3 * Implementation of symbolic matrices */
6 * GiNaC Copyright (C) 1999-2003 Johannes Gutenberg University Mainz, Germany
8 * This program is free software; you can redistribute it and/or modify
9 * it under the terms of the GNU General Public License as published by
10 * the Free Software Foundation; either version 2 of the License, or
11 * (at your option) any later version.
13 * This program is distributed in the hope that it will be useful,
14 * but WITHOUT ANY WARRANTY; without even the implied warranty of
15 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
16 * GNU General Public License for more details.
18 * You should have received a copy of the GNU General Public License
19 * along with this program; if not, write to the Free Software
20 * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
44 GINAC_IMPLEMENT_REGISTERED_CLASS(matrix, basic)
47 // default ctor, dtor, copy ctor, assignment operator and helpers:
50 /** Default ctor. Initializes to 1 x 1-dimensional zero-matrix. */
51 matrix::matrix() : inherited(TINFO_matrix), row(1), col(1)
56 void matrix::copy(const matrix & other)
58 inherited::copy(other);
61 m = other.m; // STL's vector copying invoked here
64 DEFAULT_DESTROY(matrix)
72 /** Very common ctor. Initializes to r x c-dimensional zero-matrix.
74 * @param r number of rows
75 * @param c number of cols */
76 matrix::matrix(unsigned r, unsigned c)
77 : inherited(TINFO_matrix), row(r), col(c)
84 /** Ctor from representation, for internal use only. */
85 matrix::matrix(unsigned r, unsigned c, const exvector & m2)
86 : inherited(TINFO_matrix), row(r), col(c), m(m2) {}
88 /** Construct matrix from (flat) list of elements. If the list has fewer
89 * elements than the matrix, the remaining matrix elements are set to zero.
90 * If the list has more elements than the matrix, the excessive elements are
92 matrix::matrix(unsigned r, unsigned c, const lst & l)
93 : inherited(TINFO_matrix), row(r), col(c)
97 for (unsigned i=0; i<l.nops(); i++) {
101 break; // matrix smaller than list: throw away excessive elements
110 matrix::matrix(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
112 if (!(n.find_unsigned("row", row)) || !(n.find_unsigned("col", col)))
113 throw (std::runtime_error("unknown matrix dimensions in archive"));
114 m.reserve(row * col);
115 for (unsigned int i=0; true; i++) {
117 if (n.find_ex("m", e, sym_lst, i))
124 void matrix::archive(archive_node &n) const
126 inherited::archive(n);
127 n.add_unsigned("row", row);
128 n.add_unsigned("col", col);
129 exvector::const_iterator i = m.begin(), iend = m.end();
136 DEFAULT_UNARCHIVE(matrix)
139 // functions overriding virtual functions from base classes
144 void matrix::print(const print_context & c, unsigned level) const
146 if (is_a<print_tree>(c)) {
148 inherited::print(c, level);
152 if (is_a<print_python_repr>(c))
153 c.s << class_name() << '(';
155 if (is_a<print_latex>(c))
156 c.s << "\\left(\\begin{array}{" << std::string(col,'c') << "}";
160 for (unsigned ro=0; ro<row; ++ro) {
161 if (!is_a<print_latex>(c))
163 for (unsigned co=0; co<col; ++co) {
164 m[ro*col+co].print(c);
166 if (is_a<print_latex>(c))
171 if (!is_a<print_latex>(c))
176 if (is_a<print_latex>(c))
183 if (is_a<print_latex>(c))
184 c.s << "\\end{array}\\right)";
188 if (is_a<print_python_repr>(c))
194 /** nops is defined to be rows x columns. */
195 unsigned matrix::nops() const
200 /** returns matrix entry at position (i/col, i%col). */
201 ex matrix::op(int i) const
206 /** returns matrix entry at position (i/col, i%col). */
207 ex & matrix::let_op(int i)
210 GINAC_ASSERT(i<nops());
215 /** Evaluate matrix entry by entry. */
216 ex matrix::eval(int level) const
218 // check if we have to do anything at all
219 if ((level==1)&&(flags & status_flags::evaluated))
223 if (level == -max_recursion_level)
224 throw (std::runtime_error("matrix::eval(): recursion limit exceeded"));
226 // eval() entry by entry
227 exvector m2(row*col);
229 for (unsigned r=0; r<row; ++r)
230 for (unsigned c=0; c<col; ++c)
231 m2[r*col+c] = m[r*col+c].eval(level);
233 return (new matrix(row, col, m2))->setflag(status_flags::dynallocated |
234 status_flags::evaluated);
237 ex matrix::subs(const lst & ls, const lst & lr, bool no_pattern) const
239 exvector m2(row * col);
240 for (unsigned r=0; r<row; ++r)
241 for (unsigned c=0; c<col; ++c)
242 m2[r*col+c] = m[r*col+c].subs(ls, lr, no_pattern);
244 return matrix(row, col, m2).basic::subs(ls, lr, no_pattern);
249 int matrix::compare_same_type(const basic & other) const
251 GINAC_ASSERT(is_exactly_a<matrix>(other));
252 const matrix &o = static_cast<const matrix &>(other);
254 // compare number of rows
256 return row < o.rows() ? -1 : 1;
258 // compare number of columns
260 return col < o.cols() ? -1 : 1;
262 // equal number of rows and columns, compare individual elements
264 for (unsigned r=0; r<row; ++r) {
265 for (unsigned c=0; c<col; ++c) {
266 cmpval = ((*this)(r,c)).compare(o(r,c));
267 if (cmpval!=0) return cmpval;
270 // all elements are equal => matrices are equal;
274 bool matrix::match_same_type(const basic & other) const
276 GINAC_ASSERT(is_exactly_a<matrix>(other));
277 const matrix & o = static_cast<const matrix &>(other);
279 // The number of rows and columns must be the same. This is necessary to
280 // prevent a 2x3 matrix from matching a 3x2 one.
281 return row == o.rows() && col == o.cols();
284 /** Automatic symbolic evaluation of an indexed matrix. */
285 ex matrix::eval_indexed(const basic & i) const
287 GINAC_ASSERT(is_a<indexed>(i));
288 GINAC_ASSERT(is_a<matrix>(i.op(0)));
290 bool all_indices_unsigned = static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint);
295 // One index, must be one-dimensional vector
296 if (row != 1 && col != 1)
297 throw (std::runtime_error("matrix::eval_indexed(): vector must have exactly 1 index"));
299 const idx & i1 = ex_to<idx>(i.op(1));
304 if (!i1.get_dim().is_equal(row))
305 throw (std::runtime_error("matrix::eval_indexed(): dimension of index must match number of vector elements"));
307 // Index numeric -> return vector element
308 if (all_indices_unsigned) {
309 unsigned n1 = ex_to<numeric>(i1.get_value()).to_int();
311 throw (std::runtime_error("matrix::eval_indexed(): value of index exceeds number of vector elements"));
312 return (*this)(n1, 0);
318 if (!i1.get_dim().is_equal(col))
319 throw (std::runtime_error("matrix::eval_indexed(): dimension of index must match number of vector elements"));
321 // Index numeric -> return vector element
322 if (all_indices_unsigned) {
323 unsigned n1 = ex_to<numeric>(i1.get_value()).to_int();
325 throw (std::runtime_error("matrix::eval_indexed(): value of index exceeds number of vector elements"));
326 return (*this)(0, n1);
330 } else if (i.nops() == 3) {
333 const idx & i1 = ex_to<idx>(i.op(1));
334 const idx & i2 = ex_to<idx>(i.op(2));
336 if (!i1.get_dim().is_equal(row))
337 throw (std::runtime_error("matrix::eval_indexed(): dimension of first index must match number of rows"));
338 if (!i2.get_dim().is_equal(col))
339 throw (std::runtime_error("matrix::eval_indexed(): dimension of second index must match number of columns"));
341 // Pair of dummy indices -> compute trace
342 if (is_dummy_pair(i1, i2))
345 // Both indices numeric -> return matrix element
346 if (all_indices_unsigned) {
347 unsigned n1 = ex_to<numeric>(i1.get_value()).to_int(), n2 = ex_to<numeric>(i2.get_value()).to_int();
349 throw (std::runtime_error("matrix::eval_indexed(): value of first index exceeds number of rows"));
351 throw (std::runtime_error("matrix::eval_indexed(): value of second index exceeds number of columns"));
352 return (*this)(n1, n2);
356 throw (std::runtime_error("matrix::eval_indexed(): matrix must have exactly 2 indices"));
361 /** Sum of two indexed matrices. */
362 ex matrix::add_indexed(const ex & self, const ex & other) const
364 GINAC_ASSERT(is_a<indexed>(self));
365 GINAC_ASSERT(is_a<matrix>(self.op(0)));
366 GINAC_ASSERT(is_a<indexed>(other));
367 GINAC_ASSERT(self.nops() == 2 || self.nops() == 3);
369 // Only add two matrices
370 if (is_ex_of_type(other.op(0), matrix)) {
371 GINAC_ASSERT(other.nops() == 2 || other.nops() == 3);
373 const matrix &self_matrix = ex_to<matrix>(self.op(0));
374 const matrix &other_matrix = ex_to<matrix>(other.op(0));
376 if (self.nops() == 2 && other.nops() == 2) { // vector + vector
378 if (self_matrix.row == other_matrix.row)
379 return indexed(self_matrix.add(other_matrix), self.op(1));
380 else if (self_matrix.row == other_matrix.col)
381 return indexed(self_matrix.add(other_matrix.transpose()), self.op(1));
383 } else if (self.nops() == 3 && other.nops() == 3) { // matrix + matrix
385 if (self.op(1).is_equal(other.op(1)) && self.op(2).is_equal(other.op(2)))
386 return indexed(self_matrix.add(other_matrix), self.op(1), self.op(2));
387 else if (self.op(1).is_equal(other.op(2)) && self.op(2).is_equal(other.op(1)))
388 return indexed(self_matrix.add(other_matrix.transpose()), self.op(1), self.op(2));
393 // Don't know what to do, return unevaluated sum
397 /** Product of an indexed matrix with a number. */
398 ex matrix::scalar_mul_indexed(const ex & self, const numeric & other) const
400 GINAC_ASSERT(is_a<indexed>(self));
401 GINAC_ASSERT(is_a<matrix>(self.op(0)));
402 GINAC_ASSERT(self.nops() == 2 || self.nops() == 3);
404 const matrix &self_matrix = ex_to<matrix>(self.op(0));
406 if (self.nops() == 2)
407 return indexed(self_matrix.mul(other), self.op(1));
408 else // self.nops() == 3
409 return indexed(self_matrix.mul(other), self.op(1), self.op(2));
412 /** Contraction of an indexed matrix with something else. */
413 bool matrix::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
415 GINAC_ASSERT(is_a<indexed>(*self));
416 GINAC_ASSERT(is_a<indexed>(*other));
417 GINAC_ASSERT(self->nops() == 2 || self->nops() == 3);
418 GINAC_ASSERT(is_a<matrix>(self->op(0)));
420 // Only contract with other matrices
421 if (!is_ex_of_type(other->op(0), matrix))
424 GINAC_ASSERT(other->nops() == 2 || other->nops() == 3);
426 const matrix &self_matrix = ex_to<matrix>(self->op(0));
427 const matrix &other_matrix = ex_to<matrix>(other->op(0));
429 if (self->nops() == 2) {
431 if (other->nops() == 2) { // vector * vector (scalar product)
433 if (self_matrix.col == 1) {
434 if (other_matrix.col == 1) {
435 // Column vector * column vector, transpose first vector
436 *self = self_matrix.transpose().mul(other_matrix)(0, 0);
438 // Column vector * row vector, swap factors
439 *self = other_matrix.mul(self_matrix)(0, 0);
442 if (other_matrix.col == 1) {
443 // Row vector * column vector, perfect
444 *self = self_matrix.mul(other_matrix)(0, 0);
446 // Row vector * row vector, transpose second vector
447 *self = self_matrix.mul(other_matrix.transpose())(0, 0);
453 } else { // vector * matrix
455 // B_i * A_ij = (B*A)_j (B is row vector)
456 if (is_dummy_pair(self->op(1), other->op(1))) {
457 if (self_matrix.row == 1)
458 *self = indexed(self_matrix.mul(other_matrix), other->op(2));
460 *self = indexed(self_matrix.transpose().mul(other_matrix), other->op(2));
465 // B_j * A_ij = (A*B)_i (B is column vector)
466 if (is_dummy_pair(self->op(1), other->op(2))) {
467 if (self_matrix.col == 1)
468 *self = indexed(other_matrix.mul(self_matrix), other->op(1));
470 *self = indexed(other_matrix.mul(self_matrix.transpose()), other->op(1));
476 } else if (other->nops() == 3) { // matrix * matrix
478 // A_ij * B_jk = (A*B)_ik
479 if (is_dummy_pair(self->op(2), other->op(1))) {
480 *self = indexed(self_matrix.mul(other_matrix), self->op(1), other->op(2));
485 // A_ij * B_kj = (A*Btrans)_ik
486 if (is_dummy_pair(self->op(2), other->op(2))) {
487 *self = indexed(self_matrix.mul(other_matrix.transpose()), self->op(1), other->op(1));
492 // A_ji * B_jk = (Atrans*B)_ik
493 if (is_dummy_pair(self->op(1), other->op(1))) {
494 *self = indexed(self_matrix.transpose().mul(other_matrix), self->op(2), other->op(2));
499 // A_ji * B_kj = (B*A)_ki
500 if (is_dummy_pair(self->op(1), other->op(2))) {
501 *self = indexed(other_matrix.mul(self_matrix), other->op(1), self->op(2));
512 // non-virtual functions in this class
519 * @exception logic_error (incompatible matrices) */
520 matrix matrix::add(const matrix & other) const
522 if (col != other.col || row != other.row)
523 throw std::logic_error("matrix::add(): incompatible matrices");
525 exvector sum(this->m);
526 exvector::iterator i = sum.begin(), end = sum.end();
527 exvector::const_iterator ci = other.m.begin();
531 return matrix(row,col,sum);
535 /** Difference of matrices.
537 * @exception logic_error (incompatible matrices) */
538 matrix matrix::sub(const matrix & other) const
540 if (col != other.col || row != other.row)
541 throw std::logic_error("matrix::sub(): incompatible matrices");
543 exvector dif(this->m);
544 exvector::iterator i = dif.begin(), end = dif.end();
545 exvector::const_iterator ci = other.m.begin();
549 return matrix(row,col,dif);
553 /** Product of matrices.
555 * @exception logic_error (incompatible matrices) */
556 matrix matrix::mul(const matrix & other) const
558 if (this->cols() != other.rows())
559 throw std::logic_error("matrix::mul(): incompatible matrices");
561 exvector prod(this->rows()*other.cols());
563 for (unsigned r1=0; r1<this->rows(); ++r1) {
564 for (unsigned c=0; c<this->cols(); ++c) {
565 if (m[r1*col+c].is_zero())
567 for (unsigned r2=0; r2<other.cols(); ++r2)
568 prod[r1*other.col+r2] += (m[r1*col+c] * other.m[c*other.col+r2]).expand();
571 return matrix(row, other.col, prod);
575 /** Product of matrix and scalar. */
576 matrix matrix::mul(const numeric & other) const
578 exvector prod(row * col);
580 for (unsigned r=0; r<row; ++r)
581 for (unsigned c=0; c<col; ++c)
582 prod[r*col+c] = m[r*col+c] * other;
584 return matrix(row, col, prod);
588 /** Product of matrix and scalar expression. */
589 matrix matrix::mul_scalar(const ex & other) const
591 if (other.return_type() != return_types::commutative)
592 throw std::runtime_error("matrix::mul_scalar(): non-commutative scalar");
594 exvector prod(row * col);
596 for (unsigned r=0; r<row; ++r)
597 for (unsigned c=0; c<col; ++c)
598 prod[r*col+c] = m[r*col+c] * other;
600 return matrix(row, col, prod);
604 /** Power of a matrix. Currently handles integer exponents only. */
605 matrix matrix::pow(const ex & expn) const
608 throw (std::logic_error("matrix::pow(): matrix not square"));
610 if (is_ex_exactly_of_type(expn, numeric)) {
611 // Integer cases are computed by successive multiplication, using the
612 // obvious shortcut of storing temporaries, like A^4 == (A*A)*(A*A).
613 if (expn.info(info_flags::integer)) {
614 numeric b = ex_to<numeric>(expn);
616 if (expn.info(info_flags::negative)) {
623 for (unsigned r=0; r<row; ++r)
627 // This loop computes the representation of b in base 2 from right
628 // to left and multiplies the factors whenever needed. Note
629 // that this is not entirely optimal but close to optimal and
630 // "better" algorithms are much harder to implement. (See Knuth,
631 // TAoCP2, section "Evaluation of Powers" for a good discussion.)
637 b /= _num2; // still integer.
643 throw (std::runtime_error("matrix::pow(): don't know how to handle exponent"));
647 /** operator() to access elements for reading.
649 * @param ro row of element
650 * @param co column of element
651 * @exception range_error (index out of range) */
652 const ex & matrix::operator() (unsigned ro, unsigned co) const
654 if (ro>=row || co>=col)
655 throw (std::range_error("matrix::operator(): index out of range"));
661 /** operator() to access elements for writing.
663 * @param ro row of element
664 * @param co column of element
665 * @exception range_error (index out of range) */
666 ex & matrix::operator() (unsigned ro, unsigned co)
668 if (ro>=row || co>=col)
669 throw (std::range_error("matrix::operator(): index out of range"));
671 ensure_if_modifiable();
676 /** Transposed of an m x n matrix, producing a new n x m matrix object that
677 * represents the transposed. */
678 matrix matrix::transpose(void) const
680 exvector trans(this->cols()*this->rows());
682 for (unsigned r=0; r<this->cols(); ++r)
683 for (unsigned c=0; c<this->rows(); ++c)
684 trans[r*this->rows()+c] = m[c*this->cols()+r];
686 return matrix(this->cols(),this->rows(),trans);
689 /** Determinant of square matrix. This routine doesn't actually calculate the
690 * determinant, it only implements some heuristics about which algorithm to
691 * run. If all the elements of the matrix are elements of an integral domain
692 * the determinant is also in that integral domain and the result is expanded
693 * only. If one or more elements are from a quotient field the determinant is
694 * usually also in that quotient field and the result is normalized before it
695 * is returned. This implies that the determinant of the symbolic 2x2 matrix
696 * [[a/(a-b),1],[b/(a-b),1]] is returned as unity. (In this respect, it
697 * behaves like MapleV and unlike Mathematica.)
699 * @param algo allows to chose an algorithm
700 * @return the determinant as a new expression
701 * @exception logic_error (matrix not square)
702 * @see determinant_algo */
703 ex matrix::determinant(unsigned algo) const
706 throw (std::logic_error("matrix::determinant(): matrix not square"));
707 GINAC_ASSERT(row*col==m.capacity());
709 // Gather some statistical information about this matrix:
710 bool numeric_flag = true;
711 bool normal_flag = false;
712 unsigned sparse_count = 0; // counts non-zero elements
713 exvector::const_iterator r = m.begin(), rend = m.end();
715 lst srl; // symbol replacement list
716 ex rtest = r->to_rational(srl);
717 if (!rtest.is_zero())
719 if (!rtest.info(info_flags::numeric))
720 numeric_flag = false;
721 if (!rtest.info(info_flags::crational_polynomial) &&
722 rtest.info(info_flags::rational_function))
727 // Here is the heuristics in case this routine has to decide:
728 if (algo == determinant_algo::automatic) {
729 // Minor expansion is generally a good guess:
730 algo = determinant_algo::laplace;
731 // Does anybody know when a matrix is really sparse?
732 // Maybe <~row/2.236 nonzero elements average in a row?
733 if (row>3 && 5*sparse_count<=row*col)
734 algo = determinant_algo::bareiss;
735 // Purely numeric matrix can be handled by Gauss elimination.
736 // This overrides any prior decisions.
738 algo = determinant_algo::gauss;
741 // Trap the trivial case here, since some algorithms don't like it
743 // for consistency with non-trivial determinants...
745 return m[0].normal();
747 return m[0].expand();
750 // Compute the determinant
752 case determinant_algo::gauss: {
755 int sign = tmp.gauss_elimination(true);
756 for (unsigned d=0; d<row; ++d)
757 det *= tmp.m[d*col+d];
759 return (sign*det).normal();
761 return (sign*det).normal().expand();
763 case determinant_algo::bareiss: {
766 sign = tmp.fraction_free_elimination(true);
768 return (sign*tmp.m[row*col-1]).normal();
770 return (sign*tmp.m[row*col-1]).expand();
772 case determinant_algo::divfree: {
775 sign = tmp.division_free_elimination(true);
778 ex det = tmp.m[row*col-1];
779 // factor out accumulated bogus slag
780 for (unsigned d=0; d<row-2; ++d)
781 for (unsigned j=0; j<row-d-2; ++j)
782 det = (det/tmp.m[d*col+d]).normal();
785 case determinant_algo::laplace:
787 // This is the minor expansion scheme. We always develop such
788 // that the smallest minors (i.e, the trivial 1x1 ones) are on the
789 // rightmost column. For this to be efficient, empirical tests
790 // have shown that the emptiest columns (i.e. the ones with most
791 // zeros) should be the ones on the right hand side -- although
792 // this might seem counter-intuitive (and in contradiction to some
793 // literature like the FORM manual). Please go ahead and test it
794 // if you don't believe me! Therefore we presort the columns of
796 typedef std::pair<unsigned,unsigned> uintpair;
797 std::vector<uintpair> c_zeros; // number of zeros in column
798 for (unsigned c=0; c<col; ++c) {
800 for (unsigned r=0; r<row; ++r)
801 if (m[r*col+c].is_zero())
803 c_zeros.push_back(uintpair(acc,c));
805 std::sort(c_zeros.begin(),c_zeros.end());
806 std::vector<unsigned> pre_sort;
807 for (std::vector<uintpair>::const_iterator i=c_zeros.begin(); i!=c_zeros.end(); ++i)
808 pre_sort.push_back(i->second);
809 std::vector<unsigned> pre_sort_test(pre_sort); // permutation_sign() modifies the vector so we make a copy here
810 int sign = permutation_sign(pre_sort_test.begin(), pre_sort_test.end());
811 exvector result(row*col); // represents sorted matrix
813 for (std::vector<unsigned>::const_iterator i=pre_sort.begin();
816 for (unsigned r=0; r<row; ++r)
817 result[r*col+c] = m[r*col+(*i)];
821 return (sign*matrix(row,col,result).determinant_minor()).normal();
823 return sign*matrix(row,col,result).determinant_minor();
829 /** Trace of a matrix. The result is normalized if it is in some quotient
830 * field and expanded only otherwise. This implies that the trace of the
831 * symbolic 2x2 matrix [[a/(a-b),x],[y,b/(b-a)]] is recognized to be unity.
833 * @return the sum of diagonal elements
834 * @exception logic_error (matrix not square) */
835 ex matrix::trace(void) const
838 throw (std::logic_error("matrix::trace(): matrix not square"));
841 for (unsigned r=0; r<col; ++r)
844 if (tr.info(info_flags::rational_function) &&
845 !tr.info(info_flags::crational_polynomial))
852 /** Characteristic Polynomial. Following mathematica notation the
853 * characteristic polynomial of a matrix M is defined as the determiant of
854 * (M - lambda * 1) where 1 stands for the unit matrix of the same dimension
855 * as M. Note that some CASs define it with a sign inside the determinant
856 * which gives rise to an overall sign if the dimension is odd. This method
857 * returns the characteristic polynomial collected in powers of lambda as a
860 * @return characteristic polynomial as new expression
861 * @exception logic_error (matrix not square)
862 * @see matrix::determinant() */
863 ex matrix::charpoly(const symbol & lambda) const
866 throw (std::logic_error("matrix::charpoly(): matrix not square"));
868 bool numeric_flag = true;
869 exvector::const_iterator r = m.begin(), rend = m.end();
870 while (r!=rend && numeric_flag==true) {
871 if (!r->info(info_flags::numeric))
872 numeric_flag = false;
876 // The pure numeric case is traditionally rather common. Hence, it is
877 // trapped and we use Leverrier's algorithm which goes as row^3 for
878 // every coefficient. The expensive part is the matrix multiplication.
882 ex poly = power(lambda,row)-c*power(lambda,row-1);
883 for (unsigned i=1; i<row; ++i) {
884 for (unsigned j=0; j<row; ++j)
887 c = B.trace()/ex(i+1);
888 poly -= c*power(lambda,row-i-1);
897 for (unsigned r=0; r<col; ++r)
898 M.m[r*col+r] -= lambda;
900 return M.determinant().collect(lambda);
904 /** Inverse of this matrix.
906 * @return the inverted matrix
907 * @exception logic_error (matrix not square)
908 * @exception runtime_error (singular matrix) */
909 matrix matrix::inverse(void) const
912 throw (std::logic_error("matrix::inverse(): matrix not square"));
914 // This routine actually doesn't do anything fancy at all. We compute the
915 // inverse of the matrix A by solving the system A * A^{-1} == Id.
917 // First populate the identity matrix supposed to become the right hand side.
918 matrix identity(row,col);
919 for (unsigned i=0; i<row; ++i)
920 identity(i,i) = _ex1;
922 // Populate a dummy matrix of variables, just because of compatibility with
923 // matrix::solve() which wants this (for compatibility with under-determined
924 // systems of equations).
925 matrix vars(row,col);
926 for (unsigned r=0; r<row; ++r)
927 for (unsigned c=0; c<col; ++c)
928 vars(r,c) = symbol();
932 sol = this->solve(vars,identity);
933 } catch (const std::runtime_error & e) {
934 if (e.what()==std::string("matrix::solve(): inconsistent linear system"))
935 throw (std::runtime_error("matrix::inverse(): singular matrix"));
943 /** Solve a linear system consisting of a m x n matrix and a m x p right hand
944 * side by applying an elimination scheme to the augmented matrix.
946 * @param vars n x p matrix, all elements must be symbols
947 * @param rhs m x p matrix
948 * @return n x p solution matrix
949 * @exception logic_error (incompatible matrices)
950 * @exception invalid_argument (1st argument must be matrix of symbols)
951 * @exception runtime_error (inconsistent linear system)
953 matrix matrix::solve(const matrix & vars,
957 const unsigned m = this->rows();
958 const unsigned n = this->cols();
959 const unsigned p = rhs.cols();
962 if ((rhs.rows() != m) || (vars.rows() != n) || (vars.col != p))
963 throw (std::logic_error("matrix::solve(): incompatible matrices"));
964 for (unsigned ro=0; ro<n; ++ro)
965 for (unsigned co=0; co<p; ++co)
966 if (!vars(ro,co).info(info_flags::symbol))
967 throw (std::invalid_argument("matrix::solve(): 1st argument must be matrix of symbols"));
969 // build the augmented matrix of *this with rhs attached to the right
971 for (unsigned r=0; r<m; ++r) {
972 for (unsigned c=0; c<n; ++c)
973 aug.m[r*(n+p)+c] = this->m[r*n+c];
974 for (unsigned c=0; c<p; ++c)
975 aug.m[r*(n+p)+c+n] = rhs.m[r*p+c];
978 // Gather some statistical information about the augmented matrix:
979 bool numeric_flag = true;
980 exvector::const_iterator r = aug.m.begin(), rend = aug.m.end();
981 while (r!=rend && numeric_flag==true) {
982 if (!r->info(info_flags::numeric))
983 numeric_flag = false;
987 // Here is the heuristics in case this routine has to decide:
988 if (algo == solve_algo::automatic) {
989 // Bareiss (fraction-free) elimination is generally a good guess:
990 algo = solve_algo::bareiss;
991 // For m<3, Bareiss elimination is equivalent to division free
992 // elimination but has more logistic overhead
994 algo = solve_algo::divfree;
995 // This overrides any prior decisions.
997 algo = solve_algo::gauss;
1000 // Eliminate the augmented matrix:
1002 case solve_algo::gauss:
1003 aug.gauss_elimination();
1005 case solve_algo::divfree:
1006 aug.division_free_elimination();
1008 case solve_algo::bareiss:
1010 aug.fraction_free_elimination();
1013 // assemble the solution matrix:
1015 for (unsigned co=0; co<p; ++co) {
1016 unsigned last_assigned_sol = n+1;
1017 for (int r=m-1; r>=0; --r) {
1018 unsigned fnz = 1; // first non-zero in row
1019 while ((fnz<=n) && (aug.m[r*(n+p)+(fnz-1)].is_zero()))
1022 // row consists only of zeros, corresponding rhs must be 0, too
1023 if (!aug.m[r*(n+p)+n+co].is_zero()) {
1024 throw (std::runtime_error("matrix::solve(): inconsistent linear system"));
1027 // assign solutions for vars between fnz+1 and
1028 // last_assigned_sol-1: free parameters
1029 for (unsigned c=fnz; c<last_assigned_sol-1; ++c)
1030 sol(c,co) = vars.m[c*p+co];
1031 ex e = aug.m[r*(n+p)+n+co];
1032 for (unsigned c=fnz; c<n; ++c)
1033 e -= aug.m[r*(n+p)+c]*sol.m[c*p+co];
1034 sol(fnz-1,co) = (e/(aug.m[r*(n+p)+(fnz-1)])).normal();
1035 last_assigned_sol = fnz;
1038 // assign solutions for vars between 1 and
1039 // last_assigned_sol-1: free parameters
1040 for (unsigned ro=0; ro<last_assigned_sol-1; ++ro)
1041 sol(ro,co) = vars(ro,co);
1050 /** Recursive determinant for small matrices having at least one symbolic
1051 * entry. The basic algorithm, known as Laplace-expansion, is enhanced by
1052 * some bookkeeping to avoid calculation of the same submatrices ("minors")
1053 * more than once. According to W.M.Gentleman and S.C.Johnson this algorithm
1054 * is better than elimination schemes for matrices of sparse multivariate
1055 * polynomials and also for matrices of dense univariate polynomials if the
1056 * matrix' dimesion is larger than 7.
1058 * @return the determinant as a new expression (in expanded form)
1059 * @see matrix::determinant() */
1060 ex matrix::determinant_minor(void) const
1062 // for small matrices the algorithm does not make any sense:
1063 const unsigned n = this->cols();
1065 return m[0].expand();
1067 return (m[0]*m[3]-m[2]*m[1]).expand();
1069 return (m[0]*m[4]*m[8]-m[0]*m[5]*m[7]-
1070 m[1]*m[3]*m[8]+m[2]*m[3]*m[7]+
1071 m[1]*m[5]*m[6]-m[2]*m[4]*m[6]).expand();
1073 // This algorithm can best be understood by looking at a naive
1074 // implementation of Laplace-expansion, like this one:
1076 // matrix minorM(this->rows()-1,this->cols()-1);
1077 // for (unsigned r1=0; r1<this->rows(); ++r1) {
1078 // // shortcut if element(r1,0) vanishes
1079 // if (m[r1*col].is_zero())
1081 // // assemble the minor matrix
1082 // for (unsigned r=0; r<minorM.rows(); ++r) {
1083 // for (unsigned c=0; c<minorM.cols(); ++c) {
1085 // minorM(r,c) = m[r*col+c+1];
1087 // minorM(r,c) = m[(r+1)*col+c+1];
1090 // // recurse down and care for sign:
1092 // det -= m[r1*col] * minorM.determinant_minor();
1094 // det += m[r1*col] * minorM.determinant_minor();
1096 // return det.expand();
1097 // What happens is that while proceeding down many of the minors are
1098 // computed more than once. In particular, there are binomial(n,k)
1099 // kxk minors and each one is computed factorial(n-k) times. Therefore
1100 // it is reasonable to store the results of the minors. We proceed from
1101 // right to left. At each column c we only need to retrieve the minors
1102 // calculated in step c-1. We therefore only have to store at most
1103 // 2*binomial(n,n/2) minors.
1105 // Unique flipper counter for partitioning into minors
1106 std::vector<unsigned> Pkey;
1108 // key for minor determinant (a subpartition of Pkey)
1109 std::vector<unsigned> Mkey;
1111 // we store our subminors in maps, keys being the rows they arise from
1112 typedef std::map<std::vector<unsigned>,class ex> Rmap;
1113 typedef std::map<std::vector<unsigned>,class ex>::value_type Rmap_value;
1117 // initialize A with last column:
1118 for (unsigned r=0; r<n; ++r) {
1119 Pkey.erase(Pkey.begin(),Pkey.end());
1121 A.insert(Rmap_value(Pkey,m[n*(r+1)-1]));
1123 // proceed from right to left through matrix
1124 for (int c=n-2; c>=0; --c) {
1125 Pkey.erase(Pkey.begin(),Pkey.end()); // don't change capacity
1126 Mkey.erase(Mkey.begin(),Mkey.end());
1127 for (unsigned i=0; i<n-c; ++i)
1129 unsigned fc = 0; // controls logic for our strange flipper counter
1132 for (unsigned r=0; r<n-c; ++r) {
1133 // maybe there is nothing to do?
1134 if (m[Pkey[r]*n+c].is_zero())
1136 // create the sorted key for all possible minors
1137 Mkey.erase(Mkey.begin(),Mkey.end());
1138 for (unsigned i=0; i<n-c; ++i)
1140 Mkey.push_back(Pkey[i]);
1141 // Fetch the minors and compute the new determinant
1143 det -= m[Pkey[r]*n+c]*A[Mkey];
1145 det += m[Pkey[r]*n+c]*A[Mkey];
1147 // prevent build-up of deep nesting of expressions saves time:
1149 // store the new determinant at its place in B:
1151 B.insert(Rmap_value(Pkey,det));
1152 // increment our strange flipper counter
1153 for (fc=n-c; fc>0; --fc) {
1155 if (Pkey[fc-1]<fc+c)
1159 for (unsigned j=fc; j<n-c; ++j)
1160 Pkey[j] = Pkey[j-1]+1;
1162 // next column, so change the role of A and B:
1171 /** Perform the steps of an ordinary Gaussian elimination to bring the m x n
1172 * matrix into an upper echelon form. The algorithm is ok for matrices
1173 * with numeric coefficients but quite unsuited for symbolic matrices.
1175 * @param det may be set to true to save a lot of space if one is only
1176 * interested in the diagonal elements (i.e. for calculating determinants).
1177 * The others are set to zero in this case.
1178 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1179 * number of rows was swapped and 0 if the matrix is singular. */
1180 int matrix::gauss_elimination(const bool det)
1182 ensure_if_modifiable();
1183 const unsigned m = this->rows();
1184 const unsigned n = this->cols();
1185 GINAC_ASSERT(!det || n==m);
1189 for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
1190 int indx = pivot(r0, r1, true);
1194 return 0; // leaves *this in a messy state
1199 for (unsigned r2=r0+1; r2<m; ++r2) {
1200 if (!this->m[r2*n+r1].is_zero()) {
1201 // yes, there is something to do in this row
1202 ex piv = this->m[r2*n+r1] / this->m[r0*n+r1];
1203 for (unsigned c=r1+1; c<n; ++c) {
1204 this->m[r2*n+c] -= piv * this->m[r0*n+c];
1205 if (!this->m[r2*n+c].info(info_flags::numeric))
1206 this->m[r2*n+c] = this->m[r2*n+c].normal();
1209 // fill up left hand side with zeros
1210 for (unsigned c=0; c<=r1; ++c)
1211 this->m[r2*n+c] = _ex0;
1214 // save space by deleting no longer needed elements
1215 for (unsigned c=r0+1; c<n; ++c)
1216 this->m[r0*n+c] = _ex0;
1226 /** Perform the steps of division free elimination to bring the m x n matrix
1227 * into an upper echelon form.
1229 * @param det may be set to true to save a lot of space if one is only
1230 * interested in the diagonal elements (i.e. for calculating determinants).
1231 * The others are set to zero in this case.
1232 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1233 * number of rows was swapped and 0 if the matrix is singular. */
1234 int matrix::division_free_elimination(const bool det)
1236 ensure_if_modifiable();
1237 const unsigned m = this->rows();
1238 const unsigned n = this->cols();
1239 GINAC_ASSERT(!det || n==m);
1243 for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
1244 int indx = pivot(r0, r1, true);
1248 return 0; // leaves *this in a messy state
1253 for (unsigned r2=r0+1; r2<m; ++r2) {
1254 for (unsigned c=r1+1; c<n; ++c)
1255 this->m[r2*n+c] = (this->m[r0*n+r1]*this->m[r2*n+c] - this->m[r2*n+r1]*this->m[r0*n+c]).expand();
1256 // fill up left hand side with zeros
1257 for (unsigned c=0; c<=r1; ++c)
1258 this->m[r2*n+c] = _ex0;
1261 // save space by deleting no longer needed elements
1262 for (unsigned c=r0+1; c<n; ++c)
1263 this->m[r0*n+c] = _ex0;
1273 /** Perform the steps of Bareiss' one-step fraction free elimination to bring
1274 * the matrix into an upper echelon form. Fraction free elimination means
1275 * that divide is used straightforwardly, without computing GCDs first. This
1276 * is possible, since we know the divisor at each step.
1278 * @param det may be set to true to save a lot of space if one is only
1279 * interested in the last element (i.e. for calculating determinants). The
1280 * others are set to zero in this case.
1281 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1282 * number of rows was swapped and 0 if the matrix is singular. */
1283 int matrix::fraction_free_elimination(const bool det)
1286 // (single-step fraction free elimination scheme, already known to Jordan)
1288 // Usual division-free elimination sets m[0](r,c) = m(r,c) and then sets
1289 // m[k+1](r,c) = m[k](k,k) * m[k](r,c) - m[k](r,k) * m[k](k,c).
1291 // Bareiss (fraction-free) elimination in addition divides that element
1292 // by m[k-1](k-1,k-1) for k>1, where it can be shown by means of the
1293 // Sylvester determinant that this really divides m[k+1](r,c).
1295 // We also allow rational functions where the original prove still holds.
1296 // However, we must care for numerator and denominator separately and
1297 // "manually" work in the integral domains because of subtle cancellations
1298 // (see below). This blows up the bookkeeping a bit and the formula has
1299 // to be modified to expand like this (N{x} stands for numerator of x,
1300 // D{x} for denominator of x):
1301 // 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)}
1302 // -N{m[k](r,k)}*N{m[k](k,c)}*D{m[k](k,k)}*D{m[k](r,c)}
1303 // 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)}
1304 // where for k>1 we now divide N{m[k+1](r,c)} by
1305 // N{m[k-1](k-1,k-1)}
1306 // and D{m[k+1](r,c)} by
1307 // D{m[k-1](k-1,k-1)}.
1309 ensure_if_modifiable();
1310 const unsigned m = this->rows();
1311 const unsigned n = this->cols();
1312 GINAC_ASSERT(!det || n==m);
1321 // We populate temporary matrices to subsequently operate on. There is
1322 // one holding numerators and another holding denominators of entries.
1323 // This is a must since the evaluator (or even earlier mul's constructor)
1324 // might cancel some trivial element which causes divide() to fail. The
1325 // elements are normalized first (yes, even though this algorithm doesn't
1326 // need GCDs) since the elements of *this might be unnormalized, which
1327 // makes things more complicated than they need to be.
1328 matrix tmp_n(*this);
1329 matrix tmp_d(m,n); // for denominators, if needed
1330 lst srl; // symbol replacement list
1331 exvector::const_iterator cit = this->m.begin(), citend = this->m.end();
1332 exvector::iterator tmp_n_it = tmp_n.m.begin(), tmp_d_it = tmp_d.m.begin();
1333 while (cit != citend) {
1334 ex nd = cit->normal().to_rational(srl).numer_denom();
1336 *tmp_n_it++ = nd.op(0);
1337 *tmp_d_it++ = nd.op(1);
1341 for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
1342 int indx = tmp_n.pivot(r0, r1, true);
1351 // tmp_n's rows r0 and indx were swapped, do the same in tmp_d:
1352 for (unsigned c=r1; c<n; ++c)
1353 tmp_d.m[n*indx+c].swap(tmp_d.m[n*r0+c]);
1355 for (unsigned r2=r0+1; r2<m; ++r2) {
1356 for (unsigned c=r1+1; c<n; ++c) {
1357 dividend_n = (tmp_n.m[r0*n+r1]*tmp_n.m[r2*n+c]*
1358 tmp_d.m[r2*n+r1]*tmp_d.m[r0*n+c]
1359 -tmp_n.m[r2*n+r1]*tmp_n.m[r0*n+c]*
1360 tmp_d.m[r0*n+r1]*tmp_d.m[r2*n+c]).expand();
1361 dividend_d = (tmp_d.m[r2*n+r1]*tmp_d.m[r0*n+c]*
1362 tmp_d.m[r0*n+r1]*tmp_d.m[r2*n+c]).expand();
1363 bool check = divide(dividend_n, divisor_n,
1364 tmp_n.m[r2*n+c], true);
1365 check &= divide(dividend_d, divisor_d,
1366 tmp_d.m[r2*n+c], true);
1367 GINAC_ASSERT(check);
1369 // fill up left hand side with zeros
1370 for (unsigned c=0; c<=r1; ++c)
1371 tmp_n.m[r2*n+c] = _ex0;
1373 if ((r1<n-1)&&(r0<m-1)) {
1374 // compute next iteration's divisor
1375 divisor_n = tmp_n.m[r0*n+r1].expand();
1376 divisor_d = tmp_d.m[r0*n+r1].expand();
1378 // save space by deleting no longer needed elements
1379 for (unsigned c=0; c<n; ++c) {
1380 tmp_n.m[r0*n+c] = _ex0;
1381 tmp_d.m[r0*n+c] = _ex1;
1388 // repopulate *this matrix:
1389 exvector::iterator it = this->m.begin(), itend = this->m.end();
1390 tmp_n_it = tmp_n.m.begin();
1391 tmp_d_it = tmp_d.m.begin();
1393 *it++ = ((*tmp_n_it++)/(*tmp_d_it++)).subs(srl);
1399 /** Partial pivoting method for matrix elimination schemes.
1400 * Usual pivoting (symbolic==false) returns the index to the element with the
1401 * largest absolute value in column ro and swaps the current row with the one
1402 * where the element was found. With (symbolic==true) it does the same thing
1403 * with the first non-zero element.
1405 * @param ro is the row from where to begin
1406 * @param co is the column to be inspected
1407 * @param symbolic signal if we want the first non-zero element to be pivoted
1408 * (true) or the one with the largest absolute value (false).
1409 * @return 0 if no interchange occured, -1 if all are zero (usually signaling
1410 * a degeneracy) and positive integer k means that rows ro and k were swapped.
1412 int matrix::pivot(unsigned ro, unsigned co, bool symbolic)
1416 // search first non-zero element in column co beginning at row ro
1417 while ((k<row) && (this->m[k*col+co].expand().is_zero()))
1420 // search largest element in column co beginning at row ro
1421 GINAC_ASSERT(is_exactly_a<numeric>(this->m[k*col+co]));
1422 unsigned kmax = k+1;
1423 numeric mmax = abs(ex_to<numeric>(m[kmax*col+co]));
1425 GINAC_ASSERT(is_exactly_a<numeric>(this->m[kmax*col+co]));
1426 numeric tmp = ex_to<numeric>(this->m[kmax*col+co]);
1427 if (abs(tmp) > mmax) {
1433 if (!mmax.is_zero())
1437 // all elements in column co below row ro vanish
1440 // matrix needs no pivoting
1442 // matrix needs pivoting, so swap rows k and ro
1443 ensure_if_modifiable();
1444 for (unsigned c=0; c<col; ++c)
1445 this->m[k*col+c].swap(this->m[ro*col+c]);
1450 ex lst_to_matrix(const lst & l)
1452 // Find number of rows and columns
1453 unsigned rows = l.nops(), cols = 0, i, j;
1454 for (i=0; i<rows; i++)
1455 if (l.op(i).nops() > cols)
1456 cols = l.op(i).nops();
1458 // Allocate and fill matrix
1459 matrix &M = *new matrix(rows, cols);
1460 M.setflag(status_flags::dynallocated);
1461 for (i=0; i<rows; i++)
1462 for (j=0; j<cols; j++)
1463 if (l.op(i).nops() > j)
1464 M(i, j) = l.op(i).op(j);
1470 ex diag_matrix(const lst & l)
1472 unsigned dim = l.nops();
1474 matrix &m = *new matrix(dim, dim);
1475 m.setflag(status_flags::dynallocated);
1476 for (unsigned i=0; i<dim; i++)
1482 ex unit_matrix(unsigned r, unsigned c)
1484 matrix &Id = *new matrix(r, c);
1485 Id.setflag(status_flags::dynallocated);
1486 for (unsigned i=0; i<r && i<c; i++)
1492 ex symbolic_matrix(unsigned r, unsigned c, const std::string & base_name, const std::string & tex_base_name)
1494 matrix &M = *new matrix(r, c);
1495 M.setflag(status_flags::dynallocated | status_flags::evaluated);
1497 bool long_format = (r > 10 || c > 10);
1498 bool single_row = (r == 1 || c == 1);
1500 for (unsigned i=0; i<r; i++) {
1501 for (unsigned j=0; j<c; j++) {
1502 std::ostringstream s1, s2;
1504 s2 << tex_base_name << "_{";
1515 s1 << '_' << i << '_' << j;
1516 s2 << i << ';' << j << "}";
1519 s2 << i << j << '}';
1522 M(i, j) = symbol(s1.str(), s2.str());
1529 } // namespace GiNaC