x2 matrix [[a/(a-b),1],[b/(a-b),1]] is not immediatly
- *  recognized to be unity.  (This is Mathematica's default behaviour, it
- *  should be used with care.)
- *
- *  @param     normalized may be set to false if no normalization of the
- *             result is desired (i.e. to force Mathematica behavior, Maple
- *             does normalize the result).
- *  @return    the determinant as a new expression
- *  @exception logic_error (matrix not square) */
-ex matrix::determinant(bool normalized) const
-{
-    if (row != col) {
-        throw (std::logic_error("matrix::determinant(): matrix not square"));
-    }
-
-    // check, if there are non-numeric entries in the matrix:
-    for (vector<ex>::const_iterator r=m.begin(); r!=m.end(); ++r) {
-        if (!(*r).info(info_flags::numeric)) {
-            if (normalized) {
-                return determinant_symbolic_minor(*this).normal();
-            } else {
-                return determinant_symbolic_perm(*this);
+        case determinant_algo::divfree: {
+            matrix tmp(*this);
+            int sign;
+            sign = tmp.division_free_elimination(true);
+            if (sign==0)
+                return _ex0();
+            ex det = tmp.m[row*col-1];
+            // factor out accumulated bogus slag
+            for (unsigned d=0; d<row-2; ++d)
+                for (unsigned j=0; j<row-d-2; ++j)
+                    det = (det/tmp.m[d*col+d]).normal();
+            return (sign*det);
+        }
+        case determinant_algo::laplace:
+        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:
+            typedef std::pair<unsigned,unsigned> uintpair;
+            std::vector<uintpair> c_zeros;  // number of zeros in column
+            for (unsigned c=0; c<col; ++c) {
+                unsigned acc = 0;
+                for (unsigned r=0; r<row; ++r)
+                    if (m[r*col+c].is_zero())
+                        ++acc;
+                c_zeros.push_back(uintpair(acc,c));
+            }
+            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);
+            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 (unsigned r=0; r<row; ++r)
+                    result[r*col+c] = m[r*col+(*i)];
             }
+            
+            if (normal_flag)
+                return (sign*matrix(row,col,result).determinant_minor()).normal();
+            else
+                return sign*matrix(row,col,result).determinant_minor();
         }
     }
-    // if it turns out that all elements are numeric
-    return determinant_numeric(*this);
 }
 
-/** Trace of a matrix.
+
+/** Trace of a matrix.  The result is normalized if it is in some quotient
+ *  field and expanded only otherwise.  This implies that the trace of the
+ *  symbolic 2x2 matrix [[a/(a-b),x],[y,b/(b-a)]] is recognized to be unity.
  *
  *  @return    the sum of diagonal elements
  *  @exception logic_error (matrix not square) */
 ex matrix::trace(void) const
 {
-    if (row != col) {
+    if (row != col)
         throw (std::logic_error("matrix::trace(): matrix not square"));
-    }
     
     ex tr;
-    for (int r=0; r<col; ++r) {
+    for (unsigned r=0; r<col; ++r)
         tr += m[r*col+r];
-    }
-    return tr;
+    
+    if (tr.info(info_flags::rational_function) &&
+        !tr.info(info_flags::crational_polynomial))
+        return tr.normal();
+    else
+        return tr.expand();
 }
 
-/** Characteristic Polynomial.  The characteristic polynomial of a matrix M is
- *  defined as the determiant of (M - lambda * 1) where 1 stands for the unit
- *  matrix of the same dimension as M.  This method returns the characteristic
- *  polynomial as a new expression.
+
+/** Characteristic Polynomial.  Following mathematica notation the
+ *  characteristic polynomial of a matrix M is defined as the determiant 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
+ *  returns the characteristic polynomial collected in powers of lambda as a
+ *  new expression.
  *
  *  @return    characteristic polynomial as new expression
  *  @exception logic_error (matrix not square)
  *  @see       matrix::determinant() */
-ex matrix::charpoly(ex const & lambda) const
+ex matrix::charpoly(const symbol & lambda) const
 {
-    if (row != col) {
+    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)) {
+            numeric_flag = false;
+        }
+    }
+    
+    // The pure numeric case is traditionally rather common.  Hence, it is
+    // 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);
+        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);
+        }
+        if (row%2)
+            return -poly;
+        else
+            return poly;
     }
     
     matrix M(*this);
-    for (int r=0; r<col; ++r) {
+    for (unsigned r=0; r<col; ++r)
         M.m[r*col+r] -= lambda;
-    }
-    return (M.determinant());
+    
+    return M.determinant().collect(lambda);
 }
 
+
 /** Inverse of this matrix.
  *
  *  @return    the inverted matrix
@@ -561,305 +643,575 @@ ex matrix::charpoly(ex const & lambda) const
  *  @exception runtime_error (singular matrix) */
 matrix matrix::inverse(void) const
 {
-    if (row != col) {
+    if (row != col)
         throw (std::logic_error("matrix::inverse(): matrix not square"));
-    }
     
+    // NOTE: the Gauss-Jordan elimination used here can in principle be
+    // replaced this 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 (int i=0; i<col; ++i) {
-        tmp.m[i*col+i] = exONE();
-    }
+    for (unsigned i=0; i<col; ++i)
+        tmp.m[i*col+i] = _ex1();
+    
     // create a copy of this matrix
     matrix cpy(*this);
-    for (int r1=0; r1<row; ++r1) {
-        int indx = cpy.pivot(r1);
+    for (unsigned r1=0; r1<row; ++r1) {
+        int indx = cpy.pivot(r1, r1);
         if (indx == -1) {
             throw (std::runtime_error("matrix::inverse(): singular matrix"));
         }
         if (indx != 0) {  // swap rows r and indx of matrix tmp
-            for (int i=0; i<col; ++i) {
+            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 (int c=0; c<col; ++c) {
+        for (unsigned c=0; c<col; ++c) {
             cpy.m[r1*col+c] /= a1;
             tmp.m[r1*col+c] /= a1;
         }
-        for (int r2=0; r2<row; ++r2) {
+        for (unsigned r2=0; r2<row; ++r2) {
             if (r2 != r1) {
                 ex a2 = cpy.m[r2*col+r1];
-                for (int c=0; c<col; ++c) {
+                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();
                 }
             }
         }
     }
-    return tmp;
-}
-
-void matrix::ffe_swap(int r1, int c1, int r2 ,int c2)
-{
-    ensure_if_modifiable();
     
-    ex tmp=ffe_get(r1,c1);
-    ffe_set(r1,c1,ffe_get(r2,c2));
-    ffe_set(r2,c2,tmp);
-}
-
-void matrix::ffe_set(int r, int c, ex e)
-{
-    set(r-1,c-1,e);
+    return tmp;
 }
 
-ex matrix::ffe_get(int r, int c) const
-{
-    return operator()(r-1,c-1);
-}
 
-/** Solve a set of equations for an m x n matrix by fraction-free Gaussian
- *  elimination. Based on algorithm 9.1 from 'Algorithms for Computer Algebra'
- *  by Keith O. Geddes et al.
+/** 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
+ *  @param vars n x p matrix, all elements must be symbols 
  *  @param rhs m x p matrix
+ *  @return n x p solution matrix
  *  @exception logic_error (incompatible matrices)
- *  @exception runtime_error (singular matrix) */
-matrix matrix::fraction_free_elim(matrix const & vars,
-                                  matrix const & rhs) const
+ *  @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
 {
-    if ((row != rhs.row) || (col != vars.row) || (rhs.col != vars.col)) {
+    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))
         throw (std::logic_error("matrix::solve(): incompatible matrices"));
+    for (unsigned ro=0; ro<n; ++ro)
+        for (unsigned co=0; co<p; ++co)
+            if (!vars(ro,co).info(info_flags::symbol))
+                throw (std::invalid_argument("matrix::solve(): 1st argument must be matrix of symbols"));
+    
+    // build the augmented matrix of *this with rhs attached to the right
+    matrix aug(m,n+p);
+    for (unsigned r=0; r<m; ++r) {
+        for (unsigned c=0; c<n; ++c)
+            aug.m[r*(n+p)+c] = this->m[r*n+c];
+        for (unsigned c=0; c<p; ++c)
+            aug.m[r*(n+p)+c+n] = rhs.m[r*p+c];
     }
     
-    matrix a(*this); // make a copy of the matrix
-    matrix b(rhs);     // make a copy of the rhs vector
-    
-    // given an m x n matrix a, reduce it to upper echelon form
-    int m=a.row;
-    int n=a.col;
-    int sign=1;
-    ex divisor=1;
-    int r=1;
-    
-    // eliminate below row r, with pivot in column k
-    for (int k=1; (k<=n)&&(r<=m); ++k) {
-        // find a nonzero pivot
-        int p;
-        for (p=r; (p<=m)&&(a.ffe_get(p,k).is_equal(exZERO())); ++p) {}
-        // pivot is in row p
-        if (p<=m) {
-            if (p!=r) {
-                // switch rows p and r
-                for (int j=k; j<=n; ++j) {
-                    a.ffe_swap(p,j,r,j);
-                }
-                b.ffe_swap(p,1,r,1);
-                // keep track of sign changes due to row exchange
-                sign=-sign;
-            }
-            for (int i=r+1; i<=m; ++i) {
-                for (int j=k+1; j<=n; ++j) {
-                    a.ffe_set(i,j,(a.ffe_get(r,k)*a.ffe_get(i,j)
-                                  -a.ffe_get(r,j)*a.ffe_get(i,k))/divisor);
-                    a.ffe_set(i,j,a.ffe_get(i,j).normal() /*.normal() */ );
-                }
-                b.ffe_set(i,1,(a.ffe_get(r,k)*b.ffe_get(i,1)
-                              -b.ffe_get(r,1)*a.ffe_get(i,k))/divisor);
-                b.ffe_set(i,1,b.ffe_get(i,1).normal() /*.normal() */ );
-                a.ffe_set(i,k,0);
-            }
-            divisor=a.ffe_get(r,k);
-            r++;
-        }
+    // 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;
     }
-    // optionally compute the determinant for square or augmented matrices
-    // if (r==m+1) { det=sign*divisor; } else { det=0; }
     
-    /*
-    for (int r=1; r<=m; ++r) {
-        for (int c=1; c<=n; ++c) {
-            cout << a.ffe_get(r,c) << "\t";
-        }
-        cout << " | " <<  b.ffe_get(r,1) << endl;
+    // 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();
+        case solve_algo::divfree:
+            aug.division_free_elimination();
+        case solve_algo::bareiss:
+        default:
+            aug.fraction_free_elimination();
     }
-    */
-    
-#ifdef DOASSERT
-    // test if we really have an upper echelon matrix
-    int zero_in_last_row=-1;
-    for (int r=1; r<=m; ++r) {
-        int zero_in_this_row=0;
-        for (int c=1; c<=n; ++c) {
-            if (a.ffe_get(r,c).is_equal(exZERO())) {
-               zero_in_this_row++;
+    
+    // assemble the solution matrix:
+    matrix sol(n,p);
+    for (unsigned co=0; co<p; ++co) {
+        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()))
+                ++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()) {
+                    throw (std::runtime_error("matrix::solve(): inconsistent linear system"));
+                }
             } else {
-                break;
+                // 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]);
+                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());
+                last_assigned_sol = fnz;
             }
         }
-        ASSERT((zero_in_this_row>zero_in_last_row)||(zero_in_this_row=n));
-        zero_in_last_row=zero_in_this_row;
+        // 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));
     }
-#endif // def DOASSERT
-    
-    // assemble solution
-    matrix sol(n,1);
-    int last_assigned_sol=n+1;
-    for (int r=m; r>0; --r) {
-        int first_non_zero=1;
-        while ((first_non_zero<=n)&&(a.ffe_get(r,first_non_zero).is_zero())) {
-            first_non_zero++;
-        }
-        if (first_non_zero>n) {
-            // row consists only of zeroes, corresponding rhs must be 0 as well
-            if (!b.ffe_get(r,1).is_zero()) {
-                throw (std::runtime_error("matrix::fraction_free_elim(): singular matrix"));
-            }
-        } else {
-            // assign solutions for vars between first_non_zero+1 and
-            // last_assigned_sol-1: free parameters
-            for (int c=first_non_zero+1; c<=last_assigned_sol-1; ++c) {
-                sol.ffe_set(c,1,vars.ffe_get(c,1));
+    
+    return sol;
+}
+
+
+// protected
+
+/** Recursive determinant for small matrices having at least one symbolic
+ *  entry.  The basic algorithm, known as Laplace-expansion, is enhanced by
+ *  some bookkeeping to avoid calculation of the same submatrices ("minors")
+ *  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.
+ *
+ *  @return the determinant as a new expression (in expanded form)
+ *  @see matrix::determinant() */
+ex matrix::determinant_minor(void) 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;
+    // matrix minorM(this->rows()-1,this->cols()-1);
+    // for (unsigned r1=0; r1<this->rows(); ++r1) {
+    //     // shortcut if element(r1,0) vanishes
+    //     if (m[r1*col].is_zero())
+    //         continue;
+    //     // assemble the minor matrix
+    //     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]);
+    //             else
+    //                 minorM.set(r,c,m[(r+1)*col+c+1]);
+    //         }
+    //     }
+    //     // recurse down and care for sign:
+    //     if (r1%2)
+    //         det -= m[r1*col] * minorM.determinant_minor();
+    //     else
+    //         det += m[r1*col] * minorM.determinant_minor();
+    // }
+    // return det.expand();
+    // What happens is that while proceeding down many of the minors are
+    // computed more than once.  In particular, there are binomial(n,k)
+    // kxk minors and each one is computed factorial(n-k) times.  Therefore
+    // it is reasonable to store the results of the minors.  We proceed from
+    // right to left.  At each column c we only need to retrieve the minors
+    // 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;
+    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;
+    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 (unsigned i=0; i<n-c; ++i)
+            Pkey.push_back(i);
+        unsigned fc = 0;  // controls logic for our strange flipper counter
+        do {
+            det = _ex0();
+            for (unsigned r=0; r<n-c; ++r) {
+                // maybe there is nothing to do?
+                if (m[Pkey[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
+                if (r%2)
+                    det -= m[Pkey[r]*n+c]*A[Mkey];
+                else
+                    det += m[Pkey[r]*n+c]*A[Mkey];
             }
-            ex e=b.ffe_get(r,1);
-            for (int c=first_non_zero+1; c<=n; ++c) {
-                e=e-a.ffe_get(r,c)*sol.ffe_get(c,1);
+            // prevent build-up of deep nesting of expressions saves time:
+            det = det.expand();
+            // store the new determinant at its place in B:
+            if (!det.is_zero())
+                B.insert(Rmap_value(Pkey,det));
+            // increment our strange flipper counter
+            for (fc=n-c; fc>0; --fc) {
+                ++Pkey[fc-1];
+                if (Pkey[fc-1]<fc+c)
+                    break;
             }
-            sol.ffe_set(first_non_zero,1,
-                        (e/a.ffe_get(r,first_non_zero)).normal());
-            last_assigned_sol=first_non_zero;
-        }
-    }
-    // assign solutions for vars between 1 and
-    // last_assigned_sol-1: free parameters
-    for (int c=1; c<=last_assigned_sol-1; ++c) {
-        sol.ffe_set(c,1,vars.ffe_get(c,1));
+            if (fc<n-c)
+                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();
     }
+    
+    return det;
+}
 
-    /*
-    for (int c=1; c<=n; ++c) {
-        cout << vars.ffe_get(c,1) << "->" << sol.ffe_get(c,1) << endl;
-    }
-    */
-    
-#ifdef DOASSERT
-    // test solution with echelon matrix
-    for (int r=1; r<=m; ++r) {
-        ex e=0;
-        for (int c=1; c<=n; ++c) {
-            e=e+a.ffe_get(r,c)*sol.ffe_get(c,1);
+
+/** 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
+ *  with numeric coefficients but quite unsuited for symbolic matrices.
+ *
+ *  @param det may be set to true to save a lot of space if one is only
+ *  interested in the diagonal elements (i.e. for calculating determinants).
+ *  The others are set to zero in this case.
+ *  @return sign is 1 if an even number of rows was swapped, -1 if an odd
+ *  number of rows was swapped and 0 if the matrix is singular. */
+int matrix::gauss_elimination(const bool det)
+{
+    ensure_if_modifiable();
+    const unsigned m = this->rows();
+    const unsigned n = this->cols();
+    GINAC_ASSERT(!det || n==m);
+    int sign = 1;
+    
+    unsigned r0 = 0;
+    for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
+        int indx = pivot(r0, r1, true);
+        if (indx == -1) {
+            sign = 0;
+            if (det)
+                return 0;  // leaves *this in a messy state
         }
-        if (!(e-b.ffe_get(r,1)).normal().is_zero()) {
-            cout << "e=" << e;
-            cout << "b.ffe_get(" << r<<",1)=" << b.ffe_get(r,1) << endl;
-            cout << "diff=" << (e-b.ffe_get(r,1)).normal() << endl;
+        if (indx>=0) {
+            if (indx > 0)
+                sign = -sign;
+            for (unsigned r2=r0+1; r2<m; ++r2) {
+                ex piv = this->m[r2*n+r1] / this->m[r0*n+r1];
+                for (unsigned c=r1+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();
+            }
+            if (det) {
+                // save space by deleting no longer needed elements
+                for (unsigned c=r0+1; c<n; ++c)
+                    this->m[r0*n+c] = _ex0();
+            }
+            ++r0;
         }
-        ASSERT((e-b.ffe_get(r,1)).normal().is_zero());
     }
+    
+    return sign;
+}
 
-    // test solution with original matrix
-    for (int r=1; r<=m; ++r) {
-        ex e=0;
-        for (int c=1; c<=n; ++c) {
-            e=e+ffe_get(r,c)*sol.ffe_get(c,1);
-        }
-        try {
-        if (!(e-rhs.ffe_get(r,1)).normal().is_zero()) {
-            cout << "e=" << e << endl;
-            e.printtree(cout);
-            ex en=e.normal();
-            cout << "e.normal()=" << en << endl;
-            en.printtree(cout);
-            cout << "rhs.ffe_get(" << r<<",1)=" << rhs.ffe_get(r,1) << endl;
-            cout << "diff=" << (e-rhs.ffe_get(r,1)).normal() << endl;
+
+/** Perform the steps of division free elimination to bring the m x n matrix
+ *  into an upper echelon form.
+ *
+ *  @param det may be set to true to save a lot of space if one is only
+ *  interested in the diagonal elements (i.e. for calculating determinants).
+ *  The others are set to zero in this case.
+ *  @return sign is 1 if an even number of rows was swapped, -1 if an odd
+ *  number of rows was swapped and 0 if the matrix is singular. */
+int matrix::division_free_elimination(const bool det)
+{
+    ensure_if_modifiable();
+    const unsigned m = this->rows();
+    const unsigned n = this->cols();
+    GINAC_ASSERT(!det || n==m);
+    int sign = 1;
+    
+    unsigned r0 = 0;
+    for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
+        int indx = pivot(r0, r1, true);
+        if (indx==-1) {
+            sign = 0;
+            if (det)
+                return 0;  // leaves *this in a messy state
         }
-        } catch (...) {
-            ex xxx=e-rhs.ffe_get(r,1);
-            cerr << "xxx=" << xxx << endl << endl;
+        if (indx>=0) {
+            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();
+                // fill up left hand side with zeros
+                for (unsigned c=0; c<=r1; ++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();
+            }
+            ++r0;
         }
-        ASSERT((e-rhs.ffe_get(r,1)).normal().is_zero());
     }
-#endif // def DOASSERT
     
-    return sol;
-}   
-    
-/** Solve simultaneous set of equations. */
-matrix matrix::solve(matrix const & v) const
+    return sign;
+}
+
+
+/** Perform the steps of Bareiss' one-step fraction free elimination to bring
+ *  the matrix into an upper echelon form.  Fraction free elimination means
+ *  that divide is used straightforwardly, without computing GCDs first.  This
+ *  is possible, since we know the divisor at each step.
+ *  
+ *  @param det may be set to true to save a lot of space if one is only
+ *  interested in the last element (i.e. for calculating determinants). The
+ *  others are set to zero in this case.
+ *  @return sign is 1 if an even number of rows was swapped, -1 if an odd
+ *  number of rows was swapped and 0 if the matrix is singular. */
+int matrix::fraction_free_elimination(const bool det)
 {
-    if (!(row == col && col == v.row)) {
-        throw (std::logic_error("matrix::solve(): incompatible matrices"));
-    }
+    // Method:
+    // (single-step fraction free elimination scheme, already known to Jordan)
+    //
+    // Usual division-free elimination sets m[0](r,c) = m(r,c) and then sets
+    //     m[k+1](r,c) = m[k](k,k) * m[k](r,c) - m[k](r,k) * m[k](k,c).
+    //
+    // 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).
+    //
+    // We also allow rational functions where the original prove still holds.
+    // However, we must care for numerator and denominator separately and
+    // "manually" work in the integral domains because of subtle cancellations
+    // (see below).  This blows up the bookkeeping a bit and the formula has
+    // to be modified to expand like this (N{x} stands for numerator of x,
+    // D{x} for denominator of x):
+    //     N{m[k+1](r,c)} = N{m[k](k,k)}*N{m[k](r,c)}*D{m[k](r,k)}*D{m[k](k,c)}
+    //                     -N{m[k](r,k)}*N{m[k](k,c)}*D{m[k](k,k)}*D{m[k](r,c)}
+    //     D{m[k+1](r,c)} = D{m[k](k,k)}*D{m[k](r,c)}*D{m[k](r,k)}*D{m[k](k,c)}
+    // where for k>1 we now divide N{m[k+1](r,c)} by
+    //     N{m[k-1](k-1,k-1)}
+    // and D{m[k+1](r,c)} by
+    //     D{m[k-1](k-1,k-1)}.
     
-    // build the extended matrix of *this with v attached to the right
-    matrix tmp(row,col+v.col);
-    for (int r=0; r<row; ++r) {
-        for (int c=0; c<col; ++c) {
-            tmp.m[r*tmp.col+c] = m[r*col+c];
-        }
-        for (int c=0; c<v.col; ++c) {
-            tmp.m[r*tmp.col+c+col] = v.m[r*v.col+c];
-        }
+    ensure_if_modifiable();
+    const unsigned m = this->rows();
+    const unsigned n = this->cols();
+    GINAC_ASSERT(!det || n==m);
+    int sign = 1;
+    if (m==1)
+        return 1;
+    ex divisor_n = 1;
+    ex divisor_d = 1;
+    ex dividend_n;
+    ex dividend_d;
+    
+    // We populate temporary matrices to subsequently operate on.  There is
+    // one holding numerators and another holding denominators of entries.
+    // This is a must since the evaluator (or even earlier mul's constructor)
+    // might cancel some trivial element which causes divide() to fail.  The
+    // elements are normalized first (yes, even though this algorithm doesn't
+    // need GCDs) since the elements of *this might be unnormalized, which
+    // 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();
     }
-    for (int r1=0; r1<row; ++r1) {
-        int indx = tmp.pivot(r1);
-        if (indx == -1) {
-            throw (std::runtime_error("matrix::solve(): singular matrix"));
-        }
-        for (int c=r1; c<tmp.col; ++c) {
-            tmp.m[r1*tmp.col+c] /= tmp.m[r1*tmp.col+r1];
+    
+    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) {
+            sign = 0;
+            if (det)
+                return 0;
         }
-        for (int r2=r1+1; r2<row; ++r2) {
-            for (int c=r1; c<tmp.col; ++c) {
-                tmp.m[r2*tmp.col+c]
-                    -= tmp.m[r2*tmp.col+r1] * tmp.m[r1*tmp.col+c];
+        if (indx>=0) {
+            if (indx>0) {
+                sign = -sign;
+                // tmp_n's rows r0 and indx were swapped, do the same in tmp_d:
+                for (unsigned c=r1; c<n; ++c)
+                    tmp_d.m[n*indx+c].swap(tmp_d.m[n*r0+c]);
             }
-        }
-    }
-    
-    // assemble the solution matrix
-    vector<ex> sol(v.row*v.col);
-    for (int c=0; c<v.col; ++c) {
-        for (int r=col-1; r>=0; --r) {
-            sol[r*v.col+c] = tmp[r*tmp.col+c];
-            for (int i=r+1; i<col; ++i) {
-                sol[r*v.col+c]
-                    -= tmp[r*tmp.col+i] * sol[i*v.col+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();
+                    bool check = divide(dividend_n, divisor_n,
+                                        tmp_n.m[r2*n+c], true);
+                    check &= divide(dividend_d, divisor_d,
+                                    tmp_d.m[r2*n+c], true);
+                    GINAC_ASSERT(check);
+                }
+                // fill up left hand side with zeros
+                for (unsigned c=0; c<=r1; ++c)
+                    tmp_n.m[r2*n+c] = _ex0();
             }
+            if ((r1<n-1)&&(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();
+                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();
+                    }
+                }
+            }
+            ++r0;
         }
     }
-    return matrix(v.row, v.col, sol);
+    // 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);
+    
+    return sign;
 }
 
-// protected
 
-/** Partial pivoting method.
- *  Usual pivoting returns the index to the element with the largest absolute
- *  value and swaps the current row with the one where the element was found.
- *  Here it does the same with the first non-zero element. (This works fine,
- *  but may be far from optimal for numerics.) */
-int matrix::pivot(int ro)
+/** Partial pivoting method for matrix elimination schemes.
+ *  Usual pivoting (symbolic==false) returns the index to the element with the
+ *  largest absolute value in column ro and swaps the current row with the one
+ *  where the element was found.  With (symbolic==true) it does the same thing
+ *  with the first non-zero element.
+ *
+ *  @param ro is the row from where to begin
+ *  @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
+ *  a degeneracy) and positive integer k means that rows ro and k were swapped.
+ */
+int matrix::pivot(unsigned ro, unsigned co, bool symbolic)
 {
-    int k=ro;
-    
-    for (int r=ro; r<row; ++r) {
-        if (!m[r*col+ro].is_zero()) {
-            k = r;
-            break;
+    unsigned k = ro;
+    if (symbolic) {
+        // search first non-zero element in column co beginning at row ro
+        while ((k<row) && (this->m[k*col+co].expand().is_zero()))
+            ++k;
+    } else {
+        // search largest element in column co beginning at row ro
+        GINAC_ASSERT(is_ex_of_type(this->m[k*col+co],numeric));
+        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));
+            numeric tmp = ex_to_numeric(this->m[kmax*col+co]);
+            if (abs(tmp) > mmax) {
+                mmax = tmp;
+                k = kmax;
+            }
+            ++kmax;
         }
+        if (!mmax.is_zero())
+            k = kmax;
     }
-    if (m[k*col+ro].is_zero()) {
+    if (k==row)
+        // all elements in column co below row ro vanish
         return -1;
-    }
-    if (k!=ro) {  // swap rows
-        for (int c=0; c<col; ++c) {
-            m[k*col+c].swap(m[ro*col+c]);
-        }
-        return k;
-    }
-    return 0;
+    if (k==ro)
+        // matrix needs no pivoting
+        return 0;
+    // matrix needs pivoting, so swap rows k and ro
+    ensure_if_modifiable();
+    for (unsigned c=0; c<col; ++c)
+        m[k*col+c].swap(m[ro*col+c]);
+    
+    return k;
+}
+
+/** Convert list of lists to matrix. */
+ex lst_to_matrix(const ex &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();
+    
+    // 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;
 }
 
 //////////
@@ -867,4 +1219,8 @@ int matrix::pivot(int ro)
 //////////
 
 const matrix some_matrix;
-type_info const & typeid_matrix=typeid(some_matrix);
+const type_info & typeid_matrix=typeid(some_matrix);
+
+#ifndef NO_NAMESPACE_GINAC
+} // namespace GiNaC
+#endif // ndef NO_NAMESPACE_GINAC