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 bases 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)) {
603 matrix prod(row,col);
604 if (expn.info(info_flags::negative)) {
605 k = -ex_to<numeric>(expn);
606 prod = this->inverse();
608 k = ex_to<numeric>(expn);
611 matrix result(row,col);
612 for (unsigned r=0; r<row; ++r)
613 result(r,r) = _ex1();
615 // this loop computes the representation of k in base 2 and
616 // multiplies the factors whenever needed:
617 while (b.compare(k)<=0) {
622 result = result.mul(prod);
625 prod = prod.mul(prod);
630 throw (std::runtime_error("matrix::pow(): don't know how to handle exponent"));
634 /** operator() to access elements for reading.
636 * @param ro row of element
637 * @param co column of element
638 * @exception range_error (index out of range) */
639 const ex & matrix::operator() (unsigned ro, unsigned co) const
641 if (ro>=row || co>=col)
642 throw (std::range_error("matrix::operator(): index out of range"));
648 /** operator() to access elements for writing.
650 * @param ro row of element
651 * @param co column of element
652 * @exception range_error (index out of range) */
653 ex & matrix::operator() (unsigned ro, unsigned co)
655 if (ro>=row || co>=col)
656 throw (std::range_error("matrix::operator(): index out of range"));
658 ensure_if_modifiable();
663 /** Transposed of an m x n matrix, producing a new n x m matrix object that
664 * represents the transposed. */
665 matrix matrix::transpose(void) const
667 exvector trans(this->cols()*this->rows());
669 for (unsigned r=0; r<this->cols(); ++r)
670 for (unsigned c=0; c<this->rows(); ++c)
671 trans[r*this->rows()+c] = m[c*this->cols()+r];
673 return matrix(this->cols(),this->rows(),trans);
676 /** Determinant of square matrix. This routine doesn't actually calculate the
677 * determinant, it only implements some heuristics about which algorithm to
678 * run. If all the elements of the matrix are elements of an integral domain
679 * the determinant is also in that integral domain and the result is expanded
680 * only. If one or more elements are from a quotient field the determinant is
681 * usually also in that quotient field and the result is normalized before it
682 * is returned. This implies that the determinant of the symbolic 2x2 matrix
683 * [[a/(a-b),1],[b/(a-b),1]] is returned as unity. (In this respect, it
684 * behaves like MapleV and unlike Mathematica.)
686 * @param algo allows to chose an algorithm
687 * @return the determinant as a new expression
688 * @exception logic_error (matrix not square)
689 * @see determinant_algo */
690 ex matrix::determinant(unsigned algo) const
693 throw (std::logic_error("matrix::determinant(): matrix not square"));
694 GINAC_ASSERT(row*col==m.capacity());
696 // Gather some statistical information about this matrix:
697 bool numeric_flag = true;
698 bool normal_flag = false;
699 unsigned sparse_count = 0; // counts non-zero elements
700 exvector::const_iterator r = m.begin(), rend = m.end();
702 lst srl; // symbol replacement list
703 ex rtest = r->to_rational(srl);
704 if (!rtest.is_zero())
706 if (!rtest.info(info_flags::numeric))
707 numeric_flag = false;
708 if (!rtest.info(info_flags::crational_polynomial) &&
709 rtest.info(info_flags::rational_function))
714 // Here is the heuristics in case this routine has to decide:
715 if (algo == determinant_algo::automatic) {
716 // Minor expansion is generally a good guess:
717 algo = determinant_algo::laplace;
718 // Does anybody know when a matrix is really sparse?
719 // Maybe <~row/2.236 nonzero elements average in a row?
720 if (row>3 && 5*sparse_count<=row*col)
721 algo = determinant_algo::bareiss;
722 // Purely numeric matrix can be handled by Gauss elimination.
723 // This overrides any prior decisions.
725 algo = determinant_algo::gauss;
728 // Trap the trivial case here, since some algorithms don't like it
730 // for consistency with non-trivial determinants...
732 return m[0].normal();
734 return m[0].expand();
737 // Compute the determinant
739 case determinant_algo::gauss: {
742 int sign = tmp.gauss_elimination(true);
743 for (unsigned d=0; d<row; ++d)
744 det *= tmp.m[d*col+d];
746 return (sign*det).normal();
748 return (sign*det).normal().expand();
750 case determinant_algo::bareiss: {
753 sign = tmp.fraction_free_elimination(true);
755 return (sign*tmp.m[row*col-1]).normal();
757 return (sign*tmp.m[row*col-1]).expand();
759 case determinant_algo::divfree: {
762 sign = tmp.division_free_elimination(true);
765 ex det = tmp.m[row*col-1];
766 // factor out accumulated bogus slag
767 for (unsigned d=0; d<row-2; ++d)
768 for (unsigned j=0; j<row-d-2; ++j)
769 det = (det/tmp.m[d*col+d]).normal();
772 case determinant_algo::laplace:
774 // This is the minor expansion scheme. We always develop such
775 // that the smallest minors (i.e, the trivial 1x1 ones) are on the
776 // rightmost column. For this to be efficient it turns out that
777 // the emptiest columns (i.e. the ones with most zeros) should be
778 // the ones on the right hand side. Therefore we presort the
779 // columns of the matrix:
780 typedef std::pair<unsigned,unsigned> uintpair;
781 std::vector<uintpair> c_zeros; // number of zeros in column
782 for (unsigned c=0; c<col; ++c) {
784 for (unsigned r=0; r<row; ++r)
785 if (m[r*col+c].is_zero())
787 c_zeros.push_back(uintpair(acc,c));
789 sort(c_zeros.begin(),c_zeros.end());
790 std::vector<unsigned> pre_sort;
791 for (std::vector<uintpair>::const_iterator i=c_zeros.begin(); i!=c_zeros.end(); ++i)
792 pre_sort.push_back(i->second);
793 std::vector<unsigned> pre_sort_test(pre_sort); // permutation_sign() modifies the vector so we make a copy here
794 int sign = permutation_sign(pre_sort_test.begin(), pre_sort_test.end());
795 exvector result(row*col); // represents sorted matrix
797 for (std::vector<unsigned>::const_iterator i=pre_sort.begin();
800 for (unsigned r=0; r<row; ++r)
801 result[r*col+c] = m[r*col+(*i)];
805 return (sign*matrix(row,col,result).determinant_minor()).normal();
807 return sign*matrix(row,col,result).determinant_minor();
813 /** Trace of a matrix. The result is normalized if it is in some quotient
814 * field and expanded only otherwise. This implies that the trace of the
815 * symbolic 2x2 matrix [[a/(a-b),x],[y,b/(b-a)]] is recognized to be unity.
817 * @return the sum of diagonal elements
818 * @exception logic_error (matrix not square) */
819 ex matrix::trace(void) const
822 throw (std::logic_error("matrix::trace(): matrix not square"));
825 for (unsigned r=0; r<col; ++r)
828 if (tr.info(info_flags::rational_function) &&
829 !tr.info(info_flags::crational_polynomial))
836 /** Characteristic Polynomial. Following mathematica notation the
837 * characteristic polynomial of a matrix M is defined as the determiant of
838 * (M - lambda * 1) where 1 stands for the unit matrix of the same dimension
839 * as M. Note that some CASs define it with a sign inside the determinant
840 * which gives rise to an overall sign if the dimension is odd. This method
841 * returns the characteristic polynomial collected in powers of lambda as a
844 * @return characteristic polynomial as new expression
845 * @exception logic_error (matrix not square)
846 * @see matrix::determinant() */
847 ex matrix::charpoly(const symbol & lambda) const
850 throw (std::logic_error("matrix::charpoly(): matrix not square"));
852 bool numeric_flag = true;
853 exvector::const_iterator r = m.begin(), rend = m.end();
855 if (!r->info(info_flags::numeric))
856 numeric_flag = false;
860 // The pure numeric case is traditionally rather common. Hence, it is
861 // trapped and we use Leverrier's algorithm which goes as row^3 for
862 // every coefficient. The expensive part is the matrix multiplication.
866 ex poly = power(lambda,row)-c*power(lambda,row-1);
867 for (unsigned i=1; i<row; ++i) {
868 for (unsigned j=0; j<row; ++j)
871 c = B.trace()/ex(i+1);
872 poly -= c*power(lambda,row-i-1);
881 for (unsigned r=0; r<col; ++r)
882 M.m[r*col+r] -= lambda;
884 return M.determinant().collect(lambda);
888 /** Inverse of this matrix.
890 * @return the inverted matrix
891 * @exception logic_error (matrix not square)
892 * @exception runtime_error (singular matrix) */
893 matrix matrix::inverse(void) const
896 throw (std::logic_error("matrix::inverse(): matrix not square"));
898 // This routine actually doesn't do anything fancy at all. We compute the
899 // inverse of the matrix A by solving the system A * A^{-1} == Id.
901 // First populate the identity matrix supposed to become the right hand side.
902 matrix identity(row,col);
903 for (unsigned i=0; i<row; ++i)
904 identity(i,i) = _ex1();
906 // Populate a dummy matrix of variables, just because of compatibility with
907 // matrix::solve() which wants this (for compatibility with under-determined
908 // systems of equations).
909 matrix vars(row,col);
910 for (unsigned r=0; r<row; ++r)
911 for (unsigned c=0; c<col; ++c)
912 vars(r,c) = symbol();
916 sol = this->solve(vars,identity);
917 } catch (const std::runtime_error & e) {
918 if (e.what()==std::string("matrix::solve(): inconsistent linear system"))
919 throw (std::runtime_error("matrix::inverse(): singular matrix"));
927 /** Solve a linear system consisting of a m x n matrix and a m x p right hand
928 * side by applying an elimination scheme to the augmented matrix.
930 * @param vars n x p matrix, all elements must be symbols
931 * @param rhs m x p matrix
932 * @return n x p solution matrix
933 * @exception logic_error (incompatible matrices)
934 * @exception invalid_argument (1st argument must be matrix of symbols)
935 * @exception runtime_error (inconsistent linear system)
937 matrix matrix::solve(const matrix & vars,
941 const unsigned m = this->rows();
942 const unsigned n = this->cols();
943 const unsigned p = rhs.cols();
946 if ((rhs.rows() != m) || (vars.rows() != n) || (vars.col != p))
947 throw (std::logic_error("matrix::solve(): incompatible matrices"));
948 for (unsigned ro=0; ro<n; ++ro)
949 for (unsigned co=0; co<p; ++co)
950 if (!vars(ro,co).info(info_flags::symbol))
951 throw (std::invalid_argument("matrix::solve(): 1st argument must be matrix of symbols"));
953 // build the augmented matrix of *this with rhs attached to the right
955 for (unsigned r=0; r<m; ++r) {
956 for (unsigned c=0; c<n; ++c)
957 aug.m[r*(n+p)+c] = this->m[r*n+c];
958 for (unsigned c=0; c<p; ++c)
959 aug.m[r*(n+p)+c+n] = rhs.m[r*p+c];
962 // Gather some statistical information about the augmented matrix:
963 bool numeric_flag = true;
964 exvector::const_iterator r = aug.m.begin(), rend = aug.m.end();
966 if (!r->info(info_flags::numeric))
967 numeric_flag = false;
971 // Here is the heuristics in case this routine has to decide:
972 if (algo == solve_algo::automatic) {
973 // Bareiss (fraction-free) elimination is generally a good guess:
974 algo = solve_algo::bareiss;
975 // For m<3, Bareiss elimination is equivalent to division free
976 // elimination but has more logistic overhead
978 algo = solve_algo::divfree;
979 // This overrides any prior decisions.
981 algo = solve_algo::gauss;
984 // Eliminate the augmented matrix:
986 case solve_algo::gauss:
987 aug.gauss_elimination();
989 case solve_algo::divfree:
990 aug.division_free_elimination();
992 case solve_algo::bareiss:
994 aug.fraction_free_elimination();
997 // assemble the solution matrix:
999 for (unsigned co=0; co<p; ++co) {
1000 unsigned last_assigned_sol = n+1;
1001 for (int r=m-1; r>=0; --r) {
1002 unsigned fnz = 1; // first non-zero in row
1003 while ((fnz<=n) && (aug.m[r*(n+p)+(fnz-1)].is_zero()))
1006 // row consists only of zeros, corresponding rhs must be 0, too
1007 if (!aug.m[r*(n+p)+n+co].is_zero()) {
1008 throw (std::runtime_error("matrix::solve(): inconsistent linear system"));
1011 // assign solutions for vars between fnz+1 and
1012 // last_assigned_sol-1: free parameters
1013 for (unsigned c=fnz; c<last_assigned_sol-1; ++c)
1014 sol(c,co) = vars.m[c*p+co];
1015 ex e = aug.m[r*(n+p)+n+co];
1016 for (unsigned c=fnz; c<n; ++c)
1017 e -= aug.m[r*(n+p)+c]*sol.m[c*p+co];
1018 sol(fnz-1,co) = (e/(aug.m[r*(n+p)+(fnz-1)])).normal();
1019 last_assigned_sol = fnz;
1022 // assign solutions for vars between 1 and
1023 // last_assigned_sol-1: free parameters
1024 for (unsigned ro=0; ro<last_assigned_sol-1; ++ro)
1025 sol(ro,co) = vars(ro,co);
1034 /** Recursive determinant for small matrices having at least one symbolic
1035 * entry. The basic algorithm, known as Laplace-expansion, is enhanced by
1036 * some bookkeeping to avoid calculation of the same submatrices ("minors")
1037 * more than once. According to W.M.Gentleman and S.C.Johnson this algorithm
1038 * is better than elimination schemes for matrices of sparse multivariate
1039 * polynomials and also for matrices of dense univariate polynomials if the
1040 * matrix' dimesion is larger than 7.
1042 * @return the determinant as a new expression (in expanded form)
1043 * @see matrix::determinant() */
1044 ex matrix::determinant_minor(void) const
1046 // for small matrices the algorithm does not make any sense:
1047 const unsigned n = this->cols();
1049 return m[0].expand();
1051 return (m[0]*m[3]-m[2]*m[1]).expand();
1053 return (m[0]*m[4]*m[8]-m[0]*m[5]*m[7]-
1054 m[1]*m[3]*m[8]+m[2]*m[3]*m[7]+
1055 m[1]*m[5]*m[6]-m[2]*m[4]*m[6]).expand();
1057 // This algorithm can best be understood by looking at a naive
1058 // implementation of Laplace-expansion, like this one:
1060 // matrix minorM(this->rows()-1,this->cols()-1);
1061 // for (unsigned r1=0; r1<this->rows(); ++r1) {
1062 // // shortcut if element(r1,0) vanishes
1063 // if (m[r1*col].is_zero())
1065 // // assemble the minor matrix
1066 // for (unsigned r=0; r<minorM.rows(); ++r) {
1067 // for (unsigned c=0; c<minorM.cols(); ++c) {
1069 // minorM(r,c) = m[r*col+c+1];
1071 // minorM(r,c) = m[(r+1)*col+c+1];
1074 // // recurse down and care for sign:
1076 // det -= m[r1*col] * minorM.determinant_minor();
1078 // det += m[r1*col] * minorM.determinant_minor();
1080 // return det.expand();
1081 // What happens is that while proceeding down many of the minors are
1082 // computed more than once. In particular, there are binomial(n,k)
1083 // kxk minors and each one is computed factorial(n-k) times. Therefore
1084 // it is reasonable to store the results of the minors. We proceed from
1085 // right to left. At each column c we only need to retrieve the minors
1086 // calculated in step c-1. We therefore only have to store at most
1087 // 2*binomial(n,n/2) minors.
1089 // Unique flipper counter for partitioning into minors
1090 std::vector<unsigned> Pkey;
1092 // key for minor determinant (a subpartition of Pkey)
1093 std::vector<unsigned> Mkey;
1095 // we store our subminors in maps, keys being the rows they arise from
1096 typedef std::map<std::vector<unsigned>,class ex> Rmap;
1097 typedef std::map<std::vector<unsigned>,class ex>::value_type Rmap_value;
1101 // initialize A with last column:
1102 for (unsigned r=0; r<n; ++r) {
1103 Pkey.erase(Pkey.begin(),Pkey.end());
1105 A.insert(Rmap_value(Pkey,m[n*(r+1)-1]));
1107 // proceed from right to left through matrix
1108 for (int c=n-2; c>=0; --c) {
1109 Pkey.erase(Pkey.begin(),Pkey.end()); // don't change capacity
1110 Mkey.erase(Mkey.begin(),Mkey.end());
1111 for (unsigned i=0; i<n-c; ++i)
1113 unsigned fc = 0; // controls logic for our strange flipper counter
1116 for (unsigned r=0; r<n-c; ++r) {
1117 // maybe there is nothing to do?
1118 if (m[Pkey[r]*n+c].is_zero())
1120 // create the sorted key for all possible minors
1121 Mkey.erase(Mkey.begin(),Mkey.end());
1122 for (unsigned i=0; i<n-c; ++i)
1124 Mkey.push_back(Pkey[i]);
1125 // Fetch the minors and compute the new determinant
1127 det -= m[Pkey[r]*n+c]*A[Mkey];
1129 det += m[Pkey[r]*n+c]*A[Mkey];
1131 // prevent build-up of deep nesting of expressions saves time:
1133 // store the new determinant at its place in B:
1135 B.insert(Rmap_value(Pkey,det));
1136 // increment our strange flipper counter
1137 for (fc=n-c; fc>0; --fc) {
1139 if (Pkey[fc-1]<fc+c)
1143 for (unsigned j=fc; j<n-c; ++j)
1144 Pkey[j] = Pkey[j-1]+1;
1146 // next column, so change the role of A and B:
1155 /** Perform the steps of an ordinary Gaussian elimination to bring the m x n
1156 * matrix into an upper echelon form. The algorithm is ok for matrices
1157 * with numeric coefficients but quite unsuited for symbolic matrices.
1159 * @param det may be set to true to save a lot of space if one is only
1160 * interested in the diagonal elements (i.e. for calculating determinants).
1161 * The others are set to zero in this case.
1162 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1163 * number of rows was swapped and 0 if the matrix is singular. */
1164 int matrix::gauss_elimination(const bool det)
1166 ensure_if_modifiable();
1167 const unsigned m = this->rows();
1168 const unsigned n = this->cols();
1169 GINAC_ASSERT(!det || n==m);
1173 for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
1174 int indx = pivot(r0, r1, true);
1178 return 0; // leaves *this in a messy state
1183 for (unsigned r2=r0+1; r2<m; ++r2) {
1184 if (!this->m[r2*n+r1].is_zero()) {
1185 // yes, there is something to do in this row
1186 ex piv = this->m[r2*n+r1] / this->m[r0*n+r1];
1187 for (unsigned c=r1+1; c<n; ++c) {
1188 this->m[r2*n+c] -= piv * this->m[r0*n+c];
1189 if (!this->m[r2*n+c].info(info_flags::numeric))
1190 this->m[r2*n+c] = this->m[r2*n+c].normal();
1193 // fill up left hand side with zeros
1194 for (unsigned c=0; c<=r1; ++c)
1195 this->m[r2*n+c] = _ex0();
1198 // save space by deleting no longer needed elements
1199 for (unsigned c=r0+1; c<n; ++c)
1200 this->m[r0*n+c] = _ex0();
1210 /** Perform the steps of division free elimination to bring the m x n matrix
1211 * into an upper echelon form.
1213 * @param det may be set to true to save a lot of space if one is only
1214 * interested in the diagonal elements (i.e. for calculating determinants).
1215 * The others are set to zero in this case.
1216 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1217 * number of rows was swapped and 0 if the matrix is singular. */
1218 int matrix::division_free_elimination(const bool det)
1220 ensure_if_modifiable();
1221 const unsigned m = this->rows();
1222 const unsigned n = this->cols();
1223 GINAC_ASSERT(!det || n==m);
1227 for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
1228 int indx = pivot(r0, r1, true);
1232 return 0; // leaves *this in a messy state
1237 for (unsigned r2=r0+1; r2<m; ++r2) {
1238 for (unsigned c=r1+1; c<n; ++c)
1239 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();
1240 // fill up left hand side with zeros
1241 for (unsigned c=0; c<=r1; ++c)
1242 this->m[r2*n+c] = _ex0();
1245 // save space by deleting no longer needed elements
1246 for (unsigned c=r0+1; c<n; ++c)
1247 this->m[r0*n+c] = _ex0();
1257 /** Perform the steps of Bareiss' one-step fraction free elimination to bring
1258 * the matrix into an upper echelon form. Fraction free elimination means
1259 * that divide is used straightforwardly, without computing GCDs first. This
1260 * is possible, since we know the divisor at each step.
1262 * @param det may be set to true to save a lot of space if one is only
1263 * interested in the last element (i.e. for calculating determinants). The
1264 * others are set to zero in this case.
1265 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1266 * number of rows was swapped and 0 if the matrix is singular. */
1267 int matrix::fraction_free_elimination(const bool det)
1270 // (single-step fraction free elimination scheme, already known to Jordan)
1272 // Usual division-free elimination sets m[0](r,c) = m(r,c) and then sets
1273 // m[k+1](r,c) = m[k](k,k) * m[k](r,c) - m[k](r,k) * m[k](k,c).
1275 // Bareiss (fraction-free) elimination in addition divides that element
1276 // by m[k-1](k-1,k-1) for k>1, where it can be shown by means of the
1277 // Sylvester determinant that this really divides m[k+1](r,c).
1279 // We also allow rational functions where the original prove still holds.
1280 // However, we must care for numerator and denominator separately and
1281 // "manually" work in the integral domains because of subtle cancellations
1282 // (see below). This blows up the bookkeeping a bit and the formula has
1283 // to be modified to expand like this (N{x} stands for numerator of x,
1284 // D{x} for denominator of x):
1285 // 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)}
1286 // -N{m[k](r,k)}*N{m[k](k,c)}*D{m[k](k,k)}*D{m[k](r,c)}
1287 // 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)}
1288 // where for k>1 we now divide N{m[k+1](r,c)} by
1289 // N{m[k-1](k-1,k-1)}
1290 // and D{m[k+1](r,c)} by
1291 // D{m[k-1](k-1,k-1)}.
1293 ensure_if_modifiable();
1294 const unsigned m = this->rows();
1295 const unsigned n = this->cols();
1296 GINAC_ASSERT(!det || n==m);
1305 // We populate temporary matrices to subsequently operate on. There is
1306 // one holding numerators and another holding denominators of entries.
1307 // This is a must since the evaluator (or even earlier mul's constructor)
1308 // might cancel some trivial element which causes divide() to fail. The
1309 // elements are normalized first (yes, even though this algorithm doesn't
1310 // need GCDs) since the elements of *this might be unnormalized, which
1311 // makes things more complicated than they need to be.
1312 matrix tmp_n(*this);
1313 matrix tmp_d(m,n); // for denominators, if needed
1314 lst srl; // symbol replacement list
1315 exvector::const_iterator cit = this->m.begin(), citend = this->m.end();
1316 exvector::iterator tmp_n_it = tmp_n.m.begin(), tmp_d_it = tmp_d.m.begin();
1317 while (cit != citend) {
1318 ex nd = cit->normal().to_rational(srl).numer_denom();
1320 *tmp_n_it++ = nd.op(0);
1321 *tmp_d_it++ = nd.op(1);
1325 for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
1326 int indx = tmp_n.pivot(r0, r1, true);
1335 // tmp_n's rows r0 and indx were swapped, do the same in tmp_d:
1336 for (unsigned c=r1; c<n; ++c)
1337 tmp_d.m[n*indx+c].swap(tmp_d.m[n*r0+c]);
1339 for (unsigned r2=r0+1; r2<m; ++r2) {
1340 for (unsigned c=r1+1; c<n; ++c) {
1341 dividend_n = (tmp_n.m[r0*n+r1]*tmp_n.m[r2*n+c]*
1342 tmp_d.m[r2*n+r1]*tmp_d.m[r0*n+c]
1343 -tmp_n.m[r2*n+r1]*tmp_n.m[r0*n+c]*
1344 tmp_d.m[r0*n+r1]*tmp_d.m[r2*n+c]).expand();
1345 dividend_d = (tmp_d.m[r2*n+r1]*tmp_d.m[r0*n+c]*
1346 tmp_d.m[r0*n+r1]*tmp_d.m[r2*n+c]).expand();
1347 bool check = divide(dividend_n, divisor_n,
1348 tmp_n.m[r2*n+c], true);
1349 check &= divide(dividend_d, divisor_d,
1350 tmp_d.m[r2*n+c], true);
1351 GINAC_ASSERT(check);
1353 // fill up left hand side with zeros
1354 for (unsigned c=0; c<=r1; ++c)
1355 tmp_n.m[r2*n+c] = _ex0();
1357 if ((r1<n-1)&&(r0<m-1)) {
1358 // compute next iteration's divisor
1359 divisor_n = tmp_n.m[r0*n+r1].expand();
1360 divisor_d = tmp_d.m[r0*n+r1].expand();
1362 // save space by deleting no longer needed elements
1363 for (unsigned c=0; c<n; ++c) {
1364 tmp_n.m[r0*n+c] = _ex0();
1365 tmp_d.m[r0*n+c] = _ex1();
1372 // repopulate *this matrix:
1373 exvector::iterator it = this->m.begin(), itend = this->m.end();
1374 tmp_n_it = tmp_n.m.begin();
1375 tmp_d_it = tmp_d.m.begin();
1377 *it++ = ((*tmp_n_it++)/(*tmp_d_it++)).subs(srl);
1383 /** Partial pivoting method for matrix elimination schemes.
1384 * Usual pivoting (symbolic==false) returns the index to the element with the
1385 * largest absolute value in column ro and swaps the current row with the one
1386 * where the element was found. With (symbolic==true) it does the same thing
1387 * with the first non-zero element.
1389 * @param ro is the row from where to begin
1390 * @param co is the column to be inspected
1391 * @param symbolic signal if we want the first non-zero element to be pivoted
1392 * (true) or the one with the largest absolute value (false).
1393 * @return 0 if no interchange occured, -1 if all are zero (usually signaling
1394 * a degeneracy) and positive integer k means that rows ro and k were swapped.
1396 int matrix::pivot(unsigned ro, unsigned co, bool symbolic)
1400 // search first non-zero element in column co beginning at row ro
1401 while ((k<row) && (this->m[k*col+co].expand().is_zero()))
1404 // search largest element in column co beginning at row ro
1405 GINAC_ASSERT(is_ex_of_type(this->m[k*col+co],numeric));
1406 unsigned kmax = k+1;
1407 numeric mmax = abs(ex_to<numeric>(m[kmax*col+co]));
1409 GINAC_ASSERT(is_ex_of_type(this->m[kmax*col+co],numeric));
1410 numeric tmp = ex_to<numeric>(this->m[kmax*col+co]);
1411 if (abs(tmp) > mmax) {
1417 if (!mmax.is_zero())
1421 // all elements in column co below row ro vanish
1424 // matrix needs no pivoting
1426 // matrix needs pivoting, so swap rows k and ro
1427 ensure_if_modifiable();
1428 for (unsigned c=0; c<col; ++c)
1429 this->m[k*col+c].swap(this->m[ro*col+c]);
1434 ex lst_to_matrix(const lst & l)
1436 // Find number of rows and columns
1437 unsigned rows = l.nops(), cols = 0, i, j;
1438 for (i=0; i<rows; i++)
1439 if (l.op(i).nops() > cols)
1440 cols = l.op(i).nops();
1442 // Allocate and fill matrix
1443 matrix &m = *new matrix(rows, cols);
1444 m.setflag(status_flags::dynallocated);
1445 for (i=0; i<rows; i++)
1446 for (j=0; j<cols; j++)
1447 if (l.op(i).nops() > j)
1448 m(i, j) = l.op(i).op(j);
1454 ex diag_matrix(const lst & l)
1456 unsigned dim = l.nops();
1458 matrix &m = *new matrix(dim, dim);
1459 m.setflag(status_flags::dynallocated);
1460 for (unsigned i=0; i<dim; i++)
1466 } // namespace GiNaC