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
41 GINAC_IMPLEMENT_REGISTERED_CLASS(matrix, basic)
44 // default ctor, dtor, copy ctor, assignment operator and helpers:
49 /** Default ctor. Initializes to 1 x 1-dimensional zero-matrix. */
50 matrix::matrix() : inherited(TINFO_matrix), row(1), col(1)
52 debugmsg("matrix default ctor",LOGLEVEL_CONSTRUCT);
58 /** For use by copy ctor and assignment operator. */
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 ctor 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 ctor from unsigned,unsigned,exvector",LOGLEVEL_CONSTRUCT);
98 /** Construct matrix from (flat) list of elements. If the list has fewer
99 * elements than the matrix, the remaining matrix elements are set to zero.
100 * If the list has more elements than the matrix, the excessive elements are
102 matrix::matrix(unsigned r, unsigned c, const lst & l)
103 : inherited(TINFO_matrix), row(r), col(c)
105 debugmsg("matrix ctor from unsigned,unsigned,lst",LOGLEVEL_CONSTRUCT);
106 m.resize(r*c, _ex0());
108 for (unsigned i=0; i<l.nops(); i++) {
112 break; // matrix smaller than list: throw away excessive elements
121 /** Construct object from archive_node. */
122 matrix::matrix(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
124 debugmsg("matrix ctor from archive_node", LOGLEVEL_CONSTRUCT);
125 if (!(n.find_unsigned("row", row)) || !(n.find_unsigned("col", col)))
126 throw (std::runtime_error("unknown matrix dimensions in archive"));
127 m.reserve(row * col);
128 for (unsigned int i=0; true; i++) {
130 if (n.find_ex("m", e, sym_lst, i))
137 /** Unarchive the object. */
138 ex matrix::unarchive(const archive_node &n, const lst &sym_lst)
140 return (new matrix(n, sym_lst))->setflag(status_flags::dynallocated);
143 /** Archive the object. */
144 void matrix::archive(archive_node &n) const
146 inherited::archive(n);
147 n.add_unsigned("row", row);
148 n.add_unsigned("col", col);
149 exvector::const_iterator i = m.begin(), iend = m.end();
157 // functions overriding virtual functions from bases classes
162 void matrix::print(std::ostream & os, unsigned upper_precedence) const
164 debugmsg("matrix print",LOGLEVEL_PRINT);
166 for (unsigned r=0; r<row-1; ++r) {
168 for (unsigned c=0; c<col-1; ++c)
169 os << m[r*col+c] << ",";
170 os << m[col*(r+1)-1] << "]], ";
173 for (unsigned c=0; c<col-1; ++c)
174 os << m[(row-1)*col+c] << ",";
175 os << m[row*col-1] << "]] ]]";
178 void matrix::printraw(std::ostream & os) const
180 debugmsg("matrix printraw",LOGLEVEL_PRINT);
181 os << class_name() << "(" << row << "," << col <<",";
182 for (unsigned r=0; r<row-1; ++r) {
184 for (unsigned c=0; c<col-1; ++c)
185 os << m[r*col+c] << ",";
186 os << m[col*(r-1)-1] << "),";
189 for (unsigned c=0; c<col-1; ++c)
190 os << m[(row-1)*col+c] << ",";
191 os << m[row*col-1] << "))";
194 /** nops is defined to be rows x columns. */
195 unsigned matrix::nops() const
200 /** returns matrix entry at position (i/col, i%col). */
201 ex matrix::op(int i) const
206 /** returns matrix entry at position (i/col, i%col). */
207 ex & matrix::let_op(int i)
210 GINAC_ASSERT(i<nops());
215 /** expands the elements of a matrix entry by entry. */
216 ex matrix::expand(unsigned options) const
218 exvector tmp(row*col);
219 for (unsigned i=0; i<row*col; ++i)
220 tmp[i] = m[i].expand(options);
222 return matrix(row, col, tmp);
225 /** Search ocurrences. A matrix 'has' an expression if it is the expression
226 * itself or one of the elements 'has' it. */
227 bool matrix::has(const ex & other) const
229 GINAC_ASSERT(other.bp!=0);
231 // tautology: it is the expression itself
232 if (is_equal(*other.bp)) return true;
234 // search all the elements
235 for (exvector::const_iterator r=m.begin(); r!=m.end(); ++r)
236 if ((*r).has(other)) return true;
241 /** Evaluate matrix entry by entry. */
242 ex matrix::eval(int level) const
244 debugmsg("matrix eval",LOGLEVEL_MEMBER_FUNCTION);
246 // check if we have to do anything at all
247 if ((level==1)&&(flags & status_flags::evaluated))
251 if (level == -max_recursion_level)
252 throw (std::runtime_error("matrix::eval(): recursion limit exceeded"));
254 // eval() entry by entry
255 exvector m2(row*col);
257 for (unsigned r=0; r<row; ++r)
258 for (unsigned c=0; c<col; ++c)
259 m2[r*col+c] = m[r*col+c].eval(level);
261 return (new matrix(row, col, m2))->setflag(status_flags::dynallocated |
262 status_flags::evaluated );
265 /** Evaluate matrix numerically entry by entry. */
266 ex matrix::evalf(int level) const
268 debugmsg("matrix evalf",LOGLEVEL_MEMBER_FUNCTION);
270 // check if we have to do anything at all
275 if (level == -max_recursion_level) {
276 throw (std::runtime_error("matrix::evalf(): recursion limit exceeded"));
279 // evalf() entry by entry
280 exvector m2(row*col);
282 for (unsigned r=0; r<row; ++r)
283 for (unsigned c=0; c<col; ++c)
284 m2[r*col+c] = m[r*col+c].evalf(level);
286 return matrix(row, col, m2);
289 ex matrix::subs(const lst & ls, const lst & lr) const
291 exvector m2(row * col);
292 for (unsigned r=0; r<row; ++r)
293 for (unsigned c=0; c<col; ++c)
294 m2[r*col+c] = m[r*col+c].subs(ls, lr);
296 return matrix(row, col, m2);
301 int matrix::compare_same_type(const basic & other) const
303 GINAC_ASSERT(is_exactly_of_type(other, matrix));
304 const matrix & o = static_cast<matrix &>(const_cast<basic &>(other));
306 // compare number of rows
308 return row < o.rows() ? -1 : 1;
310 // compare number of columns
312 return col < o.cols() ? -1 : 1;
314 // equal number of rows and columns, compare individual elements
316 for (unsigned r=0; r<row; ++r) {
317 for (unsigned c=0; c<col; ++c) {
318 cmpval = ((*this)(r,c)).compare(o(r,c));
319 if (cmpval!=0) return cmpval;
322 // all elements are equal => matrices are equal;
326 /** Automatic symbolic evaluation of an indexed matrix. */
327 ex matrix::eval_indexed(const basic & i) const
329 GINAC_ASSERT(is_of_type(i, indexed));
330 GINAC_ASSERT(is_ex_of_type(i.op(0), matrix));
332 bool all_indices_unsigned = static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint);
337 // One index, must be one-dimensional vector
338 if (row != 1 && col != 1)
339 throw (std::runtime_error("matrix::eval_indexed(): vector must have exactly 1 index"));
341 const idx & i1 = ex_to_idx(i.op(1));
346 if (!i1.get_dim().is_equal(row))
347 throw (std::runtime_error("matrix::eval_indexed(): dimension of index must match number of vector elements"));
349 // Index numeric -> return vector element
350 if (all_indices_unsigned) {
351 unsigned n1 = ex_to_numeric(i1.get_value()).to_int();
353 throw (std::runtime_error("matrix::eval_indexed(): value of index exceeds number of vector elements"));
354 return (*this)(n1, 0);
360 if (!i1.get_dim().is_equal(col))
361 throw (std::runtime_error("matrix::eval_indexed(): dimension of index must match number of vector elements"));
363 // Index numeric -> return vector element
364 if (all_indices_unsigned) {
365 unsigned n1 = ex_to_numeric(i1.get_value()).to_int();
367 throw (std::runtime_error("matrix::eval_indexed(): value of index exceeds number of vector elements"));
368 return (*this)(0, n1);
372 } else if (i.nops() == 3) {
375 const idx & i1 = ex_to_idx(i.op(1));
376 const idx & i2 = ex_to_idx(i.op(2));
378 if (!i1.get_dim().is_equal(row))
379 throw (std::runtime_error("matrix::eval_indexed(): dimension of first index must match number of rows"));
380 if (!i2.get_dim().is_equal(col))
381 throw (std::runtime_error("matrix::eval_indexed(): dimension of second index must match number of columns"));
383 // Pair of dummy indices -> compute trace
384 if (is_dummy_pair(i1, i2))
387 // Both indices numeric -> return matrix element
388 if (all_indices_unsigned) {
389 unsigned n1 = ex_to_numeric(i1.get_value()).to_int(), n2 = ex_to_numeric(i2.get_value()).to_int();
391 throw (std::runtime_error("matrix::eval_indexed(): value of first index exceeds number of rows"));
393 throw (std::runtime_error("matrix::eval_indexed(): value of second index exceeds number of columns"));
394 return (*this)(n1, n2);
398 throw (std::runtime_error("matrix::eval_indexed(): matrix must have exactly 2 indices"));
403 /** Sum of two indexed matrices. */
404 ex matrix::add_indexed(const ex & self, const ex & other) const
406 GINAC_ASSERT(is_ex_of_type(self, indexed));
407 GINAC_ASSERT(is_ex_of_type(self.op(0), matrix));
408 GINAC_ASSERT(is_ex_of_type(other, indexed));
409 GINAC_ASSERT(self.nops() == 2 || self.nops() == 3);
411 // Only add two matrices
412 if (is_ex_of_type(other.op(0), matrix)) {
413 GINAC_ASSERT(other.nops() == 2 || other.nops() == 3);
415 const matrix &self_matrix = ex_to_matrix(self.op(0));
416 const matrix &other_matrix = ex_to_matrix(other.op(0));
418 if (self.nops() == 2 && other.nops() == 2) { // vector + vector
420 if (self_matrix.row == other_matrix.row)
421 return indexed(self_matrix.add(other_matrix), self.op(1));
422 else if (self_matrix.row == other_matrix.col)
423 return indexed(self_matrix.add(other_matrix.transpose()), self.op(1));
425 } else if (self.nops() == 3 && other.nops() == 3) { // matrix + matrix
427 if (self.op(1).is_equal(other.op(1)) && self.op(2).is_equal(other.op(2)))
428 return indexed(self_matrix.add(other_matrix), self.op(1), self.op(2));
429 else if (self.op(1).is_equal(other.op(2)) && self.op(2).is_equal(other.op(1)))
430 return indexed(self_matrix.add(other_matrix.transpose()), self.op(1), self.op(2));
435 // Don't know what to do, return unevaluated sum
439 /** Product of an indexed matrix with a number. */
440 ex matrix::scalar_mul_indexed(const ex & self, const numeric & other) const
442 GINAC_ASSERT(is_ex_of_type(self, indexed));
443 GINAC_ASSERT(is_ex_of_type(self.op(0), matrix));
444 GINAC_ASSERT(self.nops() == 2 || self.nops() == 3);
446 const matrix &self_matrix = ex_to_matrix(self.op(0));
448 if (self.nops() == 2)
449 return indexed(self_matrix.mul(other), self.op(1));
450 else // self.nops() == 3
451 return indexed(self_matrix.mul(other), self.op(1), self.op(2));
454 /** Contraction of an indexed matrix with something else. */
455 bool matrix::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
457 GINAC_ASSERT(is_ex_of_type(*self, indexed));
458 GINAC_ASSERT(is_ex_of_type(*other, indexed));
459 GINAC_ASSERT(self->nops() == 2 || self->nops() == 3);
460 GINAC_ASSERT(is_ex_of_type(self->op(0), matrix));
462 // Only contract with other matrices
463 if (!is_ex_of_type(other->op(0), matrix))
466 GINAC_ASSERT(other->nops() == 2 || other->nops() == 3);
468 const matrix &self_matrix = ex_to_matrix(self->op(0));
469 const matrix &other_matrix = ex_to_matrix(other->op(0));
471 if (self->nops() == 2) {
472 unsigned self_dim = (self_matrix.col == 1) ? self_matrix.row : self_matrix.col;
474 if (other->nops() == 2) { // vector * vector (scalar product)
475 unsigned other_dim = (other_matrix.col == 1) ? other_matrix.row : other_matrix.col;
477 if (self_matrix.col == 1) {
478 if (other_matrix.col == 1) {
479 // Column vector * column vector, transpose first vector
480 *self = self_matrix.transpose().mul(other_matrix)(0, 0);
482 // Column vector * row vector, swap factors
483 *self = other_matrix.mul(self_matrix)(0, 0);
486 if (other_matrix.col == 1) {
487 // Row vector * column vector, perfect
488 *self = self_matrix.mul(other_matrix)(0, 0);
490 // Row vector * row vector, transpose second vector
491 *self = self_matrix.mul(other_matrix.transpose())(0, 0);
497 } else { // vector * matrix
499 // B_i * A_ij = (B*A)_j (B is row vector)
500 if (is_dummy_pair(self->op(1), other->op(1))) {
501 if (self_matrix.row == 1)
502 *self = indexed(self_matrix.mul(other_matrix), other->op(2));
504 *self = indexed(self_matrix.transpose().mul(other_matrix), other->op(2));
509 // B_j * A_ij = (A*B)_i (B is column vector)
510 if (is_dummy_pair(self->op(1), other->op(2))) {
511 if (self_matrix.col == 1)
512 *self = indexed(other_matrix.mul(self_matrix), other->op(1));
514 *self = indexed(other_matrix.mul(self_matrix.transpose()), other->op(1));
520 } else if (other->nops() == 3) { // matrix * matrix
522 // A_ij * B_jk = (A*B)_ik
523 if (is_dummy_pair(self->op(2), other->op(1))) {
524 *self = indexed(self_matrix.mul(other_matrix), self->op(1), other->op(2));
529 // A_ij * B_kj = (A*Btrans)_ik
530 if (is_dummy_pair(self->op(2), other->op(2))) {
531 *self = indexed(self_matrix.mul(other_matrix.transpose()), self->op(1), other->op(1));
536 // A_ji * B_jk = (Atrans*B)_ik
537 if (is_dummy_pair(self->op(1), other->op(1))) {
538 *self = indexed(self_matrix.transpose().mul(other_matrix), self->op(2), other->op(2));
543 // A_ji * B_kj = (B*A)_ki
544 if (is_dummy_pair(self->op(1), other->op(2))) {
545 *self = indexed(other_matrix.mul(self_matrix), other->op(1), self->op(2));
556 // non-virtual functions in this class
563 * @exception logic_error (incompatible matrices) */
564 matrix matrix::add(const matrix & other) const
566 if (col != other.col || row != other.row)
567 throw (std::logic_error("matrix::add(): incompatible matrices"));
569 exvector sum(this->m);
570 exvector::iterator i;
571 exvector::const_iterator ci;
572 for (i=sum.begin(), ci=other.m.begin(); i!=sum.end(); ++i, ++ci)
575 return matrix(row,col,sum);
579 /** Difference of matrices.
581 * @exception logic_error (incompatible matrices) */
582 matrix matrix::sub(const matrix & other) const
584 if (col != other.col || row != other.row)
585 throw (std::logic_error("matrix::sub(): incompatible matrices"));
587 exvector dif(this->m);
588 exvector::iterator i;
589 exvector::const_iterator ci;
590 for (i=dif.begin(), ci=other.m.begin(); i!=dif.end(); ++i, ++ci)
593 return matrix(row,col,dif);
597 /** Product of matrices.
599 * @exception logic_error (incompatible matrices) */
600 matrix matrix::mul(const matrix & other) const
602 if (this->cols() != other.rows())
603 throw (std::logic_error("matrix::mul(): incompatible matrices"));
605 exvector prod(this->rows()*other.cols());
607 for (unsigned r1=0; r1<this->rows(); ++r1) {
608 for (unsigned c=0; c<this->cols(); ++c) {
609 if (m[r1*col+c].is_zero())
611 for (unsigned r2=0; r2<other.cols(); ++r2)
612 prod[r1*other.col+r2] += (m[r1*col+c] * other.m[c*other.col+r2]).expand();
615 return matrix(row, other.col, prod);
619 /** Product of matrix and scalar. */
620 matrix matrix::mul(const numeric & other) const
622 exvector prod(row * col);
624 for (unsigned r=0; r<row; ++r)
625 for (unsigned c=0; c<col; ++c)
626 prod[r*col+c] = m[r*col+c] * other;
628 return matrix(row, col, prod);
632 /** operator() to access elements.
634 * @param ro row of element
635 * @param co column of element
636 * @exception range_error (index out of range) */
637 const ex & matrix::operator() (unsigned ro, unsigned co) const
639 if (ro>=row || co>=col)
640 throw (std::range_error("matrix::operator(): index out of range"));
646 /** Set individual elements manually.
648 * @exception range_error (index out of range) */
649 matrix & matrix::set(unsigned ro, unsigned co, ex value)
651 if (ro>=row || co>=col)
652 throw (std::range_error("matrix::set(): index out of range"));
654 ensure_if_modifiable();
655 m[ro*col+co] = value;
660 /** Transposed of an m x n matrix, producing a new n x m matrix object that
661 * represents the transposed. */
662 matrix matrix::transpose(void) const
664 exvector trans(this->cols()*this->rows());
666 for (unsigned r=0; r<this->cols(); ++r)
667 for (unsigned c=0; c<this->rows(); ++c)
668 trans[r*this->rows()+c] = m[c*this->cols()+r];
670 return matrix(this->cols(),this->rows(),trans);
674 /** Determinant of square matrix. This routine doesn't actually calculate the
675 * determinant, it only implements some heuristics about which algorithm to
676 * run. If all the elements of the matrix are elements of an integral domain
677 * the determinant is also in that integral domain and the result is expanded
678 * only. If one or more elements are from a quotient field the determinant is
679 * usually also in that quotient field and the result is normalized before it
680 * is returned. This implies that the determinant of the symbolic 2x2 matrix
681 * [[a/(a-b),1],[b/(a-b),1]] is returned as unity. (In this respect, it
682 * behaves like MapleV and unlike Mathematica.)
684 * @param algo allows to chose an algorithm
685 * @return the determinant as a new expression
686 * @exception logic_error (matrix not square)
687 * @see determinant_algo */
688 ex matrix::determinant(unsigned algo) const
691 throw (std::logic_error("matrix::determinant(): matrix not square"));
692 GINAC_ASSERT(row*col==m.capacity());
694 // Gather some statistical information about this matrix:
695 bool numeric_flag = true;
696 bool normal_flag = false;
697 unsigned sparse_count = 0; // counts non-zero elements
698 for (exvector::const_iterator r=m.begin(); r!=m.end(); ++r) {
699 lst srl; // symbol replacement list
700 ex rtest = (*r).to_rational(srl);
701 if (!rtest.is_zero())
703 if (!rtest.info(info_flags::numeric))
704 numeric_flag = false;
705 if (!rtest.info(info_flags::crational_polynomial) &&
706 rtest.info(info_flags::rational_function))
710 // Here is the heuristics in case this routine has to decide:
711 if (algo == determinant_algo::automatic) {
712 // Minor expansion is generally a good guess:
713 algo = determinant_algo::laplace;
714 // Does anybody know when a matrix is really sparse?
715 // Maybe <~row/2.236 nonzero elements average in a row?
716 if (row>3 && 5*sparse_count<=row*col)
717 algo = determinant_algo::bareiss;
718 // Purely numeric matrix can be handled by Gauss elimination.
719 // This overrides any prior decisions.
721 algo = determinant_algo::gauss;
724 // Trap the trivial case here, since some algorithms don't like it
726 // for consistency with non-trivial determinants...
728 return m[0].normal();
730 return m[0].expand();
733 // Compute the determinant
735 case determinant_algo::gauss: {
738 int sign = tmp.gauss_elimination(true);
739 for (unsigned d=0; d<row; ++d)
740 det *= tmp.m[d*col+d];
742 return (sign*det).normal();
744 return (sign*det).normal().expand();
746 case determinant_algo::bareiss: {
749 sign = tmp.fraction_free_elimination(true);
751 return (sign*tmp.m[row*col-1]).normal();
753 return (sign*tmp.m[row*col-1]).expand();
755 case determinant_algo::divfree: {
758 sign = tmp.division_free_elimination(true);
761 ex det = tmp.m[row*col-1];
762 // factor out accumulated bogus slag
763 for (unsigned d=0; d<row-2; ++d)
764 for (unsigned j=0; j<row-d-2; ++j)
765 det = (det/tmp.m[d*col+d]).normal();
768 case determinant_algo::laplace:
770 // This is the minor expansion scheme. We always develop such
771 // that the smallest minors (i.e, the trivial 1x1 ones) are on the
772 // rightmost column. For this to be efficient it turns out that
773 // the emptiest columns (i.e. the ones with most zeros) should be
774 // the ones on the right hand side. Therefore we presort the
775 // columns of the matrix:
776 typedef std::pair<unsigned,unsigned> uintpair;
777 std::vector<uintpair> c_zeros; // number of zeros in column
778 for (unsigned c=0; c<col; ++c) {
780 for (unsigned r=0; r<row; ++r)
781 if (m[r*col+c].is_zero())
783 c_zeros.push_back(uintpair(acc,c));
785 sort(c_zeros.begin(),c_zeros.end());
786 std::vector<unsigned> pre_sort;
787 for (std::vector<uintpair>::iterator i=c_zeros.begin(); i!=c_zeros.end(); ++i)
788 pre_sort.push_back(i->second);
789 int sign = permutation_sign(pre_sort);
790 exvector result(row*col); // represents sorted matrix
792 for (std::vector<unsigned>::iterator i=pre_sort.begin();
795 for (unsigned r=0; r<row; ++r)
796 result[r*col+c] = m[r*col+(*i)];
800 return (sign*matrix(row,col,result).determinant_minor()).normal();
802 return sign*matrix(row,col,result).determinant_minor();
808 /** Trace of a matrix. The result is normalized if it is in some quotient
809 * field and expanded only otherwise. This implies that the trace of the
810 * symbolic 2x2 matrix [[a/(a-b),x],[y,b/(b-a)]] is recognized to be unity.
812 * @return the sum of diagonal elements
813 * @exception logic_error (matrix not square) */
814 ex matrix::trace(void) const
817 throw (std::logic_error("matrix::trace(): matrix not square"));
820 for (unsigned r=0; r<col; ++r)
823 if (tr.info(info_flags::rational_function) &&
824 !tr.info(info_flags::crational_polynomial))
831 /** Characteristic Polynomial. Following mathematica notation the
832 * characteristic polynomial of a matrix M is defined as the determiant of
833 * (M - lambda * 1) where 1 stands for the unit matrix of the same dimension
834 * as M. Note that some CASs define it with a sign inside the determinant
835 * which gives rise to an overall sign if the dimension is odd. This method
836 * returns the characteristic polynomial collected in powers of lambda as a
839 * @return characteristic polynomial as new expression
840 * @exception logic_error (matrix not square)
841 * @see matrix::determinant() */
842 ex matrix::charpoly(const symbol & lambda) const
845 throw (std::logic_error("matrix::charpoly(): matrix not square"));
847 bool numeric_flag = true;
848 for (exvector::const_iterator r=m.begin(); r!=m.end(); ++r) {
849 if (!(*r).info(info_flags::numeric)) {
850 numeric_flag = false;
854 // The pure numeric case is traditionally rather common. Hence, it is
855 // trapped and we use Leverrier's algorithm which goes as row^3 for
856 // every coefficient. The expensive part is the matrix multiplication.
860 ex poly = power(lambda,row)-c*power(lambda,row-1);
861 for (unsigned i=1; i<row; ++i) {
862 for (unsigned j=0; j<row; ++j)
865 c = B.trace()/ex(i+1);
866 poly -= c*power(lambda,row-i-1);
875 for (unsigned r=0; r<col; ++r)
876 M.m[r*col+r] -= lambda;
878 return M.determinant().collect(lambda);
882 /** Inverse of this matrix.
884 * @return the inverted matrix
885 * @exception logic_error (matrix not square)
886 * @exception runtime_error (singular matrix) */
887 matrix matrix::inverse(void) const
890 throw (std::logic_error("matrix::inverse(): matrix not square"));
892 // NOTE: the Gauss-Jordan elimination used here can in principle be
893 // replaced by two clever calls to gauss_elimination() and some to
894 // transpose(). Wouldn't be more efficient (maybe less?), just more
897 // set tmp to the unit matrix
898 for (unsigned i=0; i<col; ++i)
899 tmp.m[i*col+i] = _ex1();
901 // create a copy of this matrix
903 for (unsigned r1=0; r1<row; ++r1) {
904 int indx = cpy.pivot(r1, r1);
906 throw (std::runtime_error("matrix::inverse(): singular matrix"));
908 if (indx != 0) { // swap rows r and indx of matrix tmp
909 for (unsigned i=0; i<col; ++i)
910 tmp.m[r1*col+i].swap(tmp.m[indx*col+i]);
912 ex a1 = cpy.m[r1*col+r1];
913 for (unsigned c=0; c<col; ++c) {
914 cpy.m[r1*col+c] /= a1;
915 tmp.m[r1*col+c] /= a1;
917 for (unsigned r2=0; r2<row; ++r2) {
919 if (!cpy.m[r2*col+r1].is_zero()) {
920 ex a2 = cpy.m[r2*col+r1];
921 // yes, there is something to do in this column
922 for (unsigned c=0; c<col; ++c) {
923 cpy.m[r2*col+c] -= a2 * cpy.m[r1*col+c];
924 if (!cpy.m[r2*col+c].info(info_flags::numeric))
925 cpy.m[r2*col+c] = cpy.m[r2*col+c].normal();
926 tmp.m[r2*col+c] -= a2 * tmp.m[r1*col+c];
927 if (!tmp.m[r2*col+c].info(info_flags::numeric))
928 tmp.m[r2*col+c] = tmp.m[r2*col+c].normal();
939 /** Solve a linear system consisting of a m x n matrix and a m x p right hand
940 * side by applying an elimination scheme to the augmented matrix.
942 * @param vars n x p matrix, all elements must be symbols
943 * @param rhs m x p matrix
944 * @return n x p solution matrix
945 * @exception logic_error (incompatible matrices)
946 * @exception invalid_argument (1st argument must be matrix of symbols)
947 * @exception runtime_error (inconsistent linear system)
949 matrix matrix::solve(const matrix & vars,
953 const unsigned m = this->rows();
954 const unsigned n = this->cols();
955 const unsigned p = rhs.cols();
958 if ((rhs.rows() != m) || (vars.rows() != n) || (vars.col != p))
959 throw (std::logic_error("matrix::solve(): incompatible matrices"));
960 for (unsigned ro=0; ro<n; ++ro)
961 for (unsigned co=0; co<p; ++co)
962 if (!vars(ro,co).info(info_flags::symbol))
963 throw (std::invalid_argument("matrix::solve(): 1st argument must be matrix of symbols"));
965 // build the augmented matrix of *this with rhs attached to the right
967 for (unsigned r=0; r<m; ++r) {
968 for (unsigned c=0; c<n; ++c)
969 aug.m[r*(n+p)+c] = this->m[r*n+c];
970 for (unsigned c=0; c<p; ++c)
971 aug.m[r*(n+p)+c+n] = rhs.m[r*p+c];
974 // Gather some statistical information about the augmented matrix:
975 bool numeric_flag = true;
976 for (exvector::const_iterator r=aug.m.begin(); r!=aug.m.end(); ++r) {
977 if (!(*r).info(info_flags::numeric))
978 numeric_flag = false;
981 // Here is the heuristics in case this routine has to decide:
982 if (algo == solve_algo::automatic) {
983 // Bareiss (fraction-free) elimination is generally a good guess:
984 algo = solve_algo::bareiss;
985 // For m<3, Bareiss elimination is equivalent to division free
986 // elimination but has more logistic overhead
988 algo = solve_algo::divfree;
989 // This overrides any prior decisions.
991 algo = solve_algo::gauss;
994 // Eliminate the augmented matrix:
996 case solve_algo::gauss:
997 aug.gauss_elimination();
998 case solve_algo::divfree:
999 aug.division_free_elimination();
1000 case solve_algo::bareiss:
1002 aug.fraction_free_elimination();
1005 // assemble the solution matrix:
1007 for (unsigned co=0; co<p; ++co) {
1008 unsigned last_assigned_sol = n+1;
1009 for (int r=m-1; r>=0; --r) {
1010 unsigned fnz = 1; // first non-zero in row
1011 while ((fnz<=n) && (aug.m[r*(n+p)+(fnz-1)].is_zero()))
1014 // row consists only of zeros, corresponding rhs must be 0, too
1015 if (!aug.m[r*(n+p)+n+co].is_zero()) {
1016 throw (std::runtime_error("matrix::solve(): inconsistent linear system"));
1019 // assign solutions for vars between fnz+1 and
1020 // last_assigned_sol-1: free parameters
1021 for (unsigned c=fnz; c<last_assigned_sol-1; ++c)
1022 sol.set(c,co,vars.m[c*p+co]);
1023 ex e = aug.m[r*(n+p)+n+co];
1024 for (unsigned c=fnz; c<n; ++c)
1025 e -= aug.m[r*(n+p)+c]*sol.m[c*p+co];
1027 (e/(aug.m[r*(n+p)+(fnz-1)])).normal());
1028 last_assigned_sol = fnz;
1031 // assign solutions for vars between 1 and
1032 // last_assigned_sol-1: free parameters
1033 for (unsigned ro=0; ro<last_assigned_sol-1; ++ro)
1034 sol.set(ro,co,vars(ro,co));
1043 /** Recursive determinant for small matrices having at least one symbolic
1044 * entry. The basic algorithm, known as Laplace-expansion, is enhanced by
1045 * some bookkeeping to avoid calculation of the same submatrices ("minors")
1046 * more than once. According to W.M.Gentleman and S.C.Johnson this algorithm
1047 * is better than elimination schemes for matrices of sparse multivariate
1048 * polynomials and also for matrices of dense univariate polynomials if the
1049 * matrix' dimesion is larger than 7.
1051 * @return the determinant as a new expression (in expanded form)
1052 * @see matrix::determinant() */
1053 ex matrix::determinant_minor(void) const
1055 // for small matrices the algorithm does not make any sense:
1056 const unsigned n = this->cols();
1058 return m[0].expand();
1060 return (m[0]*m[3]-m[2]*m[1]).expand();
1062 return (m[0]*m[4]*m[8]-m[0]*m[5]*m[7]-
1063 m[1]*m[3]*m[8]+m[2]*m[3]*m[7]+
1064 m[1]*m[5]*m[6]-m[2]*m[4]*m[6]).expand();
1066 // This algorithm can best be understood by looking at a naive
1067 // implementation of Laplace-expansion, like this one:
1069 // matrix minorM(this->rows()-1,this->cols()-1);
1070 // for (unsigned r1=0; r1<this->rows(); ++r1) {
1071 // // shortcut if element(r1,0) vanishes
1072 // if (m[r1*col].is_zero())
1074 // // assemble the minor matrix
1075 // for (unsigned r=0; r<minorM.rows(); ++r) {
1076 // for (unsigned c=0; c<minorM.cols(); ++c) {
1078 // minorM.set(r,c,m[r*col+c+1]);
1080 // minorM.set(r,c,m[(r+1)*col+c+1]);
1083 // // recurse down and care for sign:
1085 // det -= m[r1*col] * minorM.determinant_minor();
1087 // det += m[r1*col] * minorM.determinant_minor();
1089 // return det.expand();
1090 // What happens is that while proceeding down many of the minors are
1091 // computed more than once. In particular, there are binomial(n,k)
1092 // kxk minors and each one is computed factorial(n-k) times. Therefore
1093 // it is reasonable to store the results of the minors. We proceed from
1094 // right to left. At each column c we only need to retrieve the minors
1095 // calculated in step c-1. We therefore only have to store at most
1096 // 2*binomial(n,n/2) minors.
1098 // Unique flipper counter for partitioning into minors
1099 std::vector<unsigned> Pkey;
1101 // key for minor determinant (a subpartition of Pkey)
1102 std::vector<unsigned> Mkey;
1104 // we store our subminors in maps, keys being the rows they arise from
1105 typedef std::map<std::vector<unsigned>,class ex> Rmap;
1106 typedef std::map<std::vector<unsigned>,class ex>::value_type Rmap_value;
1110 // initialize A with last column:
1111 for (unsigned r=0; r<n; ++r) {
1112 Pkey.erase(Pkey.begin(),Pkey.end());
1114 A.insert(Rmap_value(Pkey,m[n*(r+1)-1]));
1116 // proceed from right to left through matrix
1117 for (int c=n-2; c>=0; --c) {
1118 Pkey.erase(Pkey.begin(),Pkey.end()); // don't change capacity
1119 Mkey.erase(Mkey.begin(),Mkey.end());
1120 for (unsigned i=0; i<n-c; ++i)
1122 unsigned fc = 0; // controls logic for our strange flipper counter
1125 for (unsigned r=0; r<n-c; ++r) {
1126 // maybe there is nothing to do?
1127 if (m[Pkey[r]*n+c].is_zero())
1129 // create the sorted key for all possible minors
1130 Mkey.erase(Mkey.begin(),Mkey.end());
1131 for (unsigned i=0; i<n-c; ++i)
1133 Mkey.push_back(Pkey[i]);
1134 // Fetch the minors and compute the new determinant
1136 det -= m[Pkey[r]*n+c]*A[Mkey];
1138 det += m[Pkey[r]*n+c]*A[Mkey];
1140 // prevent build-up of deep nesting of expressions saves time:
1142 // store the new determinant at its place in B:
1144 B.insert(Rmap_value(Pkey,det));
1145 // increment our strange flipper counter
1146 for (fc=n-c; fc>0; --fc) {
1148 if (Pkey[fc-1]<fc+c)
1152 for (unsigned j=fc; j<n-c; ++j)
1153 Pkey[j] = Pkey[j-1]+1;
1155 // next column, so change the role of A and B:
1164 /** Perform the steps of an ordinary Gaussian elimination to bring the m x n
1165 * matrix into an upper echelon form. The algorithm is ok for matrices
1166 * with numeric coefficients but quite unsuited for symbolic matrices.
1168 * @param det may be set to true to save a lot of space if one is only
1169 * interested in the diagonal elements (i.e. for calculating determinants).
1170 * The others are set to zero in this case.
1171 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1172 * number of rows was swapped and 0 if the matrix is singular. */
1173 int matrix::gauss_elimination(const bool det)
1175 ensure_if_modifiable();
1176 const unsigned m = this->rows();
1177 const unsigned n = this->cols();
1178 GINAC_ASSERT(!det || n==m);
1182 for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
1183 int indx = pivot(r0, r1, true);
1187 return 0; // leaves *this in a messy state
1192 for (unsigned r2=r0+1; r2<m; ++r2) {
1193 if (!this->m[r2*n+r1].is_zero()) {
1194 // yes, there is something to do in this row
1195 ex piv = this->m[r2*n+r1] / this->m[r0*n+r1];
1196 for (unsigned c=r1+1; c<n; ++c) {
1197 this->m[r2*n+c] -= piv * this->m[r0*n+c];
1198 if (!this->m[r2*n+c].info(info_flags::numeric))
1199 this->m[r2*n+c] = this->m[r2*n+c].normal();
1202 // fill up left hand side with zeros
1203 for (unsigned c=0; c<=r1; ++c)
1204 this->m[r2*n+c] = _ex0();
1207 // save space by deleting no longer needed elements
1208 for (unsigned c=r0+1; c<n; ++c)
1209 this->m[r0*n+c] = _ex0();
1219 /** Perform the steps of division free elimination to bring the m x n matrix
1220 * into an upper echelon form.
1222 * @param det may be set to true to save a lot of space if one is only
1223 * interested in the diagonal elements (i.e. for calculating determinants).
1224 * The others are set to zero in this case.
1225 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1226 * number of rows was swapped and 0 if the matrix is singular. */
1227 int matrix::division_free_elimination(const bool det)
1229 ensure_if_modifiable();
1230 const unsigned m = this->rows();
1231 const unsigned n = this->cols();
1232 GINAC_ASSERT(!det || n==m);
1236 for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
1237 int indx = pivot(r0, r1, true);
1241 return 0; // leaves *this in a messy state
1246 for (unsigned r2=r0+1; r2<m; ++r2) {
1247 for (unsigned c=r1+1; c<n; ++c)
1248 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();
1249 // fill up left hand side with zeros
1250 for (unsigned c=0; c<=r1; ++c)
1251 this->m[r2*n+c] = _ex0();
1254 // save space by deleting no longer needed elements
1255 for (unsigned c=r0+1; c<n; ++c)
1256 this->m[r0*n+c] = _ex0();
1266 /** Perform the steps of Bareiss' one-step fraction free elimination to bring
1267 * the matrix into an upper echelon form. Fraction free elimination means
1268 * that divide is used straightforwardly, without computing GCDs first. This
1269 * is possible, since we know the divisor at each step.
1271 * @param det may be set to true to save a lot of space if one is only
1272 * interested in the last element (i.e. for calculating determinants). The
1273 * others are set to zero in this case.
1274 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1275 * number of rows was swapped and 0 if the matrix is singular. */
1276 int matrix::fraction_free_elimination(const bool det)
1279 // (single-step fraction free elimination scheme, already known to Jordan)
1281 // Usual division-free elimination sets m[0](r,c) = m(r,c) and then sets
1282 // m[k+1](r,c) = m[k](k,k) * m[k](r,c) - m[k](r,k) * m[k](k,c).
1284 // Bareiss (fraction-free) elimination in addition divides that element
1285 // by m[k-1](k-1,k-1) for k>1, where it can be shown by means of the
1286 // Sylvester determinant that this really divides m[k+1](r,c).
1288 // We also allow rational functions where the original prove still holds.
1289 // However, we must care for numerator and denominator separately and
1290 // "manually" work in the integral domains because of subtle cancellations
1291 // (see below). This blows up the bookkeeping a bit and the formula has
1292 // to be modified to expand like this (N{x} stands for numerator of x,
1293 // D{x} for denominator of x):
1294 // 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)}
1295 // -N{m[k](r,k)}*N{m[k](k,c)}*D{m[k](k,k)}*D{m[k](r,c)}
1296 // 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)}
1297 // where for k>1 we now divide N{m[k+1](r,c)} by
1298 // N{m[k-1](k-1,k-1)}
1299 // and D{m[k+1](r,c)} by
1300 // D{m[k-1](k-1,k-1)}.
1302 ensure_if_modifiable();
1303 const unsigned m = this->rows();
1304 const unsigned n = this->cols();
1305 GINAC_ASSERT(!det || n==m);
1314 // We populate temporary matrices to subsequently operate on. There is
1315 // one holding numerators and another holding denominators of entries.
1316 // This is a must since the evaluator (or even earlier mul's constructor)
1317 // might cancel some trivial element which causes divide() to fail. The
1318 // elements are normalized first (yes, even though this algorithm doesn't
1319 // need GCDs) since the elements of *this might be unnormalized, which
1320 // makes things more complicated than they need to be.
1321 matrix tmp_n(*this);
1322 matrix tmp_d(m,n); // for denominators, if needed
1323 lst srl; // symbol replacement list
1324 exvector::iterator it = this->m.begin();
1325 exvector::iterator tmp_n_it = tmp_n.m.begin();
1326 exvector::iterator tmp_d_it = tmp_d.m.begin();
1327 for (; it!= this->m.end(); ++it, ++tmp_n_it, ++tmp_d_it) {
1328 (*tmp_n_it) = (*it).normal().to_rational(srl);
1329 (*tmp_d_it) = (*tmp_n_it).denom();
1330 (*tmp_n_it) = (*tmp_n_it).numer();
1334 for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
1335 int indx = tmp_n.pivot(r0, r1, true);
1344 // tmp_n's rows r0 and indx were swapped, do the same in tmp_d:
1345 for (unsigned c=r1; c<n; ++c)
1346 tmp_d.m[n*indx+c].swap(tmp_d.m[n*r0+c]);
1348 for (unsigned r2=r0+1; r2<m; ++r2) {
1349 for (unsigned c=r1+1; c<n; ++c) {
1350 dividend_n = (tmp_n.m[r0*n+r1]*tmp_n.m[r2*n+c]*
1351 tmp_d.m[r2*n+r1]*tmp_d.m[r0*n+c]
1352 -tmp_n.m[r2*n+r1]*tmp_n.m[r0*n+c]*
1353 tmp_d.m[r0*n+r1]*tmp_d.m[r2*n+c]).expand();
1354 dividend_d = (tmp_d.m[r2*n+r1]*tmp_d.m[r0*n+c]*
1355 tmp_d.m[r0*n+r1]*tmp_d.m[r2*n+c]).expand();
1356 bool check = divide(dividend_n, divisor_n,
1357 tmp_n.m[r2*n+c], true);
1358 check &= divide(dividend_d, divisor_d,
1359 tmp_d.m[r2*n+c], true);
1360 GINAC_ASSERT(check);
1362 // fill up left hand side with zeros
1363 for (unsigned c=0; c<=r1; ++c)
1364 tmp_n.m[r2*n+c] = _ex0();
1366 if ((r1<n-1)&&(r0<m-1)) {
1367 // compute next iteration's divisor
1368 divisor_n = tmp_n.m[r0*n+r1].expand();
1369 divisor_d = tmp_d.m[r0*n+r1].expand();
1371 // save space by deleting no longer needed elements
1372 for (unsigned c=0; c<n; ++c) {
1373 tmp_n.m[r0*n+c] = _ex0();
1374 tmp_d.m[r0*n+c] = _ex1();
1381 // repopulate *this matrix:
1382 it = this->m.begin();
1383 tmp_n_it = tmp_n.m.begin();
1384 tmp_d_it = tmp_d.m.begin();
1385 for (; it!= this->m.end(); ++it, ++tmp_n_it, ++tmp_d_it)
1386 (*it) = ((*tmp_n_it)/(*tmp_d_it)).subs(srl);
1392 /** Partial pivoting method for matrix elimination schemes.
1393 * Usual pivoting (symbolic==false) returns the index to the element with the
1394 * largest absolute value in column ro and swaps the current row with the one
1395 * where the element was found. With (symbolic==true) it does the same thing
1396 * with the first non-zero element.
1398 * @param ro is the row from where to begin
1399 * @param co is the column to be inspected
1400 * @param symbolic signal if we want the first non-zero element to be pivoted
1401 * (true) or the one with the largest absolute value (false).
1402 * @return 0 if no interchange occured, -1 if all are zero (usually signaling
1403 * a degeneracy) and positive integer k means that rows ro and k were swapped.
1405 int matrix::pivot(unsigned ro, unsigned co, bool symbolic)
1409 // search first non-zero element in column co beginning at row ro
1410 while ((k<row) && (this->m[k*col+co].expand().is_zero()))
1413 // search largest element in column co beginning at row ro
1414 GINAC_ASSERT(is_ex_of_type(this->m[k*col+co],numeric));
1415 unsigned kmax = k+1;
1416 numeric mmax = abs(ex_to_numeric(m[kmax*col+co]));
1418 GINAC_ASSERT(is_ex_of_type(this->m[kmax*col+co],numeric));
1419 numeric tmp = ex_to_numeric(this->m[kmax*col+co]);
1420 if (abs(tmp) > mmax) {
1426 if (!mmax.is_zero())
1430 // all elements in column co below row ro vanish
1433 // matrix needs no pivoting
1435 // matrix needs pivoting, so swap rows k and ro
1436 ensure_if_modifiable();
1437 for (unsigned c=0; c<col; ++c)
1438 this->m[k*col+c].swap(this->m[ro*col+c]);
1443 ex lst_to_matrix(const lst & l)
1445 // Find number of rows and columns
1446 unsigned rows = l.nops(), cols = 0, i, j;
1447 for (i=0; i<rows; i++)
1448 if (l.op(i).nops() > cols)
1449 cols = l.op(i).nops();
1451 // Allocate and fill matrix
1452 matrix &m = *new matrix(rows, cols);
1453 m.setflag(status_flags::dynallocated);
1454 for (i=0; i<rows; i++)
1455 for (j=0; j<cols; j++)
1456 if (l.op(i).nops() > j)
1457 m.set(i, j, l.op(i).op(j));
1463 ex diag_matrix(const lst & l)
1465 unsigned dim = l.nops();
1467 matrix &m = *new matrix(dim, dim);
1468 m.setflag(status_flags::dynallocated);
1469 for (unsigned i=0; i<dim; i++)
1470 m.set(i, i, l.op(i));
1475 } // namespace GiNaC