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 /** expands the elements of a matrix entry by entry. */
202 ex matrix::expand(unsigned options) const
204 exvector tmp(row*col);
205 for (unsigned i=0; i<row*col; ++i)
206 tmp[i] = m[i].expand(options);
208 return matrix(row, col, tmp);
211 /** Search ocurrences. A matrix 'has' an expression if it is the expression
212 * itself or one of the elements 'has' it. */
213 bool matrix::has(const ex & other) const
215 GINAC_ASSERT(other.bp!=0);
217 // tautology: it is the expression itself
218 if (is_equal(*other.bp)) return true;
220 // search all the elements
221 for (exvector::const_iterator r=m.begin(); r!=m.end(); ++r)
222 if ((*r).has(other)) return true;
227 /** Evaluate matrix entry by entry. */
228 ex matrix::eval(int level) const
230 debugmsg("matrix eval",LOGLEVEL_MEMBER_FUNCTION);
232 // check if we have to do anything at all
233 if ((level==1)&&(flags & status_flags::evaluated))
237 if (level == -max_recursion_level)
238 throw (std::runtime_error("matrix::eval(): recursion limit exceeded"));
240 // eval() entry by entry
241 exvector m2(row*col);
243 for (unsigned r=0; r<row; ++r)
244 for (unsigned c=0; c<col; ++c)
245 m2[r*col+c] = m[r*col+c].eval(level);
247 return (new matrix(row, col, m2))->setflag(status_flags::dynallocated |
248 status_flags::evaluated );
251 /** Evaluate matrix numerically entry by entry. */
252 ex matrix::evalf(int level) const
254 debugmsg("matrix evalf",LOGLEVEL_MEMBER_FUNCTION);
256 // check if we have to do anything at all
261 if (level == -max_recursion_level) {
262 throw (std::runtime_error("matrix::evalf(): recursion limit exceeded"));
265 // evalf() entry by entry
266 exvector m2(row*col);
268 for (unsigned r=0; r<row; ++r)
269 for (unsigned c=0; c<col; ++c)
270 m2[r*col+c] = m[r*col+c].evalf(level);
272 return matrix(row, col, m2);
275 ex matrix::subs(const lst & ls, const lst & lr) const
277 exvector m2(row * col);
278 for (unsigned r=0; r<row; ++r)
279 for (unsigned c=0; c<col; ++c)
280 m2[r*col+c] = m[r*col+c].subs(ls, lr);
282 return matrix(row, col, m2);
287 int matrix::compare_same_type(const basic & other) const
289 GINAC_ASSERT(is_exactly_of_type(other, matrix));
290 const matrix & o = static_cast<matrix &>(const_cast<basic &>(other));
292 // compare number of rows
294 return row < o.rows() ? -1 : 1;
296 // compare number of columns
298 return col < o.cols() ? -1 : 1;
300 // equal number of rows and columns, compare individual elements
302 for (unsigned r=0; r<row; ++r) {
303 for (unsigned c=0; c<col; ++c) {
304 cmpval = ((*this)(r,c)).compare(o(r,c));
305 if (cmpval!=0) return cmpval;
308 // all elements are equal => matrices are equal;
312 /** Automatic symbolic evaluation of an indexed matrix. */
313 ex matrix::eval_indexed(const basic & i) const
315 GINAC_ASSERT(is_of_type(i, indexed));
316 GINAC_ASSERT(is_ex_of_type(i.op(0), matrix));
318 bool all_indices_unsigned = static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint);
323 // One index, must be one-dimensional vector
324 if (row != 1 && col != 1)
325 throw (std::runtime_error("matrix::eval_indexed(): vector must have exactly 1 index"));
327 const idx & i1 = ex_to_idx(i.op(1));
332 if (!i1.get_dim().is_equal(row))
333 throw (std::runtime_error("matrix::eval_indexed(): dimension of index must match number of vector elements"));
335 // Index numeric -> return vector element
336 if (all_indices_unsigned) {
337 unsigned n1 = ex_to_numeric(i1.get_value()).to_int();
339 throw (std::runtime_error("matrix::eval_indexed(): value of index exceeds number of vector elements"));
340 return (*this)(n1, 0);
346 if (!i1.get_dim().is_equal(col))
347 throw (std::runtime_error("matrix::eval_indexed(): dimension of index must match number of vector elements"));
349 // Index numeric -> return vector element
350 if (all_indices_unsigned) {
351 unsigned n1 = ex_to_numeric(i1.get_value()).to_int();
353 throw (std::runtime_error("matrix::eval_indexed(): value of index exceeds number of vector elements"));
354 return (*this)(0, n1);
358 } else if (i.nops() == 3) {
361 const idx & i1 = ex_to_idx(i.op(1));
362 const idx & i2 = ex_to_idx(i.op(2));
364 if (!i1.get_dim().is_equal(row))
365 throw (std::runtime_error("matrix::eval_indexed(): dimension of first index must match number of rows"));
366 if (!i2.get_dim().is_equal(col))
367 throw (std::runtime_error("matrix::eval_indexed(): dimension of second index must match number of columns"));
369 // Pair of dummy indices -> compute trace
370 if (is_dummy_pair(i1, i2))
373 // Both indices numeric -> return matrix element
374 if (all_indices_unsigned) {
375 unsigned n1 = ex_to_numeric(i1.get_value()).to_int(), n2 = ex_to_numeric(i2.get_value()).to_int();
377 throw (std::runtime_error("matrix::eval_indexed(): value of first index exceeds number of rows"));
379 throw (std::runtime_error("matrix::eval_indexed(): value of second index exceeds number of columns"));
380 return (*this)(n1, n2);
384 throw (std::runtime_error("matrix::eval_indexed(): matrix must have exactly 2 indices"));
389 /** Sum of two indexed matrices. */
390 ex matrix::add_indexed(const ex & self, const ex & other) const
392 GINAC_ASSERT(is_ex_of_type(self, indexed));
393 GINAC_ASSERT(is_ex_of_type(self.op(0), matrix));
394 GINAC_ASSERT(is_ex_of_type(other, indexed));
395 GINAC_ASSERT(self.nops() == 2 || self.nops() == 3);
397 // Only add two matrices
398 if (is_ex_of_type(other.op(0), matrix)) {
399 GINAC_ASSERT(other.nops() == 2 || other.nops() == 3);
401 const matrix &self_matrix = ex_to_matrix(self.op(0));
402 const matrix &other_matrix = ex_to_matrix(other.op(0));
404 if (self.nops() == 2 && other.nops() == 2) { // vector + vector
406 if (self_matrix.row == other_matrix.row)
407 return indexed(self_matrix.add(other_matrix), self.op(1));
408 else if (self_matrix.row == other_matrix.col)
409 return indexed(self_matrix.add(other_matrix.transpose()), self.op(1));
411 } else if (self.nops() == 3 && other.nops() == 3) { // matrix + matrix
413 if (self.op(1).is_equal(other.op(1)) && self.op(2).is_equal(other.op(2)))
414 return indexed(self_matrix.add(other_matrix), self.op(1), self.op(2));
415 else if (self.op(1).is_equal(other.op(2)) && self.op(2).is_equal(other.op(1)))
416 return indexed(self_matrix.add(other_matrix.transpose()), self.op(1), self.op(2));
421 // Don't know what to do, return unevaluated sum
425 /** Product of an indexed matrix with a number. */
426 ex matrix::scalar_mul_indexed(const ex & self, const numeric & other) const
428 GINAC_ASSERT(is_ex_of_type(self, indexed));
429 GINAC_ASSERT(is_ex_of_type(self.op(0), matrix));
430 GINAC_ASSERT(self.nops() == 2 || self.nops() == 3);
432 const matrix &self_matrix = ex_to_matrix(self.op(0));
434 if (self.nops() == 2)
435 return indexed(self_matrix.mul(other), self.op(1));
436 else // self.nops() == 3
437 return indexed(self_matrix.mul(other), self.op(1), self.op(2));
440 /** Contraction of an indexed matrix with something else. */
441 bool matrix::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
443 GINAC_ASSERT(is_ex_of_type(*self, indexed));
444 GINAC_ASSERT(is_ex_of_type(*other, indexed));
445 GINAC_ASSERT(self->nops() == 2 || self->nops() == 3);
446 GINAC_ASSERT(is_ex_of_type(self->op(0), matrix));
448 // Only contract with other matrices
449 if (!is_ex_of_type(other->op(0), matrix))
452 GINAC_ASSERT(other->nops() == 2 || other->nops() == 3);
454 const matrix &self_matrix = ex_to_matrix(self->op(0));
455 const matrix &other_matrix = ex_to_matrix(other->op(0));
457 if (self->nops() == 2) {
458 unsigned self_dim = (self_matrix.col == 1) ? self_matrix.row : self_matrix.col;
460 if (other->nops() == 2) { // vector * vector (scalar product)
461 unsigned other_dim = (other_matrix.col == 1) ? other_matrix.row : other_matrix.col;
463 if (self_matrix.col == 1) {
464 if (other_matrix.col == 1) {
465 // Column vector * column vector, transpose first vector
466 *self = self_matrix.transpose().mul(other_matrix)(0, 0);
468 // Column vector * row vector, swap factors
469 *self = other_matrix.mul(self_matrix)(0, 0);
472 if (other_matrix.col == 1) {
473 // Row vector * column vector, perfect
474 *self = self_matrix.mul(other_matrix)(0, 0);
476 // Row vector * row vector, transpose second vector
477 *self = self_matrix.mul(other_matrix.transpose())(0, 0);
483 } else { // vector * matrix
485 // B_i * A_ij = (B*A)_j (B is row vector)
486 if (is_dummy_pair(self->op(1), other->op(1))) {
487 if (self_matrix.row == 1)
488 *self = indexed(self_matrix.mul(other_matrix), other->op(2));
490 *self = indexed(self_matrix.transpose().mul(other_matrix), other->op(2));
495 // B_j * A_ij = (A*B)_i (B is column vector)
496 if (is_dummy_pair(self->op(1), other->op(2))) {
497 if (self_matrix.col == 1)
498 *self = indexed(other_matrix.mul(self_matrix), other->op(1));
500 *self = indexed(other_matrix.mul(self_matrix.transpose()), other->op(1));
506 } else if (other->nops() == 3) { // matrix * matrix
508 // A_ij * B_jk = (A*B)_ik
509 if (is_dummy_pair(self->op(2), other->op(1))) {
510 *self = indexed(self_matrix.mul(other_matrix), self->op(1), other->op(2));
515 // A_ij * B_kj = (A*Btrans)_ik
516 if (is_dummy_pair(self->op(2), other->op(2))) {
517 *self = indexed(self_matrix.mul(other_matrix.transpose()), self->op(1), other->op(1));
522 // A_ji * B_jk = (Atrans*B)_ik
523 if (is_dummy_pair(self->op(1), other->op(1))) {
524 *self = indexed(self_matrix.transpose().mul(other_matrix), self->op(2), other->op(2));
529 // A_ji * B_kj = (B*A)_ki
530 if (is_dummy_pair(self->op(1), other->op(2))) {
531 *self = indexed(other_matrix.mul(self_matrix), other->op(1), self->op(2));
542 // non-virtual functions in this class
549 * @exception logic_error (incompatible matrices) */
550 matrix matrix::add(const matrix & other) const
552 if (col != other.col || row != other.row)
553 throw (std::logic_error("matrix::add(): incompatible matrices"));
555 exvector sum(this->m);
556 exvector::iterator i;
557 exvector::const_iterator ci;
558 for (i=sum.begin(), ci=other.m.begin(); i!=sum.end(); ++i, ++ci)
561 return matrix(row,col,sum);
565 /** Difference of matrices.
567 * @exception logic_error (incompatible matrices) */
568 matrix matrix::sub(const matrix & other) const
570 if (col != other.col || row != other.row)
571 throw (std::logic_error("matrix::sub(): incompatible matrices"));
573 exvector dif(this->m);
574 exvector::iterator i;
575 exvector::const_iterator ci;
576 for (i=dif.begin(), ci=other.m.begin(); i!=dif.end(); ++i, ++ci)
579 return matrix(row,col,dif);
583 /** Product of matrices.
585 * @exception logic_error (incompatible matrices) */
586 matrix matrix::mul(const matrix & other) const
588 if (this->cols() != other.rows())
589 throw (std::logic_error("matrix::mul(): incompatible matrices"));
591 exvector prod(this->rows()*other.cols());
593 for (unsigned r1=0; r1<this->rows(); ++r1) {
594 for (unsigned c=0; c<this->cols(); ++c) {
595 if (m[r1*col+c].is_zero())
597 for (unsigned r2=0; r2<other.cols(); ++r2)
598 prod[r1*other.col+r2] += (m[r1*col+c] * other.m[c*other.col+r2]).expand();
601 return matrix(row, other.col, prod);
605 /** Product of matrix and scalar. */
606 matrix matrix::mul(const numeric & other) const
608 exvector prod(row * col);
610 for (unsigned r=0; r<row; ++r)
611 for (unsigned c=0; c<col; ++c)
612 prod[r*col+c] = m[r*col+c] * other;
614 return matrix(row, col, prod);
618 /** operator() to access elements.
620 * @param ro row of element
621 * @param co column of element
622 * @exception range_error (index out of range) */
623 const ex & matrix::operator() (unsigned ro, unsigned co) const
625 if (ro>=row || co>=col)
626 throw (std::range_error("matrix::operator(): index out of range"));
632 /** Set individual elements manually.
634 * @exception range_error (index out of range) */
635 matrix & matrix::set(unsigned ro, unsigned co, ex value)
637 if (ro>=row || co>=col)
638 throw (std::range_error("matrix::set(): index out of range"));
640 ensure_if_modifiable();
641 m[ro*col+co] = value;
646 /** Transposed of an m x n matrix, producing a new n x m matrix object that
647 * represents the transposed. */
648 matrix matrix::transpose(void) const
650 exvector trans(this->cols()*this->rows());
652 for (unsigned r=0; r<this->cols(); ++r)
653 for (unsigned c=0; c<this->rows(); ++c)
654 trans[r*this->rows()+c] = m[c*this->cols()+r];
656 return matrix(this->cols(),this->rows(),trans);
660 /** Determinant of square matrix. This routine doesn't actually calculate the
661 * determinant, it only implements some heuristics about which algorithm to
662 * run. If all the elements of the matrix are elements of an integral domain
663 * the determinant is also in that integral domain and the result is expanded
664 * only. If one or more elements are from a quotient field the determinant is
665 * usually also in that quotient field and the result is normalized before it
666 * is returned. This implies that the determinant of the symbolic 2x2 matrix
667 * [[a/(a-b),1],[b/(a-b),1]] is returned as unity. (In this respect, it
668 * behaves like MapleV and unlike Mathematica.)
670 * @param algo allows to chose an algorithm
671 * @return the determinant as a new expression
672 * @exception logic_error (matrix not square)
673 * @see determinant_algo */
674 ex matrix::determinant(unsigned algo) const
677 throw (std::logic_error("matrix::determinant(): matrix not square"));
678 GINAC_ASSERT(row*col==m.capacity());
680 // Gather some statistical information about this matrix:
681 bool numeric_flag = true;
682 bool normal_flag = false;
683 unsigned sparse_count = 0; // counts non-zero elements
684 for (exvector::const_iterator r=m.begin(); r!=m.end(); ++r) {
685 lst srl; // symbol replacement list
686 ex rtest = (*r).to_rational(srl);
687 if (!rtest.is_zero())
689 if (!rtest.info(info_flags::numeric))
690 numeric_flag = false;
691 if (!rtest.info(info_flags::crational_polynomial) &&
692 rtest.info(info_flags::rational_function))
696 // Here is the heuristics in case this routine has to decide:
697 if (algo == determinant_algo::automatic) {
698 // Minor expansion is generally a good guess:
699 algo = determinant_algo::laplace;
700 // Does anybody know when a matrix is really sparse?
701 // Maybe <~row/2.236 nonzero elements average in a row?
702 if (row>3 && 5*sparse_count<=row*col)
703 algo = determinant_algo::bareiss;
704 // Purely numeric matrix can be handled by Gauss elimination.
705 // This overrides any prior decisions.
707 algo = determinant_algo::gauss;
710 // Trap the trivial case here, since some algorithms don't like it
712 // for consistency with non-trivial determinants...
714 return m[0].normal();
716 return m[0].expand();
719 // Compute the determinant
721 case determinant_algo::gauss: {
724 int sign = tmp.gauss_elimination(true);
725 for (unsigned d=0; d<row; ++d)
726 det *= tmp.m[d*col+d];
728 return (sign*det).normal();
730 return (sign*det).normal().expand();
732 case determinant_algo::bareiss: {
735 sign = tmp.fraction_free_elimination(true);
737 return (sign*tmp.m[row*col-1]).normal();
739 return (sign*tmp.m[row*col-1]).expand();
741 case determinant_algo::divfree: {
744 sign = tmp.division_free_elimination(true);
747 ex det = tmp.m[row*col-1];
748 // factor out accumulated bogus slag
749 for (unsigned d=0; d<row-2; ++d)
750 for (unsigned j=0; j<row-d-2; ++j)
751 det = (det/tmp.m[d*col+d]).normal();
754 case determinant_algo::laplace:
756 // This is the minor expansion scheme. We always develop such
757 // that the smallest minors (i.e, the trivial 1x1 ones) are on the
758 // rightmost column. For this to be efficient it turns out that
759 // the emptiest columns (i.e. the ones with most zeros) should be
760 // the ones on the right hand side. Therefore we presort the
761 // columns of the matrix:
762 typedef std::pair<unsigned,unsigned> uintpair;
763 std::vector<uintpair> c_zeros; // number of zeros in column
764 for (unsigned c=0; c<col; ++c) {
766 for (unsigned r=0; r<row; ++r)
767 if (m[r*col+c].is_zero())
769 c_zeros.push_back(uintpair(acc,c));
771 sort(c_zeros.begin(),c_zeros.end());
772 std::vector<unsigned> pre_sort;
773 for (std::vector<uintpair>::iterator i=c_zeros.begin(); i!=c_zeros.end(); ++i)
774 pre_sort.push_back(i->second);
775 int sign = permutation_sign(pre_sort);
776 exvector result(row*col); // represents sorted matrix
778 for (std::vector<unsigned>::iterator i=pre_sort.begin();
781 for (unsigned r=0; r<row; ++r)
782 result[r*col+c] = m[r*col+(*i)];
786 return (sign*matrix(row,col,result).determinant_minor()).normal();
788 return sign*matrix(row,col,result).determinant_minor();
794 /** Trace of a matrix. The result is normalized if it is in some quotient
795 * field and expanded only otherwise. This implies that the trace of the
796 * symbolic 2x2 matrix [[a/(a-b),x],[y,b/(b-a)]] is recognized to be unity.
798 * @return the sum of diagonal elements
799 * @exception logic_error (matrix not square) */
800 ex matrix::trace(void) const
803 throw (std::logic_error("matrix::trace(): matrix not square"));
806 for (unsigned r=0; r<col; ++r)
809 if (tr.info(info_flags::rational_function) &&
810 !tr.info(info_flags::crational_polynomial))
817 /** Characteristic Polynomial. Following mathematica notation the
818 * characteristic polynomial of a matrix M is defined as the determiant of
819 * (M - lambda * 1) where 1 stands for the unit matrix of the same dimension
820 * as M. Note that some CASs define it with a sign inside the determinant
821 * which gives rise to an overall sign if the dimension is odd. This method
822 * returns the characteristic polynomial collected in powers of lambda as a
825 * @return characteristic polynomial as new expression
826 * @exception logic_error (matrix not square)
827 * @see matrix::determinant() */
828 ex matrix::charpoly(const symbol & lambda) const
831 throw (std::logic_error("matrix::charpoly(): matrix not square"));
833 bool numeric_flag = true;
834 for (exvector::const_iterator r=m.begin(); r!=m.end(); ++r) {
835 if (!(*r).info(info_flags::numeric)) {
836 numeric_flag = false;
840 // The pure numeric case is traditionally rather common. Hence, it is
841 // trapped and we use Leverrier's algorithm which goes as row^3 for
842 // every coefficient. The expensive part is the matrix multiplication.
846 ex poly = power(lambda,row)-c*power(lambda,row-1);
847 for (unsigned i=1; i<row; ++i) {
848 for (unsigned j=0; j<row; ++j)
851 c = B.trace()/ex(i+1);
852 poly -= c*power(lambda,row-i-1);
861 for (unsigned r=0; r<col; ++r)
862 M.m[r*col+r] -= lambda;
864 return M.determinant().collect(lambda);
868 /** Inverse of this matrix.
870 * @return the inverted matrix
871 * @exception logic_error (matrix not square)
872 * @exception runtime_error (singular matrix) */
873 matrix matrix::inverse(void) const
876 throw (std::logic_error("matrix::inverse(): matrix not square"));
878 // NOTE: the Gauss-Jordan elimination used here can in principle be
879 // replaced by two clever calls to gauss_elimination() and some to
880 // transpose(). Wouldn't be more efficient (maybe less?), just more
883 // set tmp to the unit matrix
884 for (unsigned i=0; i<col; ++i)
885 tmp.m[i*col+i] = _ex1();
887 // create a copy of this matrix
889 for (unsigned r1=0; r1<row; ++r1) {
890 int indx = cpy.pivot(r1, r1);
892 throw (std::runtime_error("matrix::inverse(): singular matrix"));
894 if (indx != 0) { // swap rows r and indx of matrix tmp
895 for (unsigned i=0; i<col; ++i)
896 tmp.m[r1*col+i].swap(tmp.m[indx*col+i]);
898 ex a1 = cpy.m[r1*col+r1];
899 for (unsigned c=0; c<col; ++c) {
900 cpy.m[r1*col+c] /= a1;
901 tmp.m[r1*col+c] /= a1;
903 for (unsigned r2=0; r2<row; ++r2) {
905 if (!cpy.m[r2*col+r1].is_zero()) {
906 ex a2 = cpy.m[r2*col+r1];
907 // yes, there is something to do in this column
908 for (unsigned c=0; c<col; ++c) {
909 cpy.m[r2*col+c] -= a2 * cpy.m[r1*col+c];
910 if (!cpy.m[r2*col+c].info(info_flags::numeric))
911 cpy.m[r2*col+c] = cpy.m[r2*col+c].normal();
912 tmp.m[r2*col+c] -= a2 * tmp.m[r1*col+c];
913 if (!tmp.m[r2*col+c].info(info_flags::numeric))
914 tmp.m[r2*col+c] = tmp.m[r2*col+c].normal();
925 /** Solve a linear system consisting of a m x n matrix and a m x p right hand
926 * side by applying an elimination scheme to the augmented matrix.
928 * @param vars n x p matrix, all elements must be symbols
929 * @param rhs m x p matrix
930 * @return n x p solution matrix
931 * @exception logic_error (incompatible matrices)
932 * @exception invalid_argument (1st argument must be matrix of symbols)
933 * @exception runtime_error (inconsistent linear system)
935 matrix matrix::solve(const matrix & vars,
939 const unsigned m = this->rows();
940 const unsigned n = this->cols();
941 const unsigned p = rhs.cols();
944 if ((rhs.rows() != m) || (vars.rows() != n) || (vars.col != p))
945 throw (std::logic_error("matrix::solve(): incompatible matrices"));
946 for (unsigned ro=0; ro<n; ++ro)
947 for (unsigned co=0; co<p; ++co)
948 if (!vars(ro,co).info(info_flags::symbol))
949 throw (std::invalid_argument("matrix::solve(): 1st argument must be matrix of symbols"));
951 // build the augmented matrix of *this with rhs attached to the right
953 for (unsigned r=0; r<m; ++r) {
954 for (unsigned c=0; c<n; ++c)
955 aug.m[r*(n+p)+c] = this->m[r*n+c];
956 for (unsigned c=0; c<p; ++c)
957 aug.m[r*(n+p)+c+n] = rhs.m[r*p+c];
960 // Gather some statistical information about the augmented matrix:
961 bool numeric_flag = true;
962 for (exvector::const_iterator r=aug.m.begin(); r!=aug.m.end(); ++r) {
963 if (!(*r).info(info_flags::numeric))
964 numeric_flag = false;
967 // Here is the heuristics in case this routine has to decide:
968 if (algo == solve_algo::automatic) {
969 // Bareiss (fraction-free) elimination is generally a good guess:
970 algo = solve_algo::bareiss;
971 // For m<3, Bareiss elimination is equivalent to division free
972 // elimination but has more logistic overhead
974 algo = solve_algo::divfree;
975 // This overrides any prior decisions.
977 algo = solve_algo::gauss;
980 // Eliminate the augmented matrix:
982 case solve_algo::gauss:
983 aug.gauss_elimination();
984 case solve_algo::divfree:
985 aug.division_free_elimination();
986 case solve_algo::bareiss:
988 aug.fraction_free_elimination();
991 // assemble the solution matrix:
993 for (unsigned co=0; co<p; ++co) {
994 unsigned last_assigned_sol = n+1;
995 for (int r=m-1; r>=0; --r) {
996 unsigned fnz = 1; // first non-zero in row
997 while ((fnz<=n) && (aug.m[r*(n+p)+(fnz-1)].is_zero()))
1000 // row consists only of zeros, corresponding rhs must be 0, too
1001 if (!aug.m[r*(n+p)+n+co].is_zero()) {
1002 throw (std::runtime_error("matrix::solve(): inconsistent linear system"));
1005 // assign solutions for vars between fnz+1 and
1006 // last_assigned_sol-1: free parameters
1007 for (unsigned c=fnz; c<last_assigned_sol-1; ++c)
1008 sol.set(c,co,vars.m[c*p+co]);
1009 ex e = aug.m[r*(n+p)+n+co];
1010 for (unsigned c=fnz; c<n; ++c)
1011 e -= aug.m[r*(n+p)+c]*sol.m[c*p+co];
1013 (e/(aug.m[r*(n+p)+(fnz-1)])).normal());
1014 last_assigned_sol = fnz;
1017 // assign solutions for vars between 1 and
1018 // last_assigned_sol-1: free parameters
1019 for (unsigned ro=0; ro<last_assigned_sol-1; ++ro)
1020 sol.set(ro,co,vars(ro,co));
1029 /** Recursive determinant for small matrices having at least one symbolic
1030 * entry. The basic algorithm, known as Laplace-expansion, is enhanced by
1031 * some bookkeeping to avoid calculation of the same submatrices ("minors")
1032 * more than once. According to W.M.Gentleman and S.C.Johnson this algorithm
1033 * is better than elimination schemes for matrices of sparse multivariate
1034 * polynomials and also for matrices of dense univariate polynomials if the
1035 * matrix' dimesion is larger than 7.
1037 * @return the determinant as a new expression (in expanded form)
1038 * @see matrix::determinant() */
1039 ex matrix::determinant_minor(void) const
1041 // for small matrices the algorithm does not make any sense:
1042 const unsigned n = this->cols();
1044 return m[0].expand();
1046 return (m[0]*m[3]-m[2]*m[1]).expand();
1048 return (m[0]*m[4]*m[8]-m[0]*m[5]*m[7]-
1049 m[1]*m[3]*m[8]+m[2]*m[3]*m[7]+
1050 m[1]*m[5]*m[6]-m[2]*m[4]*m[6]).expand();
1052 // This algorithm can best be understood by looking at a naive
1053 // implementation of Laplace-expansion, like this one:
1055 // matrix minorM(this->rows()-1,this->cols()-1);
1056 // for (unsigned r1=0; r1<this->rows(); ++r1) {
1057 // // shortcut if element(r1,0) vanishes
1058 // if (m[r1*col].is_zero())
1060 // // assemble the minor matrix
1061 // for (unsigned r=0; r<minorM.rows(); ++r) {
1062 // for (unsigned c=0; c<minorM.cols(); ++c) {
1064 // minorM.set(r,c,m[r*col+c+1]);
1066 // minorM.set(r,c,m[(r+1)*col+c+1]);
1069 // // recurse down and care for sign:
1071 // det -= m[r1*col] * minorM.determinant_minor();
1073 // det += m[r1*col] * minorM.determinant_minor();
1075 // return det.expand();
1076 // What happens is that while proceeding down many of the minors are
1077 // computed more than once. In particular, there are binomial(n,k)
1078 // kxk minors and each one is computed factorial(n-k) times. Therefore
1079 // it is reasonable to store the results of the minors. We proceed from
1080 // right to left. At each column c we only need to retrieve the minors
1081 // calculated in step c-1. We therefore only have to store at most
1082 // 2*binomial(n,n/2) minors.
1084 // Unique flipper counter for partitioning into minors
1085 std::vector<unsigned> Pkey;
1087 // key for minor determinant (a subpartition of Pkey)
1088 std::vector<unsigned> Mkey;
1090 // we store our subminors in maps, keys being the rows they arise from
1091 typedef std::map<std::vector<unsigned>,class ex> Rmap;
1092 typedef std::map<std::vector<unsigned>,class ex>::value_type Rmap_value;
1096 // initialize A with last column:
1097 for (unsigned r=0; r<n; ++r) {
1098 Pkey.erase(Pkey.begin(),Pkey.end());
1100 A.insert(Rmap_value(Pkey,m[n*(r+1)-1]));
1102 // proceed from right to left through matrix
1103 for (int c=n-2; c>=0; --c) {
1104 Pkey.erase(Pkey.begin(),Pkey.end()); // don't change capacity
1105 Mkey.erase(Mkey.begin(),Mkey.end());
1106 for (unsigned i=0; i<n-c; ++i)
1108 unsigned fc = 0; // controls logic for our strange flipper counter
1111 for (unsigned r=0; r<n-c; ++r) {
1112 // maybe there is nothing to do?
1113 if (m[Pkey[r]*n+c].is_zero())
1115 // create the sorted key for all possible minors
1116 Mkey.erase(Mkey.begin(),Mkey.end());
1117 for (unsigned i=0; i<n-c; ++i)
1119 Mkey.push_back(Pkey[i]);
1120 // Fetch the minors and compute the new determinant
1122 det -= m[Pkey[r]*n+c]*A[Mkey];
1124 det += m[Pkey[r]*n+c]*A[Mkey];
1126 // prevent build-up of deep nesting of expressions saves time:
1128 // store the new determinant at its place in B:
1130 B.insert(Rmap_value(Pkey,det));
1131 // increment our strange flipper counter
1132 for (fc=n-c; fc>0; --fc) {
1134 if (Pkey[fc-1]<fc+c)
1138 for (unsigned j=fc; j<n-c; ++j)
1139 Pkey[j] = Pkey[j-1]+1;
1141 // next column, so change the role of A and B:
1150 /** Perform the steps of an ordinary Gaussian elimination to bring the m x n
1151 * matrix into an upper echelon form. The algorithm is ok for matrices
1152 * with numeric coefficients but quite unsuited for symbolic matrices.
1154 * @param det may be set to true to save a lot of space if one is only
1155 * interested in the diagonal elements (i.e. for calculating determinants).
1156 * The others are set to zero in this case.
1157 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1158 * number of rows was swapped and 0 if the matrix is singular. */
1159 int matrix::gauss_elimination(const bool det)
1161 ensure_if_modifiable();
1162 const unsigned m = this->rows();
1163 const unsigned n = this->cols();
1164 GINAC_ASSERT(!det || n==m);
1168 for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
1169 int indx = pivot(r0, r1, true);
1173 return 0; // leaves *this in a messy state
1178 for (unsigned r2=r0+1; r2<m; ++r2) {
1179 if (!this->m[r2*n+r1].is_zero()) {
1180 // yes, there is something to do in this row
1181 ex piv = this->m[r2*n+r1] / this->m[r0*n+r1];
1182 for (unsigned c=r1+1; c<n; ++c) {
1183 this->m[r2*n+c] -= piv * this->m[r0*n+c];
1184 if (!this->m[r2*n+c].info(info_flags::numeric))
1185 this->m[r2*n+c] = this->m[r2*n+c].normal();
1188 // fill up left hand side with zeros
1189 for (unsigned c=0; c<=r1; ++c)
1190 this->m[r2*n+c] = _ex0();
1193 // save space by deleting no longer needed elements
1194 for (unsigned c=r0+1; c<n; ++c)
1195 this->m[r0*n+c] = _ex0();
1205 /** Perform the steps of division free elimination to bring the m x n matrix
1206 * into an upper echelon form.
1208 * @param det may be set to true to save a lot of space if one is only
1209 * interested in the diagonal elements (i.e. for calculating determinants).
1210 * The others are set to zero in this case.
1211 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1212 * number of rows was swapped and 0 if the matrix is singular. */
1213 int matrix::division_free_elimination(const bool det)
1215 ensure_if_modifiable();
1216 const unsigned m = this->rows();
1217 const unsigned n = this->cols();
1218 GINAC_ASSERT(!det || n==m);
1222 for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
1223 int indx = pivot(r0, r1, true);
1227 return 0; // leaves *this in a messy state
1232 for (unsigned r2=r0+1; r2<m; ++r2) {
1233 for (unsigned c=r1+1; c<n; ++c)
1234 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();
1235 // fill up left hand side with zeros
1236 for (unsigned c=0; c<=r1; ++c)
1237 this->m[r2*n+c] = _ex0();
1240 // save space by deleting no longer needed elements
1241 for (unsigned c=r0+1; c<n; ++c)
1242 this->m[r0*n+c] = _ex0();
1252 /** Perform the steps of Bareiss' one-step fraction free elimination to bring
1253 * the matrix into an upper echelon form. Fraction free elimination means
1254 * that divide is used straightforwardly, without computing GCDs first. This
1255 * is possible, since we know the divisor at each step.
1257 * @param det may be set to true to save a lot of space if one is only
1258 * interested in the last element (i.e. for calculating determinants). The
1259 * others are set to zero in this case.
1260 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1261 * number of rows was swapped and 0 if the matrix is singular. */
1262 int matrix::fraction_free_elimination(const bool det)
1265 // (single-step fraction free elimination scheme, already known to Jordan)
1267 // Usual division-free elimination sets m[0](r,c) = m(r,c) and then sets
1268 // m[k+1](r,c) = m[k](k,k) * m[k](r,c) - m[k](r,k) * m[k](k,c).
1270 // Bareiss (fraction-free) elimination in addition divides that element
1271 // by m[k-1](k-1,k-1) for k>1, where it can be shown by means of the
1272 // Sylvester determinant that this really divides m[k+1](r,c).
1274 // We also allow rational functions where the original prove still holds.
1275 // However, we must care for numerator and denominator separately and
1276 // "manually" work in the integral domains because of subtle cancellations
1277 // (see below). This blows up the bookkeeping a bit and the formula has
1278 // to be modified to expand like this (N{x} stands for numerator of x,
1279 // D{x} for denominator of x):
1280 // 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)}
1281 // -N{m[k](r,k)}*N{m[k](k,c)}*D{m[k](k,k)}*D{m[k](r,c)}
1282 // 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)}
1283 // where for k>1 we now divide N{m[k+1](r,c)} by
1284 // N{m[k-1](k-1,k-1)}
1285 // and D{m[k+1](r,c)} by
1286 // D{m[k-1](k-1,k-1)}.
1288 ensure_if_modifiable();
1289 const unsigned m = this->rows();
1290 const unsigned n = this->cols();
1291 GINAC_ASSERT(!det || n==m);
1300 // We populate temporary matrices to subsequently operate on. There is
1301 // one holding numerators and another holding denominators of entries.
1302 // This is a must since the evaluator (or even earlier mul's constructor)
1303 // might cancel some trivial element which causes divide() to fail. The
1304 // elements are normalized first (yes, even though this algorithm doesn't
1305 // need GCDs) since the elements of *this might be unnormalized, which
1306 // makes things more complicated than they need to be.
1307 matrix tmp_n(*this);
1308 matrix tmp_d(m,n); // for denominators, if needed
1309 lst srl; // symbol replacement list
1310 exvector::iterator it = this->m.begin();
1311 exvector::iterator tmp_n_it = tmp_n.m.begin();
1312 exvector::iterator tmp_d_it = tmp_d.m.begin();
1313 for (; it!= this->m.end(); ++it, ++tmp_n_it, ++tmp_d_it) {
1314 (*tmp_n_it) = (*it).normal().to_rational(srl);
1315 (*tmp_d_it) = (*tmp_n_it).denom();
1316 (*tmp_n_it) = (*tmp_n_it).numer();
1320 for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
1321 int indx = tmp_n.pivot(r0, r1, true);
1330 // tmp_n's rows r0 and indx were swapped, do the same in tmp_d:
1331 for (unsigned c=r1; c<n; ++c)
1332 tmp_d.m[n*indx+c].swap(tmp_d.m[n*r0+c]);
1334 for (unsigned r2=r0+1; r2<m; ++r2) {
1335 for (unsigned c=r1+1; c<n; ++c) {
1336 dividend_n = (tmp_n.m[r0*n+r1]*tmp_n.m[r2*n+c]*
1337 tmp_d.m[r2*n+r1]*tmp_d.m[r0*n+c]
1338 -tmp_n.m[r2*n+r1]*tmp_n.m[r0*n+c]*
1339 tmp_d.m[r0*n+r1]*tmp_d.m[r2*n+c]).expand();
1340 dividend_d = (tmp_d.m[r2*n+r1]*tmp_d.m[r0*n+c]*
1341 tmp_d.m[r0*n+r1]*tmp_d.m[r2*n+c]).expand();
1342 bool check = divide(dividend_n, divisor_n,
1343 tmp_n.m[r2*n+c], true);
1344 check &= divide(dividend_d, divisor_d,
1345 tmp_d.m[r2*n+c], true);
1346 GINAC_ASSERT(check);
1348 // fill up left hand side with zeros
1349 for (unsigned c=0; c<=r1; ++c)
1350 tmp_n.m[r2*n+c] = _ex0();
1352 if ((r1<n-1)&&(r0<m-1)) {
1353 // compute next iteration's divisor
1354 divisor_n = tmp_n.m[r0*n+r1].expand();
1355 divisor_d = tmp_d.m[r0*n+r1].expand();
1357 // save space by deleting no longer needed elements
1358 for (unsigned c=0; c<n; ++c) {
1359 tmp_n.m[r0*n+c] = _ex0();
1360 tmp_d.m[r0*n+c] = _ex1();
1367 // repopulate *this matrix:
1368 it = this->m.begin();
1369 tmp_n_it = tmp_n.m.begin();
1370 tmp_d_it = tmp_d.m.begin();
1371 for (; it!= this->m.end(); ++it, ++tmp_n_it, ++tmp_d_it)
1372 (*it) = ((*tmp_n_it)/(*tmp_d_it)).subs(srl);
1378 /** Partial pivoting method for matrix elimination schemes.
1379 * Usual pivoting (symbolic==false) returns the index to the element with the
1380 * largest absolute value in column ro and swaps the current row with the one
1381 * where the element was found. With (symbolic==true) it does the same thing
1382 * with the first non-zero element.
1384 * @param ro is the row from where to begin
1385 * @param co is the column to be inspected
1386 * @param symbolic signal if we want the first non-zero element to be pivoted
1387 * (true) or the one with the largest absolute value (false).
1388 * @return 0 if no interchange occured, -1 if all are zero (usually signaling
1389 * a degeneracy) and positive integer k means that rows ro and k were swapped.
1391 int matrix::pivot(unsigned ro, unsigned co, bool symbolic)
1395 // search first non-zero element in column co beginning at row ro
1396 while ((k<row) && (this->m[k*col+co].expand().is_zero()))
1399 // search largest element in column co beginning at row ro
1400 GINAC_ASSERT(is_ex_of_type(this->m[k*col+co],numeric));
1401 unsigned kmax = k+1;
1402 numeric mmax = abs(ex_to_numeric(m[kmax*col+co]));
1404 GINAC_ASSERT(is_ex_of_type(this->m[kmax*col+co],numeric));
1405 numeric tmp = ex_to_numeric(this->m[kmax*col+co]);
1406 if (abs(tmp) > mmax) {
1412 if (!mmax.is_zero())
1416 // all elements in column co below row ro vanish
1419 // matrix needs no pivoting
1421 // matrix needs pivoting, so swap rows k and ro
1422 ensure_if_modifiable();
1423 for (unsigned c=0; c<col; ++c)
1424 this->m[k*col+c].swap(this->m[ro*col+c]);
1429 ex lst_to_matrix(const lst & l)
1431 // Find number of rows and columns
1432 unsigned rows = l.nops(), cols = 0, i, j;
1433 for (i=0; i<rows; i++)
1434 if (l.op(i).nops() > cols)
1435 cols = l.op(i).nops();
1437 // Allocate and fill matrix
1438 matrix &m = *new matrix(rows, cols);
1439 m.setflag(status_flags::dynallocated);
1440 for (i=0; i<rows; i++)
1441 for (j=0; j<cols; j++)
1442 if (l.op(i).nops() > j)
1443 m.set(i, j, l.op(i).op(j));
1449 ex diag_matrix(const lst & l)
1451 unsigned dim = l.nops();
1453 matrix &m = *new matrix(dim, dim);
1454 m.setflag(status_flags::dynallocated);
1455 for (unsigned i=0; i<dim; i++)
1456 m.set(i, i, l.op(i));
1461 } // namespace GiNaC