3 * Implementation of symbolic matrices */
6 * GiNaC Copyright (C) 1999-2016 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
31 #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() : 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) : row(r), col(c), m(r*c, _ex0)
73 setflag(status_flags::not_shareable);
76 /** Construct matrix from (flat) list of elements. If the list has fewer
77 * elements than the matrix, the remaining matrix elements are set to zero.
78 * If the list has more elements than the matrix, the excessive elements are
80 matrix::matrix(unsigned r, unsigned c, const lst & l)
81 : row(r), col(c), m(r*c, _ex0)
83 setflag(status_flags::not_shareable);
90 break; // matrix smaller than list: throw away excessive elements
96 /** Construct a matrix from an 2 dimensional initializer list.
97 * Throws an exception if some row has a different length than all the others.
99 matrix::matrix(std::initializer_list<std::initializer_list<ex>> l)
100 : row(l.size()), col(l.begin()->size())
102 setflag(status_flags::not_shareable);
105 for (const auto & r : l) {
107 for (const auto & e : r) {
112 throw std::invalid_argument("matrix::matrix{{}}: wrong dimension");
118 /** Ctor from representation, for internal use only. */
119 matrix::matrix(unsigned r, unsigned c, const exvector & m2)
120 : row(r), col(c), m(m2)
122 setflag(status_flags::not_shareable);
124 matrix::matrix(unsigned r, unsigned c, exvector && m2)
125 : row(r), col(c), m(std::move(m2))
127 setflag(status_flags::not_shareable);
134 void matrix::read_archive(const archive_node &n, lst &sym_lst)
136 inherited::read_archive(n, sym_lst);
138 if (!(n.find_unsigned("row", row)) || !(n.find_unsigned("col", col)))
139 throw (std::runtime_error("unknown matrix dimensions in archive"));
140 m.reserve(row * col);
141 // XXX: default ctor inserts a zero element, we need to erase it here.
143 auto first = n.find_first("m");
144 auto last = n.find_last("m");
146 for (auto i=first; i != last; ++i) {
148 n.find_ex_by_loc(i, e, sym_lst);
152 GINAC_BIND_UNARCHIVER(matrix);
154 void matrix::archive(archive_node &n) const
156 inherited::archive(n);
157 n.add_unsigned("row", row);
158 n.add_unsigned("col", col);
165 // functions overriding virtual functions from base classes
170 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
172 for (unsigned ro=0; ro<row; ++ro) {
174 for (unsigned co=0; co<col; ++co) {
175 m[ro*col+co].print(c);
186 void matrix::do_print(const print_context & c, unsigned level) const
189 print_elements(c, "[", "]", ",", ",");
193 void matrix::do_print_latex(const print_latex & c, unsigned level) const
195 c.s << "\\left(\\begin{array}{" << std::string(col,'c') << "}";
196 print_elements(c, "", "", "\\\\", "&");
197 c.s << "\\end{array}\\right)";
200 void matrix::do_print_python_repr(const print_python_repr & c, unsigned level) const
202 c.s << class_name() << '(';
203 print_elements(c, "[", "]", ",", ",");
207 /** nops is defined to be rows x columns. */
208 size_t matrix::nops() const
210 return static_cast<size_t>(row) * static_cast<size_t>(col);
213 /** returns matrix entry at position (i/col, i%col). */
214 ex matrix::op(size_t i) const
216 GINAC_ASSERT(i<nops());
221 /** returns writable matrix entry at position (i/col, i%col). */
222 ex & matrix::let_op(size_t i)
224 GINAC_ASSERT(i<nops());
226 ensure_if_modifiable();
230 ex matrix::subs(const exmap & mp, unsigned options) const
232 exvector m2(row * col);
233 for (unsigned r=0; r<row; ++r)
234 for (unsigned c=0; c<col; ++c)
235 m2[r*col+c] = m[r*col+c].subs(mp, options);
237 return matrix(row, col, std::move(m2)).subs_one_level(mp, options);
240 /** Complex conjugate every matrix entry. */
241 ex matrix::conjugate() const
243 std::unique_ptr<exvector> ev(nullptr);
244 for (auto i=m.begin(); i!=m.end(); ++i) {
245 ex x = i->conjugate();
250 if (are_ex_trivially_equal(x, *i)) {
253 ev.reset(new exvector);
254 ev->reserve(m.size());
255 for (auto j=m.begin(); j!=i; ++j) {
261 return matrix(row, col, std::move(*ev));
266 ex matrix::real_part() const
271 v.push_back(i.real_part());
272 return matrix(row, col, std::move(v));
275 ex matrix::imag_part() const
280 v.push_back(i.imag_part());
281 return matrix(row, col, std::move(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 auto ci = other.m.begin();
567 return matrix(row, col, std::move(sum));
571 /** Difference of matrices.
573 * @exception logic_error (incompatible matrices) */
574 matrix matrix::sub(const matrix & other) const
576 if (col != other.col || row != other.row)
577 throw std::logic_error("matrix::sub(): incompatible matrices");
579 exvector dif(this->m);
580 auto ci = other.m.begin();
584 return matrix(row, col, std::move(dif));
588 /** Product of matrices.
590 * @exception logic_error (incompatible matrices) */
591 matrix matrix::mul(const matrix & other) const
593 if (this->cols() != other.rows())
594 throw std::logic_error("matrix::mul(): incompatible matrices");
596 exvector prod(this->rows()*other.cols());
598 for (unsigned r1=0; r1<this->rows(); ++r1) {
599 for (unsigned c=0; c<this->cols(); ++c) {
600 // Quick test: can we shortcut?
601 if (m[r1*col+c].is_zero())
603 for (unsigned r2=0; r2<other.cols(); ++r2)
604 prod[r1*other.col+r2] += (m[r1*col+c] * other.m[c*other.col+r2]);
607 return matrix(row, other.col, std::move(prod));
611 /** Product of matrix and scalar. */
612 matrix matrix::mul(const numeric & other) const
614 exvector prod(row * col);
616 for (unsigned r=0; r<row; ++r)
617 for (unsigned c=0; c<col; ++c)
618 prod[r*col+c] = m[r*col+c] * other;
620 return matrix(row, col, std::move(prod));
624 /** Product of matrix and scalar expression. */
625 matrix matrix::mul_scalar(const ex & other) const
627 if (other.return_type() != return_types::commutative)
628 throw std::runtime_error("matrix::mul_scalar(): non-commutative scalar");
630 exvector prod(row * col);
632 for (unsigned r=0; r<row; ++r)
633 for (unsigned c=0; c<col; ++c)
634 prod[r*col+c] = m[r*col+c] * other;
636 return matrix(row, col, std::move(prod));
640 /** Power of a matrix. Currently handles integer exponents only. */
641 matrix matrix::pow(const ex & expn) const
644 throw (std::logic_error("matrix::pow(): matrix not square"));
646 if (is_exactly_a<numeric>(expn)) {
647 // Integer cases are computed by successive multiplication, using the
648 // obvious shortcut of storing temporaries, like A^4 == (A*A)*(A*A).
649 if (expn.info(info_flags::integer)) {
650 numeric b = ex_to<numeric>(expn);
652 if (expn.info(info_flags::negative)) {
659 for (unsigned r=0; r<row; ++r)
663 // This loop computes the representation of b in base 2 from right
664 // to left and multiplies the factors whenever needed. Note
665 // that this is not entirely optimal but close to optimal and
666 // "better" algorithms are much harder to implement. (See Knuth,
667 // TAoCP2, section "Evaluation of Powers" for a good discussion.)
668 while (b!=*_num1_p) {
673 b /= *_num2_p; // still integer.
679 throw (std::runtime_error("matrix::pow(): don't know how to handle exponent"));
683 /** operator() to access elements for reading.
685 * @param ro row of element
686 * @param co column of element
687 * @exception range_error (index out of range) */
688 const ex & matrix::operator() (unsigned ro, unsigned co) const
690 if (ro>=row || co>=col)
691 throw (std::range_error("matrix::operator(): index out of range"));
697 /** operator() to access elements for writing.
699 * @param ro row of element
700 * @param co column of element
701 * @exception range_error (index out of range) */
702 ex & matrix::operator() (unsigned ro, unsigned co)
704 if (ro>=row || co>=col)
705 throw (std::range_error("matrix::operator(): index out of range"));
707 ensure_if_modifiable();
712 /** Transposed of an m x n matrix, producing a new n x m matrix object that
713 * represents the transposed. */
714 matrix matrix::transpose() const
716 exvector trans(this->cols()*this->rows());
718 for (unsigned r=0; r<this->cols(); ++r)
719 for (unsigned c=0; c<this->rows(); ++c)
720 trans[r*this->rows()+c] = m[c*this->cols()+r];
722 return matrix(this->cols(), this->rows(), std::move(trans));
725 /** Determinant of square matrix. This routine doesn't actually calculate the
726 * determinant, it only implements some heuristics about which algorithm to
727 * run. If all the elements of the matrix are elements of an integral domain
728 * the determinant is also in that integral domain and the result is expanded
729 * only. If one or more elements are from a quotient field the determinant is
730 * usually also in that quotient field and the result is normalized before it
731 * is returned. This implies that the determinant of the symbolic 2x2 matrix
732 * [[a/(a-b),1],[b/(a-b),1]] is returned as unity. (In this respect, it
733 * behaves like MapleV and unlike Mathematica.)
735 * @param algo allows to chose an algorithm
736 * @return the determinant as a new expression
737 * @exception logic_error (matrix not square)
738 * @see determinant_algo */
739 ex matrix::determinant(unsigned algo) const
742 throw (std::logic_error("matrix::determinant(): matrix not square"));
743 GINAC_ASSERT(row*col==m.capacity());
745 // Gather some statistical information about this matrix:
746 bool numeric_flag = true;
747 bool normal_flag = false;
748 unsigned sparse_count = 0; // counts non-zero elements
750 if (!r.info(info_flags::numeric))
751 numeric_flag = false;
752 exmap srl; // symbol replacement list
753 ex rtest = r.to_rational(srl);
754 if (!rtest.is_zero())
756 if (!rtest.info(info_flags::crational_polynomial) &&
757 rtest.info(info_flags::rational_function))
761 // Here is the heuristics in case this routine has to decide:
762 if (algo == determinant_algo::automatic) {
763 // Minor expansion is generally a good guess:
764 algo = determinant_algo::laplace;
765 // Does anybody know when a matrix is really sparse?
766 // Maybe <~row/2.236 nonzero elements average in a row?
767 if (row>3 && 5*sparse_count<=row*col)
768 algo = determinant_algo::bareiss;
769 // Purely numeric matrix can be handled by Gauss elimination.
770 // This overrides any prior decisions.
772 algo = determinant_algo::gauss;
775 // Trap the trivial case here, since some algorithms don't like it
777 // for consistency with non-trivial determinants...
779 return m[0].normal();
781 return m[0].expand();
784 // Compute the determinant
786 case determinant_algo::gauss: {
789 int sign = tmp.gauss_elimination(true);
790 for (unsigned d=0; d<row; ++d)
791 det *= tmp.m[d*col+d];
793 return (sign*det).normal();
795 return (sign*det).normal().expand();
797 case determinant_algo::bareiss: {
800 sign = tmp.fraction_free_elimination(true);
802 return (sign*tmp.m[row*col-1]).normal();
804 return (sign*tmp.m[row*col-1]).expand();
806 case determinant_algo::divfree: {
809 sign = tmp.division_free_elimination(true);
812 ex det = tmp.m[row*col-1];
813 // factor out accumulated bogus slag
814 for (unsigned d=0; d<row-2; ++d)
815 for (unsigned j=0; j<row-d-2; ++j)
816 det = (det/tmp.m[d*col+d]).normal();
819 case determinant_algo::laplace:
821 // This is the minor expansion scheme. We always develop such
822 // that the smallest minors (i.e, the trivial 1x1 ones) are on the
823 // rightmost column. For this to be efficient, empirical tests
824 // have shown that the emptiest columns (i.e. the ones with most
825 // zeros) should be the ones on the right hand side -- although
826 // this might seem counter-intuitive (and in contradiction to some
827 // literature like the FORM manual). Please go ahead and test it
828 // if you don't believe me! Therefore we presort the columns of
830 typedef std::pair<unsigned,unsigned> uintpair;
831 std::vector<uintpair> c_zeros; // number of zeros in column
832 for (unsigned c=0; c<col; ++c) {
834 for (unsigned r=0; r<row; ++r)
835 if (m[r*col+c].is_zero())
837 c_zeros.push_back(uintpair(acc,c));
839 std::sort(c_zeros.begin(),c_zeros.end());
840 std::vector<unsigned> pre_sort;
841 for (auto & i : c_zeros)
842 pre_sort.push_back(i.second);
843 std::vector<unsigned> pre_sort_test(pre_sort); // permutation_sign() modifies the vector so we make a copy here
844 int sign = permutation_sign(pre_sort_test.begin(), pre_sort_test.end());
845 exvector result(row*col); // represents sorted matrix
847 for (auto & it : pre_sort) {
848 for (unsigned r=0; r<row; ++r)
849 result[r*col+c] = m[r*col+it];
854 return (sign*matrix(row, col, std::move(result)).determinant_minor()).normal();
856 return sign*matrix(row, col, std::move(result)).determinant_minor();
862 /** Trace of a matrix. The result is normalized if it is in some quotient
863 * field and expanded only otherwise. This implies that the trace of the
864 * symbolic 2x2 matrix [[a/(a-b),x],[y,b/(b-a)]] is recognized to be unity.
866 * @return the sum of diagonal elements
867 * @exception logic_error (matrix not square) */
868 ex matrix::trace() const
871 throw (std::logic_error("matrix::trace(): matrix not square"));
874 for (unsigned r=0; r<col; ++r)
877 if (tr.info(info_flags::rational_function) &&
878 !tr.info(info_flags::crational_polynomial))
885 /** Characteristic Polynomial. Following mathematica notation the
886 * characteristic polynomial of a matrix M is defined as the determinant of
887 * (M - lambda * 1) where 1 stands for the unit matrix of the same dimension
888 * as M. Note that some CASs define it with a sign inside the determinant
889 * which gives rise to an overall sign if the dimension is odd. This method
890 * returns the characteristic polynomial collected in powers of lambda as a
893 * @return characteristic polynomial as new expression
894 * @exception logic_error (matrix not square)
895 * @see matrix::determinant() */
896 ex matrix::charpoly(const ex & lambda) const
899 throw (std::logic_error("matrix::charpoly(): matrix not square"));
901 bool numeric_flag = true;
903 if (!r.info(info_flags::numeric)) {
904 numeric_flag = false;
909 // The pure numeric case is traditionally rather common. Hence, it is
910 // trapped and we use Leverrier's algorithm which goes as row^3 for
911 // every coefficient. The expensive part is the matrix multiplication.
916 ex poly = power(lambda, row) - c*power(lambda, row-1);
917 for (unsigned i=1; i<row; ++i) {
918 for (unsigned j=0; j<row; ++j)
921 c = B.trace() / ex(i+1);
922 poly -= c*power(lambda, row-i-1);
932 for (unsigned r=0; r<col; ++r)
933 M.m[r*col+r] -= lambda;
935 return M.determinant().collect(lambda);
940 /** Inverse of this matrix.
942 * @return the inverted matrix
943 * @exception logic_error (matrix not square)
944 * @exception runtime_error (singular matrix) */
945 matrix matrix::inverse() const
948 throw (std::logic_error("matrix::inverse(): matrix not square"));
950 // This routine actually doesn't do anything fancy at all. We compute the
951 // inverse of the matrix A by solving the system A * A^{-1} == Id.
953 // First populate the identity matrix supposed to become the right hand side.
954 matrix identity(row,col);
955 for (unsigned i=0; i<row; ++i)
956 identity(i,i) = _ex1;
958 // Populate a dummy matrix of variables, just because of compatibility with
959 // matrix::solve() which wants this (for compatibility with under-determined
960 // systems of equations).
961 matrix vars(row,col);
962 for (unsigned r=0; r<row; ++r)
963 for (unsigned c=0; c<col; ++c)
964 vars(r,c) = symbol();
968 sol = this->solve(vars,identity);
969 } catch (const std::runtime_error & e) {
970 if (e.what()==std::string("matrix::solve(): inconsistent linear system"))
971 throw (std::runtime_error("matrix::inverse(): singular matrix"));
979 /** Solve a linear system consisting of a m x n matrix and a m x p right hand
980 * side by applying an elimination scheme to the augmented matrix.
982 * @param vars n x p matrix, all elements must be symbols
983 * @param rhs m x p matrix
984 * @param algo selects the solving algorithm
985 * @return n x p solution matrix
986 * @exception logic_error (incompatible matrices)
987 * @exception invalid_argument (1st argument must be matrix of symbols)
988 * @exception runtime_error (inconsistent linear system)
990 matrix matrix::solve(const matrix & vars,
994 const unsigned m = this->rows();
995 const unsigned n = this->cols();
996 const unsigned p = rhs.cols();
999 if ((rhs.rows() != m) || (vars.rows() != n) || (vars.col != p))
1000 throw (std::logic_error("matrix::solve(): incompatible matrices"));
1001 for (unsigned ro=0; ro<n; ++ro)
1002 for (unsigned co=0; co<p; ++co)
1003 if (!vars(ro,co).info(info_flags::symbol))
1004 throw (std::invalid_argument("matrix::solve(): 1st argument must be matrix of symbols"));
1006 // build the augmented matrix of *this with rhs attached to the right
1008 for (unsigned r=0; r<m; ++r) {
1009 for (unsigned c=0; c<n; ++c)
1010 aug.m[r*(n+p)+c] = this->m[r*n+c];
1011 for (unsigned c=0; c<p; ++c)
1012 aug.m[r*(n+p)+c+n] = rhs.m[r*p+c];
1015 // Gather some statistical information about the augmented matrix:
1016 bool numeric_flag = true;
1017 for (auto & r : aug.m) {
1018 if (!r.info(info_flags::numeric)) {
1019 numeric_flag = false;
1024 // Here is the heuristics in case this routine has to decide:
1025 if (algo == solve_algo::automatic) {
1026 // Bareiss (fraction-free) elimination is generally a good guess:
1027 algo = solve_algo::bareiss;
1028 // For m<3, Bareiss elimination is equivalent to division free
1029 // elimination but has more logistic overhead
1031 algo = solve_algo::divfree;
1032 // This overrides any prior decisions.
1034 algo = solve_algo::gauss;
1037 // Eliminate the augmented matrix:
1039 case solve_algo::gauss:
1040 aug.gauss_elimination();
1042 case solve_algo::divfree:
1043 aug.division_free_elimination();
1045 case solve_algo::bareiss:
1047 aug.fraction_free_elimination();
1050 // assemble the solution matrix:
1052 for (unsigned co=0; co<p; ++co) {
1053 unsigned last_assigned_sol = n+1;
1054 for (int r=m-1; r>=0; --r) {
1055 unsigned fnz = 1; // first non-zero in row
1056 while ((fnz<=n) && (aug.m[r*(n+p)+(fnz-1)].is_zero()))
1059 // row consists only of zeros, corresponding rhs must be 0, too
1060 if (!aug.m[r*(n+p)+n+co].is_zero()) {
1061 throw (std::runtime_error("matrix::solve(): inconsistent linear system"));
1064 // assign solutions for vars between fnz+1 and
1065 // last_assigned_sol-1: free parameters
1066 for (unsigned c=fnz; c<last_assigned_sol-1; ++c)
1067 sol(c,co) = vars.m[c*p+co];
1068 ex e = aug.m[r*(n+p)+n+co];
1069 for (unsigned c=fnz; c<n; ++c)
1070 e -= aug.m[r*(n+p)+c]*sol.m[c*p+co];
1071 sol(fnz-1,co) = (e/(aug.m[r*(n+p)+(fnz-1)])).normal();
1072 last_assigned_sol = fnz;
1075 // assign solutions for vars between 1 and
1076 // last_assigned_sol-1: free parameters
1077 for (unsigned ro=0; ro<last_assigned_sol-1; ++ro)
1078 sol(ro,co) = vars(ro,co);
1085 /** Compute the rank of this matrix. */
1086 unsigned matrix::rank() const
1089 // Transform this matrix into upper echelon form and then count the
1090 // number of non-zero rows.
1092 GINAC_ASSERT(row*col==m.capacity());
1094 // Actually, any elimination scheme will do since we are only
1095 // interested in the echelon matrix' zeros.
1096 matrix to_eliminate = *this;
1097 to_eliminate.fraction_free_elimination();
1099 unsigned r = row*col; // index of last non-zero element
1101 if (!to_eliminate.m[r].is_zero())
1110 /** Recursive determinant for small matrices having at least one symbolic
1111 * entry. The basic algorithm, known as Laplace-expansion, is enhanced by
1112 * some bookkeeping to avoid calculation of the same submatrices ("minors")
1113 * more than once. According to W.M.Gentleman and S.C.Johnson this algorithm
1114 * is better than elimination schemes for matrices of sparse multivariate
1115 * polynomials and also for matrices of dense univariate polynomials if the
1116 * matrix' dimension is larger than 7.
1118 * @return the determinant as a new expression (in expanded form)
1119 * @see matrix::determinant() */
1120 ex matrix::determinant_minor() const
1122 // for small matrices the algorithm does not make any sense:
1123 const unsigned n = this->cols();
1125 return m[0].expand();
1127 return (m[0]*m[3]-m[2]*m[1]).expand();
1129 return (m[0]*m[4]*m[8]-m[0]*m[5]*m[7]-
1130 m[1]*m[3]*m[8]+m[2]*m[3]*m[7]+
1131 m[1]*m[5]*m[6]-m[2]*m[4]*m[6]).expand();
1133 // This algorithm can best be understood by looking at a naive
1134 // implementation of Laplace-expansion, like this one:
1136 // matrix minorM(this->rows()-1,this->cols()-1);
1137 // for (unsigned r1=0; r1<this->rows(); ++r1) {
1138 // // shortcut if element(r1,0) vanishes
1139 // if (m[r1*col].is_zero())
1141 // // assemble the minor matrix
1142 // for (unsigned r=0; r<minorM.rows(); ++r) {
1143 // for (unsigned c=0; c<minorM.cols(); ++c) {
1145 // minorM(r,c) = m[r*col+c+1];
1147 // minorM(r,c) = m[(r+1)*col+c+1];
1150 // // recurse down and care for sign:
1152 // det -= m[r1*col] * minorM.determinant_minor();
1154 // det += m[r1*col] * minorM.determinant_minor();
1156 // return det.expand();
1157 // What happens is that while proceeding down many of the minors are
1158 // computed more than once. In particular, there are binomial(n,k)
1159 // kxk minors and each one is computed factorial(n-k) times. Therefore
1160 // it is reasonable to store the results of the minors. We proceed from
1161 // right to left. At each column c we only need to retrieve the minors
1162 // calculated in step c-1. We therefore only have to store at most
1163 // 2*binomial(n,n/2) minors.
1165 // Unique flipper counter for partitioning into minors
1166 std::vector<unsigned> Pkey;
1168 // key for minor determinant (a subpartition of Pkey)
1169 std::vector<unsigned> Mkey;
1171 // we store our subminors in maps, keys being the rows they arise from
1172 typedef std::map<std::vector<unsigned>,class ex> Rmap;
1173 typedef std::map<std::vector<unsigned>,class ex>::value_type Rmap_value;
1177 // initialize A with last column:
1178 for (unsigned r=0; r<n; ++r) {
1179 Pkey.erase(Pkey.begin(),Pkey.end());
1181 A.insert(Rmap_value(Pkey,m[n*(r+1)-1]));
1183 // proceed from right to left through matrix
1184 for (int c=n-2; c>=0; --c) {
1185 Pkey.erase(Pkey.begin(),Pkey.end()); // don't change capacity
1186 Mkey.erase(Mkey.begin(),Mkey.end());
1187 for (unsigned i=0; i<n-c; ++i)
1189 unsigned fc = 0; // controls logic for our strange flipper counter
1192 for (unsigned r=0; r<n-c; ++r) {
1193 // maybe there is nothing to do?
1194 if (m[Pkey[r]*n+c].is_zero())
1196 // create the sorted key for all possible minors
1197 Mkey.erase(Mkey.begin(),Mkey.end());
1198 for (unsigned i=0; i<n-c; ++i)
1200 Mkey.push_back(Pkey[i]);
1201 // Fetch the minors and compute the new determinant
1203 det -= m[Pkey[r]*n+c]*A[Mkey];
1205 det += m[Pkey[r]*n+c]*A[Mkey];
1207 // prevent build-up of deep nesting of expressions saves time:
1209 // store the new determinant at its place in B:
1211 B.insert(Rmap_value(Pkey,det));
1212 // increment our strange flipper counter
1213 for (fc=n-c; fc>0; --fc) {
1215 if (Pkey[fc-1]<fc+c)
1219 for (unsigned j=fc; j<n-c; ++j)
1220 Pkey[j] = Pkey[j-1]+1;
1222 // next column, clear B and change the role of A and B:
1230 /** Perform the steps of an ordinary Gaussian elimination to bring the m x n
1231 * matrix into an upper echelon form. The algorithm is ok for matrices
1232 * with numeric coefficients but quite unsuited for symbolic matrices.
1234 * @param det may be set to true to save a lot of space if one is only
1235 * interested in the diagonal elements (i.e. for calculating determinants).
1236 * The others are set to zero in this case.
1237 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1238 * number of rows was swapped and 0 if the matrix is singular. */
1239 int matrix::gauss_elimination(const bool det)
1241 ensure_if_modifiable();
1242 const unsigned m = this->rows();
1243 const unsigned n = this->cols();
1244 GINAC_ASSERT(!det || n==m);
1248 for (unsigned c0=0; c0<n && r0<m-1; ++c0) {
1249 int indx = pivot(r0, c0, true);
1253 return 0; // leaves *this in a messy state
1258 for (unsigned r2=r0+1; r2<m; ++r2) {
1259 if (!this->m[r2*n+c0].is_zero()) {
1260 // yes, there is something to do in this row
1261 ex piv = this->m[r2*n+c0] / this->m[r0*n+c0];
1262 for (unsigned c=c0+1; c<n; ++c) {
1263 this->m[r2*n+c] -= piv * this->m[r0*n+c];
1264 if (!this->m[r2*n+c].info(info_flags::numeric))
1265 this->m[r2*n+c] = this->m[r2*n+c].normal();
1268 // fill up left hand side with zeros
1269 for (unsigned c=r0; c<=c0; ++c)
1270 this->m[r2*n+c] = _ex0;
1273 // save space by deleting no longer needed elements
1274 for (unsigned c=r0+1; c<n; ++c)
1275 this->m[r0*n+c] = _ex0;
1280 // clear remaining rows
1281 for (unsigned r=r0+1; r<m; ++r) {
1282 for (unsigned c=0; c<n; ++c)
1283 this->m[r*n+c] = _ex0;
1290 /** Perform the steps of division free elimination to bring the m x n matrix
1291 * into an upper echelon form.
1293 * @param det may be set to true to save a lot of space if one is only
1294 * interested in the diagonal elements (i.e. for calculating determinants).
1295 * The others are set to zero in this case.
1296 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1297 * number of rows was swapped and 0 if the matrix is singular. */
1298 int matrix::division_free_elimination(const bool det)
1300 ensure_if_modifiable();
1301 const unsigned m = this->rows();
1302 const unsigned n = this->cols();
1303 GINAC_ASSERT(!det || n==m);
1307 for (unsigned c0=0; c0<n && r0<m-1; ++c0) {
1308 int indx = pivot(r0, c0, true);
1312 return 0; // leaves *this in a messy state
1317 for (unsigned r2=r0+1; r2<m; ++r2) {
1318 for (unsigned c=c0+1; c<n; ++c)
1319 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();
1320 // fill up left hand side with zeros
1321 for (unsigned c=r0; c<=c0; ++c)
1322 this->m[r2*n+c] = _ex0;
1325 // save space by deleting no longer needed elements
1326 for (unsigned c=r0+1; c<n; ++c)
1327 this->m[r0*n+c] = _ex0;
1332 // clear remaining rows
1333 for (unsigned r=r0+1; r<m; ++r) {
1334 for (unsigned c=0; c<n; ++c)
1335 this->m[r*n+c] = _ex0;
1342 /** Perform the steps of Bareiss' one-step fraction free elimination to bring
1343 * the matrix into an upper echelon form. Fraction free elimination means
1344 * that divide is used straightforwardly, without computing GCDs first. This
1345 * is possible, since we know the divisor at each step.
1347 * @param det may be set to true to save a lot of space if one is only
1348 * interested in the last element (i.e. for calculating determinants). The
1349 * others are set to zero in this case.
1350 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1351 * number of rows was swapped and 0 if the matrix is singular. */
1352 int matrix::fraction_free_elimination(const bool det)
1355 // (single-step fraction free elimination scheme, already known to Jordan)
1357 // Usual division-free elimination sets m[0](r,c) = m(r,c) and then sets
1358 // m[k+1](r,c) = m[k](k,k) * m[k](r,c) - m[k](r,k) * m[k](k,c).
1360 // Bareiss (fraction-free) elimination in addition divides that element
1361 // by m[k-1](k-1,k-1) for k>1, where it can be shown by means of the
1362 // Sylvester identity that this really divides m[k+1](r,c).
1364 // We also allow rational functions where the original prove still holds.
1365 // However, we must care for numerator and denominator separately and
1366 // "manually" work in the integral domains because of subtle cancellations
1367 // (see below). This blows up the bookkeeping a bit and the formula has
1368 // to be modified to expand like this (N{x} stands for numerator of x,
1369 // D{x} for denominator of x):
1370 // 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)}
1371 // -N{m[k](r,k)}*N{m[k](k,c)}*D{m[k](k,k)}*D{m[k](r,c)}
1372 // 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)}
1373 // where for k>1 we now divide N{m[k+1](r,c)} by
1374 // N{m[k-1](k-1,k-1)}
1375 // and D{m[k+1](r,c)} by
1376 // D{m[k-1](k-1,k-1)}.
1378 ensure_if_modifiable();
1379 const unsigned m = this->rows();
1380 const unsigned n = this->cols();
1381 GINAC_ASSERT(!det || n==m);
1390 // We populate temporary matrices to subsequently operate on. There is
1391 // one holding numerators and another holding denominators of entries.
1392 // This is a must since the evaluator (or even earlier mul's constructor)
1393 // might cancel some trivial element which causes divide() to fail. The
1394 // elements are normalized first (yes, even though this algorithm doesn't
1395 // need GCDs) since the elements of *this might be unnormalized, which
1396 // makes things more complicated than they need to be.
1397 matrix tmp_n(*this);
1398 matrix tmp_d(m,n); // for denominators, if needed
1399 exmap srl; // symbol replacement list
1400 auto tmp_n_it = tmp_n.m.begin(), tmp_d_it = tmp_d.m.begin();
1401 for (auto & it : this->m) {
1402 ex nd = it.normal().to_rational(srl).numer_denom();
1403 *tmp_n_it++ = nd.op(0);
1404 *tmp_d_it++ = nd.op(1);
1408 for (unsigned c0=0; c0<n && r0<m-1; ++c0) {
1409 // When trying to find a pivot, we should try a bit harder than expand().
1410 // Searching the first non-zero element in-place here instead of calling
1411 // pivot() allows us to do no more substitutions and back-substitutions
1412 // than are actually necessary.
1415 (tmp_n[indx*n+c0].subs(srl, subs_options::no_pattern).expand().is_zero()))
1418 // all elements in column c0 below row r0 vanish
1424 // Matrix needs pivoting, swap rows r0 and indx of tmp_n and tmp_d.
1426 for (unsigned c=c0; c<n; ++c) {
1427 tmp_n.m[n*indx+c].swap(tmp_n.m[n*r0+c]);
1428 tmp_d.m[n*indx+c].swap(tmp_d.m[n*r0+c]);
1431 for (unsigned r2=r0+1; r2<m; ++r2) {
1432 for (unsigned c=c0+1; c<n; ++c) {
1433 dividend_n = (tmp_n.m[r0*n+c0]*tmp_n.m[r2*n+c]*
1434 tmp_d.m[r2*n+c0]*tmp_d.m[r0*n+c]
1435 -tmp_n.m[r2*n+c0]*tmp_n.m[r0*n+c]*
1436 tmp_d.m[r0*n+c0]*tmp_d.m[r2*n+c]).expand();
1437 dividend_d = (tmp_d.m[r2*n+c0]*tmp_d.m[r0*n+c]*
1438 tmp_d.m[r0*n+c0]*tmp_d.m[r2*n+c]).expand();
1439 bool check = divide(dividend_n, divisor_n,
1440 tmp_n.m[r2*n+c], true);
1441 check &= divide(dividend_d, divisor_d,
1442 tmp_d.m[r2*n+c], true);
1443 GINAC_ASSERT(check);
1445 // fill up left hand side with zeros
1446 for (unsigned c=r0; c<=c0; ++c)
1447 tmp_n.m[r2*n+c] = _ex0;
1449 if (c0<n && r0<m-1) {
1450 // compute next iteration's divisor
1451 divisor_n = tmp_n.m[r0*n+c0].expand();
1452 divisor_d = tmp_d.m[r0*n+c0].expand();
1454 // save space by deleting no longer needed elements
1455 for (unsigned c=0; c<n; ++c) {
1456 tmp_n.m[r0*n+c] = _ex0;
1457 tmp_d.m[r0*n+c] = _ex1;
1464 // clear remaining rows
1465 for (unsigned r=r0+1; r<m; ++r) {
1466 for (unsigned c=0; c<n; ++c)
1467 tmp_n.m[r*n+c] = _ex0;
1470 // repopulate *this matrix:
1471 tmp_n_it = tmp_n.m.begin();
1472 tmp_d_it = tmp_d.m.begin();
1473 for (auto & it : this->m)
1474 it = ((*tmp_n_it++)/(*tmp_d_it++)).subs(srl, subs_options::no_pattern);
1480 /** Partial pivoting method for matrix elimination schemes.
1481 * Usual pivoting (symbolic==false) returns the index to the element with the
1482 * largest absolute value in column ro and swaps the current row with the one
1483 * where the element was found. With (symbolic==true) it does the same thing
1484 * with the first non-zero element.
1486 * @param ro is the row from where to begin
1487 * @param co is the column to be inspected
1488 * @param symbolic signal if we want the first non-zero element to be pivoted
1489 * (true) or the one with the largest absolute value (false).
1490 * @return 0 if no interchange occurred, -1 if all are zero (usually signaling
1491 * a degeneracy) and positive integer k means that rows ro and k were swapped.
1493 int matrix::pivot(unsigned ro, unsigned co, bool symbolic)
1497 // search first non-zero element in column co beginning at row ro
1498 while ((k<row) && (this->m[k*col+co].expand().is_zero()))
1501 // search largest element in column co beginning at row ro
1502 GINAC_ASSERT(is_exactly_a<numeric>(this->m[k*col+co]));
1503 unsigned kmax = k+1;
1504 numeric mmax = abs(ex_to<numeric>(m[kmax*col+co]));
1506 GINAC_ASSERT(is_exactly_a<numeric>(this->m[kmax*col+co]));
1507 numeric tmp = ex_to<numeric>(this->m[kmax*col+co]);
1508 if (abs(tmp) > mmax) {
1514 if (!mmax.is_zero())
1518 // all elements in column co below row ro vanish
1521 // matrix needs no pivoting
1523 // matrix needs pivoting, so swap rows k and ro
1524 ensure_if_modifiable();
1525 for (unsigned c=0; c<col; ++c)
1526 this->m[k*col+c].swap(this->m[ro*col+c]);
1531 /** Function to check that all elements of the matrix are zero.
1533 bool matrix::is_zero_matrix() const
1541 ex lst_to_matrix(const lst & l)
1543 // Find number of rows and columns
1544 size_t rows = l.nops(), cols = 0;
1545 for (auto & itr : l) {
1546 if (!is_a<lst>(itr))
1547 throw (std::invalid_argument("lst_to_matrix: argument must be a list of lists"));
1548 if (itr.nops() > cols)
1552 // Allocate and fill matrix
1553 matrix & M = dynallocate<matrix>(rows, cols);
1556 for (auto & itr : l) {
1558 for (auto & itc : ex_to<lst>(itr)) {
1568 ex diag_matrix(const lst & l)
1570 size_t dim = l.nops();
1572 // Allocate and fill matrix
1573 matrix & M = dynallocate<matrix>(dim, dim);
1576 for (auto & it : l) {
1584 ex diag_matrix(std::initializer_list<ex> l)
1586 size_t dim = l.size();
1588 // Allocate and fill matrix
1589 matrix & M = dynallocate<matrix>(dim, dim);
1592 for (auto & it : l) {
1600 ex unit_matrix(unsigned r, unsigned c)
1602 matrix & Id = dynallocate<matrix>(r, c);
1603 Id.setflag(status_flags::evaluated);
1604 for (unsigned i=0; i<r && i<c; i++)
1610 ex symbolic_matrix(unsigned r, unsigned c, const std::string & base_name, const std::string & tex_base_name)
1612 matrix & M = dynallocate<matrix>(r, c);
1613 M.setflag(status_flags::evaluated);
1615 bool long_format = (r > 10 || c > 10);
1616 bool single_row = (r == 1 || c == 1);
1618 for (unsigned i=0; i<r; i++) {
1619 for (unsigned j=0; j<c; j++) {
1620 std::ostringstream s1, s2;
1622 s2 << tex_base_name << "_{";
1633 s1 << '_' << i << '_' << j;
1634 s2 << i << ';' << j << "}";
1637 s2 << i << j << '}';
1640 M(i, j) = symbol(s1.str(), s2.str());
1647 ex reduced_matrix(const matrix& m, unsigned r, unsigned c)
1649 if (r+1>m.rows() || c+1>m.cols() || m.cols()<2 || m.rows()<2)
1650 throw std::runtime_error("minor_matrix(): index out of bounds");
1652 const unsigned rows = m.rows()-1;
1653 const unsigned cols = m.cols()-1;
1654 matrix & M = dynallocate<matrix>(rows, cols);
1655 M.setflag(status_flags::evaluated);
1667 M(ro2,co2) = m(ro, co);
1678 ex sub_matrix(const matrix&m, unsigned r, unsigned nr, unsigned c, unsigned nc)
1680 if (r+nr>m.rows() || c+nc>m.cols())
1681 throw std::runtime_error("sub_matrix(): index out of bounds");
1683 matrix & M = dynallocate<matrix>(nr, nc);
1684 M.setflag(status_flags::evaluated);
1686 for (unsigned ro=0; ro<nr; ++ro) {
1687 for (unsigned co=0; co<nc; ++co) {
1688 M(ro,co) = m(ro+r,co+c);
1695 } // namespace GiNaC