X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fmatrix.cpp;h=1f4ed18b0a54e0e29ae2fd69aeb02a666add0c0d;hp=97b473628464e498a233c691e2d0b724f34e7491;hb=f6f99c4d47762da9e3d73a2f5ec6f062e82505b8;hpb=5f00c012381b3671df5a9fbfeec2d36af0f75b61

diff --git a/ginac/matrix.cpp b/ginac/matrix.cpp
index 97b47362..1f4ed18b 100644
--- a/ginac/matrix.cpp
+++ b/ginac/matrix.cpp
@@ -21,12 +21,14 @@
  */
 
 #include <algorithm>
+#include <map>
 #include <stdexcept>
 
 #include "matrix.h"
 #include "archive.h"
 #include "utils.h"
 #include "debugmsg.h"
+#include "numeric.h"
 
 #ifndef NO_NAMESPACE_GINAC
 namespace GiNaC {
@@ -248,14 +250,12 @@ 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
     exvector m2(row*col);
@@ -276,9 +276,8 @@ 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) {
@@ -301,23 +300,21 @@ ex matrix::evalf(int level) const
 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));
+    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 (unsigned r=0; r<row; ++r) {
         for (unsigned c=0; c<col; ++c) {
-            cmpval=((*this)(r,c)).compare(o(r,c));
+            cmpval = ((*this)(r,c)).compare(o(r,c));
             if (cmpval!=0) return cmpval;
         }
     }
@@ -336,9 +333,8 @@ int matrix::compare_same_type(const basic & other) const
  *  @exception logic_error (incompatible matrices) */
 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"));
-    }
     
     exvector sum(this->m);
     exvector::iterator i;
@@ -351,14 +347,14 @@ matrix matrix::add(const matrix & other) const
     return matrix(row,col,sum);
 }
 
+
 /** Difference of matrices.
  *
  *  @exception logic_error (incompatible matrices) */
 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"));
-    }
     
     exvector dif(this->m);
     exvector::iterator i;
@@ -371,14 +367,14 @@ matrix matrix::sub(const matrix & other) const
     return matrix(row,col,dif);
 }
 
+
 /** Product of matrices.
  *
  *  @exception logic_error (incompatible matrices) */
 matrix matrix::mul(const matrix & other) const
 {
-    if (col != other.row) {
+    if (col != other.row)
         throw (std::logic_error("matrix::mul(): incompatible matrices"));
-    }
     
     exvector prod(row*other.col);
     for (unsigned i=0; i<row; ++i) {
@@ -391,6 +387,7 @@ matrix matrix::mul(const matrix & other) const
     return matrix(row, other.col, prod);
 }
 
+
 /** operator() to access elements.
  *
  *  @param ro row of element
@@ -398,231 +395,106 @@ matrix matrix::mul(const matrix & other) const
  *  @exception range_error (index out of range) */
 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(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
 {
     exvector trans(col*row);
     
-    for (unsigned r=0; r<col; ++r) {
-        for (unsigned c=0; c<row; ++c) {
+    for (unsigned r=0; r<col; ++r)
+        for (unsigned 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)
-{
-    GINAC_ASSERT(M.rows()==M.cols());  // cannot happen, just in case...
-    matrix tmp(M);
-    ex det=_ex1();
-    ex piv;
-    
-    for (unsigned r1=0; r1<M.rows(); ++r1) {
-        int indx = tmp.pivot(r1);
-        if (indx == -1) {
-            return _ex0();
-        }
-        if (indx != 0) {
-            det *= _ex_1();
-        }
-        det = det * tmp.m[r1*M.cols()+r1];
-        for (unsigned r2=r1+1; r2<M.rows(); ++r2) {
-            piv = tmp.m[r2*M.cols()+r1] / tmp.m[r1*M.cols()+r1];
-            for (unsigned c=r1+1; c<M.cols(); c++) {
-                tmp.m[r2*M.cols()+c] -= piv * tmp.m[r1*M.cols()+c];
-            }
-        }
-    }
-    return det;
-}
-
-// Compute the sign of a permutation of a vector of things, used internally
-// by determinant_symbolic_perm() where it is instantiated for int.
-template <typename 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)
-{
-    GINAC_ASSERT(M.rows()==M.cols());  // cannot happen, just in case...
-    
-    if (M.rows()==1) {  // speed things up
-        return M(0,0);
-    }
-    
-    ex det;
-    ex term;
-    vector<unsigned> sigma(M.cols());
-    for (unsigned i=0; i<M.cols(); ++i) sigma[i]=i;
-    
-    do {
-        term = M(sigma[0],0);
-        for (unsigned 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)
-{
-    GINAC_ASSERT(M.rows()==M.cols());  // cannot happen, just in case...
-    
-    if (M.rows()==1) {  // end of recursion
-        return M(0,0);
-    }
-    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));
-    }
     
-    ex det;
-    matrix minorM(M.rows()-1,M.cols()-1);
-    for (unsigned r1=0; r1<M.rows(); ++r1) {
-        // 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,c+1));
-                } else {
-                    minorM.set(r,c,M(r+1,c+1));
-                }
-            }
-        }
-        // recurse down
-        if (r1%2) {
-            det -= M(r1,0) * determinant_symbolic_minor(minorM);
-        } else {
-            det += M(r1,0) * determinant_symbolic_minor(minorM);
-        }
-    }
-    return det;
+    return matrix(col,row,trans);
 }
 
-/*  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)
- *{
- *    GINAC_ASSERT(M.rows()==M.cols());  // cannot happen, just in case...
- *    
- *    matrix B(M);
- *    matrix I(M.row, M.col);
- *    ex c=B.trace();
- *    for (unsigned i=1; i<M.row; ++i) {
- *        for (unsigned 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.)
+ *  call.  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     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
+ex matrix::determinant(void) const
 {
-    if (row != col) {
+    if (row != col)
         throw (std::logic_error("matrix::determinant(): matrix not square"));
-    }
-
-    // check, if there are non-numeric entries in the matrix:
+    GINAC_ASSERT(row*col==m.capacity());
+    
+    ex det;
+    bool numeric_flag = true;
+    bool normal_flag = false;
     for (exvector::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);
-            }
-        }
+        if (!(*r).info(info_flags::numeric))
+            numeric_flag = false;
+        if ((*r).info(info_flags::rational_function) &&
+            !(*r).info(info_flags::crational_polynomial))
+            normal_flag = true;
     }
-    // if it turns out that all elements are numeric
-    return determinant_numeric(*this);
+    
+    if (numeric_flag)
+        det = determinant_numeric();
+    else
+        if (normal_flag)
+            det = determinant_minor().normal();
+        else
+            det = determinant_minor(); // is already expanded!
+    
+    return det;
 }
 
-/** 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"));
-    }
+    GINAC_ASSERT(row*col==m.capacity());
     
     ex tr;
-    for (unsigned 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
@@ -633,17 +505,17 @@ ex matrix::trace(void) const
  *  @see       matrix::determinant() */
 ex matrix::charpoly(const ex & lambda) const
 {
-    if (row != col) {
+    if (row != col)
         throw (std::logic_error("matrix::charpoly(): matrix not square"));
-    }
     
     matrix M(*this);
-    for (unsigned r=0; r<col; ++r) {
+    for (unsigned r=0; r<col; ++r)
         M.m[r*col+r] -= lambda;
-    }
+    
     return (M.determinant());
 }
 
+
 /** Inverse of this matrix.
  *
  *  @return    the inverted matrix
@@ -651,15 +523,14 @@ ex matrix::charpoly(const ex & 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"));
-    }
     
     matrix tmp(row,col);
     // set tmp to the unit matrix
-    for (unsigned i=0; i<col; ++i) {
+    for (unsigned i=0; i<col; ++i)
         tmp.m[i*col+i] = _ex1();
-    }
+
     // create a copy of this matrix
     matrix cpy(*this);
     for (unsigned r1=0; r1<row; ++r1) {
@@ -690,27 +561,31 @@ matrix matrix::inverse(void) const
     return tmp;
 }
 
+
+// superfluous helper function
 void matrix::ffe_swap(unsigned r1, unsigned c1, unsigned r2 ,unsigned c2)
 {
     ensure_if_modifiable();
     
-    ex tmp=ffe_get(r1,c1);
+    ex tmp = ffe_get(r1,c1);
     ffe_set(r1,c1,ffe_get(r2,c2));
     ffe_set(r2,c2,tmp);
 }
 
+// superfluous helper function
 void matrix::ffe_set(unsigned r, unsigned c, ex e)
 {
     set(r-1,c-1,e);
 }
 
+// superfluous helper function
 ex matrix::ffe_get(unsigned r, unsigned 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'
+ *  elimination.  Based on algorithm 9.1 from 'Algorithms for Computer Algebra'
  *  by Keith O. Geddes et al.
  *
  *  @param vars n x p matrix
@@ -720,19 +595,19 @@ ex matrix::ffe_get(unsigned r, unsigned c) const
 matrix matrix::fraction_free_elim(const matrix & vars,
                                   const matrix & rhs) const
 {
-    if ((row != rhs.row) || (col != vars.row) || (rhs.col != vars.col)) {
-        throw (std::logic_error("matrix::solve(): incompatible matrices"));
-    }
+    // FIXME: implement a Sasaki-Murao scheme which avoids division at all!
+    if ((row != rhs.row) || (col != vars.row) || (rhs.col != vars.col))
+        throw (std::logic_error("matrix::fraction_free_elim(): incompatible matrices"));
     
-    matrix a(*this); // make a copy of the matrix
-    matrix b(rhs);     // make a copy of the rhs vector
+    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
-    unsigned m=a.row;
-    unsigned n=a.col;
-    int sign=1;
-    ex divisor=1;
-    unsigned r=1;
+    unsigned m = a.row;
+    unsigned n = a.col;
+    int sign = 1;
+    ex divisor = 1;
+    unsigned r = 1;
     
     // eliminate below row r, with pivot in column k
     for (unsigned k=1; (k<=n)&&(r<=m); ++k) {
@@ -743,12 +618,11 @@ matrix matrix::fraction_free_elim(const matrix & vars,
         if (p<=m) {
             if (p!=r) {
                 // switch rows p and r
-                for (unsigned j=k; j<=n; ++j) {
+                for (unsigned 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;
+                sign = -sign;
             }
             for (unsigned i=r+1; i<=m; ++i) {
                 for (unsigned j=k+1; j<=n; ++j) {
@@ -761,12 +635,12 @@ matrix matrix::fraction_free_elim(const matrix & vars,
                 b.ffe_set(i,1,b.ffe_get(i,1).normal() /*.normal() */ );
                 a.ffe_set(i,k,0);
             }
-            divisor=a.ffe_get(r,k);
+            divisor = a.ffe_get(r,k);
             r++;
         }
     }
     // optionally compute the determinant for square or augmented matrices
-    // if (r==m+1) { det=sign*divisor; } else { det=0; }
+    // if (r==m+1) { det = sign*divisor; } else { det = 0; }
     
     /*
     for (unsigned r=1; r<=m; ++r) {
@@ -779,29 +653,33 @@ matrix matrix::fraction_free_elim(const matrix & vars,
     
 #ifdef DO_GINAC_ASSERT
     // test if we really have an upper echelon matrix
-    int zero_in_last_row=-1;
+    int zero_in_last_row = -1;
     for (unsigned r=1; r<=m; ++r) {
         int zero_in_this_row=0;
         for (unsigned c=1; c<=n; ++c) {
-            if (a.ffe_get(r,c).is_equal(_ex0())) {
+            if (a.ffe_get(r,c).is_equal(_ex0()))
                zero_in_this_row++;
-            } else {
+            else
                 break;
-            }
         }
         GINAC_ASSERT((zero_in_this_row>zero_in_last_row)||(zero_in_this_row=n));
-        zero_in_last_row=zero_in_this_row;
+        zero_in_last_row = zero_in_this_row;
     }
 #endif // def DO_GINAC_ASSERT
     
+    /*
+    cout << "after" << endl;
+    cout << "a=" << a << endl;
+    cout << "b=" << b << endl;
+    */
+    
     // assemble solution
     matrix sol(n,1);
-    unsigned last_assigned_sol=n+1;
+    unsigned last_assigned_sol = n+1;
     for (unsigned r=m; r>0; --r) {
-        unsigned first_non_zero=1;
-        while ((first_non_zero<=n)&&(a.ffe_get(r,first_non_zero).is_zero())) {
+        unsigned 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()) {
@@ -813,34 +691,26 @@ matrix matrix::fraction_free_elim(const matrix & vars,
             for (unsigned c=first_non_zero+1; c<=last_assigned_sol-1; ++c) {
                 sol.ffe_set(c,1,vars.ffe_get(c,1));
             }
-            ex e=b.ffe_get(r,1);
+            ex e = b.ffe_get(r,1);
             for (unsigned c=first_non_zero+1; c<=n; ++c) {
                 e=e-a.ffe_get(r,c)*sol.ffe_get(c,1);
             }
             sol.ffe_set(first_non_zero,1,
                         (e/a.ffe_get(r,first_non_zero)).normal());
-            last_assigned_sol=first_non_zero;
+            last_assigned_sol = first_non_zero;
         }
     }
     // assign solutions for vars between 1 and
     // last_assigned_sol-1: free parameters
-    for (unsigned c=1; c<=last_assigned_sol-1; ++c) {
+    for (unsigned c=1; c<=last_assigned_sol-1; ++c)
         sol.ffe_set(c,1,vars.ffe_get(c,1));
-    }
-
-    /*
-    for (unsigned c=1; c<=n; ++c) {
-        cout << vars.ffe_get(c,1) << "->" << sol.ffe_get(c,1) << endl;
-    }
-    */
     
 #ifdef DO_GINAC_ASSERT
     // test solution with echelon matrix
     for (unsigned r=1; r<=m; ++r) {
-        ex e=0;
-        for (unsigned c=1; c<=n; ++c) {
-            e=e+a.ffe_get(r,c)*sol.ffe_get(c,1);
-        }
+        ex e = 0;
+        for (unsigned c=1; c<=n; ++c)
+            e = e+a.ffe_get(r,c)*sol.ffe_get(c,1);
         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;
@@ -848,25 +718,24 @@ matrix matrix::fraction_free_elim(const matrix & vars,
         }
         GINAC_ASSERT((e-b.ffe_get(r,1)).normal().is_zero());
     }
-
+    
     // test solution with original matrix
     for (unsigned r=1; r<=m; ++r) {
-        ex e=0;
-        for (unsigned c=1; c<=n; ++c) {
-            e=e+ffe_get(r,c)*sol.ffe_get(c,1);
-        }
+        ex e = 0;
+        for (unsigned 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;
-        }
+            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;
+            }
         } catch (...) {
-            ex xxx=e-rhs.ffe_get(r,1);
+            ex xxx = e - rhs.ffe_get(r,1);
             cerr << "xxx=" << xxx << endl << endl;
         }
         GINAC_ASSERT((e-rhs.ffe_get(r,1)).normal().is_zero());
@@ -874,76 +743,319 @@ matrix matrix::fraction_free_elim(const matrix & vars,
 #endif // def DO_GINAC_ASSERT
     
     return sol;
-}   
+}
+
+/** Solve a set of equations for an m x n matrix.
+ *
+ *  @param vars n x p matrix
+ *  @param rhs m x p matrix
+ *  @exception logic_error (incompatible matrices)
+ *  @exception runtime_error (singular matrix) */
+matrix matrix::solve(const matrix & vars,
+                     const matrix & rhs) const
+{
+    if ((row != rhs.row) || (col != vars.row) || (rhs.col != vars.col))
+        throw (std::logic_error("matrix::solve(): incompatible matrices"));
     
-/** Solve simultaneous set of equations. */
-matrix matrix::solve(const matrix & v) const
+    throw (std::runtime_error("FIXME: need implementation."));
+}
+
+/** Old and obsolete interface: */
+matrix matrix::old_solve(const matrix & v) const
 {
-    if (!(row == col && col == v.row)) {
+    if ((v.row != col) || (col != v.row))
         throw (std::logic_error("matrix::solve(): incompatible matrices"));
-    }
     
-    // build the extended matrix of *this with v attached to the right
+    // build the augmented matrix of *this with v attached to the right
     matrix tmp(row,col+v.col);
     for (unsigned r=0; r<row; ++r) {
-        for (unsigned c=0; c<col; ++c) {
-            tmp.m[r*tmp.col+c] = m[r*col+c];
-        }
-        for (unsigned c=0; c<v.col; ++c) {
+        for (unsigned c=0; c<col; ++c)
+            tmp.m[r*tmp.col+c] = this->m[r*col+c];
+        for (unsigned c=0; c<v.col; ++c)
             tmp.m[r*tmp.col+c+col] = v.m[r*v.col+c];
+    }
+    // cout << "augmented: " << tmp << endl;
+    tmp.gauss_elimination();
+    // cout << "degaussed: " << tmp << endl;
+    // assemble the solution matrix
+    exvector sol(v.row*v.col);
+    for (unsigned c=0; c<v.col; ++c) {
+        for (unsigned r=row; r>0; --r) {
+            for (unsigned i=r; i<col; ++i)
+                sol[(r-1)*v.col+c] -= tmp.m[(r-1)*tmp.col+i]*sol[i*v.col+c];
+            sol[(r-1)*v.col+c] += tmp.m[(r-1)*tmp.col+col+c];
+            sol[(r-1)*v.col+c] = (sol[(r-1)*v.col+c]/tmp.m[(r-1)*tmp.col+(r-1)]).normal();
         }
     }
+    return matrix(v.row, v.col, sol);
+}
+
+
+// protected
+
+/** Determinant of purely numeric matrix, using pivoting.
+ *
+ *  @see       matrix::determinant() */
+ex matrix::determinant_numeric(void) const
+{
+    matrix tmp(*this);
+    ex det = _ex1();
+    ex piv;
+    
     for (unsigned r1=0; r1<row; ++r1) {
         int indx = tmp.pivot(r1);
-        if (indx == -1) {
-            throw (std::runtime_error("matrix::solve(): singular matrix"));
-        }
-        for (unsigned c=r1; c<tmp.col; ++c) {
-            tmp.m[r1*tmp.col+c] /= tmp.m[r1*tmp.col+r1];
-        }
+        if (indx == -1)
+            return _ex0();
+        if (indx != 0)
+            det *= _ex_1();
+        det = det * tmp.m[r1*col+r1];
         for (unsigned r2=r1+1; r2<row; ++r2) {
-            for (unsigned c=r1; c<tmp.col; ++c) {
-                tmp.m[r2*tmp.col+c]
-                    -= tmp.m[r2*tmp.col+r1] * tmp.m[r1*tmp.col+c];
+            piv = tmp.m[r2*col+r1] / tmp.m[r1*col+r1];
+            for (unsigned c=r1+1; c<col; c++) {
+                tmp.m[r2*col+c] -= piv * tmp.m[r1*col+c];
             }
         }
     }
     
-    // assemble the solution matrix
-    exvector sol(v.row*v.col);
-    for (unsigned c=0; c<v.col; ++c) {
-        for (unsigned r=col-1; r>=0; --r) {
-            sol[r*v.col+c] = tmp[r*tmp.col+c];
-            for (unsigned i=r+1; i<col; ++i) {
-                sol[r*v.col+c]
-                    -= tmp[r*tmp.col+i] * sol[i*v.col+c];
+    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 very bad for symbolic matrices since it returns expressions
+ *  that are quite hard to expand. */
+/*ex matrix::determinant_leverrier(const matrix & M)
+ *{
+ *    GINAC_ASSERT(M.rows()==M.cols());  // cannot happen, just in case...
+ *    
+ *    matrix B(M);
+ *    matrix I(M.row, M.col);
+ *    ex c=B.trace();
+ *    for (unsigned i=1; i<M.row; ++i) {
+ *        for (unsigned 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;
+ *    }
+ *}*/
+
+
+/** 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:
+    if (this->row==1)
+        return m[0];
+    if (this->row==2)
+        return (m[0]*m[3]-m[2]*m[1]).expand();
+    if (this->row==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->row-1,this->col-1);
+    // for (unsigned r1=0; r1<this->row; ++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
+    vector<unsigned> Pkey;
+    Pkey.reserve(this->col);
+    // key for minor determinant (a subpartition of Pkey)
+    vector<unsigned> Mkey;
+    Mkey.reserve(this->col-1);
+    // we store our subminors in maps, keys being the rows they arise from
+    typedef map<vector<unsigned>,class ex> Rmap;
+    typedef map<vector<unsigned>,class ex>::value_type Rmap_value;
+    Rmap A;
+    Rmap B;
+    ex det;
+    // initialize A with last column:
+    for (unsigned r=0; r<this->col; ++r) {
+        Pkey.erase(Pkey.begin(),Pkey.end());
+        Pkey.push_back(r);
+        A.insert(Rmap_value(Pkey,m[this->col*r+this->col-1]));
+    }
+    // proceed from right to left through matrix
+    for (int c=this->col-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<this->col-c; ++i)
+            Pkey.push_back(i);
+        unsigned fc = 0;  // controls logic for our strange flipper counter
+        do {
+            A.insert(Rmap_value(Pkey,_ex0()));
+            det = _ex0();
+            for (unsigned r=0; r<this->col-c; ++r) {
+                // maybe there is nothing to do?
+                if (m[Pkey[r]*this->col+c].is_zero())
+                    continue;
+                // create the sorted key for all possible minors
+                Mkey.erase(Mkey.begin(),Mkey.end());
+                for (unsigned i=0; i<this->col-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]*this->col+c]*A[Mkey];
+                else
+                    det += m[Pkey[r]*this->col+c]*A[Mkey];
             }
+            // prevent build-up of deep nesting of expressions saves time:
+            det = det.expand();
+            // store the new determinant at its place in B:
+            B.insert(Rmap_value(Pkey,det));
+            // increment our strange flipper counter
+            for (fc=this->col-c; fc>0; --fc) {
+                ++Pkey[fc-1];
+                if (Pkey[fc-1]<fc+c)
+                    break;
+            }
+            if (fc<this->col-c)
+                for (unsigned j=fc; j<this->col-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;
+}
+
+
+/** Determinant built by application of the full permutation group.  This
+ *  routine is only called internally by matrix::determinant(). */
+ex matrix::determinant_perm(void) const
+{
+    if (rows()==1)  // speed things up
+        return m[0];
+    
+    ex det;
+    ex term;
+    vector<unsigned> sigma(col);
+    for (unsigned i=0; i<col; ++i)
+        sigma[i]=i;
+    
+    do {
+        term = (*this)(sigma[0],0);
+        for (unsigned i=1; i<col; ++i)
+            term *= (*this)(sigma[i],i);
+        det += permutation_sign(sigma)*term;
+    } while (next_permutation(sigma.begin(), sigma.end()));
+    
+    return det;
+}
+
+
+/** Perform the steps of an ordinary Gaussian elimination to bring the matrix
+ *  into an upper echelon form.
+ *
+ *  @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(void)
+{
+    int sign = 1;
+    ensure_if_modifiable();
+    for (unsigned r1=0; r1<row-1; ++r1) {
+        int indx = pivot(r1);
+        if (indx == -1)
+            return 0;  // Note: leaves *this in a messy state.
+        if (indx > 0)
+            sign = -sign;
+        for (unsigned r2=r1+1; r2<row; ++r2) {
+            for (unsigned c=r1+1; c<col; ++c)
+                this->m[r2*col+c] -= this->m[r2*col+r1]*this->m[r1*col+c]/this->m[r1*col+r1];
+            for (unsigned c=0; c<=r1; ++c)
+                this->m[r2*col+c] = _ex0();
         }
     }
-    return matrix(v.row, v.col, sol);
+    
+    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(unsigned ro)
+ *  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 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, bool symbolic)
 {
-    unsigned k=ro;
+    unsigned k = ro;
     
-    for (unsigned r=ro; r<row; ++r) {
-        if (!m[r*col+ro].is_zero()) {
-            k = r;
-            break;
+    if (symbolic) {  // search first non-zero
+        for (unsigned r=ro; r<row; ++r) {
+            if (!m[r*col+ro].is_zero()) {
+                k = r;
+                break;
+            }
+        }
+    } else {  // search largest
+        numeric tmp(0);
+        numeric maxn(-1);
+        for (unsigned r=ro; r<row; ++r) {
+            GINAC_ASSERT(is_ex_of_type(m[r*col+ro],numeric));
+            if ((tmp = abs(ex_to_numeric(m[r*col+ro]))) > maxn &&
+                !tmp.is_zero()) {
+                maxn = tmp;
+                k = r;
+            }
         }
     }
-    if (m[k*col+ro].is_zero()) {
+    if (m[k*col+ro].is_zero())
         return -1;
-    }
     if (k!=ro) {  // swap rows
+        ensure_if_modifiable();
         for (unsigned c=0; c<col; ++c) {
             m[k*col+c].swap(m[ro*col+c]);
         }