#include "matrix.h"
#include "archive.h"
+#include "numeric.h"
+#include "lst.h"
#include "utils.h"
#include "debugmsg.h"
-#include "numeric.h"
+#include "power.h"
+#include "symbol.h"
+#include "normal.h"
#ifndef NO_NAMESPACE_GINAC
namespace GiNaC {
void matrix::copy(const matrix & other)
{
inherited::copy(other);
- row=other.row;
- col=other.col;
- m=other.m; // use STL's vector copying
+ row = other.row;
+ col = other.col;
+ m = other.m; // STL's vector copying invoked here
}
void matrix::destroy(bool call_parent)
m.resize(r*c, _ex0());
}
-// protected
+ // protected
/** Ctor from representation, for internal use only. */
matrix::matrix(unsigned r, unsigned c, const exvector & m2)
throw (std::logic_error("matrix::mul(): incompatible matrices"));
exvector prod(row*other.col);
- for (unsigned i=0; i<row; ++i) {
- for (unsigned j=0; j<other.col; ++j) {
- for (unsigned l=0; l<col; ++l) {
- prod[i*other.col+j] += m[i*col+l] * other.m[l*other.col+j];
- }
+
+ for (unsigned r1=0; r1<row; ++r1) {
+ for (unsigned c=0; c<col; ++c) {
+ if (m[r1*col+c].is_zero())
+ continue;
+ for (unsigned r2=0; r2<other.col; ++r2)
+ prod[r1*other.col+r2] += m[r1*col+c] * other.m[c*other.col+r2];
}
}
return matrix(row, other.col, prod);
* @exception logic_error (matrix not square) */
ex matrix::determinant(void) const
{
- if (row != col)
+ if (row!=col)
throw (std::logic_error("matrix::determinant(): matrix not square"));
GINAC_ASSERT(row*col==m.capacity());
+ if (this->row==1) // continuation would be pointless
+ return m[0];
- ex det;
+ // Gather some information about the matrix:
bool numeric_flag = true;
bool normal_flag = false;
+ unsigned sparse_count = 0; // count non-zero elements
for (exvector::const_iterator r=m.begin(); r!=m.end(); ++r) {
+ if (!(*r).is_zero())
+ ++sparse_count;
if (!(*r).info(info_flags::numeric))
numeric_flag = false;
if ((*r).info(info_flags::rational_function) &&
normal_flag = true;
}
- if (numeric_flag)
- det = determinant_numeric();
- else
+ // Purely numeric matrix handled by Gauss elimination
+ if (numeric_flag) {
+ ex det = 1;
+ matrix tmp(*this);
+ int sign = tmp.gauss_elimination();
+ for (int d=0; d<row; ++d)
+ det *= tmp.m[d*col+d];
+ return (sign*det);
+ }
+
+ // Does anybody know when a matrix is really sparse?
+ // Maybe <~row/2.2 nonzero elements average in a row?
+ if (5*sparse_count<=row*col) {
+ // copy *this:
+ matrix tmp(*this);
+ int sign;
+ sign = tmp.fraction_free_elimination(true);
if (normal_flag)
- det = determinant_symbolic_minor().normal();
+ return (sign*tmp.m[row*col-1]).normal();
else
- det = determinant_symbolic_minor();
+ return (sign*tmp.m[row*col-1]).expand();
+ }
- return det;
+ // Now come the minor expansion schemes. 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 pair<unsigned,unsigned> uintpair; // # of zeros, column
+ 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());
+ vector<unsigned> pre_sort; // unfortunately vector<uintpair> can't be used
+ // for permutation_sign.
+ for (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 (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();
+ return sign*matrix(row,col,result).determinant_minor();
}
}
-/** 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(const ex & lambda) const
+ex matrix::charpoly(const symbol & lambda) const
{
if (row != col)
throw (std::logic_error("matrix::charpoly(): matrix not square"));
+ bool numeric_flag = true;
+ for (exvector::const_iterator r=m.begin(); r!=m.end(); ++r) {
+ if (!(*r).info(info_flags::numeric)) {
+ 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 (unsigned r=0; r<col; ++r)
M.m[r*col+r] -= lambda;
- return (M.determinant());
+ return M.determinant().collect(lambda);
}
}
-// superfluous helper function
+// superfluous helper function, to be removed:
void matrix::ffe_swap(unsigned r1, unsigned c1, unsigned r2 ,unsigned c2)
{
ensure_if_modifiable();
ffe_set(r2,c2,tmp);
}
-// superfluous helper function
+// superfluous helper function, to be removed:
void matrix::ffe_set(unsigned r, unsigned c, ex e)
{
set(r-1,c-1,e);
}
-// superfluous helper function
+// superfluous helper function, to be removed:
ex matrix::ffe_get(unsigned r, unsigned c) const
{
return operator()(r-1,c-1);
matrix matrix::fraction_free_elim(const matrix & vars,
const matrix & rhs) const
{
- // FIXME: implement a Sasaki-Murao scheme which avoids division at all!
+ // FIXME: use implementation of matrix::fraction_free_elimination
if ((row != rhs.row) || (col != vars.row) || (rhs.col != vars.col))
throw (std::logic_error("matrix::fraction_free_elim(): incompatible matrices"));
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; }
-
- /*
- for (unsigned r=1; r<=m; ++r) {
- for (unsigned c=1; c<=n; ++c) {
- cout << a.ffe_get(r,c) << "\t";
- }
- cout << " | " << b.ffe_get(r,1) << endl;
- }
- */
+ }
+// for (unsigned r=1; r<=m; ++r) {
+// for (unsigned c=1; c<=n; ++c) {
+// cout << a.ffe_get(r,c) << "\t";
+// }
+// cout << " | " << b.ffe_get(r,1) << endl;
+// }
#ifdef DO_GINAC_ASSERT
// test if we really have an upper echelon matrix
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_zero())
zero_in_this_row++;
else
break;
}
#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;
// 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)
- return _ex0();
- if (indx != 0)
- det *= _ex_1();
- det = det * tmp.m[r1*col+r1];
- for (unsigned r2=r1+1; r2<row; ++r2) {
- 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];
- }
- }
- }
-
- 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_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;
- * }
- *}*/
-
-
/** 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")
* 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_symbolic_minor(void) const
+ex matrix::determinant_minor(void) const
{
// for small matrices the algorithm does not make any sense:
if (this->row==1)
// }
// // recurse down and care for sign:
// if (r1%2)
- // det -= m[r1*col] * minorM.determinant_symbolic_minor();
+ // det -= m[r1*col] * minorM.determinant_minor();
// else
- // det += m[r1*col] * minorM.determinant_symbolic_minor();
+ // det += m[r1*col] * minorM.determinant_minor();
// }
// return det.expand();
// What happens is that while proceeding down many of the minors are
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?
else
det += m[Pkey[r]*this->col+c]*A[Mkey];
}
- // prevent build-up of deep nesting of expressions saves some time:
+ // 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));
+ if (!det.is_zero())
+ B.insert(Rmap_value(Pkey,det));
// increment our strange flipper counter
for (fc=this->col-c; fc>0; --fc) {
++Pkey[fc-1];
}
-/** Determinant built by application of the full permutation group. This
- * routine is only called internally by matrix::determinant(). */
-ex matrix::determinant_symbolic_perm(void) const
+/** 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)
{
- 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()));
+ ensure_if_modifiable();
+ int sign = 1;
+ ex piv;
+ 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) {
+ piv = this->m[r2*col+r1] / this->m[r1*col+r1];
+ for (unsigned c=r1+1; c<col; ++c)
+ this->m[r2*col+c] -= piv * this->m[r1*col+c];
+ for (unsigned c=0; c<=r1; ++c)
+ this->m[r2*col+c] = _ex0();
+ }
+ }
- return det;
+ return sign;
}
-/** Perform the steps of an ordinary Gaussian elimination to bring the matrix
+/** Perform the steps of division free 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 matrix::division_free_elimination(void)
{
int sign = 1;
ensure_if_modifiable();
for (unsigned r1=0; r1<row-1; ++r1) {
int indx = pivot(r1);
- if (indx == -1)
+ if (indx==-1)
return 0; // Note: leaves *this in a messy state.
- if (indx > 0)
+ 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];
+ this->m[r2*col+c] = this->m[r1*col+r1]*this->m[r2*col+c] - this->m[r2*col+r1]*this->m[r1*col+c];
for (unsigned c=0; c<=r1; ++c)
this->m[r2*col+c] = _ex0();
}
}
-/** Partial pivoting method.
+/** 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(bool det)
+{
+ // 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)}.
+
+ GINAC_ASSERT(det || row==col);
+ ensure_if_modifiable();
+ if (rows()==1)
+ return 1;
+
+ int sign = 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(row,col); // for denominators, if needed
+ lst srl; // symbol replacement list
+ exvector::iterator it = m.begin();
+ exvector::iterator tmp_n_it = tmp_n.m.begin();
+ exvector::iterator tmp_d_it = tmp_d.m.begin();
+ for (; it!= 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 (unsigned r1=0; r1<row-1; ++r1) {
+ int indx = tmp_n.pivot(r1);
+ if (det && indx==-1)
+ return 0; // FIXME: what to do if det is false?
+ if (indx>0) {
+ sign = -sign;
+ // rows r1 and indx were swapped, so pivot matrix tmp_d:
+ for (unsigned c=0; c<col; ++c)
+ tmp_d.m[row*indx+c].swap(tmp_d.m[row*r1+c]);
+ }
+ if (r1>0) {
+ divisor_n = tmp_n.m[(r1-1)*col+(r1-1)].expand();
+ divisor_d = tmp_d.m[(r1-1)*col+(r1-1)].expand();
+ // save space by deleting no longer needed elements:
+ if (det) {
+ for (unsigned c=0; c<col; ++c) {
+ tmp_n.m[(r1-1)*col+c] = 0;
+ tmp_d.m[(r1-1)*col+c] = 1;
+ }
+ }
+ }
+ for (unsigned r2=r1+1; r2<row; ++r2) {
+ for (unsigned c=r1+1; c<col; ++c) {
+ dividend_n = (tmp_n.m[r1*col+r1]*tmp_n.m[r2*col+c]*
+ tmp_d.m[r2*col+r1]*tmp_d.m[r1*col+c]
+ -tmp_n.m[r2*col+r1]*tmp_n.m[r1*col+c]*
+ tmp_d.m[r1*col+r1]*tmp_d.m[r2*col+c]).expand();
+ dividend_d = (tmp_d.m[r2*col+r1]*tmp_d.m[r1*col+c]*
+ tmp_d.m[r1*col+r1]*tmp_d.m[r2*col+c]).expand();
+ bool check = divide(dividend_n, divisor_n,
+ tmp_n.m[r2*col+c],true);
+ check &= divide(dividend_d, divisor_d,
+ tmp_d.m[r2*col+c],true);
+ GINAC_ASSERT(check);
+ }
+ // fill up left hand side.
+ for (unsigned c=0; c<=r1; ++c)
+ tmp_n.m[r2*col+c] = _ex0();
+ }
+ }
+ // repopulate *this matrix:
+ it = m.begin();
+ tmp_n_it = tmp_n.m.begin();
+ tmp_d_it = tmp_d.m.begin();
+ for (; it!= m.end(); ++it, ++tmp_n_it, ++tmp_d_it)
+ (*it) = ((*tmp_n_it)/(*tmp_d_it)).subs(srl);
+
+ return sign;
+}
+
+
+/** 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
return 0;
}
+/** 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;
+}
+
//////////
// global constants
//////////
git://www.ginac.de / ginac.git / blobdiff