3 * Implementation of symbolic matrices */
6 * GiNaC Copyright (C) 1999-2001 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
42 GINAC_IMPLEMENT_REGISTERED_CLASS(matrix, basic)
45 // default ctor, dtor, copy ctor, assignment operator and helpers:
48 /** Default ctor. Initializes to 1 x 1-dimensional zero-matrix. */
49 matrix::matrix() : inherited(TINFO_matrix), row(1), col(1)
51 debugmsg("matrix default ctor",LOGLEVEL_CONSTRUCT);
55 void matrix::copy(const matrix & other)
57 inherited::copy(other);
60 m = other.m; // STL's vector copying invoked here
63 DEFAULT_DESTROY(matrix)
71 /** Very common ctor. Initializes to r x c-dimensional zero-matrix.
73 * @param r number of rows
74 * @param c number of cols */
75 matrix::matrix(unsigned r, unsigned c)
76 : inherited(TINFO_matrix), row(r), col(c)
78 debugmsg("matrix ctor from unsigned,unsigned",LOGLEVEL_CONSTRUCT);
79 m.resize(r*c, _ex0());
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 debugmsg("matrix ctor from unsigned,unsigned,exvector",LOGLEVEL_CONSTRUCT);
91 /** Construct matrix from (flat) list of elements. If the list has fewer
92 * elements than the matrix, the remaining matrix elements are set to zero.
93 * If the list has more elements than the matrix, the excessive elements are
95 matrix::matrix(unsigned r, unsigned c, const lst & l)
96 : inherited(TINFO_matrix), row(r), col(c)
98 debugmsg("matrix ctor from unsigned,unsigned,lst",LOGLEVEL_CONSTRUCT);
99 m.resize(r*c, _ex0());
101 for (unsigned i=0; i<l.nops(); i++) {
105 break; // matrix smaller than list: throw away excessive elements
114 matrix::matrix(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
116 debugmsg("matrix ctor from archive_node", LOGLEVEL_CONSTRUCT);
117 if (!(n.find_unsigned("row", row)) || !(n.find_unsigned("col", col)))
118 throw (std::runtime_error("unknown matrix dimensions in archive"));
119 m.reserve(row * col);
120 for (unsigned int i=0; true; i++) {
122 if (n.find_ex("m", e, sym_lst, i))
129 void matrix::archive(archive_node &n) const
131 inherited::archive(n);
132 n.add_unsigned("row", row);
133 n.add_unsigned("col", col);
134 exvector::const_iterator i = m.begin(), iend = m.end();
141 DEFAULT_UNARCHIVE(matrix)
144 // functions overriding virtual functions from base classes
149 void matrix::print(const print_context & c, unsigned level) const
151 debugmsg("matrix print", LOGLEVEL_PRINT);
153 if (is_of_type(c, print_tree)) {
155 inherited::print(c, level);
160 for (unsigned y=0; y<row-1; ++y) {
162 for (unsigned x=0; x<col-1; ++x) {
166 m[col*(y+1)-1].print(c);
170 for (unsigned x=0; x<col-1; ++x) {
171 m[(row-1)*col+x].print(c);
174 m[row*col-1].print(c);
180 /** nops is defined to be rows x columns. */
181 unsigned matrix::nops() const
186 /** returns matrix entry at position (i/col, i%col). */
187 ex matrix::op(int i) const
192 /** returns matrix entry at position (i/col, i%col). */
193 ex & matrix::let_op(int i)
196 GINAC_ASSERT(i<nops());
201 /** Evaluate matrix entry by entry. */
202 ex matrix::eval(int level) const
204 debugmsg("matrix eval",LOGLEVEL_MEMBER_FUNCTION);
206 // check if we have to do anything at all
207 if ((level==1)&&(flags & status_flags::evaluated))
211 if (level == -max_recursion_level)
212 throw (std::runtime_error("matrix::eval(): recursion limit exceeded"));
214 // eval() entry by entry
215 exvector m2(row*col);
217 for (unsigned r=0; r<row; ++r)
218 for (unsigned c=0; c<col; ++c)
219 m2[r*col+c] = m[r*col+c].eval(level);
221 return (new matrix(row, col, m2))->setflag(status_flags::dynallocated |
222 status_flags::evaluated );
225 ex matrix::subs(const lst & ls, const lst & lr, bool no_pattern) const
227 exvector m2(row * col);
228 for (unsigned r=0; r<row; ++r)
229 for (unsigned c=0; c<col; ++c)
230 m2[r*col+c] = m[r*col+c].subs(ls, lr, no_pattern);
232 return ex(matrix(row, col, m2)).bp->basic::subs(ls, lr, no_pattern);
237 int matrix::compare_same_type(const basic & other) const
239 GINAC_ASSERT(is_exactly_of_type(other, matrix));
240 const matrix & o = static_cast<const matrix &>(other);
242 // compare number of rows
244 return row < o.rows() ? -1 : 1;
246 // compare number of columns
248 return col < o.cols() ? -1 : 1;
250 // equal number of rows and columns, compare individual elements
252 for (unsigned r=0; r<row; ++r) {
253 for (unsigned c=0; c<col; ++c) {
254 cmpval = ((*this)(r,c)).compare(o(r,c));
255 if (cmpval!=0) return cmpval;
258 // all elements are equal => matrices are equal;
262 bool matrix::match_same_type(const basic & other) const
264 GINAC_ASSERT(is_exactly_of_type(other, matrix));
265 const matrix & o = static_cast<const matrix &>(other);
267 // The number of rows and columns must be the same. This is necessary to
268 // prevent a 2x3 matrix from matching a 3x2 one.
269 return row == o.rows() && col == o.cols();
272 /** Automatic symbolic evaluation of an indexed matrix. */
273 ex matrix::eval_indexed(const basic & i) const
275 GINAC_ASSERT(is_of_type(i, indexed));
276 GINAC_ASSERT(is_ex_of_type(i.op(0), matrix));
278 bool all_indices_unsigned = static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint);
283 // One index, must be one-dimensional vector
284 if (row != 1 && col != 1)
285 throw (std::runtime_error("matrix::eval_indexed(): vector must have exactly 1 index"));
287 const idx & i1 = ex_to<idx>(i.op(1));
292 if (!i1.get_dim().is_equal(row))
293 throw (std::runtime_error("matrix::eval_indexed(): dimension of index must match number of vector elements"));
295 // Index numeric -> return vector element
296 if (all_indices_unsigned) {
297 unsigned n1 = ex_to<numeric>(i1.get_value()).to_int();
299 throw (std::runtime_error("matrix::eval_indexed(): value of index exceeds number of vector elements"));
300 return (*this)(n1, 0);
306 if (!i1.get_dim().is_equal(col))
307 throw (std::runtime_error("matrix::eval_indexed(): dimension of index must match number of vector elements"));
309 // Index numeric -> return vector element
310 if (all_indices_unsigned) {
311 unsigned n1 = ex_to<numeric>(i1.get_value()).to_int();
313 throw (std::runtime_error("matrix::eval_indexed(): value of index exceeds number of vector elements"));
314 return (*this)(0, n1);
318 } else if (i.nops() == 3) {
321 const idx & i1 = ex_to<idx>(i.op(1));
322 const idx & i2 = ex_to<idx>(i.op(2));
324 if (!i1.get_dim().is_equal(row))
325 throw (std::runtime_error("matrix::eval_indexed(): dimension of first index must match number of rows"));
326 if (!i2.get_dim().is_equal(col))
327 throw (std::runtime_error("matrix::eval_indexed(): dimension of second index must match number of columns"));
329 // Pair of dummy indices -> compute trace
330 if (is_dummy_pair(i1, i2))
333 // Both indices numeric -> return matrix element
334 if (all_indices_unsigned) {
335 unsigned n1 = ex_to<numeric>(i1.get_value()).to_int(), n2 = ex_to<numeric>(i2.get_value()).to_int();
337 throw (std::runtime_error("matrix::eval_indexed(): value of first index exceeds number of rows"));
339 throw (std::runtime_error("matrix::eval_indexed(): value of second index exceeds number of columns"));
340 return (*this)(n1, n2);
344 throw (std::runtime_error("matrix::eval_indexed(): matrix must have exactly 2 indices"));
349 /** Sum of two indexed matrices. */
350 ex matrix::add_indexed(const ex & self, const ex & other) const
352 GINAC_ASSERT(is_ex_of_type(self, indexed));
353 GINAC_ASSERT(is_ex_of_type(self.op(0), matrix));
354 GINAC_ASSERT(is_ex_of_type(other, indexed));
355 GINAC_ASSERT(self.nops() == 2 || self.nops() == 3);
357 // Only add two matrices
358 if (is_ex_of_type(other.op(0), matrix)) {
359 GINAC_ASSERT(other.nops() == 2 || other.nops() == 3);
361 const matrix &self_matrix = ex_to<matrix>(self.op(0));
362 const matrix &other_matrix = ex_to<matrix>(other.op(0));
364 if (self.nops() == 2 && other.nops() == 2) { // vector + vector
366 if (self_matrix.row == other_matrix.row)
367 return indexed(self_matrix.add(other_matrix), self.op(1));
368 else if (self_matrix.row == other_matrix.col)
369 return indexed(self_matrix.add(other_matrix.transpose()), self.op(1));
371 } else if (self.nops() == 3 && other.nops() == 3) { // matrix + matrix
373 if (self.op(1).is_equal(other.op(1)) && self.op(2).is_equal(other.op(2)))
374 return indexed(self_matrix.add(other_matrix), self.op(1), self.op(2));
375 else if (self.op(1).is_equal(other.op(2)) && self.op(2).is_equal(other.op(1)))
376 return indexed(self_matrix.add(other_matrix.transpose()), self.op(1), self.op(2));
381 // Don't know what to do, return unevaluated sum
385 /** Product of an indexed matrix with a number. */
386 ex matrix::scalar_mul_indexed(const ex & self, const numeric & other) const
388 GINAC_ASSERT(is_ex_of_type(self, indexed));
389 GINAC_ASSERT(is_ex_of_type(self.op(0), matrix));
390 GINAC_ASSERT(self.nops() == 2 || self.nops() == 3);
392 const matrix &self_matrix = ex_to<matrix>(self.op(0));
394 if (self.nops() == 2)
395 return indexed(self_matrix.mul(other), self.op(1));
396 else // self.nops() == 3
397 return indexed(self_matrix.mul(other), self.op(1), self.op(2));
400 /** Contraction of an indexed matrix with something else. */
401 bool matrix::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
403 GINAC_ASSERT(is_ex_of_type(*self, indexed));
404 GINAC_ASSERT(is_ex_of_type(*other, indexed));
405 GINAC_ASSERT(self->nops() == 2 || self->nops() == 3);
406 GINAC_ASSERT(is_ex_of_type(self->op(0), matrix));
408 // Only contract with other matrices
409 if (!is_ex_of_type(other->op(0), matrix))
412 GINAC_ASSERT(other->nops() == 2 || other->nops() == 3);
414 const matrix &self_matrix = ex_to<matrix>(self->op(0));
415 const matrix &other_matrix = ex_to<matrix>(other->op(0));
417 if (self->nops() == 2) {
419 if (other->nops() == 2) { // vector * vector (scalar product)
421 if (self_matrix.col == 1) {
422 if (other_matrix.col == 1) {
423 // Column vector * column vector, transpose first vector
424 *self = self_matrix.transpose().mul(other_matrix)(0, 0);
426 // Column vector * row vector, swap factors
427 *self = other_matrix.mul(self_matrix)(0, 0);
430 if (other_matrix.col == 1) {
431 // Row vector * column vector, perfect
432 *self = self_matrix.mul(other_matrix)(0, 0);
434 // Row vector * row vector, transpose second vector
435 *self = self_matrix.mul(other_matrix.transpose())(0, 0);
441 } else { // vector * matrix
443 // B_i * A_ij = (B*A)_j (B is row vector)
444 if (is_dummy_pair(self->op(1), other->op(1))) {
445 if (self_matrix.row == 1)
446 *self = indexed(self_matrix.mul(other_matrix), other->op(2));
448 *self = indexed(self_matrix.transpose().mul(other_matrix), other->op(2));
453 // B_j * A_ij = (A*B)_i (B is column vector)
454 if (is_dummy_pair(self->op(1), other->op(2))) {
455 if (self_matrix.col == 1)
456 *self = indexed(other_matrix.mul(self_matrix), other->op(1));
458 *self = indexed(other_matrix.mul(self_matrix.transpose()), other->op(1));
464 } else if (other->nops() == 3) { // matrix * matrix
466 // A_ij * B_jk = (A*B)_ik
467 if (is_dummy_pair(self->op(2), other->op(1))) {
468 *self = indexed(self_matrix.mul(other_matrix), self->op(1), other->op(2));
473 // A_ij * B_kj = (A*Btrans)_ik
474 if (is_dummy_pair(self->op(2), other->op(2))) {
475 *self = indexed(self_matrix.mul(other_matrix.transpose()), self->op(1), other->op(1));
480 // A_ji * B_jk = (Atrans*B)_ik
481 if (is_dummy_pair(self->op(1), other->op(1))) {
482 *self = indexed(self_matrix.transpose().mul(other_matrix), self->op(2), other->op(2));
487 // A_ji * B_kj = (B*A)_ki
488 if (is_dummy_pair(self->op(1), other->op(2))) {
489 *self = indexed(other_matrix.mul(self_matrix), other->op(1), self->op(2));
500 // non-virtual functions in this class
507 * @exception logic_error (incompatible matrices) */
508 matrix matrix::add(const matrix & other) const
510 if (col != other.col || row != other.row)
511 throw std::logic_error("matrix::add(): incompatible matrices");
513 exvector sum(this->m);
514 exvector::iterator i = sum.begin(), end = sum.end();
515 exvector::const_iterator ci = other.m.begin();
519 return matrix(row,col,sum);
523 /** Difference of matrices.
525 * @exception logic_error (incompatible matrices) */
526 matrix matrix::sub(const matrix & other) const
528 if (col != other.col || row != other.row)
529 throw std::logic_error("matrix::sub(): incompatible matrices");
531 exvector dif(this->m);
532 exvector::iterator i = dif.begin(), end = dif.end();
533 exvector::const_iterator ci = other.m.begin();
537 return matrix(row,col,dif);
541 /** Product of matrices.
543 * @exception logic_error (incompatible matrices) */
544 matrix matrix::mul(const matrix & other) const
546 if (this->cols() != other.rows())
547 throw std::logic_error("matrix::mul(): incompatible matrices");
549 exvector prod(this->rows()*other.cols());
551 for (unsigned r1=0; r1<this->rows(); ++r1) {
552 for (unsigned c=0; c<this->cols(); ++c) {
553 if (m[r1*col+c].is_zero())
555 for (unsigned r2=0; r2<other.cols(); ++r2)
556 prod[r1*other.col+r2] += (m[r1*col+c] * other.m[c*other.col+r2]).expand();
559 return matrix(row, other.col, prod);
563 /** Product of matrix and scalar. */
564 matrix matrix::mul(const numeric & other) const
566 exvector prod(row * col);
568 for (unsigned r=0; r<row; ++r)
569 for (unsigned c=0; c<col; ++c)
570 prod[r*col+c] = m[r*col+c] * other;
572 return matrix(row, col, prod);
576 /** Product of matrix and scalar expression. */
577 matrix matrix::mul_scalar(const ex & other) const
579 if (other.return_type() != return_types::commutative)
580 throw std::runtime_error("matrix::mul_scalar(): non-commutative scalar");
582 exvector prod(row * col);
584 for (unsigned r=0; r<row; ++r)
585 for (unsigned c=0; c<col; ++c)
586 prod[r*col+c] = m[r*col+c] * other;
588 return matrix(row, col, prod);
592 /** Power of a matrix. Currently handles integer exponents only. */
593 matrix matrix::pow(const ex & expn) const
596 throw (std::logic_error("matrix::pow(): matrix not square"));
598 if (is_ex_exactly_of_type(expn, numeric)) {
599 // Integer cases are computed by successive multiplication, using the
600 // obvious shortcut of storing temporaries, like A^4 == (A*A)*(A*A).
601 if (expn.info(info_flags::integer)) {
602 numeric b = ex_to<numeric>(expn);
604 if (expn.info(info_flags::negative)) {
611 for (unsigned r=0; r<row; ++r)
613 // This loop computes the representation of b in base 2 from right
614 // to left and multiplies the factors whenever needed. Note
615 // that this is not entirely optimal but close to optimal and
616 // "better" algorithms are much harder to implement. (See Knuth,
617 // TAoCP2, section "Evaluation of Powers" for a good discussion.)
623 b *= _num1_2(); // b /= 2, still integer.
629 throw (std::runtime_error("matrix::pow(): don't know how to handle exponent"));
633 /** operator() to access elements for reading.
635 * @param ro row of element
636 * @param co column of element
637 * @exception range_error (index out of range) */
638 const ex & matrix::operator() (unsigned ro, unsigned co) const
640 if (ro>=row || co>=col)
641 throw (std::range_error("matrix::operator(): index out of range"));
647 /** operator() to access elements for writing.
649 * @param ro row of element
650 * @param co column of element
651 * @exception range_error (index out of range) */
652 ex & matrix::operator() (unsigned ro, unsigned co)
654 if (ro>=row || co>=col)
655 throw (std::range_error("matrix::operator(): index out of range"));
657 ensure_if_modifiable();
662 /** Transposed of an m x n matrix, producing a new n x m matrix object that
663 * represents the transposed. */
664 matrix matrix::transpose(void) const
666 exvector trans(this->cols()*this->rows());
668 for (unsigned r=0; r<this->cols(); ++r)
669 for (unsigned c=0; c<this->rows(); ++c)
670 trans[r*this->rows()+c] = m[c*this->cols()+r];
672 return matrix(this->cols(),this->rows(),trans);
675 /** Determinant of square matrix. This routine doesn't actually calculate the
676 * determinant, it only implements some heuristics about which algorithm to
677 * run. If all the elements of the matrix are elements of an integral domain
678 * the determinant is also in that integral domain and the result is expanded
679 * only. If one or more elements are from a quotient field the determinant is
680 * usually also in that quotient field and the result is normalized before it
681 * is returned. This implies that the determinant of the symbolic 2x2 matrix
682 * [[a/(a-b),1],[b/(a-b),1]] is returned as unity. (In this respect, it
683 * behaves like MapleV and unlike Mathematica.)
685 * @param algo allows to chose an algorithm
686 * @return the determinant as a new expression
687 * @exception logic_error (matrix not square)
688 * @see determinant_algo */
689 ex matrix::determinant(unsigned algo) const
692 throw (std::logic_error("matrix::determinant(): matrix not square"));
693 GINAC_ASSERT(row*col==m.capacity());
695 // Gather some statistical information about this matrix:
696 bool numeric_flag = true;
697 bool normal_flag = false;
698 unsigned sparse_count = 0; // counts non-zero elements
699 exvector::const_iterator r = m.begin(), rend = m.end();
701 lst srl; // symbol replacement list
702 ex rtest = r->to_rational(srl);
703 if (!rtest.is_zero())
705 if (!rtest.info(info_flags::numeric))
706 numeric_flag = false;
707 if (!rtest.info(info_flags::crational_polynomial) &&
708 rtest.info(info_flags::rational_function))
713 // Here is the heuristics in case this routine has to decide:
714 if (algo == determinant_algo::automatic) {
715 // Minor expansion is generally a good guess:
716 algo = determinant_algo::laplace;
717 // Does anybody know when a matrix is really sparse?
718 // Maybe <~row/2.236 nonzero elements average in a row?
719 if (row>3 && 5*sparse_count<=row*col)
720 algo = determinant_algo::bareiss;
721 // Purely numeric matrix can be handled by Gauss elimination.
722 // This overrides any prior decisions.
724 algo = determinant_algo::gauss;
727 // Trap the trivial case here, since some algorithms don't like it
729 // for consistency with non-trivial determinants...
731 return m[0].normal();
733 return m[0].expand();
736 // Compute the determinant
738 case determinant_algo::gauss: {
741 int sign = tmp.gauss_elimination(true);
742 for (unsigned d=0; d<row; ++d)
743 det *= tmp.m[d*col+d];
745 return (sign*det).normal();
747 return (sign*det).normal().expand();
749 case determinant_algo::bareiss: {
752 sign = tmp.fraction_free_elimination(true);
754 return (sign*tmp.m[row*col-1]).normal();
756 return (sign*tmp.m[row*col-1]).expand();
758 case determinant_algo::divfree: {
761 sign = tmp.division_free_elimination(true);
764 ex det = tmp.m[row*col-1];
765 // factor out accumulated bogus slag
766 for (unsigned d=0; d<row-2; ++d)
767 for (unsigned j=0; j<row-d-2; ++j)
768 det = (det/tmp.m[d*col+d]).normal();
771 case determinant_algo::laplace:
773 // This is the minor expansion scheme. We always develop such
774 // that the smallest minors (i.e, the trivial 1x1 ones) are on the
775 // rightmost column. For this to be efficient it turns out that
776 // the emptiest columns (i.e. the ones with most zeros) should be
777 // the ones on the right hand side. Therefore we presort the
778 // columns of the matrix:
779 typedef std::pair<unsigned,unsigned> uintpair;
780 std::vector<uintpair> c_zeros; // number of zeros in column
781 for (unsigned c=0; c<col; ++c) {
783 for (unsigned r=0; r<row; ++r)
784 if (m[r*col+c].is_zero())
786 c_zeros.push_back(uintpair(acc,c));
788 sort(c_zeros.begin(),c_zeros.end());
789 std::vector<unsigned> pre_sort;
790 for (std::vector<uintpair>::const_iterator i=c_zeros.begin(); i!=c_zeros.end(); ++i)
791 pre_sort.push_back(i->second);
792 std::vector<unsigned> pre_sort_test(pre_sort); // permutation_sign() modifies the vector so we make a copy here
793 int sign = permutation_sign(pre_sort_test.begin(), pre_sort_test.end());
794 exvector result(row*col); // represents sorted matrix
796 for (std::vector<unsigned>::const_iterator i=pre_sort.begin();
799 for (unsigned r=0; r<row; ++r)
800 result[r*col+c] = m[r*col+(*i)];
804 return (sign*matrix(row,col,result).determinant_minor()).normal();
806 return sign*matrix(row,col,result).determinant_minor();
812 /** Trace of a matrix. The result is normalized if it is in some quotient
813 * field and expanded only otherwise. This implies that the trace of the
814 * symbolic 2x2 matrix [[a/(a-b),x],[y,b/(b-a)]] is recognized to be unity.
816 * @return the sum of diagonal elements
817 * @exception logic_error (matrix not square) */
818 ex matrix::trace(void) const
821 throw (std::logic_error("matrix::trace(): matrix not square"));
824 for (unsigned r=0; r<col; ++r)
827 if (tr.info(info_flags::rational_function) &&
828 !tr.info(info_flags::crational_polynomial))
835 /** Characteristic Polynomial. Following mathematica notation the
836 * characteristic polynomial of a matrix M is defined as the determiant of
837 * (M - lambda * 1) where 1 stands for the unit matrix of the same dimension
838 * as M. Note that some CASs define it with a sign inside the determinant
839 * which gives rise to an overall sign if the dimension is odd. This method
840 * returns the characteristic polynomial collected in powers of lambda as a
843 * @return characteristic polynomial as new expression
844 * @exception logic_error (matrix not square)
845 * @see matrix::determinant() */
846 ex matrix::charpoly(const symbol & lambda) const
849 throw (std::logic_error("matrix::charpoly(): matrix not square"));
851 bool numeric_flag = true;
852 exvector::const_iterator r = m.begin(), rend = m.end();
854 if (!r->info(info_flags::numeric))
855 numeric_flag = false;
859 // The pure numeric case is traditionally rather common. Hence, it is
860 // trapped and we use Leverrier's algorithm which goes as row^3 for
861 // every coefficient. The expensive part is the matrix multiplication.
865 ex poly = power(lambda,row)-c*power(lambda,row-1);
866 for (unsigned i=1; i<row; ++i) {
867 for (unsigned j=0; j<row; ++j)
870 c = B.trace()/ex(i+1);
871 poly -= c*power(lambda,row-i-1);
880 for (unsigned r=0; r<col; ++r)
881 M.m[r*col+r] -= lambda;
883 return M.determinant().collect(lambda);
887 /** Inverse of this matrix.
889 * @return the inverted matrix
890 * @exception logic_error (matrix not square)
891 * @exception runtime_error (singular matrix) */
892 matrix matrix::inverse(void) const
895 throw (std::logic_error("matrix::inverse(): matrix not square"));
897 // This routine actually doesn't do anything fancy at all. We compute the
898 // inverse of the matrix A by solving the system A * A^{-1} == Id.
900 // First populate the identity matrix supposed to become the right hand side.
901 matrix identity(row,col);
902 for (unsigned i=0; i<row; ++i)
903 identity(i,i) = _ex1();
905 // Populate a dummy matrix of variables, just because of compatibility with
906 // matrix::solve() which wants this (for compatibility with under-determined
907 // systems of equations).
908 matrix vars(row,col);
909 for (unsigned r=0; r<row; ++r)
910 for (unsigned c=0; c<col; ++c)
911 vars(r,c) = symbol();
915 sol = this->solve(vars,identity);
916 } catch (const std::runtime_error & e) {
917 if (e.what()==std::string("matrix::solve(): inconsistent linear system"))
918 throw (std::runtime_error("matrix::inverse(): singular matrix"));
926 /** Solve a linear system consisting of a m x n matrix and a m x p right hand
927 * side by applying an elimination scheme to the augmented matrix.
929 * @param vars n x p matrix, all elements must be symbols
930 * @param rhs m x p matrix
931 * @return n x p solution matrix
932 * @exception logic_error (incompatible matrices)
933 * @exception invalid_argument (1st argument must be matrix of symbols)
934 * @exception runtime_error (inconsistent linear system)
936 matrix matrix::solve(const matrix & vars,
940 const unsigned m = this->rows();
941 const unsigned n = this->cols();
942 const unsigned p = rhs.cols();
945 if ((rhs.rows() != m) || (vars.rows() != n) || (vars.col != p))
946 throw (std::logic_error("matrix::solve(): incompatible matrices"));
947 for (unsigned ro=0; ro<n; ++ro)
948 for (unsigned co=0; co<p; ++co)
949 if (!vars(ro,co).info(info_flags::symbol))
950 throw (std::invalid_argument("matrix::solve(): 1st argument must be matrix of symbols"));
952 // build the augmented matrix of *this with rhs attached to the right
954 for (unsigned r=0; r<m; ++r) {
955 for (unsigned c=0; c<n; ++c)
956 aug.m[r*(n+p)+c] = this->m[r*n+c];
957 for (unsigned c=0; c<p; ++c)
958 aug.m[r*(n+p)+c+n] = rhs.m[r*p+c];
961 // Gather some statistical information about the augmented matrix:
962 bool numeric_flag = true;
963 exvector::const_iterator r = aug.m.begin(), rend = aug.m.end();
965 if (!r->info(info_flags::numeric))
966 numeric_flag = false;
970 // Here is the heuristics in case this routine has to decide:
971 if (algo == solve_algo::automatic) {
972 // Bareiss (fraction-free) elimination is generally a good guess:
973 algo = solve_algo::bareiss;
974 // For m<3, Bareiss elimination is equivalent to division free
975 // elimination but has more logistic overhead
977 algo = solve_algo::divfree;
978 // This overrides any prior decisions.
980 algo = solve_algo::gauss;
983 // Eliminate the augmented matrix:
985 case solve_algo::gauss:
986 aug.gauss_elimination();
988 case solve_algo::divfree:
989 aug.division_free_elimination();
991 case solve_algo::bareiss:
993 aug.fraction_free_elimination();
996 // assemble the solution matrix:
998 for (unsigned co=0; co<p; ++co) {
999 unsigned last_assigned_sol = n+1;
1000 for (int r=m-1; r>=0; --r) {
1001 unsigned fnz = 1; // first non-zero in row
1002 while ((fnz<=n) && (aug.m[r*(n+p)+(fnz-1)].is_zero()))
1005 // row consists only of zeros, corresponding rhs must be 0, too
1006 if (!aug.m[r*(n+p)+n+co].is_zero()) {
1007 throw (std::runtime_error("matrix::solve(): inconsistent linear system"));
1010 // assign solutions for vars between fnz+1 and
1011 // last_assigned_sol-1: free parameters
1012 for (unsigned c=fnz; c<last_assigned_sol-1; ++c)
1013 sol(c,co) = vars.m[c*p+co];
1014 ex e = aug.m[r*(n+p)+n+co];
1015 for (unsigned c=fnz; c<n; ++c)
1016 e -= aug.m[r*(n+p)+c]*sol.m[c*p+co];
1017 sol(fnz-1,co) = (e/(aug.m[r*(n+p)+(fnz-1)])).normal();
1018 last_assigned_sol = fnz;
1021 // assign solutions for vars between 1 and
1022 // last_assigned_sol-1: free parameters
1023 for (unsigned ro=0; ro<last_assigned_sol-1; ++ro)
1024 sol(ro,co) = vars(ro,co);
1033 /** Recursive determinant for small matrices having at least one symbolic
1034 * entry. The basic algorithm, known as Laplace-expansion, is enhanced by
1035 * some bookkeeping to avoid calculation of the same submatrices ("minors")
1036 * more than once. According to W.M.Gentleman and S.C.Johnson this algorithm
1037 * is better than elimination schemes for matrices of sparse multivariate
1038 * polynomials and also for matrices of dense univariate polynomials if the
1039 * matrix' dimesion is larger than 7.
1041 * @return the determinant as a new expression (in expanded form)
1042 * @see matrix::determinant() */
1043 ex matrix::determinant_minor(void) const
1045 // for small matrices the algorithm does not make any sense:
1046 const unsigned n = this->cols();
1048 return m[0].expand();
1050 return (m[0]*m[3]-m[2]*m[1]).expand();
1052 return (m[0]*m[4]*m[8]-m[0]*m[5]*m[7]-
1053 m[1]*m[3]*m[8]+m[2]*m[3]*m[7]+
1054 m[1]*m[5]*m[6]-m[2]*m[4]*m[6]).expand();
1056 // This algorithm can best be understood by looking at a naive
1057 // implementation of Laplace-expansion, like this one:
1059 // matrix minorM(this->rows()-1,this->cols()-1);
1060 // for (unsigned r1=0; r1<this->rows(); ++r1) {
1061 // // shortcut if element(r1,0) vanishes
1062 // if (m[r1*col].is_zero())
1064 // // assemble the minor matrix
1065 // for (unsigned r=0; r<minorM.rows(); ++r) {
1066 // for (unsigned c=0; c<minorM.cols(); ++c) {
1068 // minorM(r,c) = m[r*col+c+1];
1070 // minorM(r,c) = m[(r+1)*col+c+1];
1073 // // recurse down and care for sign:
1075 // det -= m[r1*col] * minorM.determinant_minor();
1077 // det += m[r1*col] * minorM.determinant_minor();
1079 // return det.expand();
1080 // What happens is that while proceeding down many of the minors are
1081 // computed more than once. In particular, there are binomial(n,k)
1082 // kxk minors and each one is computed factorial(n-k) times. Therefore
1083 // it is reasonable to store the results of the minors. We proceed from
1084 // right to left. At each column c we only need to retrieve the minors
1085 // calculated in step c-1. We therefore only have to store at most
1086 // 2*binomial(n,n/2) minors.
1088 // Unique flipper counter for partitioning into minors
1089 std::vector<unsigned> Pkey;
1091 // key for minor determinant (a subpartition of Pkey)
1092 std::vector<unsigned> Mkey;
1094 // we store our subminors in maps, keys being the rows they arise from
1095 typedef std::map<std::vector<unsigned>,class ex> Rmap;
1096 typedef std::map<std::vector<unsigned>,class ex>::value_type Rmap_value;
1100 // initialize A with last column:
1101 for (unsigned r=0; r<n; ++r) {
1102 Pkey.erase(Pkey.begin(),Pkey.end());
1104 A.insert(Rmap_value(Pkey,m[n*(r+1)-1]));
1106 // proceed from right to left through matrix
1107 for (int c=n-2; c>=0; --c) {
1108 Pkey.erase(Pkey.begin(),Pkey.end()); // don't change capacity
1109 Mkey.erase(Mkey.begin(),Mkey.end());
1110 for (unsigned i=0; i<n-c; ++i)
1112 unsigned fc = 0; // controls logic for our strange flipper counter
1115 for (unsigned r=0; r<n-c; ++r) {
1116 // maybe there is nothing to do?
1117 if (m[Pkey[r]*n+c].is_zero())
1119 // create the sorted key for all possible minors
1120 Mkey.erase(Mkey.begin(),Mkey.end());
1121 for (unsigned i=0; i<n-c; ++i)
1123 Mkey.push_back(Pkey[i]);
1124 // Fetch the minors and compute the new determinant
1126 det -= m[Pkey[r]*n+c]*A[Mkey];
1128 det += m[Pkey[r]*n+c]*A[Mkey];
1130 // prevent build-up of deep nesting of expressions saves time:
1132 // store the new determinant at its place in B:
1134 B.insert(Rmap_value(Pkey,det));
1135 // increment our strange flipper counter
1136 for (fc=n-c; fc>0; --fc) {
1138 if (Pkey[fc-1]<fc+c)
1142 for (unsigned j=fc; j<n-c; ++j)
1143 Pkey[j] = Pkey[j-1]+1;
1145 // next column, so change the role of A and B:
1154 /** Perform the steps of an ordinary Gaussian elimination to bring the m x n
1155 * matrix into an upper echelon form. The algorithm is ok for matrices
1156 * with numeric coefficients but quite unsuited for symbolic matrices.
1158 * @param det may be set to true to save a lot of space if one is only
1159 * interested in the diagonal elements (i.e. for calculating determinants).
1160 * The others are set to zero in this case.
1161 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1162 * number of rows was swapped and 0 if the matrix is singular. */
1163 int matrix::gauss_elimination(const bool det)
1165 ensure_if_modifiable();
1166 const unsigned m = this->rows();
1167 const unsigned n = this->cols();
1168 GINAC_ASSERT(!det || n==m);
1172 for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
1173 int indx = pivot(r0, r1, true);
1177 return 0; // leaves *this in a messy state
1182 for (unsigned r2=r0+1; r2<m; ++r2) {
1183 if (!this->m[r2*n+r1].is_zero()) {
1184 // yes, there is something to do in this row
1185 ex piv = this->m[r2*n+r1] / this->m[r0*n+r1];
1186 for (unsigned c=r1+1; c<n; ++c) {
1187 this->m[r2*n+c] -= piv * this->m[r0*n+c];
1188 if (!this->m[r2*n+c].info(info_flags::numeric))
1189 this->m[r2*n+c] = this->m[r2*n+c].normal();
1192 // fill up left hand side with zeros
1193 for (unsigned c=0; c<=r1; ++c)
1194 this->m[r2*n+c] = _ex0();
1197 // save space by deleting no longer needed elements
1198 for (unsigned c=r0+1; c<n; ++c)
1199 this->m[r0*n+c] = _ex0();
1209 /** Perform the steps of division free elimination to bring the m x n matrix
1210 * into an upper echelon form.
1212 * @param det may be set to true to save a lot of space if one is only
1213 * interested in the diagonal elements (i.e. for calculating determinants).
1214 * The others are set to zero in this case.
1215 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1216 * number of rows was swapped and 0 if the matrix is singular. */
1217 int matrix::division_free_elimination(const bool det)
1219 ensure_if_modifiable();
1220 const unsigned m = this->rows();
1221 const unsigned n = this->cols();
1222 GINAC_ASSERT(!det || n==m);
1226 for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
1227 int indx = pivot(r0, r1, true);
1231 return 0; // leaves *this in a messy state
1236 for (unsigned r2=r0+1; r2<m; ++r2) {
1237 for (unsigned c=r1+1; c<n; ++c)
1238 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();
1239 // fill up left hand side with zeros
1240 for (unsigned c=0; c<=r1; ++c)
1241 this->m[r2*n+c] = _ex0();
1244 // save space by deleting no longer needed elements
1245 for (unsigned c=r0+1; c<n; ++c)
1246 this->m[r0*n+c] = _ex0();
1256 /** Perform the steps of Bareiss' one-step fraction free elimination to bring
1257 * the matrix into an upper echelon form. Fraction free elimination means
1258 * that divide is used straightforwardly, without computing GCDs first. This
1259 * is possible, since we know the divisor at each step.
1261 * @param det may be set to true to save a lot of space if one is only
1262 * interested in the last element (i.e. for calculating determinants). The
1263 * others are set to zero in this case.
1264 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1265 * number of rows was swapped and 0 if the matrix is singular. */
1266 int matrix::fraction_free_elimination(const bool det)
1269 // (single-step fraction free elimination scheme, already known to Jordan)
1271 // Usual division-free elimination sets m[0](r,c) = m(r,c) and then sets
1272 // m[k+1](r,c) = m[k](k,k) * m[k](r,c) - m[k](r,k) * m[k](k,c).
1274 // Bareiss (fraction-free) elimination in addition divides that element
1275 // by m[k-1](k-1,k-1) for k>1, where it can be shown by means of the
1276 // Sylvester determinant that this really divides m[k+1](r,c).
1278 // We also allow rational functions where the original prove still holds.
1279 // However, we must care for numerator and denominator separately and
1280 // "manually" work in the integral domains because of subtle cancellations
1281 // (see below). This blows up the bookkeeping a bit and the formula has
1282 // to be modified to expand like this (N{x} stands for numerator of x,
1283 // D{x} for denominator of x):
1284 // 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)}
1285 // -N{m[k](r,k)}*N{m[k](k,c)}*D{m[k](k,k)}*D{m[k](r,c)}
1286 // 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)}
1287 // where for k>1 we now divide N{m[k+1](r,c)} by
1288 // N{m[k-1](k-1,k-1)}
1289 // and D{m[k+1](r,c)} by
1290 // D{m[k-1](k-1,k-1)}.
1292 ensure_if_modifiable();
1293 const unsigned m = this->rows();
1294 const unsigned n = this->cols();
1295 GINAC_ASSERT(!det || n==m);
1304 // We populate temporary matrices to subsequently operate on. There is
1305 // one holding numerators and another holding denominators of entries.
1306 // This is a must since the evaluator (or even earlier mul's constructor)
1307 // might cancel some trivial element which causes divide() to fail. The
1308 // elements are normalized first (yes, even though this algorithm doesn't
1309 // need GCDs) since the elements of *this might be unnormalized, which
1310 // makes things more complicated than they need to be.
1311 matrix tmp_n(*this);
1312 matrix tmp_d(m,n); // for denominators, if needed
1313 lst srl; // symbol replacement list
1314 exvector::const_iterator cit = this->m.begin(), citend = this->m.end();
1315 exvector::iterator tmp_n_it = tmp_n.m.begin(), tmp_d_it = tmp_d.m.begin();
1316 while (cit != citend) {
1317 ex nd = cit->normal().to_rational(srl).numer_denom();
1319 *tmp_n_it++ = nd.op(0);
1320 *tmp_d_it++ = nd.op(1);
1324 for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
1325 int indx = tmp_n.pivot(r0, r1, true);
1334 // tmp_n's rows r0 and indx were swapped, do the same in tmp_d:
1335 for (unsigned c=r1; c<n; ++c)
1336 tmp_d.m[n*indx+c].swap(tmp_d.m[n*r0+c]);
1338 for (unsigned r2=r0+1; r2<m; ++r2) {
1339 for (unsigned c=r1+1; c<n; ++c) {
1340 dividend_n = (tmp_n.m[r0*n+r1]*tmp_n.m[r2*n+c]*
1341 tmp_d.m[r2*n+r1]*tmp_d.m[r0*n+c]
1342 -tmp_n.m[r2*n+r1]*tmp_n.m[r0*n+c]*
1343 tmp_d.m[r0*n+r1]*tmp_d.m[r2*n+c]).expand();
1344 dividend_d = (tmp_d.m[r2*n+r1]*tmp_d.m[r0*n+c]*
1345 tmp_d.m[r0*n+r1]*tmp_d.m[r2*n+c]).expand();
1346 bool check = divide(dividend_n, divisor_n,
1347 tmp_n.m[r2*n+c], true);
1348 check &= divide(dividend_d, divisor_d,
1349 tmp_d.m[r2*n+c], true);
1350 GINAC_ASSERT(check);
1352 // fill up left hand side with zeros
1353 for (unsigned c=0; c<=r1; ++c)
1354 tmp_n.m[r2*n+c] = _ex0();
1356 if ((r1<n-1)&&(r0<m-1)) {
1357 // compute next iteration's divisor
1358 divisor_n = tmp_n.m[r0*n+r1].expand();
1359 divisor_d = tmp_d.m[r0*n+r1].expand();
1361 // save space by deleting no longer needed elements
1362 for (unsigned c=0; c<n; ++c) {
1363 tmp_n.m[r0*n+c] = _ex0();
1364 tmp_d.m[r0*n+c] = _ex1();
1371 // repopulate *this matrix:
1372 exvector::iterator it = this->m.begin(), itend = this->m.end();
1373 tmp_n_it = tmp_n.m.begin();
1374 tmp_d_it = tmp_d.m.begin();
1376 *it++ = ((*tmp_n_it++)/(*tmp_d_it++)).subs(srl);
1382 /** Partial pivoting method for matrix elimination schemes.
1383 * Usual pivoting (symbolic==false) returns the index to the element with the
1384 * largest absolute value in column ro and swaps the current row with the one
1385 * where the element was found. With (symbolic==true) it does the same thing
1386 * with the first non-zero element.
1388 * @param ro is the row from where to begin
1389 * @param co is the column to be inspected
1390 * @param symbolic signal if we want the first non-zero element to be pivoted
1391 * (true) or the one with the largest absolute value (false).
1392 * @return 0 if no interchange occured, -1 if all are zero (usually signaling
1393 * a degeneracy) and positive integer k means that rows ro and k were swapped.
1395 int matrix::pivot(unsigned ro, unsigned co, bool symbolic)
1399 // search first non-zero element in column co beginning at row ro
1400 while ((k<row) && (this->m[k*col+co].expand().is_zero()))
1403 // search largest element in column co beginning at row ro
1404 GINAC_ASSERT(is_ex_of_type(this->m[k*col+co],numeric));
1405 unsigned kmax = k+1;
1406 numeric mmax = abs(ex_to<numeric>(m[kmax*col+co]));
1408 GINAC_ASSERT(is_ex_of_type(this->m[kmax*col+co],numeric));
1409 numeric tmp = ex_to<numeric>(this->m[kmax*col+co]);
1410 if (abs(tmp) > mmax) {
1416 if (!mmax.is_zero())
1420 // all elements in column co below row ro vanish
1423 // matrix needs no pivoting
1425 // matrix needs pivoting, so swap rows k and ro
1426 ensure_if_modifiable();
1427 for (unsigned c=0; c<col; ++c)
1428 this->m[k*col+c].swap(this->m[ro*col+c]);
1433 ex lst_to_matrix(const lst & l)
1435 // Find number of rows and columns
1436 unsigned rows = l.nops(), cols = 0, i, j;
1437 for (i=0; i<rows; i++)
1438 if (l.op(i).nops() > cols)
1439 cols = l.op(i).nops();
1441 // Allocate and fill matrix
1442 matrix &m = *new matrix(rows, cols);
1443 m.setflag(status_flags::dynallocated);
1444 for (i=0; i<rows; i++)
1445 for (j=0; j<cols; j++)
1446 if (l.op(i).nops() > j)
1447 m(i, j) = l.op(i).op(j);
1453 ex diag_matrix(const lst & l)
1455 unsigned dim = l.nops();
1457 matrix &m = *new matrix(dim, dim);
1458 m.setflag(status_flags::dynallocated);
1459 for (unsigned i=0; i<dim; i++)
1465 } // namespace GiNaC