exvector::const_iterator i = m.begin(), iend = m.end();
while (i != iend) {
n.add_ex("m", *i);
- i++;
+ ++i;
}
}
return new matrix(*this);
}
-void matrix::print(ostream & os, unsigned upper_precedence) const
+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) {
+ 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) {
+ for (unsigned c=0; c<col-1; ++c)
os << m[(row-1)*col+c] << ",";
- }
os << m[row*col-1] << "]] ]]";
}
-void matrix::printraw(ostream & os) const
+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) {
+ 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) {
+ for (unsigned c=0; c<col-1; ++c)
os << m[(row-1)*col+c] << ",";
- }
os << m[row*col-1] << "))";
}
/** 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];
}
ex matrix::expand(unsigned options) const
{
exvector tmp(row*col);
- for (unsigned i=0; i<row*col; ++i) {
- tmp[i]=m[i].expand(options);
- }
+ for (unsigned i=0; i<row*col; ++i)
+ tmp[i] = m[i].expand(options);
+
return matrix(row, col, tmp);
}
if (is_equal(*other.bp)) return true;
// search all the elements
- for (exvector::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;
}
// eval() entry by entry
exvector m2(row*col);
- --level;
- for (unsigned r=0; r<row; ++r) {
- for (unsigned c=0; c<col; ++c) {
+ --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 );
// evalf() entry by entry
exvector m2(row*col);
--level;
- for (unsigned r=0; r<row; ++r) {
- for (unsigned 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);
}
exvector sum(this->m);
exvector::iterator i;
exvector::const_iterator ci;
- for (i=sum.begin(), ci=other.m.begin();
- i!=sum.end();
- ++i, ++ci) {
+ for (i=sum.begin(), ci=other.m.begin(); i!=sum.end(); ++i, ++ci)
(*i) += (*ci);
- }
+
return matrix(row,col,sum);
}
exvector dif(this->m);
exvector::iterator i;
exvector::const_iterator ci;
- for (i=dif.begin(), ci=other.m.begin();
- i!=dif.end();
- ++i, ++ci) {
+ for (i=dif.begin(), ci=other.m.begin(); i!=dif.end(); ++i, ++ci)
(*i) -= (*ci);
- }
+
return matrix(row,col,dif);
}
exvector prod(row*other.col);
- for (unsigned r1=0; r1<row; ++r1) {
- for (unsigned c=0; c<col; ++c) {
+ for (unsigned r1=0; r1<rows(); ++r1) {
+ for (unsigned c=0; c<cols(); ++c) {
if (m[r1*col+c].is_zero())
continue;
for (unsigned r2=0; r2<other.col; ++r2)
/** 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) */
const ex & matrix::operator() (unsigned ro, unsigned co) const
{
if (ro<0 || ro>=row || co<0 || co>=col)
throw (std::range_error("matrix::operator(): index out of range"));
-
+
return m[ro*col+co];
}
/** Determinant of square matrix. This routine doesn't actually calculate the
* determinant, it only implements some heuristics about which algorithm to
- * call. If all the elements of the matrix are elements of an integral domain
+ * 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
* [[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) */
-ex matrix::determinant(void) const
+ * @exception logic_error (matrix not square)
+ * @see determinant_algo */
+ex matrix::determinant(unsigned algo) const
{
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];
-
- // Gather some information about the matrix:
+
+ // Gather some statistical information about this matrix:
bool numeric_flag = true;
bool normal_flag = false;
- unsigned sparse_count = 0; // count non-zero elements
+ unsigned sparse_count = 0; // counts non-zero elements
for (exvector::const_iterator r=m.begin(); r!=m.end(); ++r) {
- if (!(*r).is_zero())
+ lst srl; // symbol replacement list
+ ex rtest = (*r).to_rational(srl);
+ if (!rtest.is_zero())
++sparse_count;
- if (!(*r).info(info_flags::numeric))
+ if (!rtest.info(info_flags::numeric))
numeric_flag = false;
- if ((*r).info(info_flags::rational_function) &&
- !(*r).info(info_flags::crational_polynomial))
+ if (!rtest.info(info_flags::crational_polynomial) &&
+ rtest.info(info_flags::rational_function))
normal_flag = true;
}
- // 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)
- return (sign*tmp.m[row*col-1]).normal();
- else
- return (sign*tmp.m[row*col-1]).expand();
+ // Here is the heuristics in case this routine has to decide:
+ if (algo == determinant_algo::automatic) {
+ // Minor expansion is generally a good starting point:
+ algo = determinant_algo::laplace;
+ // Does anybody know when a matrix is really sparse?
+ // Maybe <~row/2.236 nonzero elements average in a row?
+ if (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;
}
- // 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)];
+ switch(algo) {
+ case determinant_algo::gauss: {
+ ex det = 1;
+ matrix tmp(*this);
+ int sign = tmp.gauss_elimination();
+ for (unsigned d=0; d<row; ++d)
+ det *= tmp.m[d*col+d];
+ if (normal_flag)
+ return (sign*det).normal();
+ else
+ return (sign*det).expand();
+ }
+ 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();
+ }
+ 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();
+ return sign*matrix(row,col,result).determinant_minor();
+ }
}
-
- if (normal_flag)
- return sign*matrix(row,col,result).determinant_minor().normal();
- return sign*matrix(row,col,result).determinant_minor();
}
throw (std::runtime_error("matrix::inverse(): singular matrix"));
}
if (indx != 0) { // swap rows r and indx of matrix tmp
- for (unsigned i=0; i<col; ++i) {
+ for (unsigned i=0; i<col; ++i)
tmp.m[r1*col+i].swap(tmp.m[indx*col+i]);
- }
}
ex a1 = cpy.m[r1*col+r1];
for (unsigned c=0; c<col; ++c) {
}
-// superfluous helper function, to be removed:
-void matrix::ffe_swap(unsigned r1, unsigned c1, unsigned r2 ,unsigned c2)
-{
- ensure_if_modifiable();
-
- ex tmp = ffe_get(r1,c1);
- ffe_set(r1,c1,ffe_get(r2,c2));
- ffe_set(r2,c2,tmp);
-}
-
-// 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, to be removed:
-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'
* by Keith O. Geddes et al.
matrix matrix::fraction_free_elim(const matrix & vars,
const matrix & rhs) const
{
- // FIXME: use implementation of matrix::fraction_free_elimination
+ // FIXME: use implementation of matrix::fraction_free_elimination instead!
if ((row != rhs.row) || (col != vars.row) || (rhs.col != vars.col))
throw (std::logic_error("matrix::fraction_free_elim(): incompatible matrices"));
unsigned n = a.col;
int sign = 1;
ex divisor = 1;
- unsigned r = 1;
+ unsigned r = 0;
// eliminate below row r, with pivot in column k
- for (unsigned k=1; (k<=n)&&(r<=m); ++k) {
+ for (unsigned k=0; (k<n)&&(r<m); ++k) {
// find a nonzero pivot
unsigned p;
- for (p=r; (p<=m)&&(a.ffe_get(p,k).is_equal(_ex0())); ++p) {}
+ for (p=r; (p<m)&&(a.m[p*a.cols()+k].is_zero()); ++p) {}
// pivot is in row p
- if (p<=m) {
+ if (p<m) {
if (p!=r) {
- // switch rows p and r
- for (unsigned j=k; j<=n; ++j)
- a.ffe_swap(p,j,r,j);
- b.ffe_swap(p,1,r,1);
+ // swap rows p and r
+ for (unsigned j=k; j<n; ++j)
+ a.m[p*a.cols()+j].swap(a.m[r*a.cols()+j]);
+ b.m[p*b.cols()].swap(b.m[r*b.cols()]);
// keep track of sign changes due to row exchange
- sign = -sign;
+ sign *= -1;
}
- for (unsigned i=r+1; i<=m; ++i) {
- for (unsigned 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() */ );
+ for (unsigned i=r+1; i<m; ++i) {
+ for (unsigned j=k+1; j<n; ++j) {
+ a.set(i,j,(a.m[r*a.cols()+k]*a.m[i*a.cols()+j]
+ -a.m[r*a.cols()+j]*a.m[i*a.cols()+k])/divisor);
+ a.set(i,j,a.m[i*a.cols()+j].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);
+ b.set(i,0,(a.m[r*a.cols()+k]*b.m[i*b.cols()]
+ -b.m[r*b.cols()]*a.m[i*a.cols()+k])/divisor);
+ b.set(i,0,b.m[i*b.cols()].normal());
+ a.set(i,k,_ex0());
}
- divisor = a.ffe_get(r,k);
- r++;
+ divisor = a.m[r*a.cols()+k];
+ ++r;
}
- }
-// 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
int zero_in_last_row = -1;
- for (unsigned r=1; r<=m; ++r) {
+ for (unsigned r=0; r<m; ++r) {
int zero_in_this_row=0;
- for (unsigned c=1; c<=n; ++c) {
- if (a.ffe_get(r,c).is_zero())
- zero_in_this_row++;
+ for (unsigned c=0; c<n; ++c) {
+ if (a.m[r*a.cols()+c].is_zero())
+ ++zero_in_this_row;
else
break;
}
// assemble solution
matrix sol(n,1);
unsigned last_assigned_sol = n+1;
- for (unsigned r=m; r>0; --r) {
+ for (int r=m-1; r>=0; --r) {
unsigned first_non_zero = 1;
- while ((first_non_zero<=n)&&(a.ffe_get(r,first_non_zero).is_zero()))
+ while ((first_non_zero<=n)&&(a(r,first_non_zero-1).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()) {
+ if (!b.m[r*b.cols()].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 (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);
- 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());
+ for (unsigned c=first_non_zero; c<last_assigned_sol-1; ++c)
+ sol.set(c,0,vars.m[c*vars.cols()]);
+ ex e = b.m[r*b.cols()];
+ for (unsigned c=first_non_zero; c<n; ++c)
+ e -= a.m[r*a.cols()+c]*sol.m[c*sol.cols()];
+ sol.set(first_non_zero-1,0,
+ (e/(a.m[r*a.cols()+(first_non_zero-1)])).normal());
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)
- sol.ffe_set(c,1,vars.ffe_get(c,1));
+ for (unsigned c=0; c<last_assigned_sol-1; ++c)
+ sol.set(c,0,vars.m[c*vars.cols()]);
#ifdef DO_GINAC_ASSERT
// test solution with echelon matrix
- for (unsigned r=1; r<=m; ++r) {
+ for (unsigned r=0; 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);
- if (!(e-b.ffe_get(r,1)).normal().is_zero()) {
+ for (unsigned c=0; c<n; ++c)
+ e += a(r,c)*sol(c,0);
+ if (!(e-b(r,0)).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;
+ cout << "b(" << r <<",0)=" << b(r,0) << endl;
+ cout << "diff=" << (e-b(r,0)).normal() << endl;
}
- GINAC_ASSERT((e-b.ffe_get(r,1)).normal().is_zero());
+ GINAC_ASSERT((e-b(r,0)).normal().is_zero());
}
// test solution with original matrix
- for (unsigned r=1; r<=m; ++r) {
+ for (unsigned r=0; r<m; ++r) {
ex e = 0;
- for (unsigned c=1; c<=n; ++c)
- e = e+ffe_get(r,c)*sol.ffe_get(c,1);
+ for (unsigned c=0; c<n; ++c)
+ e += this->m[r*cols()+c]*sol(c,0);
try {
- if (!(e-rhs.ffe_get(r,1)).normal().is_zero()) {
- cout << "e=" << e << endl;
+ if (!(e-rhs(r,0)).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;
+ cout << "rhs(" << r <<",0)=" << rhs(r,0) << endl;
+ cout << "diff=" << (e-rhs(r,0)).normal() << endl;
}
} catch (...) {
- ex xxx = e - rhs.ffe_get(r,1);
+ ex xxx = e - rhs(r,0);
cerr << "xxx=" << xxx << endl << endl;
}
- GINAC_ASSERT((e-rhs.ffe_get(r,1)).normal().is_zero());
+ GINAC_ASSERT((e-rhs(r,0)).normal().is_zero());
}
#endif // def DO_GINAC_ASSERT
// 2*binomial(n,n/2) minors.
// Unique flipper counter for partitioning into minors
- vector<unsigned> Pkey;
+ std::vector<unsigned> Pkey;
Pkey.reserve(this->col);
// key for minor determinant (a subpartition of Pkey)
- vector<unsigned> Mkey;
+ std::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;
+ 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;
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?
+ return 0; // FIXME: what to do if det is false, some day?
if (indx>0) {
sign = -sign;
// rows r1 and indx were swapped, so pivot matrix tmp_d:
if (symbolic) { // search first non-zero
for (unsigned r=ro; r<row; ++r) {
- if (!m[r*col+ro].is_zero()) {
+ if (!m[r*col+ro].expand().is_zero()) {
k = r;
break;
}
git://www.ginac.de / ginac.git / blobdiff