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
37 #ifndef NO_NAMESPACE_GINAC
39 #endif // ndef NO_NAMESPACE_GINAC
41 GINAC_IMPLEMENT_REGISTERED_CLASS(matrix, basic)
44 // default constructor, destructor, copy constructor, assignment operator
50 /** Default ctor. Initializes to 1 x 1-dimensional zero-matrix. */
52 : inherited(TINFO_matrix), row(1), col(1)
54 debugmsg("matrix default constructor",LOGLEVEL_CONSTRUCT);
60 debugmsg("matrix destructor",LOGLEVEL_DESTRUCT);
63 matrix::matrix(const matrix & other)
65 debugmsg("matrix copy constructor",LOGLEVEL_CONSTRUCT);
69 const matrix & matrix::operator=(const matrix & other)
71 debugmsg("matrix operator=",LOGLEVEL_ASSIGNMENT);
81 void matrix::copy(const matrix & other)
83 inherited::copy(other);
86 m = other.m; // STL's vector copying invoked here
89 void matrix::destroy(bool call_parent)
91 if (call_parent) inherited::destroy(call_parent);
100 /** Very common ctor. Initializes to r x c-dimensional zero-matrix.
102 * @param r number of rows
103 * @param c number of cols */
104 matrix::matrix(unsigned r, unsigned c)
105 : inherited(TINFO_matrix), row(r), col(c)
107 debugmsg("matrix constructor from unsigned,unsigned",LOGLEVEL_CONSTRUCT);
108 m.resize(r*c, _ex0());
113 /** Ctor from representation, for internal use only. */
114 matrix::matrix(unsigned r, unsigned c, const exvector & m2)
115 : inherited(TINFO_matrix), row(r), col(c), m(m2)
117 debugmsg("matrix constructor from unsigned,unsigned,exvector",LOGLEVEL_CONSTRUCT);
124 /** Construct object from archive_node. */
125 matrix::matrix(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
127 debugmsg("matrix constructor from archive_node", LOGLEVEL_CONSTRUCT);
128 if (!(n.find_unsigned("row", row)) || !(n.find_unsigned("col", col)))
129 throw (std::runtime_error("unknown matrix dimensions in archive"));
130 m.reserve(row * col);
131 for (unsigned int i=0; true; i++) {
133 if (n.find_ex("m", e, sym_lst, i))
140 /** Unarchive the object. */
141 ex matrix::unarchive(const archive_node &n, const lst &sym_lst)
143 return (new matrix(n, sym_lst))->setflag(status_flags::dynallocated);
146 /** Archive the object. */
147 void matrix::archive(archive_node &n) const
149 inherited::archive(n);
150 n.add_unsigned("row", row);
151 n.add_unsigned("col", col);
152 exvector::const_iterator i = m.begin(), iend = m.end();
160 // functions overriding virtual functions from bases classes
165 basic * matrix::duplicate() const
167 debugmsg("matrix duplicate",LOGLEVEL_DUPLICATE);
168 return new matrix(*this);
171 void matrix::print(ostream & os, unsigned upper_precedence) const
173 debugmsg("matrix print",LOGLEVEL_PRINT);
175 for (unsigned r=0; r<row-1; ++r) {
177 for (unsigned c=0; c<col-1; ++c) {
178 os << m[r*col+c] << ",";
180 os << m[col*(r+1)-1] << "]], ";
183 for (unsigned c=0; c<col-1; ++c) {
184 os << m[(row-1)*col+c] << ",";
186 os << m[row*col-1] << "]] ]]";
189 void matrix::printraw(ostream & os) const
191 debugmsg("matrix printraw",LOGLEVEL_PRINT);
192 os << "matrix(" << row << "," << col <<",";
193 for (unsigned r=0; r<row-1; ++r) {
195 for (unsigned c=0; c<col-1; ++c) {
196 os << m[r*col+c] << ",";
198 os << m[col*(r-1)-1] << "),";
201 for (unsigned c=0; c<col-1; ++c) {
202 os << m[(row-1)*col+c] << ",";
204 os << m[row*col-1] << "))";
207 /** nops is defined to be rows x columns. */
208 unsigned matrix::nops() const
213 /** returns matrix entry at position (i/col, i%col). */
214 ex matrix::op(int i) const
219 /** returns matrix entry at position (i/col, i%col). */
220 ex & matrix::let_op(int i)
225 /** expands the elements of a matrix entry by entry. */
226 ex matrix::expand(unsigned options) const
228 exvector tmp(row*col);
229 for (unsigned i=0; i<row*col; ++i) {
230 tmp[i]=m[i].expand(options);
232 return matrix(row, col, tmp);
235 /** Search ocurrences. A matrix 'has' an expression if it is the expression
236 * itself or one of the elements 'has' it. */
237 bool matrix::has(const ex & other) const
239 GINAC_ASSERT(other.bp!=0);
241 // tautology: it is the expression itself
242 if (is_equal(*other.bp)) return true;
244 // search all the elements
245 for (exvector::const_iterator r=m.begin(); r!=m.end(); ++r) {
246 if ((*r).has(other)) return true;
251 /** evaluate matrix entry by entry. */
252 ex matrix::eval(int level) const
254 debugmsg("matrix eval",LOGLEVEL_MEMBER_FUNCTION);
256 // check if we have to do anything at all
257 if ((level==1)&&(flags & status_flags::evaluated))
261 if (level == -max_recursion_level)
262 throw (std::runtime_error("matrix::eval(): recursion limit exceeded"));
264 // eval() entry by entry
265 exvector m2(row*col);
267 for (unsigned r=0; r<row; ++r) {
268 for (unsigned c=0; c<col; ++c) {
269 m2[r*col+c] = m[r*col+c].eval(level);
273 return (new matrix(row, col, m2))->setflag(status_flags::dynallocated |
274 status_flags::evaluated );
277 /** evaluate matrix numerically entry by entry. */
278 ex matrix::evalf(int level) const
280 debugmsg("matrix evalf",LOGLEVEL_MEMBER_FUNCTION);
282 // check if we have to do anything at all
287 if (level == -max_recursion_level) {
288 throw (std::runtime_error("matrix::evalf(): recursion limit exceeded"));
291 // evalf() entry by entry
292 exvector m2(row*col);
294 for (unsigned r=0; r<row; ++r) {
295 for (unsigned c=0; c<col; ++c) {
296 m2[r*col+c] = m[r*col+c].evalf(level);
299 return matrix(row, col, m2);
304 int matrix::compare_same_type(const basic & other) const
306 GINAC_ASSERT(is_exactly_of_type(other, matrix));
307 const matrix & o = static_cast<matrix &>(const_cast<basic &>(other));
309 // compare number of rows
311 return row < o.rows() ? -1 : 1;
313 // compare number of columns
315 return col < o.cols() ? -1 : 1;
317 // equal number of rows and columns, compare individual elements
319 for (unsigned r=0; r<row; ++r) {
320 for (unsigned c=0; c<col; ++c) {
321 cmpval = ((*this)(r,c)).compare(o(r,c));
322 if (cmpval!=0) return cmpval;
325 // all elements are equal => matrices are equal;
330 // non-virtual functions in this class
337 * @exception logic_error (incompatible matrices) */
338 matrix matrix::add(const matrix & other) const
340 if (col != other.col || row != other.row)
341 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);
355 /** Difference of matrices.
357 * @exception logic_error (incompatible matrices) */
358 matrix matrix::sub(const matrix & other) const
360 if (col != other.col || row != other.row)
361 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);
375 /** Product of matrices.
377 * @exception logic_error (incompatible matrices) */
378 matrix matrix::mul(const matrix & other) const
380 if (col != other.row)
381 throw (std::logic_error("matrix::mul(): incompatible matrices"));
383 exvector prod(row*other.col);
385 for (unsigned r1=0; r1<row; ++r1) {
386 for (unsigned c=0; c<col; ++c) {
387 if (m[r1*col+c].is_zero())
389 for (unsigned r2=0; r2<other.col; ++r2)
390 prod[r1*other.col+r2] += m[r1*col+c] * other.m[c*other.col+r2];
393 return matrix(row, other.col, prod);
397 /** operator() to access elements.
399 * @param ro row of element
400 * @param co column of element
401 * @exception range_error (index out of range) */
402 const ex & matrix::operator() (unsigned ro, unsigned co) const
404 if (ro<0 || ro>=row || co<0 || co>=col)
405 throw (std::range_error("matrix::operator(): index out of range"));
411 /** Set individual elements manually.
413 * @exception range_error (index out of range) */
414 matrix & matrix::set(unsigned ro, unsigned co, ex value)
416 if (ro<0 || ro>=row || co<0 || co>=col)
417 throw (std::range_error("matrix::set(): index out of range"));
419 ensure_if_modifiable();
420 m[ro*col+co] = value;
425 /** Transposed of an m x n matrix, producing a new n x m matrix object that
426 * represents the transposed. */
427 matrix matrix::transpose(void) const
429 exvector trans(col*row);
431 for (unsigned r=0; r<col; ++r)
432 for (unsigned c=0; c<row; ++c)
433 trans[r*row+c] = m[c*col+r];
435 return matrix(col,row,trans);
439 /** Determinant of square matrix. This routine doesn't actually calculate the
440 * determinant, it only implements some heuristics about which algorithm to
441 * call. If all the elements of the matrix are elements of an integral domain
442 * the determinant is also in that integral domain and the result is expanded
443 * only. If one or more elements are from a quotient field the determinant is
444 * usually also in that quotient field and the result is normalized before it
445 * is returned. This implies that the determinant of the symbolic 2x2 matrix
446 * [[a/(a-b),1],[b/(a-b),1]] is returned as unity. (In this respect, it
447 * behaves like MapleV and unlike Mathematica.)
449 * @return the determinant as a new expression
450 * @exception logic_error (matrix not square) */
451 ex matrix::determinant(void) const
454 throw (std::logic_error("matrix::determinant(): matrix not square"));
455 GINAC_ASSERT(row*col==m.capacity());
456 if (this->row==1) // continuation would be pointless
459 bool numeric_flag = true;
460 bool normal_flag = false;
461 unsigned sparse_count = 0; // count non-zero elements
462 for (exvector::const_iterator r=m.begin(); r!=m.end(); ++r) {
463 if (!(*r).is_zero()) {
466 if (!(*r).info(info_flags::numeric)) {
467 numeric_flag = false;
469 if ((*r).info(info_flags::rational_function) &&
470 !(*r).info(info_flags::crational_polynomial)) {
475 if (numeric_flag) // purely numeric matrix
476 return determinant_numeric();
478 // Does anybody really know when a matrix is sparse?
479 if (4*sparse_count<row*col) { // < row/2 nonzero elements average in a row
480 return determinant_bareiss(normal_flag);
483 // Now come the minor expansion schemes. We always develop such that the
484 // smallest minors (i.e, the trivial 1x1 ones) are on the rightmost column.
485 // For this to be efficient it turns out that the emptiest columns (i.e.
486 // the ones with most zeros) should be the ones on the right hand side.
487 // Therefore we presort the columns of the matrix:
488 typedef pair<unsigned,unsigned> uintpair; // # of zeros, column
489 vector<uintpair> c_zeros; // number of zeros in column
490 for (unsigned c=0; c<col; ++c) {
492 for (unsigned r=0; r<row; ++r)
493 if (m[r*col+c].is_zero())
495 c_zeros.push_back(uintpair(acc,c));
497 sort(c_zeros.begin(),c_zeros.end());
498 vector<unsigned> pre_sort; // unfortunately vector<uintpair> can't be used
499 // for permutation_sign.
500 for (vector<uintpair>::iterator i=c_zeros.begin(); i!=c_zeros.end(); ++i)
501 pre_sort.push_back(i->second);
502 int sign = permutation_sign(pre_sort);
503 exvector result(row*col); // represents sorted matrix
505 for (vector<unsigned>::iterator i=pre_sort.begin();
508 for (unsigned r=0; r<row; ++r)
509 result[r*col+c] = m[r*col+(*i)];
513 return sign*matrix(row,col,result).determinant_minor().normal();
514 return sign*matrix(row,col,result).determinant_minor();
518 /** Trace of a matrix. The result is normalized if it is in some quotient
519 * field and expanded only otherwise. This implies that the trace of the
520 * symbolic 2x2 matrix [[a/(a-b),x],[y,b/(b-a)]] is recognized to be unity.
522 * @return the sum of diagonal elements
523 * @exception logic_error (matrix not square) */
524 ex matrix::trace(void) const
527 throw (std::logic_error("matrix::trace(): matrix not square"));
528 GINAC_ASSERT(row*col==m.capacity());
531 for (unsigned r=0; r<col; ++r)
534 if (tr.info(info_flags::rational_function) &&
535 !tr.info(info_flags::crational_polynomial))
542 /** Characteristic Polynomial. Following mathematica notation the
543 * characteristic polynomial of a matrix M is defined as the determiant of
544 * (M - lambda * 1) where 1 stands for the unit matrix of the same dimension
545 * as M. Note that some CASs define it with a sign inside the determinant
546 * which gives rise to an overall sign if the dimension is odd. This method
547 * returns the characteristic polynomial collected in powers of lambda as a
550 * @return characteristic polynomial as new expression
551 * @exception logic_error (matrix not square)
552 * @see matrix::determinant() */
553 ex matrix::charpoly(const symbol & lambda) const
556 throw (std::logic_error("matrix::charpoly(): matrix not square"));
558 bool numeric_flag = true;
559 for (exvector::const_iterator r=m.begin(); r!=m.end(); ++r) {
560 if (!(*r).info(info_flags::numeric)) {
561 numeric_flag = false;
565 // The pure numeric case is traditionally rather common. Hence, it is
566 // trapped and we use Leverrier's algorithm which goes as row^3 for
567 // every coefficient. The expensive section is the matrix multiplication.
571 ex poly = power(lambda,row)-c*power(lambda,row-1);
572 for (unsigned i=1; i<row; ++i) {
573 for (unsigned j=0; j<row; ++j)
576 c = B.trace()/ex(i+1);
577 poly -= c*power(lambda,row-i-1);
586 for (unsigned r=0; r<col; ++r)
587 M.m[r*col+r] -= lambda;
589 return M.determinant().collect(lambda);
593 /** Inverse of this matrix.
595 * @return the inverted matrix
596 * @exception logic_error (matrix not square)
597 * @exception runtime_error (singular matrix) */
598 matrix matrix::inverse(void) const
601 throw (std::logic_error("matrix::inverse(): matrix not square"));
604 // set tmp to the unit matrix
605 for (unsigned i=0; i<col; ++i)
606 tmp.m[i*col+i] = _ex1();
608 // create a copy of this matrix
610 for (unsigned r1=0; r1<row; ++r1) {
611 int indx = cpy.pivot(r1);
613 throw (std::runtime_error("matrix::inverse(): singular matrix"));
615 if (indx != 0) { // swap rows r and indx of matrix tmp
616 for (unsigned i=0; i<col; ++i) {
617 tmp.m[r1*col+i].swap(tmp.m[indx*col+i]);
620 ex a1 = cpy.m[r1*col+r1];
621 for (unsigned c=0; c<col; ++c) {
622 cpy.m[r1*col+c] /= a1;
623 tmp.m[r1*col+c] /= a1;
625 for (unsigned r2=0; r2<row; ++r2) {
627 ex a2 = cpy.m[r2*col+r1];
628 for (unsigned c=0; c<col; ++c) {
629 cpy.m[r2*col+c] -= a2 * cpy.m[r1*col+c];
630 tmp.m[r2*col+c] -= a2 * tmp.m[r1*col+c];
639 // superfluous helper function
640 void matrix::ffe_swap(unsigned r1, unsigned c1, unsigned r2 ,unsigned c2)
642 ensure_if_modifiable();
644 ex tmp = ffe_get(r1,c1);
645 ffe_set(r1,c1,ffe_get(r2,c2));
649 // superfluous helper function
650 void matrix::ffe_set(unsigned r, unsigned c, ex e)
655 // superfluous helper function
656 ex matrix::ffe_get(unsigned r, unsigned c) const
658 return operator()(r-1,c-1);
661 /** Solve a set of equations for an m x n matrix by fraction-free Gaussian
662 * elimination. Based on algorithm 9.1 from 'Algorithms for Computer Algebra'
663 * by Keith O. Geddes et al.
665 * @param vars n x p matrix
666 * @param rhs m x p matrix
667 * @exception logic_error (incompatible matrices)
668 * @exception runtime_error (singular matrix) */
669 matrix matrix::fraction_free_elim(const matrix & vars,
670 const matrix & rhs) const
672 // FIXME: use implementation of matrix::fraction_free_elim
673 if ((row != rhs.row) || (col != vars.row) || (rhs.col != vars.col))
674 throw (std::logic_error("matrix::fraction_free_elim(): incompatible matrices"));
676 matrix a(*this); // make a copy of the matrix
677 matrix b(rhs); // make a copy of the rhs vector
679 // given an m x n matrix a, reduce it to upper echelon form
686 // eliminate below row r, with pivot in column k
687 for (unsigned k=1; (k<=n)&&(r<=m); ++k) {
688 // find a nonzero pivot
690 for (p=r; (p<=m)&&(a.ffe_get(p,k).is_equal(_ex0())); ++p) {}
694 // switch rows p and r
695 for (unsigned j=k; j<=n; ++j)
698 // keep track of sign changes due to row exchange
701 for (unsigned i=r+1; i<=m; ++i) {
702 for (unsigned j=k+1; j<=n; ++j) {
703 a.ffe_set(i,j,(a.ffe_get(r,k)*a.ffe_get(i,j)
704 -a.ffe_get(r,j)*a.ffe_get(i,k))/divisor);
705 a.ffe_set(i,j,a.ffe_get(i,j).normal() /*.normal() */ );
707 b.ffe_set(i,1,(a.ffe_get(r,k)*b.ffe_get(i,1)
708 -b.ffe_get(r,1)*a.ffe_get(i,k))/divisor);
709 b.ffe_set(i,1,b.ffe_get(i,1).normal() /*.normal() */ );
712 divisor = a.ffe_get(r,k);
716 // optionally compute the determinant for square or augmented matrices
717 // if (r==m+1) { det = sign*divisor; } else { det = 0; }
720 for (unsigned r=1; r<=m; ++r) {
721 for (unsigned c=1; c<=n; ++c) {
722 cout << a.ffe_get(r,c) << "\t";
724 cout << " | " << b.ffe_get(r,1) << endl;
728 #ifdef DO_GINAC_ASSERT
729 // test if we really have an upper echelon matrix
730 int zero_in_last_row = -1;
731 for (unsigned r=1; r<=m; ++r) {
732 int zero_in_this_row=0;
733 for (unsigned c=1; c<=n; ++c) {
734 if (a.ffe_get(r,c).is_equal(_ex0()))
739 GINAC_ASSERT((zero_in_this_row>zero_in_last_row)||(zero_in_this_row=n));
740 zero_in_last_row = zero_in_this_row;
742 #endif // def DO_GINAC_ASSERT
745 cout << "after" << endl;
746 cout << "a=" << a << endl;
747 cout << "b=" << b << endl;
752 unsigned last_assigned_sol = n+1;
753 for (unsigned r=m; r>0; --r) {
754 unsigned first_non_zero = 1;
755 while ((first_non_zero<=n)&&(a.ffe_get(r,first_non_zero).is_zero()))
757 if (first_non_zero>n) {
758 // row consists only of zeroes, corresponding rhs must be 0 as well
759 if (!b.ffe_get(r,1).is_zero()) {
760 throw (std::runtime_error("matrix::fraction_free_elim(): singular matrix"));
763 // assign solutions for vars between first_non_zero+1 and
764 // last_assigned_sol-1: free parameters
765 for (unsigned c=first_non_zero+1; c<=last_assigned_sol-1; ++c) {
766 sol.ffe_set(c,1,vars.ffe_get(c,1));
768 ex e = b.ffe_get(r,1);
769 for (unsigned c=first_non_zero+1; c<=n; ++c) {
770 e=e-a.ffe_get(r,c)*sol.ffe_get(c,1);
772 sol.ffe_set(first_non_zero,1,
773 (e/a.ffe_get(r,first_non_zero)).normal());
774 last_assigned_sol = first_non_zero;
777 // assign solutions for vars between 1 and
778 // last_assigned_sol-1: free parameters
779 for (unsigned c=1; c<=last_assigned_sol-1; ++c)
780 sol.ffe_set(c,1,vars.ffe_get(c,1));
782 #ifdef DO_GINAC_ASSERT
783 // test solution with echelon matrix
784 for (unsigned r=1; r<=m; ++r) {
786 for (unsigned c=1; c<=n; ++c)
787 e = e+a.ffe_get(r,c)*sol.ffe_get(c,1);
788 if (!(e-b.ffe_get(r,1)).normal().is_zero()) {
790 cout << "b.ffe_get(" << r<<",1)=" << b.ffe_get(r,1) << endl;
791 cout << "diff=" << (e-b.ffe_get(r,1)).normal() << endl;
793 GINAC_ASSERT((e-b.ffe_get(r,1)).normal().is_zero());
796 // test solution with original matrix
797 for (unsigned r=1; r<=m; ++r) {
799 for (unsigned c=1; c<=n; ++c)
800 e = e+ffe_get(r,c)*sol.ffe_get(c,1);
802 if (!(e-rhs.ffe_get(r,1)).normal().is_zero()) {
803 cout << "e=" << e << endl;
806 cout << "e.normal()=" << en << endl;
808 cout << "rhs.ffe_get(" << r<<",1)=" << rhs.ffe_get(r,1) << endl;
809 cout << "diff=" << (e-rhs.ffe_get(r,1)).normal() << endl;
812 ex xxx = e - rhs.ffe_get(r,1);
813 cerr << "xxx=" << xxx << endl << endl;
815 GINAC_ASSERT((e-rhs.ffe_get(r,1)).normal().is_zero());
817 #endif // def DO_GINAC_ASSERT
822 /** Solve a set of equations for an m x n matrix.
824 * @param vars n x p matrix
825 * @param rhs m x p matrix
826 * @exception logic_error (incompatible matrices)
827 * @exception runtime_error (singular matrix) */
828 matrix matrix::solve(const matrix & vars,
829 const matrix & rhs) const
831 if ((row != rhs.row) || (col != vars.row) || (rhs.col != vars.col))
832 throw (std::logic_error("matrix::solve(): incompatible matrices"));
834 throw (std::runtime_error("FIXME: need implementation."));
837 /** Old and obsolete interface: */
838 matrix matrix::old_solve(const matrix & v) const
840 if ((v.row != col) || (col != v.row))
841 throw (std::logic_error("matrix::solve(): incompatible matrices"));
843 // build the augmented matrix of *this with v attached to the right
844 matrix tmp(row,col+v.col);
845 for (unsigned r=0; r<row; ++r) {
846 for (unsigned c=0; c<col; ++c)
847 tmp.m[r*tmp.col+c] = this->m[r*col+c];
848 for (unsigned c=0; c<v.col; ++c)
849 tmp.m[r*tmp.col+c+col] = v.m[r*v.col+c];
851 // cout << "augmented: " << tmp << endl;
852 tmp.gauss_elimination();
853 // cout << "degaussed: " << tmp << endl;
854 // assemble the solution matrix
855 exvector sol(v.row*v.col);
856 for (unsigned c=0; c<v.col; ++c) {
857 for (unsigned r=row; r>0; --r) {
858 for (unsigned i=r; i<col; ++i)
859 sol[(r-1)*v.col+c] -= tmp.m[(r-1)*tmp.col+i]*sol[i*v.col+c];
860 sol[(r-1)*v.col+c] += tmp.m[(r-1)*tmp.col+col+c];
861 sol[(r-1)*v.col+c] = (sol[(r-1)*v.col+c]/tmp.m[(r-1)*tmp.col+(r-1)]).normal();
864 return matrix(v.row, v.col, sol);
870 /** Determinant of purely numeric matrix, using pivoting.
872 * @see matrix::determinant() */
873 ex matrix::determinant_numeric(void) const
879 // standard Gauss method:
880 for (unsigned r1=0; r1<row; ++r1) {
881 int indx = tmp.pivot(r1);
886 det = det * tmp.m[r1*col+r1];
887 for (unsigned r2=r1+1; r2<row; ++r2) {
888 piv = tmp.m[r2*col+r1] / tmp.m[r1*col+r1];
889 for (unsigned c=r1+1; c<col; c++) {
890 tmp.m[r2*col+c] -= piv * tmp.m[r1*col+c];
899 /** Recursive determinant for small matrices having at least one symbolic
900 * entry. The basic algorithm, known as Laplace-expansion, is enhanced by
901 * some bookkeeping to avoid calculation of the same submatrices ("minors")
902 * more than once. According to W.M.Gentleman and S.C.Johnson this algorithm
903 * is better than elimination schemes for matrices of sparse multivariate
904 * polynomials and also for matrices of dense univariate polynomials if the
905 * matrix' dimesion is larger than 7.
907 * @return the determinant as a new expression (in expanded form)
908 * @see matrix::determinant() */
909 ex matrix::determinant_minor(void) const
911 // for small matrices the algorithm does not make any sense:
915 return (m[0]*m[3]-m[2]*m[1]).expand();
917 return (m[0]*m[4]*m[8]-m[0]*m[5]*m[7]-
918 m[1]*m[3]*m[8]+m[2]*m[3]*m[7]+
919 m[1]*m[5]*m[6]-m[2]*m[4]*m[6]).expand();
921 // This algorithm can best be understood by looking at a naive
922 // implementation of Laplace-expansion, like this one:
924 // matrix minorM(this->row-1,this->col-1);
925 // for (unsigned r1=0; r1<this->row; ++r1) {
926 // // shortcut if element(r1,0) vanishes
927 // if (m[r1*col].is_zero())
929 // // assemble the minor matrix
930 // for (unsigned r=0; r<minorM.rows(); ++r) {
931 // for (unsigned c=0; c<minorM.cols(); ++c) {
933 // minorM.set(r,c,m[r*col+c+1]);
935 // minorM.set(r,c,m[(r+1)*col+c+1]);
938 // // recurse down and care for sign:
940 // det -= m[r1*col] * minorM.determinant_minor();
942 // det += m[r1*col] * minorM.determinant_minor();
944 // return det.expand();
945 // What happens is that while proceeding down many of the minors are
946 // computed more than once. In particular, there are binomial(n,k)
947 // kxk minors and each one is computed factorial(n-k) times. Therefore
948 // it is reasonable to store the results of the minors. We proceed from
949 // right to left. At each column c we only need to retrieve the minors
950 // calculated in step c-1. We therefore only have to store at most
951 // 2*binomial(n,n/2) minors.
953 // Unique flipper counter for partitioning into minors
954 vector<unsigned> Pkey;
955 Pkey.reserve(this->col);
956 // key for minor determinant (a subpartition of Pkey)
957 vector<unsigned> Mkey;
958 Mkey.reserve(this->col-1);
959 // we store our subminors in maps, keys being the rows they arise from
960 typedef map<vector<unsigned>,class ex> Rmap;
961 typedef map<vector<unsigned>,class ex>::value_type Rmap_value;
965 // initialize A with last column:
966 for (unsigned r=0; r<this->col; ++r) {
967 Pkey.erase(Pkey.begin(),Pkey.end());
969 A.insert(Rmap_value(Pkey,m[this->col*r+this->col-1]));
971 // proceed from right to left through matrix
972 for (int c=this->col-2; c>=0; --c) {
973 Pkey.erase(Pkey.begin(),Pkey.end()); // don't change capacity
974 Mkey.erase(Mkey.begin(),Mkey.end());
975 for (unsigned i=0; i<this->col-c; ++i)
977 unsigned fc = 0; // controls logic for our strange flipper counter
980 for (unsigned r=0; r<this->col-c; ++r) {
981 // maybe there is nothing to do?
982 if (m[Pkey[r]*this->col+c].is_zero())
984 // create the sorted key for all possible minors
985 Mkey.erase(Mkey.begin(),Mkey.end());
986 for (unsigned i=0; i<this->col-c; ++i)
988 Mkey.push_back(Pkey[i]);
989 // Fetch the minors and compute the new determinant
991 det -= m[Pkey[r]*this->col+c]*A[Mkey];
993 det += m[Pkey[r]*this->col+c]*A[Mkey];
995 // prevent build-up of deep nesting of expressions saves time:
997 // store the new determinant at its place in B:
999 B.insert(Rmap_value(Pkey,det));
1000 // increment our strange flipper counter
1001 for (fc=this->col-c; fc>0; --fc) {
1003 if (Pkey[fc-1]<fc+c)
1007 for (unsigned j=fc; j<this->col-c; ++j)
1008 Pkey[j] = Pkey[j-1]+1;
1010 // next column, so change the role of A and B:
1018 /** Helper function to divide rational functions, as needed in any Bareiss
1019 * elimination scheme over a quotient field.
1022 bool rat_divide(const ex & a, const ex & b, ex & q, bool check_args = true)
1026 throw(std::overflow_error("rat_divide(): division by zero"));
1029 if (is_ex_exactly_of_type(b, numeric)) {
1032 } else if (is_ex_exactly_of_type(a, numeric))
1038 ex n; // new numerator
1039 ex d; // new denominator
1041 check &= divide(a_n, b_n, n, check_args);
1042 check &= divide(a_d, b_d, d, check_args);
1048 /** Determinant computed by using fraction free elimination. This
1049 * routine is only called internally by matrix::determinant().
1051 * @param normalize may be set to false only in integral domains. */
1052 ex matrix::determinant_bareiss(bool normalize) const
1061 // we populate a tmp matrix to subsequently operate on, it should
1062 // be normalized even though this algorithm doesn't need GCDs since
1063 // the elements of *this might be unnormalized, which complicates
1066 exvector::const_iterator i = m.begin();
1067 exvector::iterator ti = tmp.m.begin();
1068 for (; i!= m.end(); ++i, ++ti) {
1070 (*ti) = (*i).normal();
1075 for (unsigned r1=0; r1<row-1; ++r1) {
1076 int indx = tmp.pivot(r1);
1082 divisor = tmp.m[(r1-1)*col+(r1-1)].expand();
1083 // delete the elements we don't need anymore:
1084 for (unsigned c=0; c<col; ++c)
1085 tmp.m[(r1-1)*col+c] = _ex0();
1087 for (unsigned r2=r1+1; r2<row; ++r2) {
1088 for (unsigned c=r1+1; c<col; ++c) {
1089 lst srl; // symbol replacement list for .to_rational()
1090 dividend = (tmp.m[r1*tmp.col+r1]*tmp.m[r2*tmp.col+c]
1091 -tmp.m[r2*tmp.col+r1]*tmp.m[r1*tmp.col+c]).expand();
1093 #ifdef DO_GINAC_ASSERT
1094 GINAC_ASSERT(rat_divide(dividend.to_rational(srl),
1095 divisor.to_rational(srl),
1096 tmp.m[r2*tmp.col+c],true));
1098 rat_divide(dividend.to_rational(srl),
1099 divisor.to_rational(srl),
1100 tmp.m[r2*tmp.col+c],false);
1104 #ifdef DO_GINAC_ASSERT
1105 GINAC_ASSERT(divide(dividend.to_rational(srl),
1106 divisor.to_rational(srl),
1107 tmp.m[r2*tmp.col+c],true));
1109 divide(dividend.to_rational(srl),
1110 divisor.to_rational(srl),
1111 tmp.m[r2*tmp.col+c],false);
1114 tmp.m[r2*tmp.col+c] = tmp.m[r2*tmp.col+c].subs(srl);
1116 for (unsigned c=0; c<=r1; ++c)
1117 tmp.m[r2*tmp.col+c] = _ex0();
1121 return sign*tmp.m[tmp.row*tmp.col-1];
1125 /** Perform the steps of an ordinary Gaussian elimination to bring the matrix
1126 * into an upper echelon form.
1128 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1129 * number of rows was swapped and 0 if the matrix is singular. */
1130 int matrix::gauss_elimination(void)
1133 ensure_if_modifiable();
1134 for (unsigned r1=0; r1<row-1; ++r1) {
1135 int indx = pivot(r1);
1137 return 0; // Note: leaves *this in a messy state.
1140 for (unsigned r2=r1+1; r2<row; ++r2) {
1141 for (unsigned c=r1+1; c<col; ++c)
1142 this->m[r2*col+c] -= this->m[r2*col+r1]*this->m[r1*col+c]/this->m[r1*col+r1];
1143 for (unsigned c=0; c<=r1; ++c)
1144 this->m[r2*col+c] = _ex0();
1152 /** Perform the steps of division free elimination to bring the matrix
1153 * into an upper echelon form.
1155 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1156 * number of rows was swapped and 0 if the matrix is singular. */
1157 int matrix::division_free_elimination(void)
1160 ensure_if_modifiable();
1161 for (unsigned r1=0; r1<row-1; ++r1) {
1162 int indx = pivot(r1);
1164 return 0; // Note: leaves *this in a messy state.
1167 for (unsigned r2=r1+1; r2<row; ++r2) {
1168 for (unsigned c=r1+1; c<col; ++c)
1169 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];
1170 for (unsigned c=0; c<=r1; ++c)
1171 this->m[r2*col+c] = _ex0();
1179 /** Perform the steps of Bareiss' one-step fraction free elimination to bring
1180 * the matrix into an upper echelon form.
1182 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1183 * number of rows was swapped and 0 if the matrix is singular. */
1184 int matrix::fraction_free_elimination(void)
1186 ensure_if_modifiable();
1188 // first normal all elements:
1189 for (exvector::iterator i=m.begin(); i!=m.end(); ++i)
1190 (*i) = (*i).normal();
1192 // FIXME: this is unfinished, once matrix::determinant_bareiss is
1193 // bulletproof, some code ought to be copy from there to here.
1197 lst srl; // symbol replacement list for .to_rational()
1199 for (unsigned r1=0; r1<row-1; ++r1) {
1200 int indx = pivot(r1);
1202 return 0; // Note: leaves *this in a messy state.
1206 divisor = this->m[(r1-1)*col+(r1-1)].expand();
1207 for (unsigned r2=r1+1; r2<row; ++r2) {
1208 for (unsigned c=r1+1; c<col; ++c) {
1209 dividend = (this->m[r1*col+r1]*this->m[r2*col+c]
1210 -this->m[r2*col+r1]*this->m[r1*col+c]).expand();
1211 #ifdef DO_GINAC_ASSERT
1212 GINAC_ASSERT(divide(dividend.to_rational(srl),
1213 divisor.to_rational(srl),
1214 this->m[r2*col+c]));
1216 divide(dividend.to_rational(srl),
1217 divisor.to_rational(srl),
1219 #endif // DO_GINAC_ASSERT
1220 this->m[r2*col+c] = this->m[r2*col+c].subs(srl);
1222 for (unsigned c=0; c<=r1; ++c)
1223 this->m[r2*col+c] = _ex0();
1231 /** Partial pivoting method.
1232 * Usual pivoting (symbolic==false) returns the index to the element with the
1233 * largest absolute value in column ro and swaps the current row with the one
1234 * where the element was found. With (symbolic==true) it does the same thing
1235 * with the first non-zero element.
1237 * @param ro is the row to be inspected
1238 * @param symbolic signal if we want the first non-zero element to be pivoted
1239 * (true) or the one with the largest absolute value (false).
1240 * @return 0 if no interchange occured, -1 if all are zero (usually signaling
1241 * a degeneracy) and positive integer k means that rows ro and k were swapped.
1243 int matrix::pivot(unsigned ro, bool symbolic)
1247 if (symbolic) { // search first non-zero
1248 for (unsigned r=ro; r<row; ++r) {
1249 if (!m[r*col+ro].is_zero()) {
1254 } else { // search largest
1257 for (unsigned r=ro; r<row; ++r) {
1258 GINAC_ASSERT(is_ex_of_type(m[r*col+ro],numeric));
1259 if ((tmp = abs(ex_to_numeric(m[r*col+ro]))) > maxn &&
1266 if (m[k*col+ro].is_zero())
1268 if (k!=ro) { // swap rows
1269 ensure_if_modifiable();
1270 for (unsigned c=0; c<col; ++c) {
1271 m[k*col+c].swap(m[ro*col+c]);
1278 /** Convert list of lists to matrix. */
1279 ex lst_to_matrix(const ex &l)
1281 if (!is_ex_of_type(l, lst))
1282 throw(std::invalid_argument("argument to lst_to_matrix() must be a lst"));
1284 // Find number of rows and columns
1285 unsigned rows = l.nops(), cols = 0, i, j;
1286 for (i=0; i<rows; i++)
1287 if (l.op(i).nops() > cols)
1288 cols = l.op(i).nops();
1290 // Allocate and fill matrix
1291 matrix &m = *new matrix(rows, cols);
1292 for (i=0; i<rows; i++)
1293 for (j=0; j<cols; j++)
1294 if (l.op(i).nops() > j)
1295 m.set(i, j, l.op(i).op(j));
1305 const matrix some_matrix;
1306 const type_info & typeid_matrix=typeid(some_matrix);
1308 #ifndef NO_NAMESPACE_GINAC
1309 } // namespace GiNaC
1310 #endif // ndef NO_NAMESPACE_GINAC