3 * Implementation of symbolic matrices */
6 * GiNaC Copyright (C) 1999-2023 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.
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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 range = n.find_property_range("m", "m");
144 for (auto i=range.begin; i != range.end; ++i) {
146 n.find_ex_by_loc(i, e, sym_lst);
150 GINAC_BIND_UNARCHIVER(matrix);
152 void matrix::archive(archive_node &n) const
154 inherited::archive(n);
155 n.add_unsigned("row", row);
156 n.add_unsigned("col", col);
163 // functions overriding virtual functions from base classes
168 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
170 for (unsigned ro=0; ro<row; ++ro) {
172 for (unsigned co=0; co<col; ++co) {
173 m[ro*col+co].print(c);
184 void matrix::do_print(const print_context & c, unsigned level) const
187 print_elements(c, "[", "]", ",", ",");
191 void matrix::do_print_latex(const print_latex & c, unsigned level) const
193 c.s << "\\left(\\begin{array}{" << std::string(col,'c') << "}";
194 print_elements(c, "", "", "\\\\", "&");
195 c.s << "\\end{array}\\right)";
198 void matrix::do_print_python_repr(const print_python_repr & c, unsigned level) const
200 c.s << class_name() << '(';
201 print_elements(c, "[", "]", ",", ",");
205 /** nops is defined to be rows x columns. */
206 size_t matrix::nops() const
208 return static_cast<size_t>(row) * static_cast<size_t>(col);
211 /** returns matrix entry at position (i/col, i%col). */
212 ex matrix::op(size_t i) const
214 GINAC_ASSERT(i<nops());
219 /** returns writable matrix entry at position (i/col, i%col). */
220 ex & matrix::let_op(size_t i)
222 GINAC_ASSERT(i<nops());
224 ensure_if_modifiable();
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, std::move(m2)).subs_one_level(mp, options);
238 /** Complex conjugate every matrix entry. */
239 ex matrix::conjugate() const
241 std::unique_ptr<exvector> ev(nullptr);
242 for (auto i=m.begin(); i!=m.end(); ++i) {
243 ex x = i->conjugate();
248 if (are_ex_trivially_equal(x, *i)) {
251 ev.reset(new exvector);
252 ev->reserve(m.size());
253 for (auto j=m.begin(); j!=i; ++j) {
259 return matrix(row, col, std::move(*ev));
264 ex matrix::real_part() const
269 v.push_back(i.real_part());
270 return matrix(row, col, std::move(v));
273 ex matrix::imag_part() const
278 v.push_back(i.imag_part());
279 return matrix(row, col, std::move(v));
284 int matrix::compare_same_type(const basic & other) const
286 GINAC_ASSERT(is_exactly_a<matrix>(other));
287 const matrix &o = static_cast<const matrix &>(other);
289 // compare number of rows
291 return row < o.rows() ? -1 : 1;
293 // compare number of columns
295 return col < o.cols() ? -1 : 1;
297 // equal number of rows and columns, compare individual elements
299 for (unsigned r=0; r<row; ++r) {
300 for (unsigned c=0; c<col; ++c) {
301 cmpval = ((*this)(r,c)).compare(o(r,c));
302 if (cmpval!=0) return cmpval;
305 // all elements are equal => matrices are equal;
309 bool matrix::match_same_type(const basic & other) const
311 GINAC_ASSERT(is_exactly_a<matrix>(other));
312 const matrix & o = static_cast<const matrix &>(other);
314 // The number of rows and columns must be the same. This is necessary to
315 // prevent a 2x3 matrix from matching a 3x2 one.
316 return row == o.rows() && col == o.cols();
319 /** Automatic symbolic evaluation of an indexed matrix. */
320 ex matrix::eval_indexed(const basic & i) const
322 GINAC_ASSERT(is_a<indexed>(i));
323 GINAC_ASSERT(is_a<matrix>(i.op(0)));
325 bool all_indices_unsigned = static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint);
330 // One index, must be one-dimensional vector
331 if (row != 1 && col != 1)
332 throw (std::runtime_error("matrix::eval_indexed(): vector must have exactly 1 index"));
334 const idx & i1 = ex_to<idx>(i.op(1));
339 if (!i1.get_dim().is_equal(row))
340 throw (std::runtime_error("matrix::eval_indexed(): dimension of index must match number of vector elements"));
342 // Index numeric -> return vector element
343 if (all_indices_unsigned) {
344 unsigned n1 = ex_to<numeric>(i1.get_value()).to_int();
346 throw (std::runtime_error("matrix::eval_indexed(): value of index exceeds number of vector elements"));
347 return (*this)(n1, 0);
353 if (!i1.get_dim().is_equal(col))
354 throw (std::runtime_error("matrix::eval_indexed(): dimension of index must match number of vector elements"));
356 // Index numeric -> return vector element
357 if (all_indices_unsigned) {
358 unsigned n1 = ex_to<numeric>(i1.get_value()).to_int();
360 throw (std::runtime_error("matrix::eval_indexed(): value of index exceeds number of vector elements"));
361 return (*this)(0, n1);
365 } else if (i.nops() == 3) {
368 const idx & i1 = ex_to<idx>(i.op(1));
369 const idx & i2 = ex_to<idx>(i.op(2));
371 if (!i1.get_dim().is_equal(row))
372 throw (std::runtime_error("matrix::eval_indexed(): dimension of first index must match number of rows"));
373 if (!i2.get_dim().is_equal(col))
374 throw (std::runtime_error("matrix::eval_indexed(): dimension of second index must match number of columns"));
376 // Pair of dummy indices -> compute trace
377 if (is_dummy_pair(i1, i2))
380 // Both indices numeric -> return matrix element
381 if (all_indices_unsigned) {
382 unsigned n1 = ex_to<numeric>(i1.get_value()).to_int(), n2 = ex_to<numeric>(i2.get_value()).to_int();
384 throw (std::runtime_error("matrix::eval_indexed(): value of first index exceeds number of rows"));
386 throw (std::runtime_error("matrix::eval_indexed(): value of second index exceeds number of columns"));
387 return (*this)(n1, n2);
391 throw (std::runtime_error("matrix::eval_indexed(): matrix must have exactly 2 indices"));
396 /** Sum of two indexed matrices. */
397 ex matrix::add_indexed(const ex & self, const ex & other) const
399 GINAC_ASSERT(is_a<indexed>(self));
400 GINAC_ASSERT(is_a<matrix>(self.op(0)));
401 GINAC_ASSERT(is_a<indexed>(other));
402 GINAC_ASSERT(self.nops() == 2 || self.nops() == 3);
404 // Only add two matrices
405 if (is_a<matrix>(other.op(0))) {
406 GINAC_ASSERT(other.nops() == 2 || other.nops() == 3);
408 const matrix &self_matrix = ex_to<matrix>(self.op(0));
409 const matrix &other_matrix = ex_to<matrix>(other.op(0));
411 if (self.nops() == 2 && other.nops() == 2) { // vector + vector
413 if (self_matrix.row == other_matrix.row)
414 return indexed(self_matrix.add(other_matrix), self.op(1));
415 else if (self_matrix.row == other_matrix.col)
416 return indexed(self_matrix.add(other_matrix.transpose()), self.op(1));
418 } else if (self.nops() == 3 && other.nops() == 3) { // matrix + matrix
420 if (self.op(1).is_equal(other.op(1)) && self.op(2).is_equal(other.op(2)))
421 return indexed(self_matrix.add(other_matrix), self.op(1), self.op(2));
422 else if (self.op(1).is_equal(other.op(2)) && self.op(2).is_equal(other.op(1)))
423 return indexed(self_matrix.add(other_matrix.transpose()), self.op(1), self.op(2));
428 // Don't know what to do, return unevaluated sum
432 /** Product of an indexed matrix with a number. */
433 ex matrix::scalar_mul_indexed(const ex & self, const numeric & other) const
435 GINAC_ASSERT(is_a<indexed>(self));
436 GINAC_ASSERT(is_a<matrix>(self.op(0)));
437 GINAC_ASSERT(self.nops() == 2 || self.nops() == 3);
439 const matrix &self_matrix = ex_to<matrix>(self.op(0));
441 if (self.nops() == 2)
442 return indexed(self_matrix.mul(other), self.op(1));
443 else // self.nops() == 3
444 return indexed(self_matrix.mul(other), self.op(1), self.op(2));
447 /** Contraction of an indexed matrix with something else. */
448 bool matrix::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
450 GINAC_ASSERT(is_a<indexed>(*self));
451 GINAC_ASSERT(is_a<indexed>(*other));
452 GINAC_ASSERT(self->nops() == 2 || self->nops() == 3);
453 GINAC_ASSERT(is_a<matrix>(self->op(0)));
455 // Only contract with other matrices
456 if (!is_a<matrix>(other->op(0)))
459 GINAC_ASSERT(other->nops() == 2 || other->nops() == 3);
461 const matrix &self_matrix = ex_to<matrix>(self->op(0));
462 const matrix &other_matrix = ex_to<matrix>(other->op(0));
464 if (self->nops() == 2) {
466 if (other->nops() == 2) { // vector * vector (scalar product)
468 if (self_matrix.col == 1) {
469 if (other_matrix.col == 1) {
470 // Column vector * column vector, transpose first vector
471 *self = self_matrix.transpose().mul(other_matrix)(0, 0);
473 // Column vector * row vector, swap factors
474 *self = other_matrix.mul(self_matrix)(0, 0);
477 if (other_matrix.col == 1) {
478 // Row vector * column vector, perfect
479 *self = self_matrix.mul(other_matrix)(0, 0);
481 // Row vector * row vector, transpose second vector
482 *self = self_matrix.mul(other_matrix.transpose())(0, 0);
488 } else { // vector * matrix
490 // B_i * A_ij = (B*A)_j (B is row vector)
491 if (is_dummy_pair(self->op(1), other->op(1))) {
492 if (self_matrix.row == 1)
493 *self = indexed(self_matrix.mul(other_matrix), other->op(2));
495 *self = indexed(self_matrix.transpose().mul(other_matrix), other->op(2));
500 // B_j * A_ij = (A*B)_i (B is column vector)
501 if (is_dummy_pair(self->op(1), other->op(2))) {
502 if (self_matrix.col == 1)
503 *self = indexed(other_matrix.mul(self_matrix), other->op(1));
505 *self = indexed(other_matrix.mul(self_matrix.transpose()), other->op(1));
511 } else if (other->nops() == 3) { // matrix * matrix
513 // A_ij * B_jk = (A*B)_ik
514 if (is_dummy_pair(self->op(2), other->op(1))) {
515 *self = indexed(self_matrix.mul(other_matrix), self->op(1), other->op(2));
520 // A_ij * B_kj = (A*Btrans)_ik
521 if (is_dummy_pair(self->op(2), other->op(2))) {
522 *self = indexed(self_matrix.mul(other_matrix.transpose()), self->op(1), other->op(1));
527 // A_ji * B_jk = (Atrans*B)_ik
528 if (is_dummy_pair(self->op(1), other->op(1))) {
529 *self = indexed(self_matrix.transpose().mul(other_matrix), self->op(2), other->op(2));
534 // A_ji * B_kj = (B*A)_ki
535 if (is_dummy_pair(self->op(1), other->op(2))) {
536 *self = indexed(other_matrix.mul(self_matrix), other->op(1), self->op(2));
547 // non-virtual functions in this class
554 * @exception logic_error (incompatible matrices) */
555 matrix matrix::add(const matrix & other) const
557 if (col != other.col || row != other.row)
558 throw std::logic_error("matrix::add(): incompatible matrices");
560 exvector sum(this->m);
561 auto ci = other.m.begin();
565 return matrix(row, col, std::move(sum));
569 /** Difference of matrices.
571 * @exception logic_error (incompatible matrices) */
572 matrix matrix::sub(const matrix & other) const
574 if (col != other.col || row != other.row)
575 throw std::logic_error("matrix::sub(): incompatible matrices");
577 exvector dif(this->m);
578 auto ci = other.m.begin();
582 return matrix(row, col, std::move(dif));
586 /** Product of matrices.
588 * @exception logic_error (incompatible matrices) */
589 matrix matrix::mul(const matrix & other) const
591 if (this->cols() != other.rows())
592 throw std::logic_error("matrix::mul(): incompatible matrices");
594 exvector prod(this->rows()*other.cols());
596 for (unsigned r1=0; r1<this->rows(); ++r1) {
597 for (unsigned c=0; c<this->cols(); ++c) {
598 // Quick test: can we shortcut?
599 if (m[r1*col+c].is_zero())
601 for (unsigned r2=0; r2<other.cols(); ++r2)
602 prod[r1*other.col+r2] += (m[r1*col+c] * other.m[c*other.col+r2]);
605 return matrix(row, other.col, std::move(prod));
609 /** Product of matrix and scalar. */
610 matrix matrix::mul(const numeric & other) const
612 exvector prod(row * col);
614 for (unsigned r=0; r<row; ++r)
615 for (unsigned c=0; c<col; ++c)
616 prod[r*col+c] = m[r*col+c] * other;
618 return matrix(row, col, std::move(prod));
622 /** Product of matrix and scalar expression. */
623 matrix matrix::mul_scalar(const ex & other) const
625 if (other.return_type() != return_types::commutative)
626 throw std::runtime_error("matrix::mul_scalar(): non-commutative scalar");
628 exvector prod(row * col);
630 for (unsigned r=0; r<row; ++r)
631 for (unsigned c=0; c<col; ++c)
632 prod[r*col+c] = m[r*col+c] * other;
634 return matrix(row, col, std::move(prod));
638 /** Power of a matrix. Currently handles integer exponents only. */
639 matrix matrix::pow(const ex & expn) const
642 throw (std::logic_error("matrix::pow(): matrix not square"));
644 if (is_exactly_a<numeric>(expn)) {
645 // Integer cases are computed by successive multiplication, using the
646 // obvious shortcut of storing temporaries, like A^4 == (A*A)*(A*A).
647 if (expn.info(info_flags::integer)) {
648 numeric b = ex_to<numeric>(expn);
650 if (expn.info(info_flags::negative)) {
657 for (unsigned r=0; r<row; ++r)
661 // This loop computes the representation of b in base 2 from right
662 // to left and multiplies the factors whenever needed. Note
663 // that this is not entirely optimal but close to optimal and
664 // "better" algorithms are much harder to implement. (See Knuth,
665 // TAoCP2, section "Evaluation of Powers" for a good discussion.)
666 while (b!=*_num1_p) {
671 b /= *_num2_p; // still integer.
677 throw (std::runtime_error("matrix::pow(): don't know how to handle exponent"));
681 /** operator() to access elements for reading.
683 * @param ro row of element
684 * @param co column of element
685 * @exception range_error (index out of range) */
686 const ex & matrix::operator() (unsigned ro, unsigned co) const
688 if (ro>=row || co>=col)
689 throw (std::range_error("matrix::operator(): index out of range"));
695 /** operator() to access elements for writing.
697 * @param ro row of element
698 * @param co column of element
699 * @exception range_error (index out of range) */
700 ex & matrix::operator() (unsigned ro, unsigned co)
702 if (ro>=row || co>=col)
703 throw (std::range_error("matrix::operator(): index out of range"));
705 ensure_if_modifiable();
710 /** Transposed of an m x n matrix, producing a new n x m matrix object that
711 * represents the transposed. */
712 matrix matrix::transpose() const
714 exvector trans(this->cols()*this->rows());
716 for (unsigned r=0; r<this->cols(); ++r)
717 for (unsigned c=0; c<this->rows(); ++c)
718 trans[r*this->rows()+c] = m[c*this->cols()+r];
720 return matrix(this->cols(), this->rows(), std::move(trans));
723 /** Determinant of square matrix. This routine doesn't actually calculate the
724 * determinant, it only implements some heuristics about which algorithm to
725 * run. If all the elements of the matrix are elements of an integral domain
726 * the determinant is also in that integral domain and the result is expanded
727 * only. If one or more elements are from a quotient field the determinant is
728 * usually also in that quotient field and the result is normalized before it
729 * is returned. This implies that the determinant of the symbolic 2x2 matrix
730 * [[a/(a-b),1],[b/(a-b),1]] is returned as unity. (In this respect, it
731 * behaves like MapleV and unlike Mathematica.)
733 * @param algo allows to chose an algorithm
734 * @return the determinant as a new expression
735 * @exception logic_error (matrix not square)
736 * @see determinant_algo */
737 ex matrix::determinant(unsigned algo) const
740 throw (std::logic_error("matrix::determinant(): matrix not square"));
741 GINAC_ASSERT(row*col==m.capacity());
743 // Gather some statistical information about this matrix:
744 bool numeric_flag = true;
745 bool normal_flag = false;
746 unsigned sparse_count = 0; // counts non-zero elements
748 if (!r.info(info_flags::numeric))
749 numeric_flag = false;
750 exmap srl; // symbol replacement list
751 ex rtest = r.to_rational(srl);
752 if (!rtest.is_zero())
754 if (!rtest.info(info_flags::crational_polynomial) &&
755 rtest.info(info_flags::rational_function))
759 // Here is the heuristics in case this routine has to decide:
760 if (algo == determinant_algo::automatic) {
761 // Minor expansion is generally a good guess:
762 algo = determinant_algo::laplace;
763 // Does anybody know when a matrix is really sparse?
764 // Maybe <~row/2.236 nonzero elements average in a row?
765 if (row>3 && 5*sparse_count<=row*col)
766 algo = determinant_algo::bareiss;
767 // Purely numeric matrix can be handled by Gauss elimination.
768 // This overrides any prior decisions.
770 algo = determinant_algo::gauss;
773 // Trap the trivial case here, since some algorithms don't like it
775 // for consistency with non-trivial determinants...
777 return m[0].normal();
779 return m[0].expand();
782 // Compute the determinant
784 case determinant_algo::gauss: {
787 int sign = tmp.gauss_elimination(true);
788 for (unsigned d=0; d<row; ++d)
789 det *= tmp.m[d*col+d];
791 return (sign*det).normal();
793 return (sign*det).normal().expand();
795 case determinant_algo::bareiss: {
798 sign = tmp.fraction_free_elimination(true);
800 return (sign*tmp.m[row*col-1]).normal();
802 return (sign*tmp.m[row*col-1]).expand();
804 case determinant_algo::divfree: {
807 sign = tmp.division_free_elimination(true);
810 ex det = tmp.m[row*col-1];
811 // factor out accumulated bogus slag
812 for (unsigned d=0; d<row-2; ++d)
813 for (unsigned j=0; j<row-d-2; ++j)
814 det = (det/tmp.m[d*col+d]).normal();
817 case determinant_algo::laplace:
819 // This is the minor expansion scheme. We always develop such
820 // that the smallest minors (i.e, the trivial 1x1 ones) are on the
821 // rightmost column. For this to be efficient, empirical tests
822 // have shown that the emptiest columns (i.e. the ones with most
823 // zeros) should be the ones on the right hand side -- although
824 // this might seem counter-intuitive (and in contradiction to some
825 // literature like the FORM manual). Please go ahead and test it
826 // if you don't believe me! Therefore we presort the columns of
828 typedef std::pair<unsigned,unsigned> uintpair;
829 std::vector<uintpair> c_zeros; // number of zeros in column
830 for (unsigned c=0; c<col; ++c) {
832 for (unsigned r=0; r<row; ++r)
833 if (m[r*col+c].is_zero())
835 c_zeros.push_back(uintpair(acc,c));
837 std::sort(c_zeros.begin(),c_zeros.end());
838 std::vector<unsigned> pre_sort;
839 for (auto & i : c_zeros)
840 pre_sort.push_back(i.second);
841 std::vector<unsigned> pre_sort_test(pre_sort); // permutation_sign() modifies the vector so we make a copy here
842 int sign = permutation_sign(pre_sort_test.begin(), pre_sort_test.end());
843 exvector result(row*col); // represents sorted matrix
845 for (auto & it : pre_sort) {
846 for (unsigned r=0; r<row; ++r)
847 result[r*col+c] = m[r*col+it];
852 return (sign*matrix(row, col, std::move(result)).determinant_minor()).normal();
854 return sign*matrix(row, col, std::move(result)).determinant_minor();
860 /** Trace of a matrix. The result is normalized if it is in some quotient
861 * field and expanded only otherwise. This implies that the trace of the
862 * symbolic 2x2 matrix [[a/(a-b),x],[y,b/(b-a)]] is recognized to be unity.
864 * @return the sum of diagonal elements
865 * @exception logic_error (matrix not square) */
866 ex matrix::trace() const
869 throw (std::logic_error("matrix::trace(): matrix not square"));
872 for (unsigned r=0; r<col; ++r)
875 if (tr.info(info_flags::rational_function) &&
876 !tr.info(info_flags::crational_polynomial))
883 /** Characteristic Polynomial. Following mathematica notation the
884 * characteristic polynomial of a matrix M is defined as the determinant of
885 * (M - lambda * 1) where 1 stands for the unit matrix of the same dimension
886 * as M. Note that some CASs define it with a sign inside the determinant
887 * which gives rise to an overall sign if the dimension is odd. This method
888 * returns the characteristic polynomial collected in powers of lambda as a
891 * @return characteristic polynomial as new expression
892 * @exception logic_error (matrix not square)
893 * @see matrix::determinant() */
894 ex matrix::charpoly(const ex & lambda) const
897 throw (std::logic_error("matrix::charpoly(): matrix not square"));
899 bool numeric_flag = true;
901 if (!r.info(info_flags::numeric)) {
902 numeric_flag = false;
907 // The pure numeric case is traditionally rather common. Hence, it is
908 // trapped and we use Leverrier's algorithm which goes as row^3 for
909 // every coefficient. The expensive part is the matrix multiplication.
914 ex poly = power(lambda, row) - c*power(lambda, row-1);
915 for (unsigned i=1; i<row; ++i) {
916 for (unsigned j=0; j<row; ++j)
919 c = B.trace() / ex(i+1);
920 poly -= c*power(lambda, row-i-1);
930 for (unsigned r=0; r<col; ++r)
931 M.m[r*col+r] -= lambda;
933 return M.determinant().collect(lambda);
938 /** Inverse of this matrix, with automatic algorithm selection. */
939 matrix matrix::inverse() const
941 return inverse(solve_algo::automatic);
944 /** Inverse of this matrix.
946 * @param algo selects the algorithm (one of solve_algo)
947 * @return the inverted matrix
948 * @exception logic_error (matrix not square)
949 * @exception runtime_error (singular matrix) */
950 matrix matrix::inverse(unsigned algo) 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, algo);
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.cols() != 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 // Eliminate the augmented matrix:
1021 auto colid = aug.echelon_form(algo, n);
1023 // assemble the solution matrix:
1025 for (unsigned co=0; co<p; ++co) {
1026 unsigned last_assigned_sol = n+1;
1027 for (int r=m-1; r>=0; --r) {
1028 unsigned fnz = 1; // first non-zero in row
1029 while ((fnz<=n) && (aug.m[r*(n+p)+(fnz-1)].normal().is_zero()))
1032 // row consists only of zeros, corresponding rhs must be 0, too
1033 if (!aug.m[r*(n+p)+n+co].normal().is_zero()) {
1034 throw (std::runtime_error("matrix::solve(): inconsistent linear system"));
1037 // assign solutions for vars between fnz+1 and
1038 // last_assigned_sol-1: free parameters
1039 for (unsigned c=fnz; c<last_assigned_sol-1; ++c)
1040 sol(colid[c],co) = vars.m[colid[c]*p+co];
1041 ex e = aug.m[r*(n+p)+n+co];
1042 for (unsigned c=fnz; c<n; ++c)
1043 e -= aug.m[r*(n+p)+c]*sol.m[colid[c]*p+co];
1044 sol(colid[fnz-1],co) = (e/(aug.m[r*(n+p)+fnz-1])).normal();
1045 last_assigned_sol = fnz;
1048 // assign solutions for vars between 1 and
1049 // last_assigned_sol-1: free parameters
1050 for (unsigned ro=0; ro<last_assigned_sol-1; ++ro)
1051 sol(colid[ro],co) = vars(colid[ro],co);
1057 /** Compute the rank of this matrix. */
1058 unsigned matrix::rank() const
1060 return rank(solve_algo::automatic);
1063 /** Compute the rank of this matrix using the given algorithm,
1064 * which should be a member of enum solve_algo. */
1065 unsigned matrix::rank(unsigned solve_algo) const
1068 // Transform this matrix into upper echelon form and then count the
1069 // number of non-zero rows.
1070 GINAC_ASSERT(row*col==m.capacity());
1072 matrix to_eliminate = *this;
1073 to_eliminate.echelon_form(solve_algo, col);
1075 unsigned r = row*col; // index of last non-zero element
1077 if (!to_eliminate.m[r].is_zero())
1086 /** Recursive determinant for small matrices having at least one symbolic
1087 * entry. The basic algorithm, known as Laplace-expansion, is enhanced by
1088 * some bookkeeping to avoid calculation of the same submatrices ("minors")
1089 * more than once. According to W.M.Gentleman and S.C.Johnson this algorithm
1090 * is better than elimination schemes for matrices of sparse multivariate
1091 * polynomials and also for matrices of dense univariate polynomials if the
1092 * matrix' dimension is larger than 7.
1094 * @return the determinant as a new expression (in expanded form)
1095 * @see matrix::determinant() */
1096 ex matrix::determinant_minor() const
1098 const unsigned n = this->cols();
1100 // This algorithm can best be understood by looking at a naive
1101 // implementation of Laplace-expansion, like this one:
1103 // matrix minorM(this->rows()-1,this->cols()-1);
1104 // for (unsigned r1=0; r1<this->rows(); ++r1) {
1105 // // shortcut if element(r1,0) vanishes
1106 // if (m[r1*col].is_zero())
1108 // // assemble the minor matrix
1109 // for (unsigned r=0; r<minorM.rows(); ++r) {
1110 // for (unsigned c=0; c<minorM.cols(); ++c) {
1112 // minorM(r,c) = m[r*col+c+1];
1114 // minorM(r,c) = m[(r+1)*col+c+1];
1117 // // recurse down and care for sign:
1119 // det -= m[r1*col] * minorM.determinant_minor();
1121 // det += m[r1*col] * minorM.determinant_minor();
1123 // return det.expand();
1124 // What happens is that while proceeding down many of the minors are
1125 // computed more than once. In particular, there are binomial(n,k)
1126 // kxk minors and each one is computed factorial(n-k) times. Therefore
1127 // it is reasonable to store the results of the minors. We proceed from
1128 // right to left. At each column c we only need to retrieve the minors
1129 // calculated in step c-1. We therefore only have to store at most
1130 // 2*binomial(n,n/2) minors.
1132 // we store the minors in maps, keyed by the rows they arise from
1133 typedef std::vector<unsigned> keyseq;
1134 typedef std::map<keyseq, ex> Rmap;
1136 Rmap M, N; // minors used in current and next column, respectively
1137 // populate M with dummy unit, to be used as factor in rightmost column
1140 // keys to identify minor of M and N (Mkey is a subsequence of Nkey)
1146 // proceed from right to left through matrix
1147 for (int c=n-1; c>=0; --c) {
1150 for (unsigned i=0; i<n-c; ++i)
1152 unsigned fc = 0; // controls logic for minor key generator
1155 for (unsigned r=0; r<n-c; ++r) {
1156 // maybe there is nothing to do?
1157 if (m[Nkey[r]*n+c].is_zero())
1159 // Mkey is same as Nkey, but with element r removed
1161 Mkey.insert(Mkey.begin(), Nkey.begin(), Nkey.begin() + r);
1162 Mkey.insert(Mkey.end(), Nkey.begin() + r + 1, Nkey.end());
1163 // add product of matrix element and minor M to determinant
1165 det -= m[Nkey[r]*n+c]*M[Mkey];
1167 det += m[Nkey[r]*n+c]*M[Mkey];
1169 // prevent nested expressions to save time
1171 // if the next computed minor is zero, don't store it in N:
1172 // (if key is not found, operator[] will just return a zero ex)
1175 // compute next minor key
1176 for (fc=n-c; fc>0; --fc) {
1178 if (Nkey[fc-1]<fc+c)
1182 for (unsigned j=fc; j<n-c; ++j)
1183 Nkey[j] = Nkey[j-1]+1;
1185 // if N contains no minors, then they all vanished
1189 // proceed to next column: switch roles of M and N, clear N
1196 std::vector<unsigned>
1197 matrix::echelon_form(unsigned algo, int n)
1199 // Here is the heuristics in case this routine has to decide:
1200 if (algo == solve_algo::automatic) {
1201 // Gather some statistical information about the augmented matrix:
1202 bool numeric_flag = true;
1203 for (const auto & r : m) {
1204 if (!r.info(info_flags::numeric)) {
1205 numeric_flag = false;
1209 unsigned density = 0;
1210 for (const auto & r : m) {
1211 density += !r.is_zero();
1213 unsigned ncells = col*row;
1215 // For numerical matrices Gauss is good, but Markowitz becomes
1216 // better for large sparse matrices.
1217 if ((ncells > 200) && (density < ncells/2)) {
1218 algo = solve_algo::markowitz;
1220 algo = solve_algo::gauss;
1223 // For symbolic matrices Markowitz is good, but Bareiss/Divfree
1224 // is better for small and dense matrices.
1225 if ((ncells < 120) && (density*5 > ncells*3)) {
1227 algo = solve_algo::divfree;
1229 algo = solve_algo::bareiss;
1232 algo = solve_algo::markowitz;
1236 // Eliminate the augmented matrix:
1237 std::vector<unsigned> colid(col);
1238 for (unsigned c = 0; c < col; c++) {
1242 case solve_algo::gauss:
1243 gauss_elimination();
1245 case solve_algo::divfree:
1246 division_free_elimination();
1248 case solve_algo::bareiss:
1249 fraction_free_elimination();
1251 case solve_algo::markowitz:
1252 colid = markowitz_elimination(n);
1255 throw std::invalid_argument("matrix::echelon_form(): 'algo' is not one of the solve_algo enum");
1260 /** Perform the steps of an ordinary Gaussian elimination to bring the m x n
1261 * matrix into an upper echelon form. The algorithm is ok for matrices
1262 * with numeric coefficients but quite unsuited for symbolic matrices.
1264 * @param det may be set to true to save a lot of space if one is only
1265 * interested in the diagonal elements (i.e. for calculating determinants).
1266 * The others are set to zero in this case.
1267 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1268 * number of rows was swapped and 0 if the matrix is singular. */
1269 int matrix::gauss_elimination(const bool det)
1271 ensure_if_modifiable();
1272 const unsigned m = this->rows();
1273 const unsigned n = this->cols();
1274 GINAC_ASSERT(!det || n==m);
1278 for (unsigned c0=0; c0<n && r0<m-1; ++c0) {
1279 int indx = pivot(r0, c0, true);
1283 return 0; // leaves *this in a messy state
1288 for (unsigned r2=r0+1; r2<m; ++r2) {
1289 if (!this->m[r2*n+c0].is_zero()) {
1290 // yes, there is something to do in this row
1291 ex piv = this->m[r2*n+c0] / this->m[r0*n+c0];
1292 for (unsigned c=c0+1; c<n; ++c) {
1293 this->m[r2*n+c] -= piv * this->m[r0*n+c];
1294 if (!this->m[r2*n+c].info(info_flags::numeric))
1295 this->m[r2*n+c] = this->m[r2*n+c].normal();
1298 // fill up left hand side with zeros
1299 for (unsigned c=r0; c<=c0; ++c)
1300 this->m[r2*n+c] = _ex0;
1303 // save space by deleting no longer needed elements
1304 for (unsigned c=r0+1; c<n; ++c)
1305 this->m[r0*n+c] = _ex0;
1310 // clear remaining rows
1311 for (unsigned r=r0+1; r<m; ++r) {
1312 for (unsigned c=0; c<n; ++c)
1313 this->m[r*n+c] = _ex0;
1319 /* Perform Markowitz-ordered Gaussian elimination (with full
1320 * pivoting) on a matrix, constraining the choice of pivots to
1321 * the first n columns (this simplifies handling of augmented
1322 * matrices). Return the column id vector v, such that v[column]
1323 * is the original number of the column before shuffling (v[i]==i
1325 std::vector<unsigned>
1326 matrix::markowitz_elimination(unsigned n)
1328 GINAC_ASSERT(n <= col);
1329 std::vector<int> rowcnt(row, 0);
1330 std::vector<int> colcnt(col, 0);
1331 // Normalize everything before start. We'll keep all the
1332 // cells normalized throughout the algorithm to properly
1333 // handle unnormal zeros.
1334 for (unsigned r = 0; r < row; r++) {
1335 for (unsigned c = 0; c < col; c++) {
1336 if (!m[r*col + c].is_zero()) {
1337 m[r*col + c] = m[r*col + c].normal();
1343 std::vector<unsigned> colid(col);
1344 for (unsigned c = 0; c < col; c++) {
1348 for (unsigned k = 0; (k < col) && (k < row - 1); k++) {
1349 // Find the pivot that minimizes (rowcnt[r]-1)*(colcnt[c]-1).
1350 unsigned pivot_r = row + 1;
1351 unsigned pivot_c = col + 1;
1352 int pivot_m = row*col;
1353 for (unsigned r = k; r < row; r++) {
1354 for (unsigned c = k; c < n; c++) {
1355 const ex &mrc = m[r*col + c];
1358 GINAC_ASSERT(rowcnt[r] > 0);
1359 GINAC_ASSERT(colcnt[c] > 0);
1360 int measure = (rowcnt[r] - 1)*(colcnt[c] - 1);
1361 if (measure < pivot_m) {
1368 if (pivot_m == row*col) {
1369 // The rest of the matrix is zero.
1372 GINAC_ASSERT(k <= pivot_r && pivot_r < row);
1373 GINAC_ASSERT(k <= pivot_c && pivot_c < col);
1374 // Swap the pivot into (k, k).
1376 for (unsigned r = 0; r < row; r++) {
1377 m[r*col + pivot_c].swap(m[r*col + k]);
1379 std::swap(colid[pivot_c], colid[k]);
1380 std::swap(colcnt[pivot_c], colcnt[k]);
1383 for (unsigned c = k; c < col; c++) {
1384 m[pivot_r*col + c].swap(m[k*col + c]);
1386 std::swap(rowcnt[pivot_r], rowcnt[k]);
1388 // No normalization before is_zero() here, because
1389 // we maintain the matrix normalized throughout the
1391 ex a = m[k*col + k];
1392 GINAC_ASSERT(!a.is_zero());
1393 // Subtract the pivot row KJI-style (so: loop by pivot, then
1394 // column, then row) to maximally exploit pivot row zeros (at
1395 // the expense of the pivot column zeros). The speedup compared
1396 // to the usual KIJ order is not really significant though...
1397 for (unsigned r = k + 1; r < row; r++) {
1398 const ex &b = m[r*col + k];
1404 colcnt[k] = rowcnt[k] = 0;
1405 for (unsigned c = k + 1; c < col; c++) {
1406 const ex &mr0c = m[k*col + c];
1410 for (unsigned r = k + 1; r < row; r++) {
1411 if (ab[r].is_zero())
1413 bool waszero = m[r*col + c].is_zero();
1414 m[r*col + c] = (m[r*col + c] - ab[r]*mr0c).normal();
1415 bool iszero = m[r*col + c].is_zero();
1416 if (waszero && !iszero) {
1420 if (!waszero && iszero) {
1426 for (unsigned r = k + 1; r < row; r++) {
1427 ab[r] = m[r*col + k] = _ex0;
1433 /** Perform the steps of division free elimination to bring the m x n matrix
1434 * into an upper echelon form.
1436 * @param det may be set to true to save a lot of space if one is only
1437 * interested in the diagonal elements (i.e. for calculating determinants).
1438 * The others are set to zero in this case.
1439 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1440 * number of rows was swapped and 0 if the matrix is singular. */
1441 int matrix::division_free_elimination(const bool det)
1443 ensure_if_modifiable();
1444 const unsigned m = this->rows();
1445 const unsigned n = this->cols();
1446 GINAC_ASSERT(!det || n==m);
1450 for (unsigned c0=0; c0<n && r0<m-1; ++c0) {
1451 int indx = pivot(r0, c0, true);
1455 return 0; // leaves *this in a messy state
1460 for (unsigned r2=r0+1; r2<m; ++r2) {
1461 for (unsigned c=c0+1; c<n; ++c)
1462 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]).normal();
1463 // fill up left hand side with zeros
1464 for (unsigned c=r0; c<=c0; ++c)
1465 this->m[r2*n+c] = _ex0;
1468 // save space by deleting no longer needed elements
1469 for (unsigned c=r0+1; c<n; ++c)
1470 this->m[r0*n+c] = _ex0;
1475 // clear remaining rows
1476 for (unsigned r=r0+1; r<m; ++r) {
1477 for (unsigned c=0; c<n; ++c)
1478 this->m[r*n+c] = _ex0;
1485 /** Perform the steps of Bareiss' one-step fraction free elimination to bring
1486 * the matrix into an upper echelon form. Fraction free elimination means
1487 * that divide is used straightforwardly, without computing GCDs first. This
1488 * is possible, since we know the divisor at each step.
1490 * @param det may be set to true to save a lot of space if one is only
1491 * interested in the last element (i.e. for calculating determinants). The
1492 * others are set to zero in this case.
1493 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1494 * number of rows was swapped and 0 if the matrix is singular. */
1495 int matrix::fraction_free_elimination(const bool det)
1498 // (single-step fraction free elimination scheme, already known to Jordan)
1500 // Usual division-free elimination sets m[0](r,c) = m(r,c) and then sets
1501 // m[k+1](r,c) = m[k](k,k) * m[k](r,c) - m[k](r,k) * m[k](k,c).
1503 // Bareiss (fraction-free) elimination in addition divides that element
1504 // by m[k-1](k-1,k-1) for k>1, where it can be shown by means of the
1505 // Sylvester identity that this really divides m[k+1](r,c).
1507 // We also allow rational functions where the original prove still holds.
1508 // However, we must care for numerator and denominator separately and
1509 // "manually" work in the integral domains because of subtle cancellations
1510 // (see below). This blows up the bookkeeping a bit and the formula has
1511 // to be modified to expand like this (N{x} stands for numerator of x,
1512 // D{x} for denominator of x):
1513 // 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)}
1514 // -N{m[k](r,k)}*N{m[k](k,c)}*D{m[k](k,k)}*D{m[k](r,c)}
1515 // 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)}
1516 // where for k>1 we now divide N{m[k+1](r,c)} by
1517 // N{m[k-1](k-1,k-1)}
1518 // and D{m[k+1](r,c)} by
1519 // D{m[k-1](k-1,k-1)}.
1521 ensure_if_modifiable();
1522 const unsigned m = this->rows();
1523 const unsigned n = this->cols();
1524 GINAC_ASSERT(!det || n==m);
1533 // We populate temporary matrices to subsequently operate on. There is
1534 // one holding numerators and another holding denominators of entries.
1535 // This is a must since the evaluator (or even earlier mul's constructor)
1536 // might cancel some trivial element which causes divide() to fail. The
1537 // elements are normalized first (yes, even though this algorithm doesn't
1538 // need GCDs) since the elements of *this might be unnormalized, which
1539 // makes things more complicated than they need to be.
1540 matrix tmp_n(*this);
1541 matrix tmp_d(m,n); // for denominators, if needed
1542 exmap srl; // symbol replacement list
1543 auto tmp_n_it = tmp_n.m.begin(), tmp_d_it = tmp_d.m.begin();
1544 for (auto & it : this->m) {
1545 ex nd = it.normal().to_rational(srl).numer_denom();
1546 *tmp_n_it++ = nd.op(0);
1547 *tmp_d_it++ = nd.op(1);
1551 for (unsigned c0=0; c0<n && r0<m-1; ++c0) {
1552 // When trying to find a pivot, we should try a bit harder than expand().
1553 // Searching the first non-zero element in-place here instead of calling
1554 // pivot() allows us to do no more substitutions and back-substitutions
1555 // than are actually necessary.
1558 (tmp_n[indx*n+c0].subs(srl, subs_options::no_pattern).expand().is_zero()))
1561 // all elements in column c0 below row r0 vanish
1567 // Matrix needs pivoting, swap rows r0 and indx of tmp_n and tmp_d.
1569 for (unsigned c=c0; c<n; ++c) {
1570 tmp_n.m[n*indx+c].swap(tmp_n.m[n*r0+c]);
1571 tmp_d.m[n*indx+c].swap(tmp_d.m[n*r0+c]);
1574 for (unsigned r2=r0+1; r2<m; ++r2) {
1575 for (unsigned c=c0+1; c<n; ++c) {
1576 dividend_n = (tmp_n.m[r0*n+c0]*tmp_n.m[r2*n+c]*
1577 tmp_d.m[r2*n+c0]*tmp_d.m[r0*n+c]
1578 -tmp_n.m[r2*n+c0]*tmp_n.m[r0*n+c]*
1579 tmp_d.m[r0*n+c0]*tmp_d.m[r2*n+c]).expand();
1580 dividend_d = (tmp_d.m[r2*n+c0]*tmp_d.m[r0*n+c]*
1581 tmp_d.m[r0*n+c0]*tmp_d.m[r2*n+c]).expand();
1582 bool check = divide(dividend_n, divisor_n,
1583 tmp_n.m[r2*n+c], true);
1584 check &= divide(dividend_d, divisor_d,
1585 tmp_d.m[r2*n+c], true);
1586 GINAC_ASSERT(check);
1588 // fill up left hand side with zeros
1589 for (unsigned c=r0; c<=c0; ++c)
1590 tmp_n.m[r2*n+c] = _ex0;
1592 if (c0<n && r0<m-1) {
1593 // compute next iteration's divisor
1594 divisor_n = tmp_n.m[r0*n+c0].expand();
1595 divisor_d = tmp_d.m[r0*n+c0].expand();
1597 // save space by deleting no longer needed elements
1598 for (unsigned c=0; c<n; ++c) {
1599 tmp_n.m[r0*n+c] = _ex0;
1600 tmp_d.m[r0*n+c] = _ex1;
1607 // clear remaining rows
1608 for (unsigned r=r0+1; r<m; ++r) {
1609 for (unsigned c=0; c<n; ++c)
1610 tmp_n.m[r*n+c] = _ex0;
1613 // repopulate *this matrix:
1614 tmp_n_it = tmp_n.m.begin();
1615 tmp_d_it = tmp_d.m.begin();
1616 for (auto & it : this->m)
1617 it = ((*tmp_n_it++)/(*tmp_d_it++)).subs(srl, subs_options::no_pattern);
1623 /** Partial pivoting method for matrix elimination schemes.
1624 * Usual pivoting (symbolic==false) returns the index to the element with the
1625 * largest absolute value in column ro and swaps the current row with the one
1626 * where the element was found. With (symbolic==true) it does the same thing
1627 * with the first non-zero element.
1629 * @param ro is the row from where to begin
1630 * @param co is the column to be inspected
1631 * @param symbolic signal if we want the first non-zero element to be pivoted
1632 * (true) or the one with the largest absolute value (false).
1633 * @return 0 if no interchange occurred, -1 if all are zero (usually signaling
1634 * a degeneracy) and positive integer k means that rows ro and k were swapped.
1636 int matrix::pivot(unsigned ro, unsigned co, bool symbolic)
1640 // search first non-zero element in column co beginning at row ro
1641 while ((k<row) && (this->m[k*col+co].expand().is_zero()))
1644 // search largest element in column co beginning at row ro
1645 GINAC_ASSERT(is_exactly_a<numeric>(this->m[k*col+co]));
1646 unsigned kmax = k+1;
1647 numeric mmax = abs(ex_to<numeric>(m[kmax*col+co]));
1649 GINAC_ASSERT(is_exactly_a<numeric>(this->m[kmax*col+co]));
1650 numeric tmp = ex_to<numeric>(this->m[kmax*col+co]);
1651 if (abs(tmp) > mmax) {
1657 if (!mmax.is_zero())
1661 // all elements in column co below row ro vanish
1664 // matrix needs no pivoting
1666 // matrix needs pivoting, so swap rows k and ro
1667 ensure_if_modifiable();
1668 for (unsigned c=0; c<col; ++c)
1669 this->m[k*col+c].swap(this->m[ro*col+c]);
1674 /** Function to check that all elements of the matrix are zero.
1676 bool matrix::is_zero_matrix() const
1684 ex lst_to_matrix(const lst & l)
1686 // Find number of rows and columns
1687 size_t rows = l.nops(), cols = 0;
1688 for (auto & itr : l) {
1689 if (!is_a<lst>(itr))
1690 throw (std::invalid_argument("lst_to_matrix: argument must be a list of lists"));
1691 if (itr.nops() > cols)
1695 // Allocate and fill matrix
1696 matrix & M = dynallocate<matrix>(rows, cols);
1699 for (auto & itr : l) {
1701 for (auto & itc : ex_to<lst>(itr)) {
1711 ex diag_matrix(const lst & l)
1713 size_t dim = l.nops();
1715 // Allocate and fill matrix
1716 matrix & M = dynallocate<matrix>(dim, dim);
1719 for (auto & it : l) {
1727 ex diag_matrix(std::initializer_list<ex> l)
1729 size_t dim = l.size();
1731 // Allocate and fill matrix
1732 matrix & M = dynallocate<matrix>(dim, dim);
1735 for (auto & it : l) {
1743 ex unit_matrix(unsigned r, unsigned c)
1745 matrix & Id = dynallocate<matrix>(r, c);
1746 Id.setflag(status_flags::evaluated);
1747 for (unsigned i=0; i<r && i<c; i++)
1753 ex symbolic_matrix(unsigned r, unsigned c, const std::string & base_name, const std::string & tex_base_name)
1755 matrix & M = dynallocate<matrix>(r, c);
1756 M.setflag(status_flags::evaluated);
1758 bool long_format = (r > 10 || c > 10);
1759 bool single_row = (r == 1 || c == 1);
1761 for (unsigned i=0; i<r; i++) {
1762 for (unsigned j=0; j<c; j++) {
1763 std::ostringstream s1, s2;
1765 s2 << tex_base_name << "_{";
1776 s1 << '_' << i << '_' << j;
1777 s2 << i << ';' << j << "}";
1780 s2 << i << j << '}';
1783 M(i, j) = symbol(s1.str(), s2.str());
1790 ex reduced_matrix(const matrix& m, unsigned r, unsigned c)
1792 if (r+1>m.rows() || c+1>m.cols() || m.cols()<2 || m.rows()<2)
1793 throw std::runtime_error("minor_matrix(): index out of bounds");
1795 const unsigned rows = m.rows()-1;
1796 const unsigned cols = m.cols()-1;
1797 matrix & M = dynallocate<matrix>(rows, cols);
1798 M.setflag(status_flags::evaluated);
1810 M(ro2,co2) = m(ro, co);
1821 ex sub_matrix(const matrix&m, unsigned r, unsigned nr, unsigned c, unsigned nc)
1823 if (r+nr>m.rows() || c+nc>m.cols())
1824 throw std::runtime_error("sub_matrix(): index out of bounds");
1826 matrix & M = dynallocate<matrix>(nr, nc);
1827 M.setflag(status_flags::evaluated);
1829 for (unsigned ro=0; ro<nr; ++ro) {
1830 for (unsigned co=0; co<nc; ++co) {
1831 M(ro,co) = m(ro+r,co+c);
1838 } // namespace GiNaC