3 * Implementation of symbolic matrices */
6 * GiNaC Copyright (C) 1999-2015 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);
78 /** Ctor from representation, for internal use only. */
79 matrix::matrix(unsigned r, unsigned c, const exvector & m2)
80 : row(r), col(c), m(m2)
82 setflag(status_flags::not_shareable);
85 /** Construct matrix from (flat) list of elements. If the list has fewer
86 * elements than the matrix, the remaining matrix elements are set to zero.
87 * If the list has more elements than the matrix, the excessive elements are
89 matrix::matrix(unsigned r, unsigned c, const lst & l)
90 : row(r), col(c), m(r*c, _ex0)
92 setflag(status_flags::not_shareable);
99 break; // matrix smaller than list: throw away excessive elements
109 void matrix::read_archive(const archive_node &n, lst &sym_lst)
111 inherited::read_archive(n, sym_lst);
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 // XXX: default ctor inserts a zero element, we need to erase it here.
118 auto first = n.find_first("m");
119 auto last = n.find_last("m");
121 for (auto i=first; i != last; ++i) {
123 n.find_ex_by_loc(i, e, sym_lst);
127 GINAC_BIND_UNARCHIVER(matrix);
129 void matrix::archive(archive_node &n) const
131 inherited::archive(n);
132 n.add_unsigned("row", row);
133 n.add_unsigned("col", col);
140 // functions overriding virtual functions from base classes
145 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
147 for (unsigned ro=0; ro<row; ++ro) {
149 for (unsigned co=0; co<col; ++co) {
150 m[ro*col+co].print(c);
161 void matrix::do_print(const print_context & c, unsigned level) const
164 print_elements(c, "[", "]", ",", ",");
168 void matrix::do_print_latex(const print_latex & c, unsigned level) const
170 c.s << "\\left(\\begin{array}{" << std::string(col,'c') << "}";
171 print_elements(c, "", "", "\\\\", "&");
172 c.s << "\\end{array}\\right)";
175 void matrix::do_print_python_repr(const print_python_repr & c, unsigned level) const
177 c.s << class_name() << '(';
178 print_elements(c, "[", "]", ",", ",");
182 /** nops is defined to be rows x columns. */
183 size_t matrix::nops() const
185 return static_cast<size_t>(row) * static_cast<size_t>(col);
188 /** returns matrix entry at position (i/col, i%col). */
189 ex matrix::op(size_t i) const
191 GINAC_ASSERT(i<nops());
196 /** returns writable matrix entry at position (i/col, i%col). */
197 ex & matrix::let_op(size_t i)
199 GINAC_ASSERT(i<nops());
201 ensure_if_modifiable();
205 /** Evaluate matrix entry by entry. */
206 ex matrix::eval(int level) const
208 // check if we have to do anything at all
209 if ((level==1)&&(flags & status_flags::evaluated))
213 if (level == -max_recursion_level)
214 throw (std::runtime_error("matrix::eval(): recursion limit exceeded"));
216 // eval() entry by entry
217 exvector m2(row*col);
219 for (unsigned r=0; r<row; ++r)
220 for (unsigned c=0; c<col; ++c)
221 m2[r*col+c] = m[r*col+c].eval(level);
223 return (new matrix(row, col, std::move(m2)))->setflag(status_flags::dynallocated |
224 status_flags::evaluated);
227 ex matrix::subs(const exmap & mp, unsigned options) const
229 exvector m2(row * col);
230 for (unsigned r=0; r<row; ++r)
231 for (unsigned c=0; c<col; ++c)
232 m2[r*col+c] = m[r*col+c].subs(mp, options);
234 return matrix(row, col, std::move(m2)).subs_one_level(mp, options);
237 /** Complex conjugate every matrix entry. */
238 ex matrix::conjugate() const
240 std::unique_ptr<exvector> ev(nullptr);
241 for (auto i=m.begin(); i!=m.end(); ++i) {
242 ex x = i->conjugate();
247 if (are_ex_trivially_equal(x, *i)) {
250 ev.reset(new exvector);
251 ev->reserve(m.size());
252 for (auto j=m.begin(); j!=i; ++j) {
258 return matrix(row, col, std::move(*ev));
263 ex matrix::real_part() const
268 v.push_back(i.real_part());
269 return matrix(row, col, std::move(v));
272 ex matrix::imag_part() const
277 v.push_back(i.imag_part());
278 return matrix(row, col, std::move(v));
283 int matrix::compare_same_type(const basic & other) const
285 GINAC_ASSERT(is_exactly_a<matrix>(other));
286 const matrix &o = static_cast<const matrix &>(other);
288 // compare number of rows
290 return row < o.rows() ? -1 : 1;
292 // compare number of columns
294 return col < o.cols() ? -1 : 1;
296 // equal number of rows and columns, compare individual elements
298 for (unsigned r=0; r<row; ++r) {
299 for (unsigned c=0; c<col; ++c) {
300 cmpval = ((*this)(r,c)).compare(o(r,c));
301 if (cmpval!=0) return cmpval;
304 // all elements are equal => matrices are equal;
308 bool matrix::match_same_type(const basic & other) const
310 GINAC_ASSERT(is_exactly_a<matrix>(other));
311 const matrix & o = static_cast<const matrix &>(other);
313 // The number of rows and columns must be the same. This is necessary to
314 // prevent a 2x3 matrix from matching a 3x2 one.
315 return row == o.rows() && col == o.cols();
318 /** Automatic symbolic evaluation of an indexed matrix. */
319 ex matrix::eval_indexed(const basic & i) const
321 GINAC_ASSERT(is_a<indexed>(i));
322 GINAC_ASSERT(is_a<matrix>(i.op(0)));
324 bool all_indices_unsigned = static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint);
329 // One index, must be one-dimensional vector
330 if (row != 1 && col != 1)
331 throw (std::runtime_error("matrix::eval_indexed(): vector must have exactly 1 index"));
333 const idx & i1 = ex_to<idx>(i.op(1));
338 if (!i1.get_dim().is_equal(row))
339 throw (std::runtime_error("matrix::eval_indexed(): dimension of index must match number of vector elements"));
341 // Index numeric -> return vector element
342 if (all_indices_unsigned) {
343 unsigned n1 = ex_to<numeric>(i1.get_value()).to_int();
345 throw (std::runtime_error("matrix::eval_indexed(): value of index exceeds number of vector elements"));
346 return (*this)(n1, 0);
352 if (!i1.get_dim().is_equal(col))
353 throw (std::runtime_error("matrix::eval_indexed(): dimension of index must match number of vector elements"));
355 // Index numeric -> return vector element
356 if (all_indices_unsigned) {
357 unsigned n1 = ex_to<numeric>(i1.get_value()).to_int();
359 throw (std::runtime_error("matrix::eval_indexed(): value of index exceeds number of vector elements"));
360 return (*this)(0, n1);
364 } else if (i.nops() == 3) {
367 const idx & i1 = ex_to<idx>(i.op(1));
368 const idx & i2 = ex_to<idx>(i.op(2));
370 if (!i1.get_dim().is_equal(row))
371 throw (std::runtime_error("matrix::eval_indexed(): dimension of first index must match number of rows"));
372 if (!i2.get_dim().is_equal(col))
373 throw (std::runtime_error("matrix::eval_indexed(): dimension of second index must match number of columns"));
375 // Pair of dummy indices -> compute trace
376 if (is_dummy_pair(i1, i2))
379 // Both indices numeric -> return matrix element
380 if (all_indices_unsigned) {
381 unsigned n1 = ex_to<numeric>(i1.get_value()).to_int(), n2 = ex_to<numeric>(i2.get_value()).to_int();
383 throw (std::runtime_error("matrix::eval_indexed(): value of first index exceeds number of rows"));
385 throw (std::runtime_error("matrix::eval_indexed(): value of second index exceeds number of columns"));
386 return (*this)(n1, n2);
390 throw (std::runtime_error("matrix::eval_indexed(): matrix must have exactly 2 indices"));
395 /** Sum of two indexed matrices. */
396 ex matrix::add_indexed(const ex & self, const ex & other) const
398 GINAC_ASSERT(is_a<indexed>(self));
399 GINAC_ASSERT(is_a<matrix>(self.op(0)));
400 GINAC_ASSERT(is_a<indexed>(other));
401 GINAC_ASSERT(self.nops() == 2 || self.nops() == 3);
403 // Only add two matrices
404 if (is_a<matrix>(other.op(0))) {
405 GINAC_ASSERT(other.nops() == 2 || other.nops() == 3);
407 const matrix &self_matrix = ex_to<matrix>(self.op(0));
408 const matrix &other_matrix = ex_to<matrix>(other.op(0));
410 if (self.nops() == 2 && other.nops() == 2) { // vector + vector
412 if (self_matrix.row == other_matrix.row)
413 return indexed(self_matrix.add(other_matrix), self.op(1));
414 else if (self_matrix.row == other_matrix.col)
415 return indexed(self_matrix.add(other_matrix.transpose()), self.op(1));
417 } else if (self.nops() == 3 && other.nops() == 3) { // matrix + matrix
419 if (self.op(1).is_equal(other.op(1)) && self.op(2).is_equal(other.op(2)))
420 return indexed(self_matrix.add(other_matrix), self.op(1), self.op(2));
421 else if (self.op(1).is_equal(other.op(2)) && self.op(2).is_equal(other.op(1)))
422 return indexed(self_matrix.add(other_matrix.transpose()), self.op(1), self.op(2));
427 // Don't know what to do, return unevaluated sum
431 /** Product of an indexed matrix with a number. */
432 ex matrix::scalar_mul_indexed(const ex & self, const numeric & other) const
434 GINAC_ASSERT(is_a<indexed>(self));
435 GINAC_ASSERT(is_a<matrix>(self.op(0)));
436 GINAC_ASSERT(self.nops() == 2 || self.nops() == 3);
438 const matrix &self_matrix = ex_to<matrix>(self.op(0));
440 if (self.nops() == 2)
441 return indexed(self_matrix.mul(other), self.op(1));
442 else // self.nops() == 3
443 return indexed(self_matrix.mul(other), self.op(1), self.op(2));
446 /** Contraction of an indexed matrix with something else. */
447 bool matrix::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
449 GINAC_ASSERT(is_a<indexed>(*self));
450 GINAC_ASSERT(is_a<indexed>(*other));
451 GINAC_ASSERT(self->nops() == 2 || self->nops() == 3);
452 GINAC_ASSERT(is_a<matrix>(self->op(0)));
454 // Only contract with other matrices
455 if (!is_a<matrix>(other->op(0)))
458 GINAC_ASSERT(other->nops() == 2 || other->nops() == 3);
460 const matrix &self_matrix = ex_to<matrix>(self->op(0));
461 const matrix &other_matrix = ex_to<matrix>(other->op(0));
463 if (self->nops() == 2) {
465 if (other->nops() == 2) { // vector * vector (scalar product)
467 if (self_matrix.col == 1) {
468 if (other_matrix.col == 1) {
469 // Column vector * column vector, transpose first vector
470 *self = self_matrix.transpose().mul(other_matrix)(0, 0);
472 // Column vector * row vector, swap factors
473 *self = other_matrix.mul(self_matrix)(0, 0);
476 if (other_matrix.col == 1) {
477 // Row vector * column vector, perfect
478 *self = self_matrix.mul(other_matrix)(0, 0);
480 // Row vector * row vector, transpose second vector
481 *self = self_matrix.mul(other_matrix.transpose())(0, 0);
487 } else { // vector * matrix
489 // B_i * A_ij = (B*A)_j (B is row vector)
490 if (is_dummy_pair(self->op(1), other->op(1))) {
491 if (self_matrix.row == 1)
492 *self = indexed(self_matrix.mul(other_matrix), other->op(2));
494 *self = indexed(self_matrix.transpose().mul(other_matrix), other->op(2));
499 // B_j * A_ij = (A*B)_i (B is column vector)
500 if (is_dummy_pair(self->op(1), other->op(2))) {
501 if (self_matrix.col == 1)
502 *self = indexed(other_matrix.mul(self_matrix), other->op(1));
504 *self = indexed(other_matrix.mul(self_matrix.transpose()), other->op(1));
510 } else if (other->nops() == 3) { // matrix * matrix
512 // A_ij * B_jk = (A*B)_ik
513 if (is_dummy_pair(self->op(2), other->op(1))) {
514 *self = indexed(self_matrix.mul(other_matrix), self->op(1), other->op(2));
519 // A_ij * B_kj = (A*Btrans)_ik
520 if (is_dummy_pair(self->op(2), other->op(2))) {
521 *self = indexed(self_matrix.mul(other_matrix.transpose()), self->op(1), other->op(1));
526 // A_ji * B_jk = (Atrans*B)_ik
527 if (is_dummy_pair(self->op(1), other->op(1))) {
528 *self = indexed(self_matrix.transpose().mul(other_matrix), self->op(2), other->op(2));
533 // A_ji * B_kj = (B*A)_ki
534 if (is_dummy_pair(self->op(1), other->op(2))) {
535 *self = indexed(other_matrix.mul(self_matrix), other->op(1), self->op(2));
546 // non-virtual functions in this class
553 * @exception logic_error (incompatible matrices) */
554 matrix matrix::add(const matrix & other) const
556 if (col != other.col || row != other.row)
557 throw std::logic_error("matrix::add(): incompatible matrices");
559 exvector sum(this->m);
560 auto ci = other.m.begin();
564 return matrix(row, col, std::move(sum));
568 /** Difference of matrices.
570 * @exception logic_error (incompatible matrices) */
571 matrix matrix::sub(const matrix & other) const
573 if (col != other.col || row != other.row)
574 throw std::logic_error("matrix::sub(): incompatible matrices");
576 exvector dif(this->m);
577 auto ci = other.m.begin();
581 return matrix(row, col, std::move(dif));
585 /** Product of matrices.
587 * @exception logic_error (incompatible matrices) */
588 matrix matrix::mul(const matrix & other) const
590 if (this->cols() != other.rows())
591 throw std::logic_error("matrix::mul(): incompatible matrices");
593 exvector prod(this->rows()*other.cols());
595 for (unsigned r1=0; r1<this->rows(); ++r1) {
596 for (unsigned c=0; c<this->cols(); ++c) {
597 // Quick test: can we shortcut?
598 if (m[r1*col+c].is_zero())
600 for (unsigned r2=0; r2<other.cols(); ++r2)
601 prod[r1*other.col+r2] += (m[r1*col+c] * other.m[c*other.col+r2]);
604 return matrix(row, other.col, std::move(prod));
608 /** Product of matrix and scalar. */
609 matrix matrix::mul(const numeric & other) const
611 exvector prod(row * col);
613 for (unsigned r=0; r<row; ++r)
614 for (unsigned c=0; c<col; ++c)
615 prod[r*col+c] = m[r*col+c] * other;
617 return matrix(row, col, std::move(prod));
621 /** Product of matrix and scalar expression. */
622 matrix matrix::mul_scalar(const ex & other) const
624 if (other.return_type() != return_types::commutative)
625 throw std::runtime_error("matrix::mul_scalar(): non-commutative scalar");
627 exvector prod(row * col);
629 for (unsigned r=0; r<row; ++r)
630 for (unsigned c=0; c<col; ++c)
631 prod[r*col+c] = m[r*col+c] * other;
633 return matrix(row, col, std::move(prod));
637 /** Power of a matrix. Currently handles integer exponents only. */
638 matrix matrix::pow(const ex & expn) const
641 throw (std::logic_error("matrix::pow(): matrix not square"));
643 if (is_exactly_a<numeric>(expn)) {
644 // Integer cases are computed by successive multiplication, using the
645 // obvious shortcut of storing temporaries, like A^4 == (A*A)*(A*A).
646 if (expn.info(info_flags::integer)) {
647 numeric b = ex_to<numeric>(expn);
649 if (expn.info(info_flags::negative)) {
656 for (unsigned r=0; r<row; ++r)
660 // This loop computes the representation of b in base 2 from right
661 // to left and multiplies the factors whenever needed. Note
662 // that this is not entirely optimal but close to optimal and
663 // "better" algorithms are much harder to implement. (See Knuth,
664 // TAoCP2, section "Evaluation of Powers" for a good discussion.)
665 while (b!=*_num1_p) {
670 b /= *_num2_p; // still integer.
676 throw (std::runtime_error("matrix::pow(): don't know how to handle exponent"));
680 /** operator() to access elements for reading.
682 * @param ro row of element
683 * @param co column of element
684 * @exception range_error (index out of range) */
685 const ex & matrix::operator() (unsigned ro, unsigned co) const
687 if (ro>=row || co>=col)
688 throw (std::range_error("matrix::operator(): index out of range"));
694 /** operator() to access elements for writing.
696 * @param ro row of element
697 * @param co column of element
698 * @exception range_error (index out of range) */
699 ex & matrix::operator() (unsigned ro, unsigned co)
701 if (ro>=row || co>=col)
702 throw (std::range_error("matrix::operator(): index out of range"));
704 ensure_if_modifiable();
709 /** Transposed of an m x n matrix, producing a new n x m matrix object that
710 * represents the transposed. */
711 matrix matrix::transpose() const
713 exvector trans(this->cols()*this->rows());
715 for (unsigned r=0; r<this->cols(); ++r)
716 for (unsigned c=0; c<this->rows(); ++c)
717 trans[r*this->rows()+c] = m[c*this->cols()+r];
719 return matrix(this->cols(), this->rows(), std::move(trans));
722 /** Determinant of square matrix. This routine doesn't actually calculate the
723 * determinant, it only implements some heuristics about which algorithm to
724 * run. If all the elements of the matrix are elements of an integral domain
725 * the determinant is also in that integral domain and the result is expanded
726 * only. If one or more elements are from a quotient field the determinant is
727 * usually also in that quotient field and the result is normalized before it
728 * is returned. This implies that the determinant of the symbolic 2x2 matrix
729 * [[a/(a-b),1],[b/(a-b),1]] is returned as unity. (In this respect, it
730 * behaves like MapleV and unlike Mathematica.)
732 * @param algo allows to chose an algorithm
733 * @return the determinant as a new expression
734 * @exception logic_error (matrix not square)
735 * @see determinant_algo */
736 ex matrix::determinant(unsigned algo) const
739 throw (std::logic_error("matrix::determinant(): matrix not square"));
740 GINAC_ASSERT(row*col==m.capacity());
742 // Gather some statistical information about this matrix:
743 bool numeric_flag = true;
744 bool normal_flag = false;
745 unsigned sparse_count = 0; // counts non-zero elements
747 if (!r.info(info_flags::numeric))
748 numeric_flag = false;
749 exmap srl; // symbol replacement list
750 ex rtest = r.to_rational(srl);
751 if (!rtest.is_zero())
753 if (!rtest.info(info_flags::crational_polynomial) &&
754 rtest.info(info_flags::rational_function))
758 // Here is the heuristics in case this routine has to decide:
759 if (algo == determinant_algo::automatic) {
760 // Minor expansion is generally a good guess:
761 algo = determinant_algo::laplace;
762 // Does anybody know when a matrix is really sparse?
763 // Maybe <~row/2.236 nonzero elements average in a row?
764 if (row>3 && 5*sparse_count<=row*col)
765 algo = determinant_algo::bareiss;
766 // Purely numeric matrix can be handled by Gauss elimination.
767 // This overrides any prior decisions.
769 algo = determinant_algo::gauss;
772 // Trap the trivial case here, since some algorithms don't like it
774 // for consistency with non-trivial determinants...
776 return m[0].normal();
778 return m[0].expand();
781 // Compute the determinant
783 case determinant_algo::gauss: {
786 int sign = tmp.gauss_elimination(true);
787 for (unsigned d=0; d<row; ++d)
788 det *= tmp.m[d*col+d];
790 return (sign*det).normal();
792 return (sign*det).normal().expand();
794 case determinant_algo::bareiss: {
797 sign = tmp.fraction_free_elimination(true);
799 return (sign*tmp.m[row*col-1]).normal();
801 return (sign*tmp.m[row*col-1]).expand();
803 case determinant_algo::divfree: {
806 sign = tmp.division_free_elimination(true);
809 ex det = tmp.m[row*col-1];
810 // factor out accumulated bogus slag
811 for (unsigned d=0; d<row-2; ++d)
812 for (unsigned j=0; j<row-d-2; ++j)
813 det = (det/tmp.m[d*col+d]).normal();
816 case determinant_algo::laplace:
818 // This is the minor expansion scheme. We always develop such
819 // that the smallest minors (i.e, the trivial 1x1 ones) are on the
820 // rightmost column. For this to be efficient, empirical tests
821 // have shown that the emptiest columns (i.e. the ones with most
822 // zeros) should be the ones on the right hand side -- although
823 // this might seem counter-intuitive (and in contradiction to some
824 // literature like the FORM manual). Please go ahead and test it
825 // if you don't believe me! Therefore we presort the columns of
827 typedef std::pair<unsigned,unsigned> uintpair;
828 std::vector<uintpair> c_zeros; // number of zeros in column
829 for (unsigned c=0; c<col; ++c) {
831 for (unsigned r=0; r<row; ++r)
832 if (m[r*col+c].is_zero())
834 c_zeros.push_back(uintpair(acc,c));
836 std::sort(c_zeros.begin(),c_zeros.end());
837 std::vector<unsigned> pre_sort;
838 for (auto & i : c_zeros)
839 pre_sort.push_back(i.second);
840 std::vector<unsigned> pre_sort_test(pre_sort); // permutation_sign() modifies the vector so we make a copy here
841 int sign = permutation_sign(pre_sort_test.begin(), pre_sort_test.end());
842 exvector result(row*col); // represents sorted matrix
844 for (auto & it : pre_sort) {
845 for (unsigned r=0; r<row; ++r)
846 result[r*col+c] = m[r*col+it];
851 return (sign*matrix(row, col, std::move(result)).determinant_minor()).normal();
853 return sign*matrix(row, col, std::move(result)).determinant_minor();
859 /** Trace of a matrix. The result is normalized if it is in some quotient
860 * field and expanded only otherwise. This implies that the trace of the
861 * symbolic 2x2 matrix [[a/(a-b),x],[y,b/(b-a)]] is recognized to be unity.
863 * @return the sum of diagonal elements
864 * @exception logic_error (matrix not square) */
865 ex matrix::trace() const
868 throw (std::logic_error("matrix::trace(): matrix not square"));
871 for (unsigned r=0; r<col; ++r)
874 if (tr.info(info_flags::rational_function) &&
875 !tr.info(info_flags::crational_polynomial))
882 /** Characteristic Polynomial. Following mathematica notation the
883 * characteristic polynomial of a matrix M is defined as the determinant of
884 * (M - lambda * 1) where 1 stands for the unit matrix of the same dimension
885 * as M. Note that some CASs define it with a sign inside the determinant
886 * which gives rise to an overall sign if the dimension is odd. This method
887 * returns the characteristic polynomial collected in powers of lambda as a
890 * @return characteristic polynomial as new expression
891 * @exception logic_error (matrix not square)
892 * @see matrix::determinant() */
893 ex matrix::charpoly(const ex & lambda) const
896 throw (std::logic_error("matrix::charpoly(): matrix not square"));
898 bool numeric_flag = true;
900 if (!r.info(info_flags::numeric)) {
901 numeric_flag = false;
906 // The pure numeric case is traditionally rather common. Hence, it is
907 // trapped and we use Leverrier's algorithm which goes as row^3 for
908 // every coefficient. The expensive part is the matrix multiplication.
913 ex poly = power(lambda, row) - c*power(lambda, row-1);
914 for (unsigned i=1; i<row; ++i) {
915 for (unsigned j=0; j<row; ++j)
918 c = B.trace() / ex(i+1);
919 poly -= c*power(lambda, row-i-1);
929 for (unsigned r=0; r<col; ++r)
930 M.m[r*col+r] -= lambda;
932 return M.determinant().collect(lambda);
937 /** Inverse of this matrix.
939 * @return the inverted matrix
940 * @exception logic_error (matrix not square)
941 * @exception runtime_error (singular matrix) */
942 matrix matrix::inverse() const
945 throw (std::logic_error("matrix::inverse(): matrix not square"));
947 // This routine actually doesn't do anything fancy at all. We compute the
948 // inverse of the matrix A by solving the system A * A^{-1} == Id.
950 // First populate the identity matrix supposed to become the right hand side.
951 matrix identity(row,col);
952 for (unsigned i=0; i<row; ++i)
953 identity(i,i) = _ex1;
955 // Populate a dummy matrix of variables, just because of compatibility with
956 // matrix::solve() which wants this (for compatibility with under-determined
957 // systems of equations).
958 matrix vars(row,col);
959 for (unsigned r=0; r<row; ++r)
960 for (unsigned c=0; c<col; ++c)
961 vars(r,c) = symbol();
965 sol = this->solve(vars,identity);
966 } catch (const std::runtime_error & e) {
967 if (e.what()==std::string("matrix::solve(): inconsistent linear system"))
968 throw (std::runtime_error("matrix::inverse(): singular matrix"));
976 /** Solve a linear system consisting of a m x n matrix and a m x p right hand
977 * side by applying an elimination scheme to the augmented matrix.
979 * @param vars n x p matrix, all elements must be symbols
980 * @param rhs m x p matrix
981 * @param algo selects the solving algorithm
982 * @return n x p solution matrix
983 * @exception logic_error (incompatible matrices)
984 * @exception invalid_argument (1st argument must be matrix of symbols)
985 * @exception runtime_error (inconsistent linear system)
987 matrix matrix::solve(const matrix & vars,
991 const unsigned m = this->rows();
992 const unsigned n = this->cols();
993 const unsigned p = rhs.cols();
996 if ((rhs.rows() != m) || (vars.rows() != n) || (vars.col != p))
997 throw (std::logic_error("matrix::solve(): incompatible matrices"));
998 for (unsigned ro=0; ro<n; ++ro)
999 for (unsigned co=0; co<p; ++co)
1000 if (!vars(ro,co).info(info_flags::symbol))
1001 throw (std::invalid_argument("matrix::solve(): 1st argument must be matrix of symbols"));
1003 // build the augmented matrix of *this with rhs attached to the right
1005 for (unsigned r=0; r<m; ++r) {
1006 for (unsigned c=0; c<n; ++c)
1007 aug.m[r*(n+p)+c] = this->m[r*n+c];
1008 for (unsigned c=0; c<p; ++c)
1009 aug.m[r*(n+p)+c+n] = rhs.m[r*p+c];
1012 // Gather some statistical information about the augmented matrix:
1013 bool numeric_flag = true;
1014 for (auto & r : aug.m) {
1015 if (!r.info(info_flags::numeric)) {
1016 numeric_flag = false;
1021 // Here is the heuristics in case this routine has to decide:
1022 if (algo == solve_algo::automatic) {
1023 // Bareiss (fraction-free) elimination is generally a good guess:
1024 algo = solve_algo::bareiss;
1025 // For m<3, Bareiss elimination is equivalent to division free
1026 // elimination but has more logistic overhead
1028 algo = solve_algo::divfree;
1029 // This overrides any prior decisions.
1031 algo = solve_algo::gauss;
1034 // Eliminate the augmented matrix:
1036 case solve_algo::gauss:
1037 aug.gauss_elimination();
1039 case solve_algo::divfree:
1040 aug.division_free_elimination();
1042 case solve_algo::bareiss:
1044 aug.fraction_free_elimination();
1047 // assemble the solution matrix:
1049 for (unsigned co=0; co<p; ++co) {
1050 unsigned last_assigned_sol = n+1;
1051 for (int r=m-1; r>=0; --r) {
1052 unsigned fnz = 1; // first non-zero in row
1053 while ((fnz<=n) && (aug.m[r*(n+p)+(fnz-1)].is_zero()))
1056 // row consists only of zeros, corresponding rhs must be 0, too
1057 if (!aug.m[r*(n+p)+n+co].is_zero()) {
1058 throw (std::runtime_error("matrix::solve(): inconsistent linear system"));
1061 // assign solutions for vars between fnz+1 and
1062 // last_assigned_sol-1: free parameters
1063 for (unsigned c=fnz; c<last_assigned_sol-1; ++c)
1064 sol(c,co) = vars.m[c*p+co];
1065 ex e = aug.m[r*(n+p)+n+co];
1066 for (unsigned c=fnz; c<n; ++c)
1067 e -= aug.m[r*(n+p)+c]*sol.m[c*p+co];
1068 sol(fnz-1,co) = (e/(aug.m[r*(n+p)+(fnz-1)])).normal();
1069 last_assigned_sol = fnz;
1072 // assign solutions for vars between 1 and
1073 // last_assigned_sol-1: free parameters
1074 for (unsigned ro=0; ro<last_assigned_sol-1; ++ro)
1075 sol(ro,co) = vars(ro,co);
1082 /** Compute the rank of this matrix. */
1083 unsigned matrix::rank() const
1086 // Transform this matrix into upper echelon form and then count the
1087 // number of non-zero rows.
1089 GINAC_ASSERT(row*col==m.capacity());
1091 // Actually, any elimination scheme will do since we are only
1092 // interested in the echelon matrix' zeros.
1093 matrix to_eliminate = *this;
1094 to_eliminate.fraction_free_elimination();
1096 unsigned r = row*col; // index of last non-zero element
1098 if (!to_eliminate.m[r].is_zero())
1107 /** Recursive determinant for small matrices having at least one symbolic
1108 * entry. The basic algorithm, known as Laplace-expansion, is enhanced by
1109 * some bookkeeping to avoid calculation of the same submatrices ("minors")
1110 * more than once. According to W.M.Gentleman and S.C.Johnson this algorithm
1111 * is better than elimination schemes for matrices of sparse multivariate
1112 * polynomials and also for matrices of dense univariate polynomials if the
1113 * matrix' dimension is larger than 7.
1115 * @return the determinant as a new expression (in expanded form)
1116 * @see matrix::determinant() */
1117 ex matrix::determinant_minor() const
1119 // for small matrices the algorithm does not make any sense:
1120 const unsigned n = this->cols();
1122 return m[0].expand();
1124 return (m[0]*m[3]-m[2]*m[1]).expand();
1126 return (m[0]*m[4]*m[8]-m[0]*m[5]*m[7]-
1127 m[1]*m[3]*m[8]+m[2]*m[3]*m[7]+
1128 m[1]*m[5]*m[6]-m[2]*m[4]*m[6]).expand();
1130 // This algorithm can best be understood by looking at a naive
1131 // implementation of Laplace-expansion, like this one:
1133 // matrix minorM(this->rows()-1,this->cols()-1);
1134 // for (unsigned r1=0; r1<this->rows(); ++r1) {
1135 // // shortcut if element(r1,0) vanishes
1136 // if (m[r1*col].is_zero())
1138 // // assemble the minor matrix
1139 // for (unsigned r=0; r<minorM.rows(); ++r) {
1140 // for (unsigned c=0; c<minorM.cols(); ++c) {
1142 // minorM(r,c) = m[r*col+c+1];
1144 // minorM(r,c) = m[(r+1)*col+c+1];
1147 // // recurse down and care for sign:
1149 // det -= m[r1*col] * minorM.determinant_minor();
1151 // det += m[r1*col] * minorM.determinant_minor();
1153 // return det.expand();
1154 // What happens is that while proceeding down many of the minors are
1155 // computed more than once. In particular, there are binomial(n,k)
1156 // kxk minors and each one is computed factorial(n-k) times. Therefore
1157 // it is reasonable to store the results of the minors. We proceed from
1158 // right to left. At each column c we only need to retrieve the minors
1159 // calculated in step c-1. We therefore only have to store at most
1160 // 2*binomial(n,n/2) minors.
1162 // Unique flipper counter for partitioning into minors
1163 std::vector<unsigned> Pkey;
1165 // key for minor determinant (a subpartition of Pkey)
1166 std::vector<unsigned> Mkey;
1168 // we store our subminors in maps, keys being the rows they arise from
1169 typedef std::map<std::vector<unsigned>,class ex> Rmap;
1170 typedef std::map<std::vector<unsigned>,class ex>::value_type Rmap_value;
1174 // initialize A with last column:
1175 for (unsigned r=0; r<n; ++r) {
1176 Pkey.erase(Pkey.begin(),Pkey.end());
1178 A.insert(Rmap_value(Pkey,m[n*(r+1)-1]));
1180 // proceed from right to left through matrix
1181 for (int c=n-2; c>=0; --c) {
1182 Pkey.erase(Pkey.begin(),Pkey.end()); // don't change capacity
1183 Mkey.erase(Mkey.begin(),Mkey.end());
1184 for (unsigned i=0; i<n-c; ++i)
1186 unsigned fc = 0; // controls logic for our strange flipper counter
1189 for (unsigned r=0; r<n-c; ++r) {
1190 // maybe there is nothing to do?
1191 if (m[Pkey[r]*n+c].is_zero())
1193 // create the sorted key for all possible minors
1194 Mkey.erase(Mkey.begin(),Mkey.end());
1195 for (unsigned i=0; i<n-c; ++i)
1197 Mkey.push_back(Pkey[i]);
1198 // Fetch the minors and compute the new determinant
1200 det -= m[Pkey[r]*n+c]*A[Mkey];
1202 det += m[Pkey[r]*n+c]*A[Mkey];
1204 // prevent build-up of deep nesting of expressions saves time:
1206 // store the new determinant at its place in B:
1208 B.insert(Rmap_value(Pkey,det));
1209 // increment our strange flipper counter
1210 for (fc=n-c; fc>0; --fc) {
1212 if (Pkey[fc-1]<fc+c)
1216 for (unsigned j=fc; j<n-c; ++j)
1217 Pkey[j] = Pkey[j-1]+1;
1219 // next column, so change the role of A and B:
1228 /** Perform the steps of an ordinary Gaussian elimination to bring the m x n
1229 * matrix into an upper echelon form. The algorithm is ok for matrices
1230 * with numeric coefficients but quite unsuited for symbolic matrices.
1232 * @param det may be set to true to save a lot of space if one is only
1233 * interested in the diagonal elements (i.e. for calculating determinants).
1234 * The others are set to zero in this case.
1235 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1236 * number of rows was swapped and 0 if the matrix is singular. */
1237 int matrix::gauss_elimination(const bool det)
1239 ensure_if_modifiable();
1240 const unsigned m = this->rows();
1241 const unsigned n = this->cols();
1242 GINAC_ASSERT(!det || n==m);
1246 for (unsigned c0=0; c0<n && r0<m-1; ++c0) {
1247 int indx = pivot(r0, c0, true);
1251 return 0; // leaves *this in a messy state
1256 for (unsigned r2=r0+1; r2<m; ++r2) {
1257 if (!this->m[r2*n+c0].is_zero()) {
1258 // yes, there is something to do in this row
1259 ex piv = this->m[r2*n+c0] / this->m[r0*n+c0];
1260 for (unsigned c=c0+1; c<n; ++c) {
1261 this->m[r2*n+c] -= piv * this->m[r0*n+c];
1262 if (!this->m[r2*n+c].info(info_flags::numeric))
1263 this->m[r2*n+c] = this->m[r2*n+c].normal();
1266 // fill up left hand side with zeros
1267 for (unsigned c=r0; c<=c0; ++c)
1268 this->m[r2*n+c] = _ex0;
1271 // save space by deleting no longer needed elements
1272 for (unsigned c=r0+1; c<n; ++c)
1273 this->m[r0*n+c] = _ex0;
1278 // clear remaining rows
1279 for (unsigned r=r0+1; r<m; ++r) {
1280 for (unsigned c=0; c<n; ++c)
1281 this->m[r*n+c] = _ex0;
1288 /** Perform the steps of division free elimination to bring the m x n matrix
1289 * into an upper echelon form.
1291 * @param det may be set to true to save a lot of space if one is only
1292 * interested in the diagonal elements (i.e. for calculating determinants).
1293 * The others are set to zero in this case.
1294 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1295 * number of rows was swapped and 0 if the matrix is singular. */
1296 int matrix::division_free_elimination(const bool det)
1298 ensure_if_modifiable();
1299 const unsigned m = this->rows();
1300 const unsigned n = this->cols();
1301 GINAC_ASSERT(!det || n==m);
1305 for (unsigned c0=0; c0<n && r0<m-1; ++c0) {
1306 int indx = pivot(r0, c0, true);
1310 return 0; // leaves *this in a messy state
1315 for (unsigned r2=r0+1; r2<m; ++r2) {
1316 for (unsigned c=c0+1; c<n; ++c)
1317 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();
1318 // fill up left hand side with zeros
1319 for (unsigned c=r0; c<=c0; ++c)
1320 this->m[r2*n+c] = _ex0;
1323 // save space by deleting no longer needed elements
1324 for (unsigned c=r0+1; c<n; ++c)
1325 this->m[r0*n+c] = _ex0;
1330 // clear remaining rows
1331 for (unsigned r=r0+1; r<m; ++r) {
1332 for (unsigned c=0; c<n; ++c)
1333 this->m[r*n+c] = _ex0;
1340 /** Perform the steps of Bareiss' one-step fraction free elimination to bring
1341 * the matrix into an upper echelon form. Fraction free elimination means
1342 * that divide is used straightforwardly, without computing GCDs first. This
1343 * is possible, since we know the divisor at each step.
1345 * @param det may be set to true to save a lot of space if one is only
1346 * interested in the last element (i.e. for calculating determinants). The
1347 * others are set to zero in this case.
1348 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1349 * number of rows was swapped and 0 if the matrix is singular. */
1350 int matrix::fraction_free_elimination(const bool det)
1353 // (single-step fraction free elimination scheme, already known to Jordan)
1355 // Usual division-free elimination sets m[0](r,c) = m(r,c) and then sets
1356 // m[k+1](r,c) = m[k](k,k) * m[k](r,c) - m[k](r,k) * m[k](k,c).
1358 // Bareiss (fraction-free) elimination in addition divides that element
1359 // by m[k-1](k-1,k-1) for k>1, where it can be shown by means of the
1360 // Sylvester identity that this really divides m[k+1](r,c).
1362 // We also allow rational functions where the original prove still holds.
1363 // However, we must care for numerator and denominator separately and
1364 // "manually" work in the integral domains because of subtle cancellations
1365 // (see below). This blows up the bookkeeping a bit and the formula has
1366 // to be modified to expand like this (N{x} stands for numerator of x,
1367 // D{x} for denominator of x):
1368 // 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)}
1369 // -N{m[k](r,k)}*N{m[k](k,c)}*D{m[k](k,k)}*D{m[k](r,c)}
1370 // 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)}
1371 // where for k>1 we now divide N{m[k+1](r,c)} by
1372 // N{m[k-1](k-1,k-1)}
1373 // and D{m[k+1](r,c)} by
1374 // D{m[k-1](k-1,k-1)}.
1376 ensure_if_modifiable();
1377 const unsigned m = this->rows();
1378 const unsigned n = this->cols();
1379 GINAC_ASSERT(!det || n==m);
1388 // We populate temporary matrices to subsequently operate on. There is
1389 // one holding numerators and another holding denominators of entries.
1390 // This is a must since the evaluator (or even earlier mul's constructor)
1391 // might cancel some trivial element which causes divide() to fail. The
1392 // elements are normalized first (yes, even though this algorithm doesn't
1393 // need GCDs) since the elements of *this might be unnormalized, which
1394 // makes things more complicated than they need to be.
1395 matrix tmp_n(*this);
1396 matrix tmp_d(m,n); // for denominators, if needed
1397 exmap srl; // symbol replacement list
1398 auto tmp_n_it = tmp_n.m.begin(), tmp_d_it = tmp_d.m.begin();
1399 for (auto & it : this->m) {
1400 ex nd = it.normal().to_rational(srl).numer_denom();
1401 *tmp_n_it++ = nd.op(0);
1402 *tmp_d_it++ = nd.op(1);
1406 for (unsigned c0=0; c0<n && r0<m-1; ++c0) {
1407 // When trying to find a pivot, we should try a bit harder than expand().
1408 // Searching the first non-zero element in-place here instead of calling
1409 // pivot() allows us to do no more substitutions and back-substitutions
1410 // than are actually necessary.
1413 (tmp_n[indx*n+c0].subs(srl, subs_options::no_pattern).expand().is_zero()))
1416 // all elements in column c0 below row r0 vanish
1422 // Matrix needs pivoting, swap rows r0 and indx of tmp_n and tmp_d.
1424 for (unsigned c=c0; c<n; ++c) {
1425 tmp_n.m[n*indx+c].swap(tmp_n.m[n*r0+c]);
1426 tmp_d.m[n*indx+c].swap(tmp_d.m[n*r0+c]);
1429 for (unsigned r2=r0+1; r2<m; ++r2) {
1430 for (unsigned c=c0+1; c<n; ++c) {
1431 dividend_n = (tmp_n.m[r0*n+c0]*tmp_n.m[r2*n+c]*
1432 tmp_d.m[r2*n+c0]*tmp_d.m[r0*n+c]
1433 -tmp_n.m[r2*n+c0]*tmp_n.m[r0*n+c]*
1434 tmp_d.m[r0*n+c0]*tmp_d.m[r2*n+c]).expand();
1435 dividend_d = (tmp_d.m[r2*n+c0]*tmp_d.m[r0*n+c]*
1436 tmp_d.m[r0*n+c0]*tmp_d.m[r2*n+c]).expand();
1437 bool check = divide(dividend_n, divisor_n,
1438 tmp_n.m[r2*n+c], true);
1439 check &= divide(dividend_d, divisor_d,
1440 tmp_d.m[r2*n+c], true);
1441 GINAC_ASSERT(check);
1443 // fill up left hand side with zeros
1444 for (unsigned c=r0; c<=c0; ++c)
1445 tmp_n.m[r2*n+c] = _ex0;
1447 if (c0<n && r0<m-1) {
1448 // compute next iteration's divisor
1449 divisor_n = tmp_n.m[r0*n+c0].expand();
1450 divisor_d = tmp_d.m[r0*n+c0].expand();
1452 // save space by deleting no longer needed elements
1453 for (unsigned c=0; c<n; ++c) {
1454 tmp_n.m[r0*n+c] = _ex0;
1455 tmp_d.m[r0*n+c] = _ex1;
1462 // clear remaining rows
1463 for (unsigned r=r0+1; r<m; ++r) {
1464 for (unsigned c=0; c<n; ++c)
1465 tmp_n.m[r*n+c] = _ex0;
1468 // repopulate *this matrix:
1469 tmp_n_it = tmp_n.m.begin();
1470 tmp_d_it = tmp_d.m.begin();
1471 for (auto & it : this->m)
1472 it = ((*tmp_n_it++)/(*tmp_d_it++)).subs(srl, subs_options::no_pattern);
1478 /** Partial pivoting method for matrix elimination schemes.
1479 * Usual pivoting (symbolic==false) returns the index to the element with the
1480 * largest absolute value in column ro and swaps the current row with the one
1481 * where the element was found. With (symbolic==true) it does the same thing
1482 * with the first non-zero element.
1484 * @param ro is the row from where to begin
1485 * @param co is the column to be inspected
1486 * @param symbolic signal if we want the first non-zero element to be pivoted
1487 * (true) or the one with the largest absolute value (false).
1488 * @return 0 if no interchange occurred, -1 if all are zero (usually signaling
1489 * a degeneracy) and positive integer k means that rows ro and k were swapped.
1491 int matrix::pivot(unsigned ro, unsigned co, bool symbolic)
1495 // search first non-zero element in column co beginning at row ro
1496 while ((k<row) && (this->m[k*col+co].expand().is_zero()))
1499 // search largest element in column co beginning at row ro
1500 GINAC_ASSERT(is_exactly_a<numeric>(this->m[k*col+co]));
1501 unsigned kmax = k+1;
1502 numeric mmax = abs(ex_to<numeric>(m[kmax*col+co]));
1504 GINAC_ASSERT(is_exactly_a<numeric>(this->m[kmax*col+co]));
1505 numeric tmp = ex_to<numeric>(this->m[kmax*col+co]);
1506 if (abs(tmp) > mmax) {
1512 if (!mmax.is_zero())
1516 // all elements in column co below row ro vanish
1519 // matrix needs no pivoting
1521 // matrix needs pivoting, so swap rows k and ro
1522 ensure_if_modifiable();
1523 for (unsigned c=0; c<col; ++c)
1524 this->m[k*col+c].swap(this->m[ro*col+c]);
1529 /** Function to check that all elements of the matrix are zero.
1531 bool matrix::is_zero_matrix() const
1539 ex lst_to_matrix(const lst & l)
1541 // Find number of rows and columns
1542 size_t rows = l.nops(), cols = 0;
1543 for (auto & itr : l) {
1544 if (!is_a<lst>(itr))
1545 throw (std::invalid_argument("lst_to_matrix: argument must be a list of lists"));
1546 if (itr.nops() > cols)
1550 // Allocate and fill matrix
1551 matrix &M = *new matrix(rows, cols);
1552 M.setflag(status_flags::dynallocated);
1555 for (auto & itr : l) {
1557 for (auto & itc : ex_to<lst>(itr)) {
1567 ex diag_matrix(const lst & l)
1569 size_t dim = l.nops();
1571 // Allocate and fill matrix
1572 matrix &M = *new matrix(dim, dim);
1573 M.setflag(status_flags::dynallocated);
1576 for (auto & it : l) {
1584 ex unit_matrix(unsigned r, unsigned c)
1586 matrix &Id = *new matrix(r, c);
1587 Id.setflag(status_flags::dynallocated);
1588 for (unsigned i=0; i<r && i<c; i++)
1594 ex symbolic_matrix(unsigned r, unsigned c, const std::string & base_name, const std::string & tex_base_name)
1596 matrix &M = *new matrix(r, c);
1597 M.setflag(status_flags::dynallocated | status_flags::evaluated);
1599 bool long_format = (r > 10 || c > 10);
1600 bool single_row = (r == 1 || c == 1);
1602 for (unsigned i=0; i<r; i++) {
1603 for (unsigned j=0; j<c; j++) {
1604 std::ostringstream s1, s2;
1606 s2 << tex_base_name << "_{";
1617 s1 << '_' << i << '_' << j;
1618 s2 << i << ';' << j << "}";
1621 s2 << i << j << '}';
1624 M(i, j) = symbol(s1.str(), s2.str());
1631 ex reduced_matrix(const matrix& m, unsigned r, unsigned c)
1633 if (r+1>m.rows() || c+1>m.cols() || m.cols()<2 || m.rows()<2)
1634 throw std::runtime_error("minor_matrix(): index out of bounds");
1636 const unsigned rows = m.rows()-1;
1637 const unsigned cols = m.cols()-1;
1638 matrix &M = *new matrix(rows, cols);
1639 M.setflag(status_flags::dynallocated | status_flags::evaluated);
1651 M(ro2,co2) = m(ro, co);
1662 ex sub_matrix(const matrix&m, unsigned r, unsigned nr, unsigned c, unsigned nc)
1664 if (r+nr>m.rows() || c+nc>m.cols())
1665 throw std::runtime_error("sub_matrix(): index out of bounds");
1667 matrix &M = *new matrix(nr, nc);
1668 M.setflag(status_flags::dynallocated | status_flags::evaluated);
1670 for (unsigned ro=0; ro<nr; ++ro) {
1671 for (unsigned co=0; co<nc; ++co) {
1672 M(ro,co) = m(ro+r,co+c);
1679 } // namespace GiNaC