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()) {
+ if (!(*r).is_zero())
++sparse_count;
- }
- if (!(*r).info(info_flags::numeric)) {
+ if (!(*r).info(info_flags::numeric))
numeric_flag = false;
- }
if ((*r).info(info_flags::rational_function) &&
- !(*r).info(info_flags::crational_polynomial)) {
+ !(*r).info(info_flags::crational_polynomial))
normal_flag = true;
- }
}
if (numeric_flag) // purely numeric matrix
return determinant_numeric();
-
// Does anybody really know when a matrix is sparse?
- if (4*sparse_count<row*col) { // < row/2 nonzero elements average in a row
- return determinant_bareiss(normal_flag);
+ // 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)
+ return (sign*tmp.m[row*col-1]).normal();
+ else
+ return (sign*tmp.m[row*col-1]).expand();
}
// Now come the minor expansion schemes. We always develop such that the
// 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 section is the matrix multiplication.
+ // every coefficient. The expensive part is the matrix multiplication.
if (numeric_flag) {
matrix B(*this);
ex c = B.trace();
matrix matrix::fraction_free_elim(const matrix & vars,
const matrix & rhs) const
{
- // FIXME: use implementation of matrix::fraction_free_elim
+ // 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;
return det;
}
-/** Helper function to divide rational functions, as needed in any Bareiss
- * elimination scheme over a quotient field.
- *
- * @see divide() */
-bool rat_divide(const ex & a, const ex & b, ex & q, bool check_args = true)
-{
- q = _ex0();
- if (b.is_zero())
- throw(std::overflow_error("rat_divide(): division by zero"));
- if (a.is_zero())
- return true;
- if (is_ex_exactly_of_type(b, numeric)) {
- q = a / b;
- return true;
- } else if (is_ex_exactly_of_type(a, numeric))
- return false;
- ex a_n = a.numer();
- ex a_d = a.denom();
- ex b_n = b.numer();
- ex b_d = b.denom();
- ex n; // new numerator
- ex d; // new denominator
- bool check = true;
- check &= divide(a_n, b_n, n, check_args);
- check &= divide(a_d, b_d, d, check_args);
- q = n/d;
- return check;
-}
-
-
-/** Determinant computed by using fraction free elimination. This
- * routine is only called internally by matrix::determinant().
- *
- * @param normalize may be set to false only in integral domains. */
-ex matrix::determinant_bareiss(bool normalize) const
-{
- if (rows()==1)
- return m[0];
-
- int sign = 1;
- ex divisor = 1;
- ex dividend;
-
- // we populate a tmp matrix to subsequently operate on, it should
- // be normalized even though this algorithm doesn't need GCDs since
- // the elements of *this might be unnormalized, which complicates
- // things:
- matrix tmp(*this);
- exvector::const_iterator i = m.begin();
- exvector::iterator ti = tmp.m.begin();
- for (; i!= m.end(); ++i, ++ti) {
- if (normalize)
- (*ti) = (*i).normal();
- else
- (*ti) = (*i);
- }
-
- for (unsigned r1=0; r1<row-1; ++r1) {
- int indx = tmp.pivot(r1);
- if (indx==-1)
- return _ex0();
- if (indx>0)
- sign = -sign;
- if (r1>0) {
- divisor = tmp.m[(r1-1)*col+(r1-1)].expand();
- // delete the elements we don't need anymore:
- for (unsigned c=0; c<col; ++c)
- tmp.m[(r1-1)*col+c] = _ex0();
- }
- for (unsigned r2=r1+1; r2<row; ++r2) {
- for (unsigned c=r1+1; c<col; ++c) {
- lst srl; // symbol replacement list for .to_rational()
- dividend = (tmp.m[r1*tmp.col+r1]*tmp.m[r2*tmp.col+c]
- -tmp.m[r2*tmp.col+r1]*tmp.m[r1*tmp.col+c]).expand();
- if (normalize) {
-#ifdef DO_GINAC_ASSERT
- GINAC_ASSERT(rat_divide(dividend.to_rational(srl),
- divisor.to_rational(srl),
- tmp.m[r2*tmp.col+c],true));
-#else
- rat_divide(dividend.to_rational(srl),
- divisor.to_rational(srl),
- tmp.m[r2*tmp.col+c],false);
-#endif
- }
- else {
-#ifdef DO_GINAC_ASSERT
- GINAC_ASSERT(divide(dividend.to_rational(srl),
- divisor.to_rational(srl),
- tmp.m[r2*tmp.col+c],true));
-#else
- divide(dividend.to_rational(srl),
- divisor.to_rational(srl),
- tmp.m[r2*tmp.col+c],false);
-#endif
- }
- tmp.m[r2*tmp.col+c] = tmp.m[r2*tmp.col+c].subs(srl);
- }
- for (unsigned c=0; c<=r1; ++c)
- tmp.m[r2*tmp.col+c] = _ex0();
- }
- }
-
- return sign*tmp.m[tmp.row*tmp.col-1];
-}
-
/** Perform the steps of an ordinary Gaussian elimination to bring the matrix
* into an upper echelon form.
/** Perform the steps of Bareiss' one-step fraction free elimination to bring
- * the matrix into an upper echelon form.
- *
+ * 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(void)
+int matrix::fraction_free_elimination(bool det)
{
- ensure_if_modifiable();
+ // 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)}.
- // first normal all elements:
- for (exvector::iterator i=m.begin(); i!=m.end(); ++i)
- (*i) = (*i).normal();
+ GINAC_ASSERT(det || row==col);
+ ensure_if_modifiable();
+ if (rows()==1)
+ return 1;
- // FIXME: this is unfinished, once matrix::determinant_bareiss is
- // bulletproof, some code ought to be copy from there to here.
int sign = 1;
- ex divisor = 1;
- ex dividend;
- lst srl; // symbol replacement list for .to_rational()
+ 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 = pivot(r1);
- if (indx==-1)
- return 0; // Note: leaves *this in a messy state.
- if (indx>0)
+// cout << "==<" << r1 << ">" << string(60,'=') << endl;
+ 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;
- if (r1>0)
- divisor = this->m[(r1-1)*col+(r1-1)].expand();
+ // 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]);
+ }
+// cout << tmp_n << endl;
+// cout << tmp_d << endl;
+ 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 = (this->m[r1*col+r1]*this->m[r2*col+c]
- -this->m[r2*col+r1]*this->m[r1*col+c]).expand();
-#ifdef DO_GINAC_ASSERT
- GINAC_ASSERT(divide(dividend.to_rational(srl),
- divisor.to_rational(srl),
- this->m[r2*col+c]));
-#else
- divide(dividend.to_rational(srl),
- divisor.to_rational(srl),
- this->m[r2*col+c]);
-#endif // DO_GINAC_ASSERT
- this->m[r2*col+c] = this->m[r2*col+c].subs(srl);
+ 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();
+// cout << "Element " << r2 << ',' << c << endl;
+// cout << "dividend_n==" << dividend_n << endl;
+// cout << "dividend_d==" << dividend_d << endl;
+// cout << " divisor_n==" << divisor_n << endl;
+// cout << " divisor_d==" << divisor_d << endl;
+// cout << string(20,'-') << endl;
+ 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)
- this->m[r2*col+c] = _ex0();
+ tmp_n.m[r2*col+c] = _ex0();
}
+// cout << tmp_n << endl;
+// cout << tmp_d << endl;
}
+ // 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;
}
numeric::numeric(const char *s) : basic(TINFO_numeric)
-{ // MISSING: treatment of complex and ints and rationals.
+{ // MISSING: treatment of complex numbers
debugmsg("numeric constructor from string",LOGLEVEL_CONSTRUCT);
if (strchr(s, '.'))
value = new cl_LF(s);
else
- value = new cl_I(s);
+ value = new cl_R(s);
calchash();
setflag(status_flags::evaluated|
status_flags::hash_calculated);
bool numeric::info(unsigned inf) const
{
switch (inf) {
- case info_flags::numeric:
- case info_flags::polynomial:
- case info_flags::rational_function:
- return true;
- case info_flags::real:
- return is_real();
- case info_flags::rational:
- case info_flags::rational_polynomial:
- return is_rational();
- case info_flags::crational:
- case info_flags::crational_polynomial:
- return is_crational();
- case info_flags::integer:
- case info_flags::integer_polynomial:
- return is_integer();
- case info_flags::cinteger:
- case info_flags::cinteger_polynomial:
- return is_cinteger();
- case info_flags::positive:
- return is_positive();
- case info_flags::negative:
- return is_negative();
- case info_flags::nonnegative:
- return !is_negative();
- case info_flags::posint:
- return is_pos_integer();
- case info_flags::negint:
- return is_integer() && is_negative();
- case info_flags::nonnegint:
- return is_nonneg_integer();
- case info_flags::even:
- return is_even();
- case info_flags::odd:
- return is_odd();
- case info_flags::prime:
- return is_prime();
+ case info_flags::numeric:
+ case info_flags::polynomial:
+ case info_flags::rational_function:
+ return true;
+ case info_flags::real:
+ return is_real();
+ case info_flags::rational:
+ case info_flags::rational_polynomial:
+ return is_rational();
+ case info_flags::crational:
+ case info_flags::crational_polynomial:
+ return is_crational();
+ case info_flags::integer:
+ case info_flags::integer_polynomial:
+ return is_integer();
+ case info_flags::cinteger:
+ case info_flags::cinteger_polynomial:
+ return is_cinteger();
+ case info_flags::positive:
+ return is_positive();
+ case info_flags::negative:
+ return is_negative();
+ case info_flags::nonnegative:
+ return !is_negative();
+ case info_flags::posint:
+ return is_pos_integer();
+ case info_flags::negint:
+ return is_integer() && is_negative();
+ case info_flags::nonnegint:
+ return is_nonneg_integer();
+ case info_flags::even:
+ return is_even();
+ case info_flags::odd:
+ return is_odd();
+ case info_flags::prime:
+ return is_prime();
+ case info_flags::algebraic:
+ return !is_real();
}
return false;
}