3 * Implementation of symbolic matrices */
6 * GiNaC Copyright (C) 1999-2000 Johannes Gutenberg University Mainz, Germany
8 * This program is free software; you can redistribute it and/or modify
9 * it under the terms of the GNU General Public License as published by
10 * the Free Software Foundation; either version 2 of the License, or
11 * (at your option) any later version.
13 * This program is distributed in the hope that it will be useful,
14 * but WITHOUT ANY WARRANTY; without even the implied warranty of
15 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
16 * GNU General Public License for more details.
18 * You should have received a copy of the GNU General Public License
19 * along with this program; if not, write to the Free Software
20 * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
33 #ifndef NO_NAMESPACE_GINAC
35 #endif // ndef NO_NAMESPACE_GINAC
37 GINAC_IMPLEMENT_REGISTERED_CLASS(matrix, basic)
40 // default constructor, destructor, copy constructor, assignment operator
46 /** Default ctor. Initializes to 1 x 1-dimensional zero-matrix. */
48 : inherited(TINFO_matrix), row(1), col(1)
50 debugmsg("matrix default constructor",LOGLEVEL_CONSTRUCT);
56 debugmsg("matrix destructor",LOGLEVEL_DESTRUCT);
59 matrix::matrix(const matrix & other)
61 debugmsg("matrix copy constructor",LOGLEVEL_CONSTRUCT);
65 const matrix & matrix::operator=(const matrix & other)
67 debugmsg("matrix operator=",LOGLEVEL_ASSIGNMENT);
77 void matrix::copy(const matrix & other)
79 inherited::copy(other);
82 m=other.m; // use STL's vector copying
85 void matrix::destroy(bool call_parent)
87 if (call_parent) inherited::destroy(call_parent);
96 /** Very common ctor. Initializes to r x c-dimensional zero-matrix.
98 * @param r number of rows
99 * @param c number of cols */
100 matrix::matrix(unsigned r, unsigned c)
101 : inherited(TINFO_matrix), row(r), col(c)
103 debugmsg("matrix constructor from unsigned,unsigned",LOGLEVEL_CONSTRUCT);
104 m.resize(r*c, _ex0());
109 /** Ctor from representation, for internal use only. */
110 matrix::matrix(unsigned r, unsigned c, const exvector & m2)
111 : inherited(TINFO_matrix), row(r), col(c), m(m2)
113 debugmsg("matrix constructor from unsigned,unsigned,exvector",LOGLEVEL_CONSTRUCT);
120 /** Construct object from archive_node. */
121 matrix::matrix(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
123 debugmsg("matrix constructor from archive_node", LOGLEVEL_CONSTRUCT);
124 if (!(n.find_unsigned("row", row)) || !(n.find_unsigned("col", col)))
125 throw (std::runtime_error("unknown matrix dimensions in archive"));
126 m.reserve(row * col);
127 for (unsigned int i=0; true; i++) {
129 if (n.find_ex("m", e, sym_lst, i))
136 /** Unarchive the object. */
137 ex matrix::unarchive(const archive_node &n, const lst &sym_lst)
139 return (new matrix(n, sym_lst))->setflag(status_flags::dynallocated);
142 /** Archive the object. */
143 void matrix::archive(archive_node &n) const
145 inherited::archive(n);
146 n.add_unsigned("row", row);
147 n.add_unsigned("col", col);
148 exvector::const_iterator i = m.begin(), iend = m.end();
156 // functions overriding virtual functions from bases classes
161 basic * matrix::duplicate() const
163 debugmsg("matrix duplicate",LOGLEVEL_DUPLICATE);
164 return new matrix(*this);
167 void matrix::print(ostream & os, unsigned upper_precedence) const
169 debugmsg("matrix print",LOGLEVEL_PRINT);
171 for (unsigned r=0; r<row-1; ++r) {
173 for (unsigned c=0; c<col-1; ++c) {
174 os << m[r*col+c] << ",";
176 os << m[col*(r+1)-1] << "]], ";
179 for (unsigned c=0; c<col-1; ++c) {
180 os << m[(row-1)*col+c] << ",";
182 os << m[row*col-1] << "]] ]]";
185 void matrix::printraw(ostream & os) const
187 debugmsg("matrix printraw",LOGLEVEL_PRINT);
188 os << "matrix(" << row << "," << col <<",";
189 for (unsigned r=0; r<row-1; ++r) {
191 for (unsigned c=0; c<col-1; ++c) {
192 os << m[r*col+c] << ",";
194 os << m[col*(r-1)-1] << "),";
197 for (unsigned c=0; c<col-1; ++c) {
198 os << m[(row-1)*col+c] << ",";
200 os << m[row*col-1] << "))";
203 /** nops is defined to be rows x columns. */
204 unsigned matrix::nops() const
209 /** returns matrix entry at position (i/col, i%col). */
210 ex matrix::op(int i) const
215 /** returns matrix entry at position (i/col, i%col). */
216 ex & matrix::let_op(int i)
221 /** expands the elements of a matrix entry by entry. */
222 ex matrix::expand(unsigned options) const
224 exvector tmp(row*col);
225 for (unsigned i=0; i<row*col; ++i) {
226 tmp[i]=m[i].expand(options);
228 return matrix(row, col, tmp);
231 /** Search ocurrences. A matrix 'has' an expression if it is the expression
232 * itself or one of the elements 'has' it. */
233 bool matrix::has(const ex & other) const
235 GINAC_ASSERT(other.bp!=0);
237 // tautology: it is the expression itself
238 if (is_equal(*other.bp)) return true;
240 // search all the elements
241 for (exvector::const_iterator r=m.begin(); r!=m.end(); ++r) {
242 if ((*r).has(other)) return true;
247 /** evaluate matrix entry by entry. */
248 ex matrix::eval(int level) const
250 debugmsg("matrix eval",LOGLEVEL_MEMBER_FUNCTION);
252 // check if we have to do anything at all
253 if ((level==1)&&(flags & status_flags::evaluated)) {
258 if (level == -max_recursion_level) {
259 throw (std::runtime_error("matrix::eval(): recursion limit exceeded"));
262 // eval() entry by entry
263 exvector m2(row*col);
265 for (unsigned r=0; r<row; ++r) {
266 for (unsigned c=0; c<col; ++c) {
267 m2[r*col+c] = m[r*col+c].eval(level);
271 return (new matrix(row, col, m2))->setflag(status_flags::dynallocated |
272 status_flags::evaluated );
275 /** evaluate matrix numerically entry by entry. */
276 ex matrix::evalf(int level) const
278 debugmsg("matrix evalf",LOGLEVEL_MEMBER_FUNCTION);
280 // check if we have to do anything at all
286 if (level == -max_recursion_level) {
287 throw (std::runtime_error("matrix::evalf(): recursion limit exceeded"));
290 // evalf() entry by entry
291 exvector m2(row*col);
293 for (unsigned r=0; r<row; ++r) {
294 for (unsigned c=0; c<col; ++c) {
295 m2[r*col+c] = m[r*col+c].evalf(level);
298 return matrix(row, col, m2);
303 int matrix::compare_same_type(const basic & other) const
305 GINAC_ASSERT(is_exactly_of_type(other, matrix));
306 const matrix & o = static_cast<matrix &>(const_cast<basic &>(other));
308 // compare number of rows
310 return row < o.rows() ? -1 : 1;
312 // compare number of columns
314 return col < o.cols() ? -1 : 1;
316 // equal number of rows and columns, compare individual elements
318 for (unsigned r=0; r<row; ++r) {
319 for (unsigned c=0; c<col; ++c) {
320 cmpval = ((*this)(r,c)).compare(o(r,c));
321 if (cmpval!=0) return cmpval;
324 // all elements are equal => matrices are equal;
329 // non-virtual functions in this class
336 * @exception logic_error (incompatible matrices) */
337 matrix matrix::add(const matrix & other) const
339 if (col != other.col || row != other.row) {
340 throw (std::logic_error("matrix::add(): incompatible matrices"));
343 exvector sum(this->m);
344 exvector::iterator i;
345 exvector::const_iterator ci;
346 for (i=sum.begin(), ci=other.m.begin();
351 return matrix(row,col,sum);
354 /** Difference of matrices.
356 * @exception logic_error (incompatible matrices) */
357 matrix matrix::sub(const matrix & other) const
359 if (col != other.col || row != other.row) {
360 throw (std::logic_error("matrix::sub(): incompatible matrices"));
363 exvector dif(this->m);
364 exvector::iterator i;
365 exvector::const_iterator ci;
366 for (i=dif.begin(), ci=other.m.begin();
371 return matrix(row,col,dif);
374 /** Product of matrices.
376 * @exception logic_error (incompatible matrices) */
377 matrix matrix::mul(const matrix & other) const
379 if (col != other.row) {
380 throw (std::logic_error("matrix::mul(): incompatible matrices"));
383 exvector prod(row*other.col);
384 for (unsigned i=0; i<row; ++i) {
385 for (unsigned j=0; j<other.col; ++j) {
386 for (unsigned l=0; l<col; ++l) {
387 prod[i*other.col+j] += m[i*col+l] * other.m[l*other.col+j];
391 return matrix(row, other.col, prod);
394 /** operator() to access elements.
396 * @param ro row of element
397 * @param co column of element
398 * @exception range_error (index out of range) */
399 const ex & matrix::operator() (unsigned ro, unsigned co) const
401 if (ro<0 || ro>=row || co<0 || co>=col) {
402 throw (std::range_error("matrix::operator(): index out of range"));
408 /** Set individual elements manually.
410 * @exception range_error (index out of range) */
411 matrix & matrix::set(unsigned ro, unsigned co, ex value)
413 if (ro<0 || ro>=row || co<0 || co>=col) {
414 throw (std::range_error("matrix::set(): index out of range"));
417 ensure_if_modifiable();
418 m[ro*col+co] = value;
422 /** Transposed of an m x n matrix, producing a new n x m matrix object that
423 * represents the transposed. */
424 matrix matrix::transpose(void) const
426 exvector trans(col*row);
428 for (unsigned r=0; r<col; ++r)
429 for (unsigned c=0; c<row; ++c)
430 trans[r*row+c] = m[c*col+r];
432 return matrix(col,row,trans);
435 /* Leverrier algorithm for large matrices having at least one symbolic entry.
436 * This routine is only called internally by matrix::determinant(). The
437 * algorithm is very bad for symbolic matrices since it returns expressions
438 * that are quite hard to expand. */
439 /*ex matrix::determinant_symbolic_leverrier(const matrix & M)
441 * GINAC_ASSERT(M.rows()==M.cols()); // cannot happen, just in case...
444 * matrix I(M.row, M.col);
446 * for (unsigned i=1; i<M.row; ++i) {
447 * for (unsigned j=0; j<M.row; ++j)
448 * I.m[j*M.col+j] = c;
449 * B = M.mul(B.sub(I));
450 * c = B.trace()/ex(i+1);
459 /** Determinant of square matrix. This routine doesn't actually calculate the
460 * determinant, it only implements some heuristics about which algorithm to
461 * call. When the parameter for normalization is explicitly turned off this
462 * method does not normalize its result at the end, which might imply that
463 * the symbolic 2x2 matrix [[a/(a-b),1],[b/(a-b),1]] is not immediatly
464 * recognized to be unity. (This is Mathematica's default behaviour, it
465 * should be used with care.)
467 * @param normalized may be set to false if no normalization of the
468 * result is desired (i.e. to force Mathematica behavior, Maple
469 * does normalize the result).
470 * @return the determinant as a new expression
471 * @exception logic_error (matrix not square) */
472 ex matrix::determinant(bool normalized) const
475 throw (std::logic_error("matrix::determinant(): matrix not square"));
478 // check, if there are non-numeric entries in the matrix:
479 for (exvector::const_iterator r=m.begin(); r!=m.end(); ++r) {
480 if (!(*r).info(info_flags::numeric)) {
482 // return determinant_symbolic_minor().normal();
483 return determinant_symbolic_minor().normal();
485 return determinant_symbolic_perm();
488 // if it turns out that all elements are numeric
489 return determinant_numeric();
492 /** Trace of a matrix.
494 * @return the sum of diagonal elements
495 * @exception logic_error (matrix not square) */
496 ex matrix::trace(void) const
499 throw (std::logic_error("matrix::trace(): matrix not square"));
503 for (unsigned r=0; r<col; ++r)
509 /** Characteristic Polynomial. The characteristic polynomial of a matrix M is
510 * defined as the determiant of (M - lambda * 1) where 1 stands for the unit
511 * matrix of the same dimension as M. This method returns the characteristic
512 * polynomial as a new expression.
514 * @return characteristic polynomial as new expression
515 * @exception logic_error (matrix not square)
516 * @see matrix::determinant() */
517 ex matrix::charpoly(const ex & lambda) const
520 throw (std::logic_error("matrix::charpoly(): matrix not square"));
524 for (unsigned r=0; r<col; ++r)
525 M.m[r*col+r] -= lambda;
527 return (M.determinant());
530 /** Inverse of this matrix.
532 * @return the inverted matrix
533 * @exception logic_error (matrix not square)
534 * @exception runtime_error (singular matrix) */
535 matrix matrix::inverse(void) const
538 throw (std::logic_error("matrix::inverse(): matrix not square"));
542 // set tmp to the unit matrix
543 for (unsigned i=0; i<col; ++i)
544 tmp.m[i*col+i] = _ex1();
546 // create a copy of this matrix
548 for (unsigned r1=0; r1<row; ++r1) {
549 int indx = cpy.pivot(r1);
551 throw (std::runtime_error("matrix::inverse(): singular matrix"));
553 if (indx != 0) { // swap rows r and indx of matrix tmp
554 for (unsigned i=0; i<col; ++i) {
555 tmp.m[r1*col+i].swap(tmp.m[indx*col+i]);
558 ex a1 = cpy.m[r1*col+r1];
559 for (unsigned c=0; c<col; ++c) {
560 cpy.m[r1*col+c] /= a1;
561 tmp.m[r1*col+c] /= a1;
563 for (unsigned r2=0; r2<row; ++r2) {
565 ex a2 = cpy.m[r2*col+r1];
566 for (unsigned c=0; c<col; ++c) {
567 cpy.m[r2*col+c] -= a2 * cpy.m[r1*col+c];
568 tmp.m[r2*col+c] -= a2 * tmp.m[r1*col+c];
576 // superfluous helper function
577 void matrix::ffe_swap(unsigned r1, unsigned c1, unsigned r2 ,unsigned c2)
579 ensure_if_modifiable();
581 ex tmp = ffe_get(r1,c1);
582 ffe_set(r1,c1,ffe_get(r2,c2));
586 // superfluous helper function
587 void matrix::ffe_set(unsigned r, unsigned c, ex e)
592 // superfluous helper function
593 ex matrix::ffe_get(unsigned r, unsigned c) const
595 return operator()(r-1,c-1);
598 /** Solve a set of equations for an m x n matrix by fraction-free Gaussian
599 * elimination. Based on algorithm 9.1 from 'Algorithms for Computer Algebra'
600 * by Keith O. Geddes et al.
602 * @param vars n x p matrix
603 * @param rhs m x p matrix
604 * @exception logic_error (incompatible matrices)
605 * @exception runtime_error (singular matrix) */
606 matrix matrix::fraction_free_elim(const matrix & vars,
607 const matrix & rhs) const
609 // FIXME: implement a Sasaki-Murao scheme which avoids division at all!
610 if ((row != rhs.row) || (col != vars.row) || (rhs.col != vars.col))
611 throw (std::logic_error("matrix::fraction_free_elim(): incompatible matrices"));
613 matrix a(*this); // make a copy of the matrix
614 matrix b(rhs); // make a copy of the rhs vector
616 // given an m x n matrix a, reduce it to upper echelon form
623 // eliminate below row r, with pivot in column k
624 for (unsigned k=1; (k<=n)&&(r<=m); ++k) {
625 // find a nonzero pivot
627 for (p=r; (p<=m)&&(a.ffe_get(p,k).is_equal(_ex0())); ++p) {}
631 // switch rows p and r
632 for (unsigned j=k; j<=n; ++j)
635 // keep track of sign changes due to row exchange
638 for (unsigned i=r+1; i<=m; ++i) {
639 for (unsigned j=k+1; j<=n; ++j) {
640 a.ffe_set(i,j,(a.ffe_get(r,k)*a.ffe_get(i,j)
641 -a.ffe_get(r,j)*a.ffe_get(i,k))/divisor);
642 a.ffe_set(i,j,a.ffe_get(i,j).normal() /*.normal() */ );
644 b.ffe_set(i,1,(a.ffe_get(r,k)*b.ffe_get(i,1)
645 -b.ffe_get(r,1)*a.ffe_get(i,k))/divisor);
646 b.ffe_set(i,1,b.ffe_get(i,1).normal() /*.normal() */ );
649 divisor = a.ffe_get(r,k);
653 // optionally compute the determinant for square or augmented matrices
654 // if (r==m+1) { det = sign*divisor; } else { det = 0; }
657 for (unsigned r=1; r<=m; ++r) {
658 for (unsigned c=1; c<=n; ++c) {
659 cout << a.ffe_get(r,c) << "\t";
661 cout << " | " << b.ffe_get(r,1) << endl;
665 #ifdef DO_GINAC_ASSERT
666 // test if we really have an upper echelon matrix
667 int zero_in_last_row = -1;
668 for (unsigned r=1; r<=m; ++r) {
669 int zero_in_this_row=0;
670 for (unsigned c=1; c<=n; ++c) {
671 if (a.ffe_get(r,c).is_equal(_ex0()))
676 GINAC_ASSERT((zero_in_this_row>zero_in_last_row)||(zero_in_this_row=n));
677 zero_in_last_row = zero_in_this_row;
679 #endif // def DO_GINAC_ASSERT
682 cout << "after" << endl;
683 cout << "a=" << a << endl;
684 cout << "b=" << b << endl;
689 unsigned last_assigned_sol = n+1;
690 for (unsigned r=m; r>0; --r) {
691 unsigned first_non_zero = 1;
692 while ((first_non_zero<=n)&&(a.ffe_get(r,first_non_zero).is_zero()))
694 if (first_non_zero>n) {
695 // row consists only of zeroes, corresponding rhs must be 0 as well
696 if (!b.ffe_get(r,1).is_zero()) {
697 throw (std::runtime_error("matrix::fraction_free_elim(): singular matrix"));
700 // assign solutions for vars between first_non_zero+1 and
701 // last_assigned_sol-1: free parameters
702 for (unsigned c=first_non_zero+1; c<=last_assigned_sol-1; ++c) {
703 sol.ffe_set(c,1,vars.ffe_get(c,1));
705 ex e = b.ffe_get(r,1);
706 for (unsigned c=first_non_zero+1; c<=n; ++c) {
707 e=e-a.ffe_get(r,c)*sol.ffe_get(c,1);
709 sol.ffe_set(first_non_zero,1,
710 (e/a.ffe_get(r,first_non_zero)).normal());
711 last_assigned_sol = first_non_zero;
714 // assign solutions for vars between 1 and
715 // last_assigned_sol-1: free parameters
716 for (unsigned c=1; c<=last_assigned_sol-1; ++c)
717 sol.ffe_set(c,1,vars.ffe_get(c,1));
719 #ifdef DO_GINAC_ASSERT
720 // test solution with echelon matrix
721 for (unsigned r=1; r<=m; ++r) {
723 for (unsigned c=1; c<=n; ++c)
724 e = e+a.ffe_get(r,c)*sol.ffe_get(c,1);
725 if (!(e-b.ffe_get(r,1)).normal().is_zero()) {
727 cout << "b.ffe_get(" << r<<",1)=" << b.ffe_get(r,1) << endl;
728 cout << "diff=" << (e-b.ffe_get(r,1)).normal() << endl;
730 GINAC_ASSERT((e-b.ffe_get(r,1)).normal().is_zero());
733 // test solution with original matrix
734 for (unsigned r=1; r<=m; ++r) {
736 for (unsigned c=1; c<=n; ++c)
737 e = e+ffe_get(r,c)*sol.ffe_get(c,1);
739 if (!(e-rhs.ffe_get(r,1)).normal().is_zero()) {
740 cout << "e=" << e << endl;
743 cout << "e.normal()=" << en << endl;
745 cout << "rhs.ffe_get(" << r<<",1)=" << rhs.ffe_get(r,1) << endl;
746 cout << "diff=" << (e-rhs.ffe_get(r,1)).normal() << endl;
749 ex xxx = e - rhs.ffe_get(r,1);
750 cerr << "xxx=" << xxx << endl << endl;
752 GINAC_ASSERT((e-rhs.ffe_get(r,1)).normal().is_zero());
754 #endif // def DO_GINAC_ASSERT
759 /** Solve a set of equations for an m x n matrix.
761 * @param vars n x p matrix
762 * @param rhs m x p matrix
763 * @exception logic_error (incompatible matrices)
764 * @exception runtime_error (singular matrix) */
765 matrix matrix::solve(const matrix & vars,
766 const matrix & rhs) const
768 if ((row != rhs.row) || (col != vars.row) || (rhs.col != vars.col))
769 throw (std::logic_error("matrix::solve(): incompatible matrices"));
771 throw (std::runtime_error("FIXME: need implementation."));
774 /** Old and obsolete interface: */
775 matrix matrix::old_solve(const matrix & v) const
777 if ((v.row != col) || (col != v.row))
778 throw (std::logic_error("matrix::solve(): incompatible matrices"));
780 // build the augmented matrix of *this with v attached to the right
781 matrix tmp(row,col+v.col);
782 for (unsigned r=0; r<row; ++r) {
783 for (unsigned c=0; c<col; ++c)
784 tmp.m[r*tmp.col+c] = this->m[r*col+c];
785 for (unsigned c=0; c<v.col; ++c)
786 tmp.m[r*tmp.col+c+col] = v.m[r*v.col+c];
788 // cout << "augmented: " << tmp << endl;
789 tmp.gauss_elimination();
790 // cout << "degaussed: " << tmp << endl;
791 // assemble the solution matrix
792 exvector sol(v.row*v.col);
793 for (unsigned c=0; c<v.col; ++c) {
794 for (unsigned r=row; r>0; --r) {
795 for (unsigned i=r; i<col; ++i)
796 sol[(r-1)*v.col+c] -= tmp.m[(r-1)*tmp.col+i]*sol[i*v.col+c];
797 sol[(r-1)*v.col+c] += tmp.m[(r-1)*tmp.col+col+c];
798 sol[(r-1)*v.col+c] = (sol[(r-1)*v.col+c]/tmp.m[(r-1)*tmp.col+(r-1)]).normal();
801 return matrix(v.row, v.col, sol);
806 /** Determinant of purely numeric matrix, using pivoting.
808 * @see matrix::determinant() */
809 ex matrix::determinant_numeric(void) const
815 for (unsigned r1=0; r1<row; ++r1) {
816 int indx = tmp.pivot(r1);
821 det = det * tmp.m[r1*col+r1];
822 for (unsigned r2=r1+1; r2<row; ++r2) {
823 piv = tmp.m[r2*col+r1] / tmp.m[r1*col+r1];
824 for (unsigned c=r1+1; c<col; c++) {
825 tmp.m[r2*col+c] -= piv * tmp.m[r1*col+c];
832 /** Recursive determinant for small matrices having at least one symbolic
833 * entry. The basic algorithm, known as Laplace-expansion, is enhanced by
834 * some bookkeeping to avoid calculation of the same submatrices ("minors")
835 * more than once. According to W.M.Gentleman and S.C.Johnson this algorithm
836 * is better than elimination schemes for sparse multivariate polynomials and
837 * also for dense univariate polynomials once the dimesion becomes larger
840 * @see matrix::determinant() */
841 ex matrix::determinant_symbolic_minor(void) const
843 // for small matrices the algorithm does not make sense:
847 return (m[0]*m[3]-m[2]*m[1]);
849 return ((m[4]*m[8]-m[5]*m[7])*m[0]-
850 (m[1]*m[8]-m[2]*m[7])*m[3]+
851 (m[1]*m[5]-m[4]*m[2])*m[6]);
853 // This algorithm can best be understood by looking at a naive
854 // implementation of Laplace-expansion, like this one:
856 // matrix minorM(this->row-1,this->col-1);
857 // for (unsigned r1=0; r1<this->row; ++r1) {
858 // // shortcut if element(r1,0) vanishes
859 // if (m[r1*col].is_zero())
861 // // assemble the minor matrix
862 // for (unsigned r=0; r<minorM.rows(); ++r) {
863 // for (unsigned c=0; c<minorM.cols(); ++c) {
865 // minorM.set(r,c,m[r*col+c+1]);
867 // minorM.set(r,c,m[(r+1)*col+c+1]);
870 // // recurse down and care for sign:
872 // det -= m[r1*col] * minorM.determinant_symbolic_minor();
874 // det += m[r1*col] * minorM.determinant_symbolic_minor();
877 // What happens is that while proceeding down many of the minors are
878 // computed more than once. In particular, there are binomial(n,k)
879 // kxk minors and each one is computed factorial(n-k) times. Therefore
880 // it is reasonable to store the results of the minors. We proceed from
881 // right to left. At each column c we only need to retrieve the minors
882 // calculated in step c-1. We therefore only have to store at most
883 // 2*binomial(n,n/2) minors.
885 // we store our subminors in maps, keys being the rows they arise from
886 typedef map<vector<unsigned>,class ex> Rmap;
887 typedef map<vector<unsigned>,class ex>::value_type Rmap_value;
890 vector<unsigned> Pkey; // Unique flipper counter for partitioning into minors
891 Pkey.reserve(this->col);
892 vector<unsigned> Mkey; // key for minor determinant (a subpartition of Pkey)
893 Mkey.reserve(this->col-1);
894 // initialize A with last column:
895 for (unsigned r=0; r<this->col; ++r) {
896 Pkey.erase(Pkey.begin(),Pkey.end());
898 A.insert(Rmap_value(Pkey,m[this->col*r+this->col-1]));
900 // proceed from right to left through matrix
901 for (int c=this->col-2; c>=0; --c) {
902 Pkey.erase(Pkey.begin(),Pkey.end()); // don't change capacity
903 Mkey.erase(Mkey.begin(),Mkey.end());
904 for (unsigned i=0; i<this->col-c; ++i)
906 unsigned fc = 0; // controls logic for our strange flipper counter
908 A.insert(Rmap_value(Pkey,_ex0()));
910 for (unsigned r=0; r<this->col-c; ++r) {
911 // maybe there is nothing to do?
912 if (m[Pkey[r]*this->col+c].is_zero())
914 // create the sorted key for all possible minors
915 Mkey.erase(Mkey.begin(),Mkey.end());
916 for (unsigned i=0; i<this->col-c; ++i)
918 Mkey.push_back(Pkey[i]);
919 // Fetch the minors and compute the new determinant
921 det -= m[Pkey[r]*this->col+c]*A[Mkey];
923 det += m[Pkey[r]*this->col+c]*A[Mkey];
925 // Store the new determinant at its place in B:
926 B.insert(Rmap_value(Pkey,det));
927 // increment our strange flipper counter
928 for (fc=this->col-c; fc>0; --fc) {
934 for (unsigned j=fc; j<this->col-c; ++j)
935 Pkey[j] = Pkey[j-1]+1;
937 // change the role of A and B:
945 /** Determinant built by application of the full permutation group. This
946 * routine is only called internally by matrix::determinant(). */
947 ex matrix::determinant_symbolic_perm(void) const
949 if (rows()==1) // speed things up
954 vector<unsigned> sigma(col);
955 for (unsigned i=0; i<col; ++i)
959 term = (*this)(sigma[0],0);
960 for (unsigned i=1; i<col; ++i)
961 term *= (*this)(sigma[i],i);
962 det += permutation_sign(sigma)*term;
963 } while (next_permutation(sigma.begin(), sigma.end()));
968 /** Perform the steps of an ordinary Gaussian elimination to bring the matrix
969 * into an upper echelon form.
971 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
972 * number of rows was swapped and 0 if the matrix is singular. */
973 int matrix::gauss_elimination(void)
976 ensure_if_modifiable();
977 for (unsigned r1=0; r1<row-1; ++r1) {
978 int indx = pivot(r1);
980 return 0; // Note: leaves *this in a messy state.
983 for (unsigned r2=r1+1; r2<row; ++r2) {
984 for (unsigned c=r1+1; c<col; ++c)
985 this->m[r2*col+c] -= this->m[r2*col+r1]*this->m[r1*col+c]/this->m[r1*col+r1];
986 for (unsigned c=0; c<=r1; ++c)
987 this->m[r2*col+c] = _ex0();
993 /** Partial pivoting method.
994 * Usual pivoting (symbolic==false) returns the index to the element with the
995 * largest absolute value in column ro and swaps the current row with the one
996 * where the element was found. With (symbolic==true) it does the same thing
997 * with the first non-zero element.
999 * @param ro is the row to be inspected
1000 * @param symbolic signal if we want the first non-zero element to be pivoted
1001 * (true) or the one with the largest absolute value (false).
1002 * @return 0 if no interchange occured, -1 if all are zero (usually signaling
1003 * a degeneracy) and positive integer k means that rows ro and k were swapped.
1005 int matrix::pivot(unsigned ro, bool symbolic)
1009 if (symbolic) { // search first non-zero
1010 for (unsigned r=ro; r<row; ++r) {
1011 if (!m[r*col+ro].is_zero()) {
1016 } else { // search largest
1019 for (unsigned r=ro; r<row; ++r) {
1020 GINAC_ASSERT(is_ex_of_type(m[r*col+ro],numeric));
1021 if ((tmp = abs(ex_to_numeric(m[r*col+ro]))) > maxn &&
1028 if (m[k*col+ro].is_zero())
1030 if (k!=ro) { // swap rows
1031 ensure_if_modifiable();
1032 for (unsigned c=0; c<col; ++c) {
1033 m[k*col+c].swap(m[ro*col+c]);
1044 const matrix some_matrix;
1045 const type_info & typeid_matrix=typeid(some_matrix);
1047 #ifndef NO_NAMESPACE_GINAC
1048 } // namespace GiNaC
1049 #endif // ndef NO_NAMESPACE_GINAC