3 * Implementation of symbolic matrices */
6 * GiNaC Copyright (C) 1999-2008 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., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
38 #include "operators.h"
45 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(matrix, basic,
46 print_func<print_context>(&matrix::do_print).
47 print_func<print_latex>(&matrix::do_print_latex).
48 print_func<print_tree>(&matrix::do_print_tree).
49 print_func<print_python_repr>(&matrix::do_print_python_repr))
52 // default constructor
55 /** Default ctor. Initializes to 1 x 1-dimensional zero-matrix. */
56 matrix::matrix() : inherited(&matrix::tinfo_static), row(1), col(1), m(1, _ex0)
58 setflag(status_flags::not_shareable);
67 /** Very common ctor. Initializes to r x c-dimensional zero-matrix.
69 * @param r number of rows
70 * @param c number of cols */
71 matrix::matrix(unsigned r, unsigned c)
72 : inherited(&matrix::tinfo_static), row(r), col(c), m(r*c, _ex0)
74 setflag(status_flags::not_shareable);
79 /** Ctor from representation, for internal use only. */
80 matrix::matrix(unsigned r, unsigned c, const exvector & m2)
81 : inherited(&matrix::tinfo_static), row(r), col(c), m(m2)
83 setflag(status_flags::not_shareable);
86 /** Construct matrix from (flat) list of elements. If the list has fewer
87 * elements than the matrix, the remaining matrix elements are set to zero.
88 * If the list has more elements than the matrix, the excessive elements are
90 matrix::matrix(unsigned r, unsigned c, const lst & l)
91 : inherited(&matrix::tinfo_static), row(r), col(c), m(r*c, _ex0)
93 setflag(status_flags::not_shareable);
96 for (lst::const_iterator it = l.begin(); it != l.end(); ++it, ++i) {
100 break; // matrix smaller than list: throw away excessive elements
109 matrix::matrix(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
111 setflag(status_flags::not_shareable);
113 if (!(n.find_unsigned("row", row)) || !(n.find_unsigned("col", col)))
114 throw (std::runtime_error("unknown matrix dimensions in archive"));
115 m.reserve(row * col);
116 archive_node::archive_node_cit first = n.find_first("m");
117 archive_node::archive_node_cit last = n.find_last("m");
119 for (archive_node::archive_node_cit i=first; i<last; ++i) {
121 n.find_ex_by_loc(i, e, sym_lst);
126 void matrix::archive(archive_node &n) const
128 inherited::archive(n);
129 n.add_unsigned("row", row);
130 n.add_unsigned("col", col);
131 exvector::const_iterator i = m.begin(), iend = m.end();
138 DEFAULT_UNARCHIVE(matrix)
141 // functions overriding virtual functions from base classes
146 void matrix::print_elements(const print_context & c, const char *row_start, const char *row_end, const char *row_sep, const char *col_sep) const
148 for (unsigned ro=0; ro<row; ++ro) {
150 for (unsigned co=0; co<col; ++co) {
151 m[ro*col+co].print(c);
162 void matrix::do_print(const print_context & c, unsigned level) const
165 print_elements(c, "[", "]", ",", ",");
169 void matrix::do_print_latex(const print_latex & c, unsigned level) const
171 c.s << "\\left(\\begin{array}{" << std::string(col,'c') << "}";
172 print_elements(c, "", "", "\\\\", "&");
173 c.s << "\\end{array}\\right)";
176 void matrix::do_print_python_repr(const print_python_repr & c, unsigned level) const
178 c.s << class_name() << '(';
179 print_elements(c, "[", "]", ",", ",");
183 /** nops is defined to be rows x columns. */
184 size_t matrix::nops() const
186 return static_cast<size_t>(row) * static_cast<size_t>(col);
189 /** returns matrix entry at position (i/col, i%col). */
190 ex matrix::op(size_t i) const
192 GINAC_ASSERT(i<nops());
197 /** returns writable matrix entry at position (i/col, i%col). */
198 ex & matrix::let_op(size_t i)
200 GINAC_ASSERT(i<nops());
202 ensure_if_modifiable();
206 /** Evaluate matrix entry by entry. */
207 ex matrix::eval(int level) const
209 // check if we have to do anything at all
210 if ((level==1)&&(flags & status_flags::evaluated))
214 if (level == -max_recursion_level)
215 throw (std::runtime_error("matrix::eval(): recursion limit exceeded"));
217 // eval() entry by entry
218 exvector m2(row*col);
220 for (unsigned r=0; r<row; ++r)
221 for (unsigned c=0; c<col; ++c)
222 m2[r*col+c] = m[r*col+c].eval(level);
224 return (new matrix(row, col, m2))->setflag(status_flags::dynallocated |
225 status_flags::evaluated);
228 ex matrix::subs(const exmap & mp, unsigned options) const
230 exvector m2(row * col);
231 for (unsigned r=0; r<row; ++r)
232 for (unsigned c=0; c<col; ++c)
233 m2[r*col+c] = m[r*col+c].subs(mp, options);
235 return matrix(row, col, m2).subs_one_level(mp, options);
238 /** Complex conjugate every matrix entry. */
239 ex matrix::conjugate() const
242 for (exvector::const_iterator i=m.begin(); i!=m.end(); ++i) {
243 ex x = i->conjugate();
248 if (are_ex_trivially_equal(x, *i)) {
252 ev->reserve(m.size());
253 for (exvector::const_iterator j=m.begin(); j!=i; ++j) {
259 ex result = matrix(row, col, *ev);
266 ex matrix::real_part() const
270 for (exvector::const_iterator i=m.begin(); i!=m.end(); ++i)
271 v.push_back(i->real_part());
272 return matrix(row, col, v);
275 ex matrix::imag_part() const
279 for (exvector::const_iterator i=m.begin(); i!=m.end(); ++i)
280 v.push_back(i->imag_part());
281 return matrix(row, col, v);
286 int matrix::compare_same_type(const basic & other) const
288 GINAC_ASSERT(is_exactly_a<matrix>(other));
289 const matrix &o = static_cast<const matrix &>(other);
291 // compare number of rows
293 return row < o.rows() ? -1 : 1;
295 // compare number of columns
297 return col < o.cols() ? -1 : 1;
299 // equal number of rows and columns, compare individual elements
301 for (unsigned r=0; r<row; ++r) {
302 for (unsigned c=0; c<col; ++c) {
303 cmpval = ((*this)(r,c)).compare(o(r,c));
304 if (cmpval!=0) return cmpval;
307 // all elements are equal => matrices are equal;
311 bool matrix::match_same_type(const basic & other) const
313 GINAC_ASSERT(is_exactly_a<matrix>(other));
314 const matrix & o = static_cast<const matrix &>(other);
316 // The number of rows and columns must be the same. This is necessary to
317 // prevent a 2x3 matrix from matching a 3x2 one.
318 return row == o.rows() && col == o.cols();
321 /** Automatic symbolic evaluation of an indexed matrix. */
322 ex matrix::eval_indexed(const basic & i) const
324 GINAC_ASSERT(is_a<indexed>(i));
325 GINAC_ASSERT(is_a<matrix>(i.op(0)));
327 bool all_indices_unsigned = static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint);
332 // One index, must be one-dimensional vector
333 if (row != 1 && col != 1)
334 throw (std::runtime_error("matrix::eval_indexed(): vector must have exactly 1 index"));
336 const idx & i1 = ex_to<idx>(i.op(1));
341 if (!i1.get_dim().is_equal(row))
342 throw (std::runtime_error("matrix::eval_indexed(): dimension of index must match number of vector elements"));
344 // Index numeric -> return vector element
345 if (all_indices_unsigned) {
346 unsigned n1 = ex_to<numeric>(i1.get_value()).to_int();
348 throw (std::runtime_error("matrix::eval_indexed(): value of index exceeds number of vector elements"));
349 return (*this)(n1, 0);
355 if (!i1.get_dim().is_equal(col))
356 throw (std::runtime_error("matrix::eval_indexed(): dimension of index must match number of vector elements"));
358 // Index numeric -> return vector element
359 if (all_indices_unsigned) {
360 unsigned n1 = ex_to<numeric>(i1.get_value()).to_int();
362 throw (std::runtime_error("matrix::eval_indexed(): value of index exceeds number of vector elements"));
363 return (*this)(0, n1);
367 } else if (i.nops() == 3) {
370 const idx & i1 = ex_to<idx>(i.op(1));
371 const idx & i2 = ex_to<idx>(i.op(2));
373 if (!i1.get_dim().is_equal(row))
374 throw (std::runtime_error("matrix::eval_indexed(): dimension of first index must match number of rows"));
375 if (!i2.get_dim().is_equal(col))
376 throw (std::runtime_error("matrix::eval_indexed(): dimension of second index must match number of columns"));
378 // Pair of dummy indices -> compute trace
379 if (is_dummy_pair(i1, i2))
382 // Both indices numeric -> return matrix element
383 if (all_indices_unsigned) {
384 unsigned n1 = ex_to<numeric>(i1.get_value()).to_int(), n2 = ex_to<numeric>(i2.get_value()).to_int();
386 throw (std::runtime_error("matrix::eval_indexed(): value of first index exceeds number of rows"));
388 throw (std::runtime_error("matrix::eval_indexed(): value of second index exceeds number of columns"));
389 return (*this)(n1, n2);
393 throw (std::runtime_error("matrix::eval_indexed(): matrix must have exactly 2 indices"));
398 /** Sum of two indexed matrices. */
399 ex matrix::add_indexed(const ex & self, const ex & other) const
401 GINAC_ASSERT(is_a<indexed>(self));
402 GINAC_ASSERT(is_a<matrix>(self.op(0)));
403 GINAC_ASSERT(is_a<indexed>(other));
404 GINAC_ASSERT(self.nops() == 2 || self.nops() == 3);
406 // Only add two matrices
407 if (is_a<matrix>(other.op(0))) {
408 GINAC_ASSERT(other.nops() == 2 || other.nops() == 3);
410 const matrix &self_matrix = ex_to<matrix>(self.op(0));
411 const matrix &other_matrix = ex_to<matrix>(other.op(0));
413 if (self.nops() == 2 && other.nops() == 2) { // vector + vector
415 if (self_matrix.row == other_matrix.row)
416 return indexed(self_matrix.add(other_matrix), self.op(1));
417 else if (self_matrix.row == other_matrix.col)
418 return indexed(self_matrix.add(other_matrix.transpose()), self.op(1));
420 } else if (self.nops() == 3 && other.nops() == 3) { // matrix + matrix
422 if (self.op(1).is_equal(other.op(1)) && self.op(2).is_equal(other.op(2)))
423 return indexed(self_matrix.add(other_matrix), self.op(1), self.op(2));
424 else if (self.op(1).is_equal(other.op(2)) && self.op(2).is_equal(other.op(1)))
425 return indexed(self_matrix.add(other_matrix.transpose()), self.op(1), self.op(2));
430 // Don't know what to do, return unevaluated sum
434 /** Product of an indexed matrix with a number. */
435 ex matrix::scalar_mul_indexed(const ex & self, const numeric & other) const
437 GINAC_ASSERT(is_a<indexed>(self));
438 GINAC_ASSERT(is_a<matrix>(self.op(0)));
439 GINAC_ASSERT(self.nops() == 2 || self.nops() == 3);
441 const matrix &self_matrix = ex_to<matrix>(self.op(0));
443 if (self.nops() == 2)
444 return indexed(self_matrix.mul(other), self.op(1));
445 else // self.nops() == 3
446 return indexed(self_matrix.mul(other), self.op(1), self.op(2));
449 /** Contraction of an indexed matrix with something else. */
450 bool matrix::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
452 GINAC_ASSERT(is_a<indexed>(*self));
453 GINAC_ASSERT(is_a<indexed>(*other));
454 GINAC_ASSERT(self->nops() == 2 || self->nops() == 3);
455 GINAC_ASSERT(is_a<matrix>(self->op(0)));
457 // Only contract with other matrices
458 if (!is_a<matrix>(other->op(0)))
461 GINAC_ASSERT(other->nops() == 2 || other->nops() == 3);
463 const matrix &self_matrix = ex_to<matrix>(self->op(0));
464 const matrix &other_matrix = ex_to<matrix>(other->op(0));
466 if (self->nops() == 2) {
468 if (other->nops() == 2) { // vector * vector (scalar product)
470 if (self_matrix.col == 1) {
471 if (other_matrix.col == 1) {
472 // Column vector * column vector, transpose first vector
473 *self = self_matrix.transpose().mul(other_matrix)(0, 0);
475 // Column vector * row vector, swap factors
476 *self = other_matrix.mul(self_matrix)(0, 0);
479 if (other_matrix.col == 1) {
480 // Row vector * column vector, perfect
481 *self = self_matrix.mul(other_matrix)(0, 0);
483 // Row vector * row vector, transpose second vector
484 *self = self_matrix.mul(other_matrix.transpose())(0, 0);
490 } else { // vector * matrix
492 // B_i * A_ij = (B*A)_j (B is row vector)
493 if (is_dummy_pair(self->op(1), other->op(1))) {
494 if (self_matrix.row == 1)
495 *self = indexed(self_matrix.mul(other_matrix), other->op(2));
497 *self = indexed(self_matrix.transpose().mul(other_matrix), other->op(2));
502 // B_j * A_ij = (A*B)_i (B is column vector)
503 if (is_dummy_pair(self->op(1), other->op(2))) {
504 if (self_matrix.col == 1)
505 *self = indexed(other_matrix.mul(self_matrix), other->op(1));
507 *self = indexed(other_matrix.mul(self_matrix.transpose()), other->op(1));
513 } else if (other->nops() == 3) { // matrix * matrix
515 // A_ij * B_jk = (A*B)_ik
516 if (is_dummy_pair(self->op(2), other->op(1))) {
517 *self = indexed(self_matrix.mul(other_matrix), self->op(1), other->op(2));
522 // A_ij * B_kj = (A*Btrans)_ik
523 if (is_dummy_pair(self->op(2), other->op(2))) {
524 *self = indexed(self_matrix.mul(other_matrix.transpose()), self->op(1), other->op(1));
529 // A_ji * B_jk = (Atrans*B)_ik
530 if (is_dummy_pair(self->op(1), other->op(1))) {
531 *self = indexed(self_matrix.transpose().mul(other_matrix), self->op(2), other->op(2));
536 // A_ji * B_kj = (B*A)_ki
537 if (is_dummy_pair(self->op(1), other->op(2))) {
538 *self = indexed(other_matrix.mul(self_matrix), other->op(1), self->op(2));
549 // non-virtual functions in this class
556 * @exception logic_error (incompatible matrices) */
557 matrix matrix::add(const matrix & other) const
559 if (col != other.col || row != other.row)
560 throw std::logic_error("matrix::add(): incompatible matrices");
562 exvector sum(this->m);
563 exvector::iterator i = sum.begin(), end = sum.end();
564 exvector::const_iterator ci = other.m.begin();
568 return matrix(row,col,sum);
572 /** Difference of matrices.
574 * @exception logic_error (incompatible matrices) */
575 matrix matrix::sub(const matrix & other) const
577 if (col != other.col || row != other.row)
578 throw std::logic_error("matrix::sub(): incompatible matrices");
580 exvector dif(this->m);
581 exvector::iterator i = dif.begin(), end = dif.end();
582 exvector::const_iterator ci = other.m.begin();
586 return matrix(row,col,dif);
590 /** Product of matrices.
592 * @exception logic_error (incompatible matrices) */
593 matrix matrix::mul(const matrix & other) const
595 if (this->cols() != other.rows())
596 throw std::logic_error("matrix::mul(): incompatible matrices");
598 exvector prod(this->rows()*other.cols());
600 for (unsigned r1=0; r1<this->rows(); ++r1) {
601 for (unsigned c=0; c<this->cols(); ++c) {
602 // Quick test: can we shortcut?
603 if (m[r1*col+c].is_zero())
605 for (unsigned r2=0; r2<other.cols(); ++r2)
606 prod[r1*other.col+r2] += (m[r1*col+c] * other.m[c*other.col+r2]);
609 return matrix(row, other.col, prod);
613 /** Product of matrix and scalar. */
614 matrix matrix::mul(const numeric & other) const
616 exvector prod(row * col);
618 for (unsigned r=0; r<row; ++r)
619 for (unsigned c=0; c<col; ++c)
620 prod[r*col+c] = m[r*col+c] * other;
622 return matrix(row, col, prod);
626 /** Product of matrix and scalar expression. */
627 matrix matrix::mul_scalar(const ex & other) const
629 if (other.return_type() != return_types::commutative)
630 throw std::runtime_error("matrix::mul_scalar(): non-commutative scalar");
632 exvector prod(row * col);
634 for (unsigned r=0; r<row; ++r)
635 for (unsigned c=0; c<col; ++c)
636 prod[r*col+c] = m[r*col+c] * other;
638 return matrix(row, col, prod);
642 /** Power of a matrix. Currently handles integer exponents only. */
643 matrix matrix::pow(const ex & expn) const
646 throw (std::logic_error("matrix::pow(): matrix not square"));
648 if (is_exactly_a<numeric>(expn)) {
649 // Integer cases are computed by successive multiplication, using the
650 // obvious shortcut of storing temporaries, like A^4 == (A*A)*(A*A).
651 if (expn.info(info_flags::integer)) {
652 numeric b = ex_to<numeric>(expn);
654 if (expn.info(info_flags::negative)) {
661 for (unsigned r=0; r<row; ++r)
665 // This loop computes the representation of b in base 2 from right
666 // to left and multiplies the factors whenever needed. Note
667 // that this is not entirely optimal but close to optimal and
668 // "better" algorithms are much harder to implement. (See Knuth,
669 // TAoCP2, section "Evaluation of Powers" for a good discussion.)
670 while (b!=*_num1_p) {
675 b /= *_num2_p; // still integer.
681 throw (std::runtime_error("matrix::pow(): don't know how to handle exponent"));
685 /** operator() to access elements for reading.
687 * @param ro row of element
688 * @param co column of element
689 * @exception range_error (index out of range) */
690 const ex & matrix::operator() (unsigned ro, unsigned co) const
692 if (ro>=row || co>=col)
693 throw (std::range_error("matrix::operator(): index out of range"));
699 /** operator() to access elements for writing.
701 * @param ro row of element
702 * @param co column of element
703 * @exception range_error (index out of range) */
704 ex & matrix::operator() (unsigned ro, unsigned co)
706 if (ro>=row || co>=col)
707 throw (std::range_error("matrix::operator(): index out of range"));
709 ensure_if_modifiable();
714 /** Transposed of an m x n matrix, producing a new n x m matrix object that
715 * represents the transposed. */
716 matrix matrix::transpose() const
718 exvector trans(this->cols()*this->rows());
720 for (unsigned r=0; r<this->cols(); ++r)
721 for (unsigned c=0; c<this->rows(); ++c)
722 trans[r*this->rows()+c] = m[c*this->cols()+r];
724 return matrix(this->cols(),this->rows(),trans);
727 /** Determinant of square matrix. This routine doesn't actually calculate the
728 * determinant, it only implements some heuristics about which algorithm to
729 * run. If all the elements of the matrix are elements of an integral domain
730 * the determinant is also in that integral domain and the result is expanded
731 * only. If one or more elements are from a quotient field the determinant is
732 * usually also in that quotient field and the result is normalized before it
733 * is returned. This implies that the determinant of the symbolic 2x2 matrix
734 * [[a/(a-b),1],[b/(a-b),1]] is returned as unity. (In this respect, it
735 * behaves like MapleV and unlike Mathematica.)
737 * @param algo allows to chose an algorithm
738 * @return the determinant as a new expression
739 * @exception logic_error (matrix not square)
740 * @see determinant_algo */
741 ex matrix::determinant(unsigned algo) const
744 throw (std::logic_error("matrix::determinant(): matrix not square"));
745 GINAC_ASSERT(row*col==m.capacity());
747 // Gather some statistical information about this matrix:
748 bool numeric_flag = true;
749 bool normal_flag = false;
750 unsigned sparse_count = 0; // counts non-zero elements
751 exvector::const_iterator r = m.begin(), rend = m.end();
753 if (!r->info(info_flags::numeric))
754 numeric_flag = false;
755 exmap srl; // symbol replacement list
756 ex rtest = r->to_rational(srl);
757 if (!rtest.is_zero())
759 if (!rtest.info(info_flags::crational_polynomial) &&
760 rtest.info(info_flags::rational_function))
765 // Here is the heuristics in case this routine has to decide:
766 if (algo == determinant_algo::automatic) {
767 // Minor expansion is generally a good guess:
768 algo = determinant_algo::laplace;
769 // Does anybody know when a matrix is really sparse?
770 // Maybe <~row/2.236 nonzero elements average in a row?
771 if (row>3 && 5*sparse_count<=row*col)
772 algo = determinant_algo::bareiss;
773 // Purely numeric matrix can be handled by Gauss elimination.
774 // This overrides any prior decisions.
776 algo = determinant_algo::gauss;
779 // Trap the trivial case here, since some algorithms don't like it
781 // for consistency with non-trivial determinants...
783 return m[0].normal();
785 return m[0].expand();
788 // Compute the determinant
790 case determinant_algo::gauss: {
793 int sign = tmp.gauss_elimination(true);
794 for (unsigned d=0; d<row; ++d)
795 det *= tmp.m[d*col+d];
797 return (sign*det).normal();
799 return (sign*det).normal().expand();
801 case determinant_algo::bareiss: {
804 sign = tmp.fraction_free_elimination(true);
806 return (sign*tmp.m[row*col-1]).normal();
808 return (sign*tmp.m[row*col-1]).expand();
810 case determinant_algo::divfree: {
813 sign = tmp.division_free_elimination(true);
816 ex det = tmp.m[row*col-1];
817 // factor out accumulated bogus slag
818 for (unsigned d=0; d<row-2; ++d)
819 for (unsigned j=0; j<row-d-2; ++j)
820 det = (det/tmp.m[d*col+d]).normal();
823 case determinant_algo::laplace:
825 // This is the minor expansion scheme. We always develop such
826 // that the smallest minors (i.e, the trivial 1x1 ones) are on the
827 // rightmost column. For this to be efficient, empirical tests
828 // have shown that the emptiest columns (i.e. the ones with most
829 // zeros) should be the ones on the right hand side -- although
830 // this might seem counter-intuitive (and in contradiction to some
831 // literature like the FORM manual). Please go ahead and test it
832 // if you don't believe me! Therefore we presort the columns of
834 typedef std::pair<unsigned,unsigned> uintpair;
835 std::vector<uintpair> c_zeros; // number of zeros in column
836 for (unsigned c=0; c<col; ++c) {
838 for (unsigned r=0; r<row; ++r)
839 if (m[r*col+c].is_zero())
841 c_zeros.push_back(uintpair(acc,c));
843 std::sort(c_zeros.begin(),c_zeros.end());
844 std::vector<unsigned> pre_sort;
845 for (std::vector<uintpair>::const_iterator i=c_zeros.begin(); i!=c_zeros.end(); ++i)
846 pre_sort.push_back(i->second);
847 std::vector<unsigned> pre_sort_test(pre_sort); // permutation_sign() modifies the vector so we make a copy here
848 int sign = permutation_sign(pre_sort_test.begin(), pre_sort_test.end());
849 exvector result(row*col); // represents sorted matrix
851 for (std::vector<unsigned>::const_iterator i=pre_sort.begin();
854 for (unsigned r=0; r<row; ++r)
855 result[r*col+c] = m[r*col+(*i)];
859 return (sign*matrix(row,col,result).determinant_minor()).normal();
861 return sign*matrix(row,col,result).determinant_minor();
867 /** Trace of a matrix. The result is normalized if it is in some quotient
868 * field and expanded only otherwise. This implies that the trace of the
869 * symbolic 2x2 matrix [[a/(a-b),x],[y,b/(b-a)]] is recognized to be unity.
871 * @return the sum of diagonal elements
872 * @exception logic_error (matrix not square) */
873 ex matrix::trace() const
876 throw (std::logic_error("matrix::trace(): matrix not square"));
879 for (unsigned r=0; r<col; ++r)
882 if (tr.info(info_flags::rational_function) &&
883 !tr.info(info_flags::crational_polynomial))
890 /** Characteristic Polynomial. Following mathematica notation the
891 * characteristic polynomial of a matrix M is defined as the determiant of
892 * (M - lambda * 1) where 1 stands for the unit matrix of the same dimension
893 * as M. Note that some CASs define it with a sign inside the determinant
894 * which gives rise to an overall sign if the dimension is odd. This method
895 * returns the characteristic polynomial collected in powers of lambda as a
898 * @return characteristic polynomial as new expression
899 * @exception logic_error (matrix not square)
900 * @see matrix::determinant() */
901 ex matrix::charpoly(const ex & lambda) const
904 throw (std::logic_error("matrix::charpoly(): matrix not square"));
906 bool numeric_flag = true;
907 exvector::const_iterator r = m.begin(), rend = m.end();
908 while (r!=rend && numeric_flag==true) {
909 if (!r->info(info_flags::numeric))
910 numeric_flag = false;
914 // The pure numeric case is traditionally rather common. Hence, it is
915 // trapped and we use Leverrier's algorithm which goes as row^3 for
916 // every coefficient. The expensive part is the matrix multiplication.
921 ex poly = power(lambda, row) - c*power(lambda, row-1);
922 for (unsigned i=1; i<row; ++i) {
923 for (unsigned j=0; j<row; ++j)
926 c = B.trace() / ex(i+1);
927 poly -= c*power(lambda, row-i-1);
937 for (unsigned r=0; r<col; ++r)
938 M.m[r*col+r] -= lambda;
940 return M.determinant().collect(lambda);
945 /** Inverse of this matrix.
947 * @return the inverted matrix
948 * @exception logic_error (matrix not square)
949 * @exception runtime_error (singular matrix) */
950 matrix matrix::inverse() const
953 throw (std::logic_error("matrix::inverse(): matrix not square"));
955 // This routine actually doesn't do anything fancy at all. We compute the
956 // inverse of the matrix A by solving the system A * A^{-1} == Id.
958 // First populate the identity matrix supposed to become the right hand side.
959 matrix identity(row,col);
960 for (unsigned i=0; i<row; ++i)
961 identity(i,i) = _ex1;
963 // Populate a dummy matrix of variables, just because of compatibility with
964 // matrix::solve() which wants this (for compatibility with under-determined
965 // systems of equations).
966 matrix vars(row,col);
967 for (unsigned r=0; r<row; ++r)
968 for (unsigned c=0; c<col; ++c)
969 vars(r,c) = symbol();
973 sol = this->solve(vars,identity);
974 } catch (const std::runtime_error & e) {
975 if (e.what()==std::string("matrix::solve(): inconsistent linear system"))
976 throw (std::runtime_error("matrix::inverse(): singular matrix"));
984 /** Solve a linear system consisting of a m x n matrix and a m x p right hand
985 * side by applying an elimination scheme to the augmented matrix.
987 * @param vars n x p matrix, all elements must be symbols
988 * @param rhs m x p matrix
989 * @param algo selects the solving algorithm
990 * @return n x p solution matrix
991 * @exception logic_error (incompatible matrices)
992 * @exception invalid_argument (1st argument must be matrix of symbols)
993 * @exception runtime_error (inconsistent linear system)
995 matrix matrix::solve(const matrix & vars,
999 const unsigned m = this->rows();
1000 const unsigned n = this->cols();
1001 const unsigned p = rhs.cols();
1004 if ((rhs.rows() != m) || (vars.rows() != n) || (vars.col != p))
1005 throw (std::logic_error("matrix::solve(): incompatible matrices"));
1006 for (unsigned ro=0; ro<n; ++ro)
1007 for (unsigned co=0; co<p; ++co)
1008 if (!vars(ro,co).info(info_flags::symbol))
1009 throw (std::invalid_argument("matrix::solve(): 1st argument must be matrix of symbols"));
1011 // build the augmented matrix of *this with rhs attached to the right
1013 for (unsigned r=0; r<m; ++r) {
1014 for (unsigned c=0; c<n; ++c)
1015 aug.m[r*(n+p)+c] = this->m[r*n+c];
1016 for (unsigned c=0; c<p; ++c)
1017 aug.m[r*(n+p)+c+n] = rhs.m[r*p+c];
1020 // Gather some statistical information about the augmented matrix:
1021 bool numeric_flag = true;
1022 exvector::const_iterator r = aug.m.begin(), rend = aug.m.end();
1023 while (r!=rend && numeric_flag==true) {
1024 if (!r->info(info_flags::numeric))
1025 numeric_flag = false;
1029 // Here is the heuristics in case this routine has to decide:
1030 if (algo == solve_algo::automatic) {
1031 // Bareiss (fraction-free) elimination is generally a good guess:
1032 algo = solve_algo::bareiss;
1033 // For m<3, Bareiss elimination is equivalent to division free
1034 // elimination but has more logistic overhead
1036 algo = solve_algo::divfree;
1037 // This overrides any prior decisions.
1039 algo = solve_algo::gauss;
1042 // Eliminate the augmented matrix:
1044 case solve_algo::gauss:
1045 aug.gauss_elimination();
1047 case solve_algo::divfree:
1048 aug.division_free_elimination();
1050 case solve_algo::bareiss:
1052 aug.fraction_free_elimination();
1055 // assemble the solution matrix:
1057 for (unsigned co=0; co<p; ++co) {
1058 unsigned last_assigned_sol = n+1;
1059 for (int r=m-1; r>=0; --r) {
1060 unsigned fnz = 1; // first non-zero in row
1061 while ((fnz<=n) && (aug.m[r*(n+p)+(fnz-1)].is_zero()))
1064 // row consists only of zeros, corresponding rhs must be 0, too
1065 if (!aug.m[r*(n+p)+n+co].is_zero()) {
1066 throw (std::runtime_error("matrix::solve(): inconsistent linear system"));
1069 // assign solutions for vars between fnz+1 and
1070 // last_assigned_sol-1: free parameters
1071 for (unsigned c=fnz; c<last_assigned_sol-1; ++c)
1072 sol(c,co) = vars.m[c*p+co];
1073 ex e = aug.m[r*(n+p)+n+co];
1074 for (unsigned c=fnz; c<n; ++c)
1075 e -= aug.m[r*(n+p)+c]*sol.m[c*p+co];
1076 sol(fnz-1,co) = (e/(aug.m[r*(n+p)+(fnz-1)])).normal();
1077 last_assigned_sol = fnz;
1080 // assign solutions for vars between 1 and
1081 // last_assigned_sol-1: free parameters
1082 for (unsigned ro=0; ro<last_assigned_sol-1; ++ro)
1083 sol(ro,co) = vars(ro,co);
1090 /** Compute the rank of this matrix. */
1091 unsigned matrix::rank() const
1094 // Transform this matrix into upper echelon form and then count the
1095 // number of non-zero rows.
1097 GINAC_ASSERT(row*col==m.capacity());
1099 // Actually, any elimination scheme will do since we are only
1100 // interested in the echelon matrix' zeros.
1101 matrix to_eliminate = *this;
1102 to_eliminate.fraction_free_elimination();
1104 unsigned r = row*col; // index of last non-zero element
1106 if (!to_eliminate.m[r].is_zero())
1115 /** Recursive determinant for small matrices having at least one symbolic
1116 * entry. The basic algorithm, known as Laplace-expansion, is enhanced by
1117 * some bookkeeping to avoid calculation of the same submatrices ("minors")
1118 * more than once. According to W.M.Gentleman and S.C.Johnson this algorithm
1119 * is better than elimination schemes for matrices of sparse multivariate
1120 * polynomials and also for matrices of dense univariate polynomials if the
1121 * matrix' dimesion is larger than 7.
1123 * @return the determinant as a new expression (in expanded form)
1124 * @see matrix::determinant() */
1125 ex matrix::determinant_minor() const
1127 // for small matrices the algorithm does not make any sense:
1128 const unsigned n = this->cols();
1130 return m[0].expand();
1132 return (m[0]*m[3]-m[2]*m[1]).expand();
1134 return (m[0]*m[4]*m[8]-m[0]*m[5]*m[7]-
1135 m[1]*m[3]*m[8]+m[2]*m[3]*m[7]+
1136 m[1]*m[5]*m[6]-m[2]*m[4]*m[6]).expand();
1138 // This algorithm can best be understood by looking at a naive
1139 // implementation of Laplace-expansion, like this one:
1141 // matrix minorM(this->rows()-1,this->cols()-1);
1142 // for (unsigned r1=0; r1<this->rows(); ++r1) {
1143 // // shortcut if element(r1,0) vanishes
1144 // if (m[r1*col].is_zero())
1146 // // assemble the minor matrix
1147 // for (unsigned r=0; r<minorM.rows(); ++r) {
1148 // for (unsigned c=0; c<minorM.cols(); ++c) {
1150 // minorM(r,c) = m[r*col+c+1];
1152 // minorM(r,c) = m[(r+1)*col+c+1];
1155 // // recurse down and care for sign:
1157 // det -= m[r1*col] * minorM.determinant_minor();
1159 // det += m[r1*col] * minorM.determinant_minor();
1161 // return det.expand();
1162 // What happens is that while proceeding down many of the minors are
1163 // computed more than once. In particular, there are binomial(n,k)
1164 // kxk minors and each one is computed factorial(n-k) times. Therefore
1165 // it is reasonable to store the results of the minors. We proceed from
1166 // right to left. At each column c we only need to retrieve the minors
1167 // calculated in step c-1. We therefore only have to store at most
1168 // 2*binomial(n,n/2) minors.
1170 // Unique flipper counter for partitioning into minors
1171 std::vector<unsigned> Pkey;
1173 // key for minor determinant (a subpartition of Pkey)
1174 std::vector<unsigned> Mkey;
1176 // we store our subminors in maps, keys being the rows they arise from
1177 typedef std::map<std::vector<unsigned>,class ex> Rmap;
1178 typedef std::map<std::vector<unsigned>,class ex>::value_type Rmap_value;
1182 // initialize A with last column:
1183 for (unsigned r=0; r<n; ++r) {
1184 Pkey.erase(Pkey.begin(),Pkey.end());
1186 A.insert(Rmap_value(Pkey,m[n*(r+1)-1]));
1188 // proceed from right to left through matrix
1189 for (int c=n-2; c>=0; --c) {
1190 Pkey.erase(Pkey.begin(),Pkey.end()); // don't change capacity
1191 Mkey.erase(Mkey.begin(),Mkey.end());
1192 for (unsigned i=0; i<n-c; ++i)
1194 unsigned fc = 0; // controls logic for our strange flipper counter
1197 for (unsigned r=0; r<n-c; ++r) {
1198 // maybe there is nothing to do?
1199 if (m[Pkey[r]*n+c].is_zero())
1201 // create the sorted key for all possible minors
1202 Mkey.erase(Mkey.begin(),Mkey.end());
1203 for (unsigned i=0; i<n-c; ++i)
1205 Mkey.push_back(Pkey[i]);
1206 // Fetch the minors and compute the new determinant
1208 det -= m[Pkey[r]*n+c]*A[Mkey];
1210 det += m[Pkey[r]*n+c]*A[Mkey];
1212 // prevent build-up of deep nesting of expressions saves time:
1214 // store the new determinant at its place in B:
1216 B.insert(Rmap_value(Pkey,det));
1217 // increment our strange flipper counter
1218 for (fc=n-c; fc>0; --fc) {
1220 if (Pkey[fc-1]<fc+c)
1224 for (unsigned j=fc; j<n-c; ++j)
1225 Pkey[j] = Pkey[j-1]+1;
1227 // next column, so change the role of A and B:
1236 /** Perform the steps of an ordinary Gaussian elimination to bring the m x n
1237 * matrix into an upper echelon form. The algorithm is ok for matrices
1238 * with numeric coefficients but quite unsuited for symbolic matrices.
1240 * @param det may be set to true to save a lot of space if one is only
1241 * interested in the diagonal elements (i.e. for calculating determinants).
1242 * The others are set to zero in this case.
1243 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1244 * number of rows was swapped and 0 if the matrix is singular. */
1245 int matrix::gauss_elimination(const bool det)
1247 ensure_if_modifiable();
1248 const unsigned m = this->rows();
1249 const unsigned n = this->cols();
1250 GINAC_ASSERT(!det || n==m);
1254 for (unsigned c0=0; c0<n && r0<m-1; ++c0) {
1255 int indx = pivot(r0, c0, true);
1259 return 0; // leaves *this in a messy state
1264 for (unsigned r2=r0+1; r2<m; ++r2) {
1265 if (!this->m[r2*n+c0].is_zero()) {
1266 // yes, there is something to do in this row
1267 ex piv = this->m[r2*n+c0] / this->m[r0*n+c0];
1268 for (unsigned c=c0+1; c<n; ++c) {
1269 this->m[r2*n+c] -= piv * this->m[r0*n+c];
1270 if (!this->m[r2*n+c].info(info_flags::numeric))
1271 this->m[r2*n+c] = this->m[r2*n+c].normal();
1274 // fill up left hand side with zeros
1275 for (unsigned c=r0; c<=c0; ++c)
1276 this->m[r2*n+c] = _ex0;
1279 // save space by deleting no longer needed elements
1280 for (unsigned c=r0+1; c<n; ++c)
1281 this->m[r0*n+c] = _ex0;
1286 // clear remaining rows
1287 for (unsigned r=r0+1; r<m; ++r) {
1288 for (unsigned c=0; c<n; ++c)
1289 this->m[r*n+c] = _ex0;
1296 /** Perform the steps of division free elimination to bring the m x n matrix
1297 * into an upper echelon form.
1299 * @param det may be set to true to save a lot of space if one is only
1300 * interested in the diagonal elements (i.e. for calculating determinants).
1301 * The others are set to zero in this case.
1302 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1303 * number of rows was swapped and 0 if the matrix is singular. */
1304 int matrix::division_free_elimination(const bool det)
1306 ensure_if_modifiable();
1307 const unsigned m = this->rows();
1308 const unsigned n = this->cols();
1309 GINAC_ASSERT(!det || n==m);
1313 for (unsigned c0=0; c0<n && r0<m-1; ++c0) {
1314 int indx = pivot(r0, c0, true);
1318 return 0; // leaves *this in a messy state
1323 for (unsigned r2=r0+1; r2<m; ++r2) {
1324 for (unsigned c=c0+1; c<n; ++c)
1325 this->m[r2*n+c] = (this->m[r0*n+c0]*this->m[r2*n+c] - this->m[r2*n+c0]*this->m[r0*n+c]).expand();
1326 // fill up left hand side with zeros
1327 for (unsigned c=r0; c<=c0; ++c)
1328 this->m[r2*n+c] = _ex0;
1331 // save space by deleting no longer needed elements
1332 for (unsigned c=r0+1; c<n; ++c)
1333 this->m[r0*n+c] = _ex0;
1338 // clear remaining rows
1339 for (unsigned r=r0+1; r<m; ++r) {
1340 for (unsigned c=0; c<n; ++c)
1341 this->m[r*n+c] = _ex0;
1348 /** Perform the steps of Bareiss' one-step fraction free elimination to bring
1349 * the matrix into an upper echelon form. Fraction free elimination means
1350 * that divide is used straightforwardly, without computing GCDs first. This
1351 * is possible, since we know the divisor at each step.
1353 * @param det may be set to true to save a lot of space if one is only
1354 * interested in the last element (i.e. for calculating determinants). The
1355 * others are set to zero in this case.
1356 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1357 * number of rows was swapped and 0 if the matrix is singular. */
1358 int matrix::fraction_free_elimination(const bool det)
1361 // (single-step fraction free elimination scheme, already known to Jordan)
1363 // Usual division-free elimination sets m[0](r,c) = m(r,c) and then sets
1364 // m[k+1](r,c) = m[k](k,k) * m[k](r,c) - m[k](r,k) * m[k](k,c).
1366 // Bareiss (fraction-free) elimination in addition divides that element
1367 // by m[k-1](k-1,k-1) for k>1, where it can be shown by means of the
1368 // Sylvester identity that this really divides m[k+1](r,c).
1370 // We also allow rational functions where the original prove still holds.
1371 // However, we must care for numerator and denominator separately and
1372 // "manually" work in the integral domains because of subtle cancellations
1373 // (see below). This blows up the bookkeeping a bit and the formula has
1374 // to be modified to expand like this (N{x} stands for numerator of x,
1375 // D{x} for denominator of x):
1376 // 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)}
1377 // -N{m[k](r,k)}*N{m[k](k,c)}*D{m[k](k,k)}*D{m[k](r,c)}
1378 // 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)}
1379 // where for k>1 we now divide N{m[k+1](r,c)} by
1380 // N{m[k-1](k-1,k-1)}
1381 // and D{m[k+1](r,c)} by
1382 // D{m[k-1](k-1,k-1)}.
1384 ensure_if_modifiable();
1385 const unsigned m = this->rows();
1386 const unsigned n = this->cols();
1387 GINAC_ASSERT(!det || n==m);
1396 // We populate temporary matrices to subsequently operate on. There is
1397 // one holding numerators and another holding denominators of entries.
1398 // This is a must since the evaluator (or even earlier mul's constructor)
1399 // might cancel some trivial element which causes divide() to fail. The
1400 // elements are normalized first (yes, even though this algorithm doesn't
1401 // need GCDs) since the elements of *this might be unnormalized, which
1402 // makes things more complicated than they need to be.
1403 matrix tmp_n(*this);
1404 matrix tmp_d(m,n); // for denominators, if needed
1405 exmap srl; // symbol replacement list
1406 exvector::const_iterator cit = this->m.begin(), citend = this->m.end();
1407 exvector::iterator tmp_n_it = tmp_n.m.begin(), tmp_d_it = tmp_d.m.begin();
1408 while (cit != citend) {
1409 ex nd = cit->normal().to_rational(srl).numer_denom();
1411 *tmp_n_it++ = nd.op(0);
1412 *tmp_d_it++ = nd.op(1);
1416 for (unsigned c0=0; c0<n && r0<m-1; ++c0) {
1417 // When trying to find a pivot, we should try a bit harder than expand().
1418 // Searching the first non-zero element in-place here instead of calling
1419 // pivot() allows us to do no more substitutions and back-substitutions
1420 // than are actually necessary.
1423 (tmp_n[indx*n+c0].subs(srl, subs_options::no_pattern).expand().is_zero()))
1426 // all elements in column c0 below row r0 vanish
1432 // Matrix needs pivoting, swap rows r0 and indx of tmp_n and tmp_d.
1434 for (unsigned c=c0; c<n; ++c) {
1435 tmp_n.m[n*indx+c].swap(tmp_n.m[n*r0+c]);
1436 tmp_d.m[n*indx+c].swap(tmp_d.m[n*r0+c]);
1439 for (unsigned r2=r0+1; r2<m; ++r2) {
1440 for (unsigned c=c0+1; c<n; ++c) {
1441 dividend_n = (tmp_n.m[r0*n+c0]*tmp_n.m[r2*n+c]*
1442 tmp_d.m[r2*n+c0]*tmp_d.m[r0*n+c]
1443 -tmp_n.m[r2*n+c0]*tmp_n.m[r0*n+c]*
1444 tmp_d.m[r0*n+c0]*tmp_d.m[r2*n+c]).expand();
1445 dividend_d = (tmp_d.m[r2*n+c0]*tmp_d.m[r0*n+c]*
1446 tmp_d.m[r0*n+c0]*tmp_d.m[r2*n+c]).expand();
1447 bool check = divide(dividend_n, divisor_n,
1448 tmp_n.m[r2*n+c], true);
1449 check &= divide(dividend_d, divisor_d,
1450 tmp_d.m[r2*n+c], true);
1451 GINAC_ASSERT(check);
1453 // fill up left hand side with zeros
1454 for (unsigned c=r0; c<=c0; ++c)
1455 tmp_n.m[r2*n+c] = _ex0;
1457 if (c0<n && r0<m-1) {
1458 // compute next iteration's divisor
1459 divisor_n = tmp_n.m[r0*n+c0].expand();
1460 divisor_d = tmp_d.m[r0*n+c0].expand();
1462 // save space by deleting no longer needed elements
1463 for (unsigned c=0; c<n; ++c) {
1464 tmp_n.m[r0*n+c] = _ex0;
1465 tmp_d.m[r0*n+c] = _ex1;
1472 // clear remaining rows
1473 for (unsigned r=r0+1; r<m; ++r) {
1474 for (unsigned c=0; c<n; ++c)
1475 tmp_n.m[r*n+c] = _ex0;
1478 // repopulate *this matrix:
1479 exvector::iterator it = this->m.begin(), itend = this->m.end();
1480 tmp_n_it = tmp_n.m.begin();
1481 tmp_d_it = tmp_d.m.begin();
1483 *it++ = ((*tmp_n_it++)/(*tmp_d_it++)).subs(srl, subs_options::no_pattern);
1489 /** Partial pivoting method for matrix elimination schemes.
1490 * Usual pivoting (symbolic==false) returns the index to the element with the
1491 * largest absolute value in column ro and swaps the current row with the one
1492 * where the element was found. With (symbolic==true) it does the same thing
1493 * with the first non-zero element.
1495 * @param ro is the row from where to begin
1496 * @param co is the column to be inspected
1497 * @param symbolic signal if we want the first non-zero element to be pivoted
1498 * (true) or the one with the largest absolute value (false).
1499 * @return 0 if no interchange occured, -1 if all are zero (usually signaling
1500 * a degeneracy) and positive integer k means that rows ro and k were swapped.
1502 int matrix::pivot(unsigned ro, unsigned co, bool symbolic)
1506 // search first non-zero element in column co beginning at row ro
1507 while ((k<row) && (this->m[k*col+co].expand().is_zero()))
1510 // search largest element in column co beginning at row ro
1511 GINAC_ASSERT(is_exactly_a<numeric>(this->m[k*col+co]));
1512 unsigned kmax = k+1;
1513 numeric mmax = abs(ex_to<numeric>(m[kmax*col+co]));
1515 GINAC_ASSERT(is_exactly_a<numeric>(this->m[kmax*col+co]));
1516 numeric tmp = ex_to<numeric>(this->m[kmax*col+co]);
1517 if (abs(tmp) > mmax) {
1523 if (!mmax.is_zero())
1527 // all elements in column co below row ro vanish
1530 // matrix needs no pivoting
1532 // matrix needs pivoting, so swap rows k and ro
1533 ensure_if_modifiable();
1534 for (unsigned c=0; c<col; ++c)
1535 this->m[k*col+c].swap(this->m[ro*col+c]);
1540 /** Function to check that all elements of the matrix are zero.
1542 bool matrix::is_zero_matrix() const
1544 for (exvector::const_iterator i=m.begin(); i!=m.end(); ++i)
1550 ex lst_to_matrix(const lst & l)
1552 lst::const_iterator itr, itc;
1554 // Find number of rows and columns
1555 size_t rows = l.nops(), cols = 0;
1556 for (itr = l.begin(); itr != l.end(); ++itr) {
1557 if (!is_a<lst>(*itr))
1558 throw (std::invalid_argument("lst_to_matrix: argument must be a list of lists"));
1559 if (itr->nops() > cols)
1563 // Allocate and fill matrix
1564 matrix &M = *new matrix(rows, cols);
1565 M.setflag(status_flags::dynallocated);
1568 for (itr = l.begin(), i = 0; itr != l.end(); ++itr, ++i) {
1570 for (itc = ex_to<lst>(*itr).begin(), j = 0; itc != ex_to<lst>(*itr).end(); ++itc, ++j)
1577 ex diag_matrix(const lst & l)
1579 lst::const_iterator it;
1580 size_t dim = l.nops();
1582 // Allocate and fill matrix
1583 matrix &M = *new matrix(dim, dim);
1584 M.setflag(status_flags::dynallocated);
1587 for (it = l.begin(), i = 0; it != l.end(); ++it, ++i)
1593 ex unit_matrix(unsigned r, unsigned c)
1595 matrix &Id = *new matrix(r, c);
1596 Id.setflag(status_flags::dynallocated);
1597 for (unsigned i=0; i<r && i<c; i++)
1603 ex symbolic_matrix(unsigned r, unsigned c, const std::string & base_name, const std::string & tex_base_name)
1605 matrix &M = *new matrix(r, c);
1606 M.setflag(status_flags::dynallocated | status_flags::evaluated);
1608 bool long_format = (r > 10 || c > 10);
1609 bool single_row = (r == 1 || c == 1);
1611 for (unsigned i=0; i<r; i++) {
1612 for (unsigned j=0; j<c; j++) {
1613 std::ostringstream s1, s2;
1615 s2 << tex_base_name << "_{";
1626 s1 << '_' << i << '_' << j;
1627 s2 << i << ';' << j << "}";
1630 s2 << i << j << '}';
1633 M(i, j) = symbol(s1.str(), s2.str());
1640 ex reduced_matrix(const matrix& m, unsigned r, unsigned c)
1642 if (r+1>m.rows() || c+1>m.cols() || m.cols()<2 || m.rows()<2)
1643 throw std::runtime_error("minor_matrix(): index out of bounds");
1645 const unsigned rows = m.rows()-1;
1646 const unsigned cols = m.cols()-1;
1647 matrix &M = *new matrix(rows, cols);
1648 M.setflag(status_flags::dynallocated | status_flags::evaluated);
1660 M(ro2,co2) = m(ro, co);
1671 ex sub_matrix(const matrix&m, unsigned r, unsigned nr, unsigned c, unsigned nc)
1673 if (r+nr>m.rows() || c+nc>m.cols())
1674 throw std::runtime_error("sub_matrix(): index out of bounds");
1676 matrix &M = *new matrix(nr, nc);
1677 M.setflag(status_flags::dynallocated | status_flags::evaluated);
1679 for (unsigned ro=0; ro<nr; ++ro) {
1680 for (unsigned co=0; co<nc; ++co) {
1681 M(ro,co) = m(ro+r,co+c);
1688 } // namespace GiNaC