3 * Implementation of symbolic matrices */
6 * GiNaC Copyright (C) 1999-2001 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. */
51 matrix::matrix() : inherited(TINFO_matrix), row(1), col(1)
53 debugmsg("matrix default constructor",LOGLEVEL_CONSTRUCT);
59 void matrix::copy(const matrix & other)
61 inherited::copy(other);
64 m = other.m; // STL's vector copying invoked here
67 void matrix::destroy(bool call_parent)
69 if (call_parent) inherited::destroy(call_parent);
78 /** Very common ctor. Initializes to r x c-dimensional zero-matrix.
80 * @param r number of rows
81 * @param c number of cols */
82 matrix::matrix(unsigned r, unsigned c)
83 : inherited(TINFO_matrix), row(r), col(c)
85 debugmsg("matrix constructor from unsigned,unsigned",LOGLEVEL_CONSTRUCT);
86 m.resize(r*c, _ex0());
91 /** Ctor from representation, for internal use only. */
92 matrix::matrix(unsigned r, unsigned c, const exvector & m2)
93 : inherited(TINFO_matrix), row(r), col(c), m(m2)
95 debugmsg("matrix constructor from unsigned,unsigned,exvector",LOGLEVEL_CONSTRUCT);
102 /** Construct object from archive_node. */
103 matrix::matrix(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
105 debugmsg("matrix constructor from archive_node", LOGLEVEL_CONSTRUCT);
106 if (!(n.find_unsigned("row", row)) || !(n.find_unsigned("col", col)))
107 throw (std::runtime_error("unknown matrix dimensions in archive"));
108 m.reserve(row * col);
109 for (unsigned int i=0; true; i++) {
111 if (n.find_ex("m", e, sym_lst, i))
118 /** Unarchive the object. */
119 ex matrix::unarchive(const archive_node &n, const lst &sym_lst)
121 return (new matrix(n, sym_lst))->setflag(status_flags::dynallocated);
124 /** Archive the object. */
125 void matrix::archive(archive_node &n) const
127 inherited::archive(n);
128 n.add_unsigned("row", row);
129 n.add_unsigned("col", col);
130 exvector::const_iterator i = m.begin(), iend = m.end();
138 // functions overriding virtual functions from bases classes
143 void matrix::print(std::ostream & os, unsigned upper_precedence) const
145 debugmsg("matrix print",LOGLEVEL_PRINT);
147 for (unsigned r=0; r<row-1; ++r) {
149 for (unsigned c=0; c<col-1; ++c)
150 os << m[r*col+c] << ",";
151 os << m[col*(r+1)-1] << "]], ";
154 for (unsigned c=0; c<col-1; ++c)
155 os << m[(row-1)*col+c] << ",";
156 os << m[row*col-1] << "]] ]]";
159 void matrix::printraw(std::ostream & os) const
161 debugmsg("matrix printraw",LOGLEVEL_PRINT);
162 os << "matrix(" << row << "," << col <<",";
163 for (unsigned r=0; r<row-1; ++r) {
165 for (unsigned c=0; c<col-1; ++c)
166 os << m[r*col+c] << ",";
167 os << m[col*(r-1)-1] << "),";
170 for (unsigned c=0; c<col-1; ++c)
171 os << m[(row-1)*col+c] << ",";
172 os << m[row*col-1] << "))";
175 /** nops is defined to be rows x columns. */
176 unsigned matrix::nops() const
181 /** returns matrix entry at position (i/col, i%col). */
182 ex matrix::op(int i) const
187 /** returns matrix entry at position (i/col, i%col). */
188 ex & matrix::let_op(int i)
191 GINAC_ASSERT(i<nops());
196 /** expands the elements of a matrix entry by entry. */
197 ex matrix::expand(unsigned options) const
199 exvector tmp(row*col);
200 for (unsigned i=0; i<row*col; ++i)
201 tmp[i] = m[i].expand(options);
203 return matrix(row, col, tmp);
206 /** Search ocurrences. A matrix 'has' an expression if it is the expression
207 * itself or one of the elements 'has' it. */
208 bool matrix::has(const ex & other) const
210 GINAC_ASSERT(other.bp!=0);
212 // tautology: it is the expression itself
213 if (is_equal(*other.bp)) return true;
215 // search all the elements
216 for (exvector::const_iterator r=m.begin(); r!=m.end(); ++r)
217 if ((*r).has(other)) return true;
222 /** evaluate matrix entry by entry. */
223 ex matrix::eval(int level) const
225 debugmsg("matrix eval",LOGLEVEL_MEMBER_FUNCTION);
227 // check if we have to do anything at all
228 if ((level==1)&&(flags & status_flags::evaluated))
232 if (level == -max_recursion_level)
233 throw (std::runtime_error("matrix::eval(): recursion limit exceeded"));
235 // eval() entry by entry
236 exvector m2(row*col);
238 for (unsigned r=0; r<row; ++r)
239 for (unsigned c=0; c<col; ++c)
240 m2[r*col+c] = m[r*col+c].eval(level);
242 return (new matrix(row, col, m2))->setflag(status_flags::dynallocated |
243 status_flags::evaluated );
246 /** evaluate matrix numerically entry by entry. */
247 ex matrix::evalf(int level) const
249 debugmsg("matrix evalf",LOGLEVEL_MEMBER_FUNCTION);
251 // check if we have to do anything at all
256 if (level == -max_recursion_level) {
257 throw (std::runtime_error("matrix::evalf(): recursion limit exceeded"));
260 // evalf() entry by entry
261 exvector m2(row*col);
263 for (unsigned r=0; r<row; ++r)
264 for (unsigned c=0; c<col; ++c)
265 m2[r*col+c] = m[r*col+c].evalf(level);
267 return matrix(row, col, m2);
272 int matrix::compare_same_type(const basic & other) const
274 GINAC_ASSERT(is_exactly_of_type(other, matrix));
275 const matrix & o = static_cast<matrix &>(const_cast<basic &>(other));
277 // compare number of rows
279 return row < o.rows() ? -1 : 1;
281 // compare number of columns
283 return col < o.cols() ? -1 : 1;
285 // equal number of rows and columns, compare individual elements
287 for (unsigned r=0; r<row; ++r) {
288 for (unsigned c=0; c<col; ++c) {
289 cmpval = ((*this)(r,c)).compare(o(r,c));
290 if (cmpval!=0) return cmpval;
293 // all elements are equal => matrices are equal;
298 // non-virtual functions in this class
305 * @exception logic_error (incompatible matrices) */
306 matrix matrix::add(const matrix & other) const
308 if (col != other.col || row != other.row)
309 throw (std::logic_error("matrix::add(): incompatible matrices"));
311 exvector sum(this->m);
312 exvector::iterator i;
313 exvector::const_iterator ci;
314 for (i=sum.begin(), ci=other.m.begin(); i!=sum.end(); ++i, ++ci)
317 return matrix(row,col,sum);
321 /** Difference of matrices.
323 * @exception logic_error (incompatible matrices) */
324 matrix matrix::sub(const matrix & other) const
326 if (col != other.col || row != other.row)
327 throw (std::logic_error("matrix::sub(): incompatible matrices"));
329 exvector dif(this->m);
330 exvector::iterator i;
331 exvector::const_iterator ci;
332 for (i=dif.begin(), ci=other.m.begin(); i!=dif.end(); ++i, ++ci)
335 return matrix(row,col,dif);
339 /** Product of matrices.
341 * @exception logic_error (incompatible matrices) */
342 matrix matrix::mul(const matrix & other) const
344 if (this->cols() != other.rows())
345 throw (std::logic_error("matrix::mul(): incompatible matrices"));
347 exvector prod(this->rows()*other.cols());
349 for (unsigned r1=0; r1<this->rows(); ++r1) {
350 for (unsigned c=0; c<this->cols(); ++c) {
351 if (m[r1*col+c].is_zero())
353 for (unsigned r2=0; r2<other.cols(); ++r2)
354 prod[r1*other.col+r2] += (m[r1*col+c] * other.m[c*other.col+r2]).expand();
357 return matrix(row, other.col, prod);
361 /** operator() to access elements.
363 * @param ro row of element
364 * @param co column of element
365 * @exception range_error (index out of range) */
366 const ex & matrix::operator() (unsigned ro, unsigned co) const
368 if (ro>=row || co>=col)
369 throw (std::range_error("matrix::operator(): index out of range"));
375 /** Set individual elements manually.
377 * @exception range_error (index out of range) */
378 matrix & matrix::set(unsigned ro, unsigned co, ex value)
380 if (ro>=row || co>=col)
381 throw (std::range_error("matrix::set(): index out of range"));
383 ensure_if_modifiable();
384 m[ro*col+co] = value;
389 /** Transposed of an m x n matrix, producing a new n x m matrix object that
390 * represents the transposed. */
391 matrix matrix::transpose(void) const
393 exvector trans(this->cols()*this->rows());
395 for (unsigned r=0; r<this->cols(); ++r)
396 for (unsigned c=0; c<this->rows(); ++c)
397 trans[r*this->rows()+c] = m[c*this->cols()+r];
399 return matrix(this->cols(),this->rows(),trans);
403 /** Determinant of square matrix. This routine doesn't actually calculate the
404 * determinant, it only implements some heuristics about which algorithm to
405 * run. If all the elements of the matrix are elements of an integral domain
406 * the determinant is also in that integral domain and the result is expanded
407 * only. If one or more elements are from a quotient field the determinant is
408 * usually also in that quotient field and the result is normalized before it
409 * is returned. This implies that the determinant of the symbolic 2x2 matrix
410 * [[a/(a-b),1],[b/(a-b),1]] is returned as unity. (In this respect, it
411 * behaves like MapleV and unlike Mathematica.)
413 * @param algo allows to chose an algorithm
414 * @return the determinant as a new expression
415 * @exception logic_error (matrix not square)
416 * @see determinant_algo */
417 ex matrix::determinant(unsigned algo) const
420 throw (std::logic_error("matrix::determinant(): matrix not square"));
421 GINAC_ASSERT(row*col==m.capacity());
423 // Gather some statistical information about this matrix:
424 bool numeric_flag = true;
425 bool normal_flag = false;
426 unsigned sparse_count = 0; // counts non-zero elements
427 for (exvector::const_iterator r=m.begin(); r!=m.end(); ++r) {
428 lst srl; // symbol replacement list
429 ex rtest = (*r).to_rational(srl);
430 if (!rtest.is_zero())
432 if (!rtest.info(info_flags::numeric))
433 numeric_flag = false;
434 if (!rtest.info(info_flags::crational_polynomial) &&
435 rtest.info(info_flags::rational_function))
439 // Here is the heuristics in case this routine has to decide:
440 if (algo == determinant_algo::automatic) {
441 // Minor expansion is generally a good guess:
442 algo = determinant_algo::laplace;
443 // Does anybody know when a matrix is really sparse?
444 // Maybe <~row/2.236 nonzero elements average in a row?
445 if (row>3 && 5*sparse_count<=row*col)
446 algo = determinant_algo::bareiss;
447 // Purely numeric matrix can be handled by Gauss elimination.
448 // This overrides any prior decisions.
450 algo = determinant_algo::gauss;
453 // Trap the trivial case here, since some algorithms don't like it
455 // for consistency with non-trivial determinants...
457 return m[0].normal();
459 return m[0].expand();
462 // Compute the determinant
464 case determinant_algo::gauss: {
467 int sign = tmp.gauss_elimination(true);
468 for (unsigned d=0; d<row; ++d)
469 det *= tmp.m[d*col+d];
471 return (sign*det).normal();
473 return (sign*det).normal().expand();
475 case determinant_algo::bareiss: {
478 sign = tmp.fraction_free_elimination(true);
480 return (sign*tmp.m[row*col-1]).normal();
482 return (sign*tmp.m[row*col-1]).expand();
484 case determinant_algo::divfree: {
487 sign = tmp.division_free_elimination(true);
490 ex det = tmp.m[row*col-1];
491 // factor out accumulated bogus slag
492 for (unsigned d=0; d<row-2; ++d)
493 for (unsigned j=0; j<row-d-2; ++j)
494 det = (det/tmp.m[d*col+d]).normal();
497 case determinant_algo::laplace:
499 // This is the minor expansion scheme. We always develop such
500 // that the smallest minors (i.e, the trivial 1x1 ones) are on the
501 // rightmost column. For this to be efficient it turns out that
502 // the emptiest columns (i.e. the ones with most zeros) should be
503 // the ones on the right hand side. Therefore we presort the
504 // columns of the matrix:
505 typedef std::pair<unsigned,unsigned> uintpair;
506 std::vector<uintpair> c_zeros; // number of zeros in column
507 for (unsigned c=0; c<col; ++c) {
509 for (unsigned r=0; r<row; ++r)
510 if (m[r*col+c].is_zero())
512 c_zeros.push_back(uintpair(acc,c));
514 sort(c_zeros.begin(),c_zeros.end());
515 std::vector<unsigned> pre_sort;
516 for (std::vector<uintpair>::iterator i=c_zeros.begin(); i!=c_zeros.end(); ++i)
517 pre_sort.push_back(i->second);
518 int sign = permutation_sign(pre_sort);
519 exvector result(row*col); // represents sorted matrix
521 for (std::vector<unsigned>::iterator i=pre_sort.begin();
524 for (unsigned r=0; r<row; ++r)
525 result[r*col+c] = m[r*col+(*i)];
529 return (sign*matrix(row,col,result).determinant_minor()).normal();
531 return sign*matrix(row,col,result).determinant_minor();
537 /** Trace of a matrix. The result is normalized if it is in some quotient
538 * field and expanded only otherwise. This implies that the trace of the
539 * symbolic 2x2 matrix [[a/(a-b),x],[y,b/(b-a)]] is recognized to be unity.
541 * @return the sum of diagonal elements
542 * @exception logic_error (matrix not square) */
543 ex matrix::trace(void) const
546 throw (std::logic_error("matrix::trace(): matrix not square"));
549 for (unsigned r=0; r<col; ++r)
552 if (tr.info(info_flags::rational_function) &&
553 !tr.info(info_flags::crational_polynomial))
560 /** Characteristic Polynomial. Following mathematica notation the
561 * characteristic polynomial of a matrix M is defined as the determiant of
562 * (M - lambda * 1) where 1 stands for the unit matrix of the same dimension
563 * as M. Note that some CASs define it with a sign inside the determinant
564 * which gives rise to an overall sign if the dimension is odd. This method
565 * returns the characteristic polynomial collected in powers of lambda as a
568 * @return characteristic polynomial as new expression
569 * @exception logic_error (matrix not square)
570 * @see matrix::determinant() */
571 ex matrix::charpoly(const symbol & lambda) const
574 throw (std::logic_error("matrix::charpoly(): matrix not square"));
576 bool numeric_flag = true;
577 for (exvector::const_iterator r=m.begin(); r!=m.end(); ++r) {
578 if (!(*r).info(info_flags::numeric)) {
579 numeric_flag = false;
583 // The pure numeric case is traditionally rather common. Hence, it is
584 // trapped and we use Leverrier's algorithm which goes as row^3 for
585 // every coefficient. The expensive part is the matrix multiplication.
589 ex poly = power(lambda,row)-c*power(lambda,row-1);
590 for (unsigned i=1; i<row; ++i) {
591 for (unsigned j=0; j<row; ++j)
594 c = B.trace()/ex(i+1);
595 poly -= c*power(lambda,row-i-1);
604 for (unsigned r=0; r<col; ++r)
605 M.m[r*col+r] -= lambda;
607 return M.determinant().collect(lambda);
611 /** Inverse of this matrix.
613 * @return the inverted matrix
614 * @exception logic_error (matrix not square)
615 * @exception runtime_error (singular matrix) */
616 matrix matrix::inverse(void) const
619 throw (std::logic_error("matrix::inverse(): matrix not square"));
621 // NOTE: the Gauss-Jordan elimination used here can in principle be
622 // replaced by two clever calls to gauss_elimination() and some to
623 // transpose(). Wouldn't be more efficient (maybe less?), just more
626 // set tmp to the unit matrix
627 for (unsigned i=0; i<col; ++i)
628 tmp.m[i*col+i] = _ex1();
630 // create a copy of this matrix
632 for (unsigned r1=0; r1<row; ++r1) {
633 int indx = cpy.pivot(r1, r1);
635 throw (std::runtime_error("matrix::inverse(): singular matrix"));
637 if (indx != 0) { // swap rows r and indx of matrix tmp
638 for (unsigned i=0; i<col; ++i)
639 tmp.m[r1*col+i].swap(tmp.m[indx*col+i]);
641 ex a1 = cpy.m[r1*col+r1];
642 for (unsigned c=0; c<col; ++c) {
643 cpy.m[r1*col+c] /= a1;
644 tmp.m[r1*col+c] /= a1;
646 for (unsigned r2=0; r2<row; ++r2) {
648 if (!cpy.m[r2*col+r1].is_zero()) {
649 ex a2 = cpy.m[r2*col+r1];
650 // yes, there is something to do in this column
651 for (unsigned c=0; c<col; ++c) {
652 cpy.m[r2*col+c] -= a2 * cpy.m[r1*col+c];
653 if (!cpy.m[r2*col+c].info(info_flags::numeric))
654 cpy.m[r2*col+c] = cpy.m[r2*col+c].normal();
655 tmp.m[r2*col+c] -= a2 * tmp.m[r1*col+c];
656 if (!tmp.m[r2*col+c].info(info_flags::numeric))
657 tmp.m[r2*col+c] = tmp.m[r2*col+c].normal();
668 /** Solve a linear system consisting of a m x n matrix and a m x p right hand
669 * side by applying an elimination scheme to the augmented matrix.
671 * @param vars n x p matrix, all elements must be symbols
672 * @param rhs m x p matrix
673 * @return n x p solution matrix
674 * @exception logic_error (incompatible matrices)
675 * @exception invalid_argument (1st argument must be matrix of symbols)
676 * @exception runtime_error (inconsistent linear system)
678 matrix matrix::solve(const matrix & vars,
682 const unsigned m = this->rows();
683 const unsigned n = this->cols();
684 const unsigned p = rhs.cols();
687 if ((rhs.rows() != m) || (vars.rows() != n) || (vars.col != p))
688 throw (std::logic_error("matrix::solve(): incompatible matrices"));
689 for (unsigned ro=0; ro<n; ++ro)
690 for (unsigned co=0; co<p; ++co)
691 if (!vars(ro,co).info(info_flags::symbol))
692 throw (std::invalid_argument("matrix::solve(): 1st argument must be matrix of symbols"));
694 // build the augmented matrix of *this with rhs attached to the right
696 for (unsigned r=0; r<m; ++r) {
697 for (unsigned c=0; c<n; ++c)
698 aug.m[r*(n+p)+c] = this->m[r*n+c];
699 for (unsigned c=0; c<p; ++c)
700 aug.m[r*(n+p)+c+n] = rhs.m[r*p+c];
703 // Gather some statistical information about the augmented matrix:
704 bool numeric_flag = true;
705 for (exvector::const_iterator r=aug.m.begin(); r!=aug.m.end(); ++r) {
706 if (!(*r).info(info_flags::numeric))
707 numeric_flag = false;
710 // Here is the heuristics in case this routine has to decide:
711 if (algo == solve_algo::automatic) {
712 // Bareiss (fraction-free) elimination is generally a good guess:
713 algo = solve_algo::bareiss;
714 // For m<3, Bareiss elimination is equivalent to division free
715 // elimination but has more logistic overhead
717 algo = solve_algo::divfree;
718 // This overrides any prior decisions.
720 algo = solve_algo::gauss;
723 // Eliminate the augmented matrix:
725 case solve_algo::gauss:
726 aug.gauss_elimination();
727 case solve_algo::divfree:
728 aug.division_free_elimination();
729 case solve_algo::bareiss:
731 aug.fraction_free_elimination();
734 // assemble the solution matrix:
736 for (unsigned co=0; co<p; ++co) {
737 unsigned last_assigned_sol = n+1;
738 for (int r=m-1; r>=0; --r) {
739 unsigned fnz = 1; // first non-zero in row
740 while ((fnz<=n) && (aug.m[r*(n+p)+(fnz-1)].is_zero()))
743 // row consists only of zeros, corresponding rhs must be 0, too
744 if (!aug.m[r*(n+p)+n+co].is_zero()) {
745 throw (std::runtime_error("matrix::solve(): inconsistent linear system"));
748 // assign solutions for vars between fnz+1 and
749 // last_assigned_sol-1: free parameters
750 for (unsigned c=fnz; c<last_assigned_sol-1; ++c)
751 sol.set(c,co,vars.m[c*p+co]);
752 ex e = aug.m[r*(n+p)+n+co];
753 for (unsigned c=fnz; c<n; ++c)
754 e -= aug.m[r*(n+p)+c]*sol.m[c*p+co];
756 (e/(aug.m[r*(n+p)+(fnz-1)])).normal());
757 last_assigned_sol = fnz;
760 // assign solutions for vars between 1 and
761 // last_assigned_sol-1: free parameters
762 for (unsigned ro=0; ro<last_assigned_sol-1; ++ro)
763 sol.set(ro,co,vars(ro,co));
772 /** Recursive determinant for small matrices having at least one symbolic
773 * entry. The basic algorithm, known as Laplace-expansion, is enhanced by
774 * some bookkeeping to avoid calculation of the same submatrices ("minors")
775 * more than once. According to W.M.Gentleman and S.C.Johnson this algorithm
776 * is better than elimination schemes for matrices of sparse multivariate
777 * polynomials and also for matrices of dense univariate polynomials if the
778 * matrix' dimesion is larger than 7.
780 * @return the determinant as a new expression (in expanded form)
781 * @see matrix::determinant() */
782 ex matrix::determinant_minor(void) const
784 // for small matrices the algorithm does not make any sense:
785 const unsigned n = this->cols();
787 return m[0].expand();
789 return (m[0]*m[3]-m[2]*m[1]).expand();
791 return (m[0]*m[4]*m[8]-m[0]*m[5]*m[7]-
792 m[1]*m[3]*m[8]+m[2]*m[3]*m[7]+
793 m[1]*m[5]*m[6]-m[2]*m[4]*m[6]).expand();
795 // This algorithm can best be understood by looking at a naive
796 // implementation of Laplace-expansion, like this one:
798 // matrix minorM(this->rows()-1,this->cols()-1);
799 // for (unsigned r1=0; r1<this->rows(); ++r1) {
800 // // shortcut if element(r1,0) vanishes
801 // if (m[r1*col].is_zero())
803 // // assemble the minor matrix
804 // for (unsigned r=0; r<minorM.rows(); ++r) {
805 // for (unsigned c=0; c<minorM.cols(); ++c) {
807 // minorM.set(r,c,m[r*col+c+1]);
809 // minorM.set(r,c,m[(r+1)*col+c+1]);
812 // // recurse down and care for sign:
814 // det -= m[r1*col] * minorM.determinant_minor();
816 // det += m[r1*col] * minorM.determinant_minor();
818 // return det.expand();
819 // What happens is that while proceeding down many of the minors are
820 // computed more than once. In particular, there are binomial(n,k)
821 // kxk minors and each one is computed factorial(n-k) times. Therefore
822 // it is reasonable to store the results of the minors. We proceed from
823 // right to left. At each column c we only need to retrieve the minors
824 // calculated in step c-1. We therefore only have to store at most
825 // 2*binomial(n,n/2) minors.
827 // Unique flipper counter for partitioning into minors
828 std::vector<unsigned> Pkey;
830 // key for minor determinant (a subpartition of Pkey)
831 std::vector<unsigned> Mkey;
833 // we store our subminors in maps, keys being the rows they arise from
834 typedef std::map<std::vector<unsigned>,class ex> Rmap;
835 typedef std::map<std::vector<unsigned>,class ex>::value_type Rmap_value;
839 // initialize A with last column:
840 for (unsigned r=0; r<n; ++r) {
841 Pkey.erase(Pkey.begin(),Pkey.end());
843 A.insert(Rmap_value(Pkey,m[n*(r+1)-1]));
845 // proceed from right to left through matrix
846 for (int c=n-2; c>=0; --c) {
847 Pkey.erase(Pkey.begin(),Pkey.end()); // don't change capacity
848 Mkey.erase(Mkey.begin(),Mkey.end());
849 for (unsigned i=0; i<n-c; ++i)
851 unsigned fc = 0; // controls logic for our strange flipper counter
854 for (unsigned r=0; r<n-c; ++r) {
855 // maybe there is nothing to do?
856 if (m[Pkey[r]*n+c].is_zero())
858 // create the sorted key for all possible minors
859 Mkey.erase(Mkey.begin(),Mkey.end());
860 for (unsigned i=0; i<n-c; ++i)
862 Mkey.push_back(Pkey[i]);
863 // Fetch the minors and compute the new determinant
865 det -= m[Pkey[r]*n+c]*A[Mkey];
867 det += m[Pkey[r]*n+c]*A[Mkey];
869 // prevent build-up of deep nesting of expressions saves time:
871 // store the new determinant at its place in B:
873 B.insert(Rmap_value(Pkey,det));
874 // increment our strange flipper counter
875 for (fc=n-c; fc>0; --fc) {
881 for (unsigned j=fc; j<n-c; ++j)
882 Pkey[j] = Pkey[j-1]+1;
884 // next column, so change the role of A and B:
893 /** Perform the steps of an ordinary Gaussian elimination to bring the m x n
894 * matrix into an upper echelon form. The algorithm is ok for matrices
895 * with numeric coefficients but quite unsuited for symbolic matrices.
897 * @param det may be set to true to save a lot of space if one is only
898 * interested in the diagonal elements (i.e. for calculating determinants).
899 * The others are set to zero in this case.
900 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
901 * number of rows was swapped and 0 if the matrix is singular. */
902 int matrix::gauss_elimination(const bool det)
904 ensure_if_modifiable();
905 const unsigned m = this->rows();
906 const unsigned n = this->cols();
907 GINAC_ASSERT(!det || n==m);
911 for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
912 int indx = pivot(r0, r1, true);
916 return 0; // leaves *this in a messy state
921 for (unsigned r2=r0+1; r2<m; ++r2) {
922 if (!this->m[r2*n+r1].is_zero()) {
923 // yes, there is something to do in this row
924 ex piv = this->m[r2*n+r1] / this->m[r0*n+r1];
925 for (unsigned c=r1+1; c<n; ++c) {
926 this->m[r2*n+c] -= piv * this->m[r0*n+c];
927 if (!this->m[r2*n+c].info(info_flags::numeric))
928 this->m[r2*n+c] = this->m[r2*n+c].normal();
931 // fill up left hand side with zeros
932 for (unsigned c=0; c<=r1; ++c)
933 this->m[r2*n+c] = _ex0();
936 // save space by deleting no longer needed elements
937 for (unsigned c=r0+1; c<n; ++c)
938 this->m[r0*n+c] = _ex0();
948 /** Perform the steps of division free elimination to bring the m x n matrix
949 * into an upper echelon form.
951 * @param det may be set to true to save a lot of space if one is only
952 * interested in the diagonal elements (i.e. for calculating determinants).
953 * The others are set to zero in this case.
954 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
955 * number of rows was swapped and 0 if the matrix is singular. */
956 int matrix::division_free_elimination(const bool det)
958 ensure_if_modifiable();
959 const unsigned m = this->rows();
960 const unsigned n = this->cols();
961 GINAC_ASSERT(!det || n==m);
965 for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
966 int indx = pivot(r0, r1, true);
970 return 0; // leaves *this in a messy state
975 for (unsigned r2=r0+1; r2<m; ++r2) {
976 for (unsigned c=r1+1; c<n; ++c)
977 this->m[r2*n+c] = (this->m[r0*n+r1]*this->m[r2*n+c] - this->m[r2*n+r1]*this->m[r0*n+c]).expand();
978 // fill up left hand side with zeros
979 for (unsigned c=0; c<=r1; ++c)
980 this->m[r2*n+c] = _ex0();
983 // save space by deleting no longer needed elements
984 for (unsigned c=r0+1; c<n; ++c)
985 this->m[r0*n+c] = _ex0();
995 /** Perform the steps of Bareiss' one-step fraction free elimination to bring
996 * the matrix into an upper echelon form. Fraction free elimination means
997 * that divide is used straightforwardly, without computing GCDs first. This
998 * is possible, since we know the divisor at each step.
1000 * @param det may be set to true to save a lot of space if one is only
1001 * interested in the last element (i.e. for calculating determinants). The
1002 * others are set to zero in this case.
1003 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1004 * number of rows was swapped and 0 if the matrix is singular. */
1005 int matrix::fraction_free_elimination(const bool det)
1008 // (single-step fraction free elimination scheme, already known to Jordan)
1010 // Usual division-free elimination sets m[0](r,c) = m(r,c) and then sets
1011 // m[k+1](r,c) = m[k](k,k) * m[k](r,c) - m[k](r,k) * m[k](k,c).
1013 // Bareiss (fraction-free) elimination in addition divides that element
1014 // by m[k-1](k-1,k-1) for k>1, where it can be shown by means of the
1015 // Sylvester determinant that this really divides m[k+1](r,c).
1017 // We also allow rational functions where the original prove still holds.
1018 // However, we must care for numerator and denominator separately and
1019 // "manually" work in the integral domains because of subtle cancellations
1020 // (see below). This blows up the bookkeeping a bit and the formula has
1021 // to be modified to expand like this (N{x} stands for numerator of x,
1022 // D{x} for denominator of x):
1023 // 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)}
1024 // -N{m[k](r,k)}*N{m[k](k,c)}*D{m[k](k,k)}*D{m[k](r,c)}
1025 // 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)}
1026 // where for k>1 we now divide N{m[k+1](r,c)} by
1027 // N{m[k-1](k-1,k-1)}
1028 // and D{m[k+1](r,c)} by
1029 // D{m[k-1](k-1,k-1)}.
1031 ensure_if_modifiable();
1032 const unsigned m = this->rows();
1033 const unsigned n = this->cols();
1034 GINAC_ASSERT(!det || n==m);
1043 // We populate temporary matrices to subsequently operate on. There is
1044 // one holding numerators and another holding denominators of entries.
1045 // This is a must since the evaluator (or even earlier mul's constructor)
1046 // might cancel some trivial element which causes divide() to fail. The
1047 // elements are normalized first (yes, even though this algorithm doesn't
1048 // need GCDs) since the elements of *this might be unnormalized, which
1049 // makes things more complicated than they need to be.
1050 matrix tmp_n(*this);
1051 matrix tmp_d(m,n); // for denominators, if needed
1052 lst srl; // symbol replacement list
1053 exvector::iterator it = this->m.begin();
1054 exvector::iterator tmp_n_it = tmp_n.m.begin();
1055 exvector::iterator tmp_d_it = tmp_d.m.begin();
1056 for (; it!= this->m.end(); ++it, ++tmp_n_it, ++tmp_d_it) {
1057 (*tmp_n_it) = (*it).normal().to_rational(srl);
1058 (*tmp_d_it) = (*tmp_n_it).denom();
1059 (*tmp_n_it) = (*tmp_n_it).numer();
1063 for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
1064 int indx = tmp_n.pivot(r0, r1, true);
1073 // tmp_n's rows r0 and indx were swapped, do the same in tmp_d:
1074 for (unsigned c=r1; c<n; ++c)
1075 tmp_d.m[n*indx+c].swap(tmp_d.m[n*r0+c]);
1077 for (unsigned r2=r0+1; r2<m; ++r2) {
1078 for (unsigned c=r1+1; c<n; ++c) {
1079 dividend_n = (tmp_n.m[r0*n+r1]*tmp_n.m[r2*n+c]*
1080 tmp_d.m[r2*n+r1]*tmp_d.m[r0*n+c]
1081 -tmp_n.m[r2*n+r1]*tmp_n.m[r0*n+c]*
1082 tmp_d.m[r0*n+r1]*tmp_d.m[r2*n+c]).expand();
1083 dividend_d = (tmp_d.m[r2*n+r1]*tmp_d.m[r0*n+c]*
1084 tmp_d.m[r0*n+r1]*tmp_d.m[r2*n+c]).expand();
1085 bool check = divide(dividend_n, divisor_n,
1086 tmp_n.m[r2*n+c], true);
1087 check &= divide(dividend_d, divisor_d,
1088 tmp_d.m[r2*n+c], true);
1089 GINAC_ASSERT(check);
1091 // fill up left hand side with zeros
1092 for (unsigned c=0; c<=r1; ++c)
1093 tmp_n.m[r2*n+c] = _ex0();
1095 if ((r1<n-1)&&(r0<m-1)) {
1096 // compute next iteration's divisor
1097 divisor_n = tmp_n.m[r0*n+r1].expand();
1098 divisor_d = tmp_d.m[r0*n+r1].expand();
1100 // save space by deleting no longer needed elements
1101 for (unsigned c=0; c<n; ++c) {
1102 tmp_n.m[r0*n+c] = _ex0();
1103 tmp_d.m[r0*n+c] = _ex1();
1110 // repopulate *this matrix:
1111 it = this->m.begin();
1112 tmp_n_it = tmp_n.m.begin();
1113 tmp_d_it = tmp_d.m.begin();
1114 for (; it!= this->m.end(); ++it, ++tmp_n_it, ++tmp_d_it)
1115 (*it) = ((*tmp_n_it)/(*tmp_d_it)).subs(srl);
1121 /** Partial pivoting method for matrix elimination schemes.
1122 * Usual pivoting (symbolic==false) returns the index to the element with the
1123 * largest absolute value in column ro and swaps the current row with the one
1124 * where the element was found. With (symbolic==true) it does the same thing
1125 * with the first non-zero element.
1127 * @param ro is the row from where to begin
1128 * @param co is the column to be inspected
1129 * @param symbolic signal if we want the first non-zero element to be pivoted
1130 * (true) or the one with the largest absolute value (false).
1131 * @return 0 if no interchange occured, -1 if all are zero (usually signaling
1132 * a degeneracy) and positive integer k means that rows ro and k were swapped.
1134 int matrix::pivot(unsigned ro, unsigned co, bool symbolic)
1138 // search first non-zero element in column co beginning at row ro
1139 while ((k<row) && (this->m[k*col+co].expand().is_zero()))
1142 // search largest element in column co beginning at row ro
1143 GINAC_ASSERT(is_ex_of_type(this->m[k*col+co],numeric));
1144 unsigned kmax = k+1;
1145 numeric mmax = abs(ex_to_numeric(m[kmax*col+co]));
1147 GINAC_ASSERT(is_ex_of_type(this->m[kmax*col+co],numeric));
1148 numeric tmp = ex_to_numeric(this->m[kmax*col+co]);
1149 if (abs(tmp) > mmax) {
1155 if (!mmax.is_zero())
1159 // all elements in column co below row ro vanish
1162 // matrix needs no pivoting
1164 // matrix needs pivoting, so swap rows k and ro
1165 ensure_if_modifiable();
1166 for (unsigned c=0; c<col; ++c)
1167 this->m[k*col+c].swap(this->m[ro*col+c]);
1172 /** Convert list of lists to matrix. */
1173 ex lst_to_matrix(const ex &l)
1175 if (!is_ex_of_type(l, lst))
1176 throw(std::invalid_argument("argument to lst_to_matrix() must be a lst"));
1178 // Find number of rows and columns
1179 unsigned rows = l.nops(), cols = 0, i, j;
1180 for (i=0; i<rows; i++)
1181 if (l.op(i).nops() > cols)
1182 cols = l.op(i).nops();
1184 // Allocate and fill matrix
1185 matrix &m = *new matrix(rows, cols);
1186 for (i=0; i<rows; i++)
1187 for (j=0; j<cols; j++)
1188 if (l.op(i).nops() > j)
1189 m.set(i, j, l.op(i).op(j));
1195 #ifndef NO_NAMESPACE_GINAC
1196 } // namespace GiNaC
1197 #endif // ndef NO_NAMESPACE_GINAC