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
41 GINAC_IMPLEMENT_REGISTERED_CLASS(matrix, basic)
44 // default ctor, dtor, copy ctor, assignment operator and helpers:
49 /** Default ctor. Initializes to 1 x 1-dimensional zero-matrix. */
50 matrix::matrix() : inherited(TINFO_matrix), row(1), col(1)
52 debugmsg("matrix default ctor",LOGLEVEL_CONSTRUCT);
58 /** For use by copy ctor and assignment operator. */
59 void matrix::copy(const matrix & other)
61 inherited::copy(other);
64 m = other.m; // STL's vector copying invoked here
67 void matrix::destroy(bool call_parent)
69 if (call_parent) inherited::destroy(call_parent);
78 /** Very common ctor. Initializes to r x c-dimensional zero-matrix.
80 * @param r number of rows
81 * @param c number of cols */
82 matrix::matrix(unsigned r, unsigned c)
83 : inherited(TINFO_matrix), row(r), col(c)
85 debugmsg("matrix ctor from unsigned,unsigned",LOGLEVEL_CONSTRUCT);
86 m.resize(r*c, _ex0());
91 /** Ctor from representation, for internal use only. */
92 matrix::matrix(unsigned r, unsigned c, const exvector & m2)
93 : inherited(TINFO_matrix), row(r), col(c), m(m2)
95 debugmsg("matrix ctor from unsigned,unsigned,exvector",LOGLEVEL_CONSTRUCT);
102 /** Construct object from archive_node. */
103 matrix::matrix(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
105 debugmsg("matrix ctor from archive_node", LOGLEVEL_CONSTRUCT);
106 if (!(n.find_unsigned("row", row)) || !(n.find_unsigned("col", col)))
107 throw (std::runtime_error("unknown matrix dimensions in archive"));
108 m.reserve(row * col);
109 for (unsigned int i=0; true; i++) {
111 if (n.find_ex("m", e, sym_lst, i))
118 /** Unarchive the object. */
119 ex matrix::unarchive(const archive_node &n, const lst &sym_lst)
121 return (new matrix(n, sym_lst))->setflag(status_flags::dynallocated);
124 /** Archive the object. */
125 void matrix::archive(archive_node &n) const
127 inherited::archive(n);
128 n.add_unsigned("row", row);
129 n.add_unsigned("col", col);
130 exvector::const_iterator i = m.begin(), iend = m.end();
138 // functions overriding virtual functions from bases classes
143 void matrix::print(std::ostream & os, unsigned upper_precedence) const
145 debugmsg("matrix print",LOGLEVEL_PRINT);
147 for (unsigned r=0; r<row-1; ++r) {
149 for (unsigned c=0; c<col-1; ++c)
150 os << m[r*col+c] << ",";
151 os << m[col*(r+1)-1] << "]], ";
154 for (unsigned c=0; c<col-1; ++c)
155 os << m[(row-1)*col+c] << ",";
156 os << m[row*col-1] << "]] ]]";
159 void matrix::printraw(std::ostream & os) const
161 debugmsg("matrix printraw",LOGLEVEL_PRINT);
162 os << class_name() << "(" << row << "," << col <<",";
163 for (unsigned r=0; r<row-1; ++r) {
165 for (unsigned c=0; c<col-1; ++c)
166 os << m[r*col+c] << ",";
167 os << m[col*(r-1)-1] << "),";
170 for (unsigned c=0; c<col-1; ++c)
171 os << m[(row-1)*col+c] << ",";
172 os << m[row*col-1] << "))";
175 /** nops is defined to be rows x columns. */
176 unsigned matrix::nops() const
181 /** returns matrix entry at position (i/col, i%col). */
182 ex matrix::op(int i) const
187 /** returns matrix entry at position (i/col, i%col). */
188 ex & matrix::let_op(int i)
191 GINAC_ASSERT(i<nops());
196 /** expands the elements of a matrix entry by entry. */
197 ex matrix::expand(unsigned options) const
199 exvector tmp(row*col);
200 for (unsigned i=0; i<row*col; ++i)
201 tmp[i] = m[i].expand(options);
203 return matrix(row, col, tmp);
206 /** Search ocurrences. A matrix 'has' an expression if it is the expression
207 * itself or one of the elements 'has' it. */
208 bool matrix::has(const ex & other) const
210 GINAC_ASSERT(other.bp!=0);
212 // tautology: it is the expression itself
213 if (is_equal(*other.bp)) return true;
215 // search all the elements
216 for (exvector::const_iterator r=m.begin(); r!=m.end(); ++r)
217 if ((*r).has(other)) return true;
222 /** Evaluate matrix entry by entry. */
223 ex matrix::eval(int level) const
225 debugmsg("matrix eval",LOGLEVEL_MEMBER_FUNCTION);
227 // check if we have to do anything at all
228 if ((level==1)&&(flags & status_flags::evaluated))
232 if (level == -max_recursion_level)
233 throw (std::runtime_error("matrix::eval(): recursion limit exceeded"));
235 // eval() entry by entry
236 exvector m2(row*col);
238 for (unsigned r=0; r<row; ++r)
239 for (unsigned c=0; c<col; ++c)
240 m2[r*col+c] = m[r*col+c].eval(level);
242 return (new matrix(row, col, m2))->setflag(status_flags::dynallocated |
243 status_flags::evaluated );
246 /** Evaluate matrix numerically entry by entry. */
247 ex matrix::evalf(int level) const
249 debugmsg("matrix evalf",LOGLEVEL_MEMBER_FUNCTION);
251 // check if we have to do anything at all
256 if (level == -max_recursion_level) {
257 throw (std::runtime_error("matrix::evalf(): recursion limit exceeded"));
260 // evalf() entry by entry
261 exvector m2(row*col);
263 for (unsigned r=0; r<row; ++r)
264 for (unsigned c=0; c<col; ++c)
265 m2[r*col+c] = m[r*col+c].evalf(level);
267 return matrix(row, col, m2);
270 ex matrix::subs(const lst & ls, const lst & lr) const
272 exvector m2(row * col);
273 for (unsigned r=0; r<row; ++r)
274 for (unsigned c=0; c<col; ++c)
275 m2[r*col+c] = m[r*col+c].subs(ls, lr);
277 return matrix(row, col, m2);
282 int matrix::compare_same_type(const basic & other) const
284 GINAC_ASSERT(is_exactly_of_type(other, matrix));
285 const matrix & o = static_cast<matrix &>(const_cast<basic &>(other));
287 // compare number of rows
289 return row < o.rows() ? -1 : 1;
291 // compare number of columns
293 return col < o.cols() ? -1 : 1;
295 // equal number of rows and columns, compare individual elements
297 for (unsigned r=0; r<row; ++r) {
298 for (unsigned c=0; c<col; ++c) {
299 cmpval = ((*this)(r,c)).compare(o(r,c));
300 if (cmpval!=0) return cmpval;
303 // all elements are equal => matrices are equal;
307 /** Automatic symbolic evaluation of an indexed matrix. */
308 ex matrix::eval_indexed(const basic & i) const
310 GINAC_ASSERT(is_of_type(i, indexed));
311 GINAC_ASSERT(is_ex_of_type(i.op(0), matrix));
313 bool all_indices_unsigned = static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint);
318 // One index, must be one-dimensional vector
319 if (row != 1 && col != 1)
320 throw (std::runtime_error("matrix::eval_indexed(): vector must have exactly 1 index"));
322 const idx & i1 = ex_to_idx(i.op(1));
327 if (!i1.get_dim().is_equal(row))
328 throw (std::runtime_error("matrix::eval_indexed(): dimension of index must match number of vector elements"));
330 // Index numeric -> return vector element
331 if (all_indices_unsigned) {
332 unsigned n1 = ex_to_numeric(i1.get_value()).to_int();
334 throw (std::runtime_error("matrix::eval_indexed(): value of index exceeds number of vector elements"));
335 return (*this)(n1, 0);
341 if (!i1.get_dim().is_equal(col))
342 throw (std::runtime_error("matrix::eval_indexed(): dimension of index must match number of vector elements"));
344 // Index numeric -> return vector element
345 if (all_indices_unsigned) {
346 unsigned n1 = ex_to_numeric(i1.get_value()).to_int();
348 throw (std::runtime_error("matrix::eval_indexed(): value of index exceeds number of vector elements"));
349 return (*this)(0, n1);
353 } else if (i.nops() == 3) {
356 const idx & i1 = ex_to_idx(i.op(1));
357 const idx & i2 = ex_to_idx(i.op(2));
359 if (!i1.get_dim().is_equal(row))
360 throw (std::runtime_error("matrix::eval_indexed(): dimension of first index must match number of rows"));
361 if (!i2.get_dim().is_equal(col))
362 throw (std::runtime_error("matrix::eval_indexed(): dimension of second index must match number of columns"));
364 // Pair of dummy indices -> compute trace
365 if (is_dummy_pair(i1, i2))
368 // Both indices numeric -> return matrix element
369 if (all_indices_unsigned) {
370 unsigned n1 = ex_to_numeric(i1.get_value()).to_int(), n2 = ex_to_numeric(i2.get_value()).to_int();
372 throw (std::runtime_error("matrix::eval_indexed(): value of first index exceeds number of rows"));
374 throw (std::runtime_error("matrix::eval_indexed(): value of second index exceeds number of columns"));
375 return (*this)(n1, n2);
379 throw (std::runtime_error("matrix::eval_indexed(): matrix must have exactly 2 indices"));
384 /** Contraction of an indexed matrix with something else. */
385 bool matrix::contract_with(ex & self, ex & other) const
387 GINAC_ASSERT(is_ex_of_type(self, indexed));
388 GINAC_ASSERT(is_ex_of_type(other, indexed));
389 GINAC_ASSERT(self.nops() == 2 || self.nops() == 3);
390 GINAC_ASSERT(is_ex_of_type(self.op(0), matrix));
392 // Only contract with other matrices
393 if (!is_ex_of_type(other.op(0), matrix))
396 GINAC_ASSERT(other.nops() == 2 || other.nops() == 3);
398 const matrix &self_matrix = ex_to_matrix(self.op(0));
399 const matrix &other_matrix = ex_to_matrix(other.op(0));
401 if (self.nops() == 2) {
402 unsigned self_dim = (self_matrix.col == 1) ? self_matrix.row : self_matrix.col;
404 if (other.nops() == 2) { // vector * vector (scalar product)
405 unsigned other_dim = (other_matrix.col == 1) ? other_matrix.row : other_matrix.col;
407 if (self_matrix.col == 1) {
408 if (other_matrix.col == 1) {
409 // Column vector * column vector, transpose first vector
410 self = self_matrix.transpose().mul(other_matrix)(0, 0);
412 // Column vector * row vector, swap factors
413 self = other_matrix.mul(self_matrix)(0, 0);
416 if (other_matrix.col == 1) {
417 // Row vector * column vector, perfect
418 self = self_matrix.mul(other_matrix)(0, 0);
420 // Row vector * row vector, transpose second vector
421 self = self_matrix.mul(other_matrix.transpose())(0, 0);
427 } else { // vector * matrix
429 GINAC_ASSERT(other.nops() == 3);
431 // B_i * A_ij = (B*A)_j (B is row vector)
432 if (is_dummy_pair(self.op(1), other.op(1))) {
433 if (self_matrix.row == 1)
434 self = indexed(self_matrix.mul(other_matrix), other.op(2));
436 self = indexed(self_matrix.transpose().mul(other_matrix), other.op(2));
441 // B_j * A_ij = (A*B)_i (B is column vector)
442 if (is_dummy_pair(self.op(1), other.op(2))) {
443 if (self_matrix.col == 1)
444 self = indexed(other_matrix.mul(self_matrix), other.op(1));
446 self = indexed(other_matrix.mul(self_matrix.transpose()), other.op(1));
452 } else if (other.nops() == 3) { // matrix * matrix
454 GINAC_ASSERT(self.nops() == 3);
455 GINAC_ASSERT(other.nops() == 3);
457 // A_ij * B_jk = (A*B)_ik
458 if (is_dummy_pair(self.op(2), other.op(1))) {
459 self = indexed(self_matrix.mul(other_matrix), self.op(1), other.op(2));
464 // A_ij * B_kj = (A*Btrans)_ik
465 if (is_dummy_pair(self.op(2), other.op(2))) {
466 self = indexed(self_matrix.mul(other_matrix.transpose()), self.op(1), other.op(1));
471 // A_ji * B_jk = (Atrans*B)_ik
472 if (is_dummy_pair(self.op(1), other.op(1))) {
473 self = indexed(self_matrix.transpose().mul(other_matrix), self.op(2), other.op(2));
478 // A_ji * B_kj = (B*A)_ki
479 if (is_dummy_pair(self.op(1), other.op(2))) {
480 self = indexed(other_matrix.mul(self_matrix), other.op(1), self.op(2));
491 // non-virtual functions in this class
498 * @exception logic_error (incompatible matrices) */
499 matrix matrix::add(const matrix & other) const
501 if (col != other.col || row != other.row)
502 throw (std::logic_error("matrix::add(): incompatible matrices"));
504 exvector sum(this->m);
505 exvector::iterator i;
506 exvector::const_iterator ci;
507 for (i=sum.begin(), ci=other.m.begin(); i!=sum.end(); ++i, ++ci)
510 return matrix(row,col,sum);
514 /** Difference of matrices.
516 * @exception logic_error (incompatible matrices) */
517 matrix matrix::sub(const matrix & other) const
519 if (col != other.col || row != other.row)
520 throw (std::logic_error("matrix::sub(): incompatible matrices"));
522 exvector dif(this->m);
523 exvector::iterator i;
524 exvector::const_iterator ci;
525 for (i=dif.begin(), ci=other.m.begin(); i!=dif.end(); ++i, ++ci)
528 return matrix(row,col,dif);
532 /** Product of matrices.
534 * @exception logic_error (incompatible matrices) */
535 matrix matrix::mul(const matrix & other) const
537 if (this->cols() != other.rows())
538 throw (std::logic_error("matrix::mul(): incompatible matrices"));
540 exvector prod(this->rows()*other.cols());
542 for (unsigned r1=0; r1<this->rows(); ++r1) {
543 for (unsigned c=0; c<this->cols(); ++c) {
544 if (m[r1*col+c].is_zero())
546 for (unsigned r2=0; r2<other.cols(); ++r2)
547 prod[r1*other.col+r2] += (m[r1*col+c] * other.m[c*other.col+r2]).expand();
550 return matrix(row, other.col, prod);
554 /** operator() to access elements.
556 * @param ro row of element
557 * @param co column of element
558 * @exception range_error (index out of range) */
559 const ex & matrix::operator() (unsigned ro, unsigned co) const
561 if (ro>=row || co>=col)
562 throw (std::range_error("matrix::operator(): index out of range"));
568 /** Set individual elements manually.
570 * @exception range_error (index out of range) */
571 matrix & matrix::set(unsigned ro, unsigned co, ex value)
573 if (ro>=row || co>=col)
574 throw (std::range_error("matrix::set(): index out of range"));
576 ensure_if_modifiable();
577 m[ro*col+co] = value;
582 /** Transposed of an m x n matrix, producing a new n x m matrix object that
583 * represents the transposed. */
584 matrix matrix::transpose(void) const
586 exvector trans(this->cols()*this->rows());
588 for (unsigned r=0; r<this->cols(); ++r)
589 for (unsigned c=0; c<this->rows(); ++c)
590 trans[r*this->rows()+c] = m[c*this->cols()+r];
592 return matrix(this->cols(),this->rows(),trans);
596 /** Determinant of square matrix. This routine doesn't actually calculate the
597 * determinant, it only implements some heuristics about which algorithm to
598 * run. If all the elements of the matrix are elements of an integral domain
599 * the determinant is also in that integral domain and the result is expanded
600 * only. If one or more elements are from a quotient field the determinant is
601 * usually also in that quotient field and the result is normalized before it
602 * is returned. This implies that the determinant of the symbolic 2x2 matrix
603 * [[a/(a-b),1],[b/(a-b),1]] is returned as unity. (In this respect, it
604 * behaves like MapleV and unlike Mathematica.)
606 * @param algo allows to chose an algorithm
607 * @return the determinant as a new expression
608 * @exception logic_error (matrix not square)
609 * @see determinant_algo */
610 ex matrix::determinant(unsigned algo) const
613 throw (std::logic_error("matrix::determinant(): matrix not square"));
614 GINAC_ASSERT(row*col==m.capacity());
616 // Gather some statistical information about this matrix:
617 bool numeric_flag = true;
618 bool normal_flag = false;
619 unsigned sparse_count = 0; // counts non-zero elements
620 for (exvector::const_iterator r=m.begin(); r!=m.end(); ++r) {
621 lst srl; // symbol replacement list
622 ex rtest = (*r).to_rational(srl);
623 if (!rtest.is_zero())
625 if (!rtest.info(info_flags::numeric))
626 numeric_flag = false;
627 if (!rtest.info(info_flags::crational_polynomial) &&
628 rtest.info(info_flags::rational_function))
632 // Here is the heuristics in case this routine has to decide:
633 if (algo == determinant_algo::automatic) {
634 // Minor expansion is generally a good guess:
635 algo = determinant_algo::laplace;
636 // Does anybody know when a matrix is really sparse?
637 // Maybe <~row/2.236 nonzero elements average in a row?
638 if (row>3 && 5*sparse_count<=row*col)
639 algo = determinant_algo::bareiss;
640 // Purely numeric matrix can be handled by Gauss elimination.
641 // This overrides any prior decisions.
643 algo = determinant_algo::gauss;
646 // Trap the trivial case here, since some algorithms don't like it
648 // for consistency with non-trivial determinants...
650 return m[0].normal();
652 return m[0].expand();
655 // Compute the determinant
657 case determinant_algo::gauss: {
660 int sign = tmp.gauss_elimination(true);
661 for (unsigned d=0; d<row; ++d)
662 det *= tmp.m[d*col+d];
664 return (sign*det).normal();
666 return (sign*det).normal().expand();
668 case determinant_algo::bareiss: {
671 sign = tmp.fraction_free_elimination(true);
673 return (sign*tmp.m[row*col-1]).normal();
675 return (sign*tmp.m[row*col-1]).expand();
677 case determinant_algo::divfree: {
680 sign = tmp.division_free_elimination(true);
683 ex det = tmp.m[row*col-1];
684 // factor out accumulated bogus slag
685 for (unsigned d=0; d<row-2; ++d)
686 for (unsigned j=0; j<row-d-2; ++j)
687 det = (det/tmp.m[d*col+d]).normal();
690 case determinant_algo::laplace:
692 // This is the minor expansion scheme. We always develop such
693 // that the smallest minors (i.e, the trivial 1x1 ones) are on the
694 // rightmost column. For this to be efficient it turns out that
695 // the emptiest columns (i.e. the ones with most zeros) should be
696 // the ones on the right hand side. Therefore we presort the
697 // columns of the matrix:
698 typedef std::pair<unsigned,unsigned> uintpair;
699 std::vector<uintpair> c_zeros; // number of zeros in column
700 for (unsigned c=0; c<col; ++c) {
702 for (unsigned r=0; r<row; ++r)
703 if (m[r*col+c].is_zero())
705 c_zeros.push_back(uintpair(acc,c));
707 sort(c_zeros.begin(),c_zeros.end());
708 std::vector<unsigned> pre_sort;
709 for (std::vector<uintpair>::iterator i=c_zeros.begin(); i!=c_zeros.end(); ++i)
710 pre_sort.push_back(i->second);
711 int sign = permutation_sign(pre_sort);
712 exvector result(row*col); // represents sorted matrix
714 for (std::vector<unsigned>::iterator i=pre_sort.begin();
717 for (unsigned r=0; r<row; ++r)
718 result[r*col+c] = m[r*col+(*i)];
722 return (sign*matrix(row,col,result).determinant_minor()).normal();
724 return sign*matrix(row,col,result).determinant_minor();
730 /** Trace of a matrix. The result is normalized if it is in some quotient
731 * field and expanded only otherwise. This implies that the trace of the
732 * symbolic 2x2 matrix [[a/(a-b),x],[y,b/(b-a)]] is recognized to be unity.
734 * @return the sum of diagonal elements
735 * @exception logic_error (matrix not square) */
736 ex matrix::trace(void) const
739 throw (std::logic_error("matrix::trace(): matrix not square"));
742 for (unsigned r=0; r<col; ++r)
745 if (tr.info(info_flags::rational_function) &&
746 !tr.info(info_flags::crational_polynomial))
753 /** Characteristic Polynomial. Following mathematica notation the
754 * characteristic polynomial of a matrix M is defined as the determiant of
755 * (M - lambda * 1) where 1 stands for the unit matrix of the same dimension
756 * as M. Note that some CASs define it with a sign inside the determinant
757 * which gives rise to an overall sign if the dimension is odd. This method
758 * returns the characteristic polynomial collected in powers of lambda as a
761 * @return characteristic polynomial as new expression
762 * @exception logic_error (matrix not square)
763 * @see matrix::determinant() */
764 ex matrix::charpoly(const symbol & lambda) const
767 throw (std::logic_error("matrix::charpoly(): matrix not square"));
769 bool numeric_flag = true;
770 for (exvector::const_iterator r=m.begin(); r!=m.end(); ++r) {
771 if (!(*r).info(info_flags::numeric)) {
772 numeric_flag = false;
776 // The pure numeric case is traditionally rather common. Hence, it is
777 // trapped and we use Leverrier's algorithm which goes as row^3 for
778 // every coefficient. The expensive part is the matrix multiplication.
782 ex poly = power(lambda,row)-c*power(lambda,row-1);
783 for (unsigned i=1; i<row; ++i) {
784 for (unsigned j=0; j<row; ++j)
787 c = B.trace()/ex(i+1);
788 poly -= c*power(lambda,row-i-1);
797 for (unsigned r=0; r<col; ++r)
798 M.m[r*col+r] -= lambda;
800 return M.determinant().collect(lambda);
804 /** Inverse of this matrix.
806 * @return the inverted matrix
807 * @exception logic_error (matrix not square)
808 * @exception runtime_error (singular matrix) */
809 matrix matrix::inverse(void) const
812 throw (std::logic_error("matrix::inverse(): matrix not square"));
814 // NOTE: the Gauss-Jordan elimination used here can in principle be
815 // replaced by two clever calls to gauss_elimination() and some to
816 // transpose(). Wouldn't be more efficient (maybe less?), just more
819 // set tmp to the unit matrix
820 for (unsigned i=0; i<col; ++i)
821 tmp.m[i*col+i] = _ex1();
823 // create a copy of this matrix
825 for (unsigned r1=0; r1<row; ++r1) {
826 int indx = cpy.pivot(r1, r1);
828 throw (std::runtime_error("matrix::inverse(): singular matrix"));
830 if (indx != 0) { // swap rows r and indx of matrix tmp
831 for (unsigned i=0; i<col; ++i)
832 tmp.m[r1*col+i].swap(tmp.m[indx*col+i]);
834 ex a1 = cpy.m[r1*col+r1];
835 for (unsigned c=0; c<col; ++c) {
836 cpy.m[r1*col+c] /= a1;
837 tmp.m[r1*col+c] /= a1;
839 for (unsigned r2=0; r2<row; ++r2) {
841 if (!cpy.m[r2*col+r1].is_zero()) {
842 ex a2 = cpy.m[r2*col+r1];
843 // yes, there is something to do in this column
844 for (unsigned c=0; c<col; ++c) {
845 cpy.m[r2*col+c] -= a2 * cpy.m[r1*col+c];
846 if (!cpy.m[r2*col+c].info(info_flags::numeric))
847 cpy.m[r2*col+c] = cpy.m[r2*col+c].normal();
848 tmp.m[r2*col+c] -= a2 * tmp.m[r1*col+c];
849 if (!tmp.m[r2*col+c].info(info_flags::numeric))
850 tmp.m[r2*col+c] = tmp.m[r2*col+c].normal();
861 /** Solve a linear system consisting of a m x n matrix and a m x p right hand
862 * side by applying an elimination scheme to the augmented matrix.
864 * @param vars n x p matrix, all elements must be symbols
865 * @param rhs m x p matrix
866 * @return n x p solution matrix
867 * @exception logic_error (incompatible matrices)
868 * @exception invalid_argument (1st argument must be matrix of symbols)
869 * @exception runtime_error (inconsistent linear system)
871 matrix matrix::solve(const matrix & vars,
875 const unsigned m = this->rows();
876 const unsigned n = this->cols();
877 const unsigned p = rhs.cols();
880 if ((rhs.rows() != m) || (vars.rows() != n) || (vars.col != p))
881 throw (std::logic_error("matrix::solve(): incompatible matrices"));
882 for (unsigned ro=0; ro<n; ++ro)
883 for (unsigned co=0; co<p; ++co)
884 if (!vars(ro,co).info(info_flags::symbol))
885 throw (std::invalid_argument("matrix::solve(): 1st argument must be matrix of symbols"));
887 // build the augmented matrix of *this with rhs attached to the right
889 for (unsigned r=0; r<m; ++r) {
890 for (unsigned c=0; c<n; ++c)
891 aug.m[r*(n+p)+c] = this->m[r*n+c];
892 for (unsigned c=0; c<p; ++c)
893 aug.m[r*(n+p)+c+n] = rhs.m[r*p+c];
896 // Gather some statistical information about the augmented matrix:
897 bool numeric_flag = true;
898 for (exvector::const_iterator r=aug.m.begin(); r!=aug.m.end(); ++r) {
899 if (!(*r).info(info_flags::numeric))
900 numeric_flag = false;
903 // Here is the heuristics in case this routine has to decide:
904 if (algo == solve_algo::automatic) {
905 // Bareiss (fraction-free) elimination is generally a good guess:
906 algo = solve_algo::bareiss;
907 // For m<3, Bareiss elimination is equivalent to division free
908 // elimination but has more logistic overhead
910 algo = solve_algo::divfree;
911 // This overrides any prior decisions.
913 algo = solve_algo::gauss;
916 // Eliminate the augmented matrix:
918 case solve_algo::gauss:
919 aug.gauss_elimination();
920 case solve_algo::divfree:
921 aug.division_free_elimination();
922 case solve_algo::bareiss:
924 aug.fraction_free_elimination();
927 // assemble the solution matrix:
929 for (unsigned co=0; co<p; ++co) {
930 unsigned last_assigned_sol = n+1;
931 for (int r=m-1; r>=0; --r) {
932 unsigned fnz = 1; // first non-zero in row
933 while ((fnz<=n) && (aug.m[r*(n+p)+(fnz-1)].is_zero()))
936 // row consists only of zeros, corresponding rhs must be 0, too
937 if (!aug.m[r*(n+p)+n+co].is_zero()) {
938 throw (std::runtime_error("matrix::solve(): inconsistent linear system"));
941 // assign solutions for vars between fnz+1 and
942 // last_assigned_sol-1: free parameters
943 for (unsigned c=fnz; c<last_assigned_sol-1; ++c)
944 sol.set(c,co,vars.m[c*p+co]);
945 ex e = aug.m[r*(n+p)+n+co];
946 for (unsigned c=fnz; c<n; ++c)
947 e -= aug.m[r*(n+p)+c]*sol.m[c*p+co];
949 (e/(aug.m[r*(n+p)+(fnz-1)])).normal());
950 last_assigned_sol = fnz;
953 // assign solutions for vars between 1 and
954 // last_assigned_sol-1: free parameters
955 for (unsigned ro=0; ro<last_assigned_sol-1; ++ro)
956 sol.set(ro,co,vars(ro,co));
965 /** Recursive determinant for small matrices having at least one symbolic
966 * entry. The basic algorithm, known as Laplace-expansion, is enhanced by
967 * some bookkeeping to avoid calculation of the same submatrices ("minors")
968 * more than once. According to W.M.Gentleman and S.C.Johnson this algorithm
969 * is better than elimination schemes for matrices of sparse multivariate
970 * polynomials and also for matrices of dense univariate polynomials if the
971 * matrix' dimesion is larger than 7.
973 * @return the determinant as a new expression (in expanded form)
974 * @see matrix::determinant() */
975 ex matrix::determinant_minor(void) const
977 // for small matrices the algorithm does not make any sense:
978 const unsigned n = this->cols();
980 return m[0].expand();
982 return (m[0]*m[3]-m[2]*m[1]).expand();
984 return (m[0]*m[4]*m[8]-m[0]*m[5]*m[7]-
985 m[1]*m[3]*m[8]+m[2]*m[3]*m[7]+
986 m[1]*m[5]*m[6]-m[2]*m[4]*m[6]).expand();
988 // This algorithm can best be understood by looking at a naive
989 // implementation of Laplace-expansion, like this one:
991 // matrix minorM(this->rows()-1,this->cols()-1);
992 // for (unsigned r1=0; r1<this->rows(); ++r1) {
993 // // shortcut if element(r1,0) vanishes
994 // if (m[r1*col].is_zero())
996 // // assemble the minor matrix
997 // for (unsigned r=0; r<minorM.rows(); ++r) {
998 // for (unsigned c=0; c<minorM.cols(); ++c) {
1000 // minorM.set(r,c,m[r*col+c+1]);
1002 // minorM.set(r,c,m[(r+1)*col+c+1]);
1005 // // recurse down and care for sign:
1007 // det -= m[r1*col] * minorM.determinant_minor();
1009 // det += m[r1*col] * minorM.determinant_minor();
1011 // return det.expand();
1012 // What happens is that while proceeding down many of the minors are
1013 // computed more than once. In particular, there are binomial(n,k)
1014 // kxk minors and each one is computed factorial(n-k) times. Therefore
1015 // it is reasonable to store the results of the minors. We proceed from
1016 // right to left. At each column c we only need to retrieve the minors
1017 // calculated in step c-1. We therefore only have to store at most
1018 // 2*binomial(n,n/2) minors.
1020 // Unique flipper counter for partitioning into minors
1021 std::vector<unsigned> Pkey;
1023 // key for minor determinant (a subpartition of Pkey)
1024 std::vector<unsigned> Mkey;
1026 // we store our subminors in maps, keys being the rows they arise from
1027 typedef std::map<std::vector<unsigned>,class ex> Rmap;
1028 typedef std::map<std::vector<unsigned>,class ex>::value_type Rmap_value;
1032 // initialize A with last column:
1033 for (unsigned r=0; r<n; ++r) {
1034 Pkey.erase(Pkey.begin(),Pkey.end());
1036 A.insert(Rmap_value(Pkey,m[n*(r+1)-1]));
1038 // proceed from right to left through matrix
1039 for (int c=n-2; c>=0; --c) {
1040 Pkey.erase(Pkey.begin(),Pkey.end()); // don't change capacity
1041 Mkey.erase(Mkey.begin(),Mkey.end());
1042 for (unsigned i=0; i<n-c; ++i)
1044 unsigned fc = 0; // controls logic for our strange flipper counter
1047 for (unsigned r=0; r<n-c; ++r) {
1048 // maybe there is nothing to do?
1049 if (m[Pkey[r]*n+c].is_zero())
1051 // create the sorted key for all possible minors
1052 Mkey.erase(Mkey.begin(),Mkey.end());
1053 for (unsigned i=0; i<n-c; ++i)
1055 Mkey.push_back(Pkey[i]);
1056 // Fetch the minors and compute the new determinant
1058 det -= m[Pkey[r]*n+c]*A[Mkey];
1060 det += m[Pkey[r]*n+c]*A[Mkey];
1062 // prevent build-up of deep nesting of expressions saves time:
1064 // store the new determinant at its place in B:
1066 B.insert(Rmap_value(Pkey,det));
1067 // increment our strange flipper counter
1068 for (fc=n-c; fc>0; --fc) {
1070 if (Pkey[fc-1]<fc+c)
1074 for (unsigned j=fc; j<n-c; ++j)
1075 Pkey[j] = Pkey[j-1]+1;
1077 // next column, so change the role of A and B:
1086 /** Perform the steps of an ordinary Gaussian elimination to bring the m x n
1087 * matrix into an upper echelon form. The algorithm is ok for matrices
1088 * with numeric coefficients but quite unsuited for symbolic matrices.
1090 * @param det may be set to true to save a lot of space if one is only
1091 * interested in the diagonal elements (i.e. for calculating determinants).
1092 * The others are set to zero in this case.
1093 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1094 * number of rows was swapped and 0 if the matrix is singular. */
1095 int matrix::gauss_elimination(const bool det)
1097 ensure_if_modifiable();
1098 const unsigned m = this->rows();
1099 const unsigned n = this->cols();
1100 GINAC_ASSERT(!det || n==m);
1104 for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
1105 int indx = pivot(r0, r1, true);
1109 return 0; // leaves *this in a messy state
1114 for (unsigned r2=r0+1; r2<m; ++r2) {
1115 if (!this->m[r2*n+r1].is_zero()) {
1116 // yes, there is something to do in this row
1117 ex piv = this->m[r2*n+r1] / this->m[r0*n+r1];
1118 for (unsigned c=r1+1; c<n; ++c) {
1119 this->m[r2*n+c] -= piv * this->m[r0*n+c];
1120 if (!this->m[r2*n+c].info(info_flags::numeric))
1121 this->m[r2*n+c] = this->m[r2*n+c].normal();
1124 // fill up left hand side with zeros
1125 for (unsigned c=0; c<=r1; ++c)
1126 this->m[r2*n+c] = _ex0();
1129 // save space by deleting no longer needed elements
1130 for (unsigned c=r0+1; c<n; ++c)
1131 this->m[r0*n+c] = _ex0();
1141 /** Perform the steps of division free elimination to bring the m x n matrix
1142 * into an upper echelon form.
1144 * @param det may be set to true to save a lot of space if one is only
1145 * interested in the diagonal elements (i.e. for calculating determinants).
1146 * The others are set to zero in this case.
1147 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1148 * number of rows was swapped and 0 if the matrix is singular. */
1149 int matrix::division_free_elimination(const bool det)
1151 ensure_if_modifiable();
1152 const unsigned m = this->rows();
1153 const unsigned n = this->cols();
1154 GINAC_ASSERT(!det || n==m);
1158 for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
1159 int indx = pivot(r0, r1, true);
1163 return 0; // leaves *this in a messy state
1168 for (unsigned r2=r0+1; r2<m; ++r2) {
1169 for (unsigned c=r1+1; c<n; ++c)
1170 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();
1171 // fill up left hand side with zeros
1172 for (unsigned c=0; c<=r1; ++c)
1173 this->m[r2*n+c] = _ex0();
1176 // save space by deleting no longer needed elements
1177 for (unsigned c=r0+1; c<n; ++c)
1178 this->m[r0*n+c] = _ex0();
1188 /** Perform the steps of Bareiss' one-step fraction free elimination to bring
1189 * the matrix into an upper echelon form. Fraction free elimination means
1190 * that divide is used straightforwardly, without computing GCDs first. This
1191 * is possible, since we know the divisor at each step.
1193 * @param det may be set to true to save a lot of space if one is only
1194 * interested in the last element (i.e. for calculating determinants). The
1195 * others are set to zero in this case.
1196 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1197 * number of rows was swapped and 0 if the matrix is singular. */
1198 int matrix::fraction_free_elimination(const bool det)
1201 // (single-step fraction free elimination scheme, already known to Jordan)
1203 // Usual division-free elimination sets m[0](r,c) = m(r,c) and then sets
1204 // m[k+1](r,c) = m[k](k,k) * m[k](r,c) - m[k](r,k) * m[k](k,c).
1206 // Bareiss (fraction-free) elimination in addition divides that element
1207 // by m[k-1](k-1,k-1) for k>1, where it can be shown by means of the
1208 // Sylvester determinant that this really divides m[k+1](r,c).
1210 // We also allow rational functions where the original prove still holds.
1211 // However, we must care for numerator and denominator separately and
1212 // "manually" work in the integral domains because of subtle cancellations
1213 // (see below). This blows up the bookkeeping a bit and the formula has
1214 // to be modified to expand like this (N{x} stands for numerator of x,
1215 // D{x} for denominator of x):
1216 // 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)}
1217 // -N{m[k](r,k)}*N{m[k](k,c)}*D{m[k](k,k)}*D{m[k](r,c)}
1218 // 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)}
1219 // where for k>1 we now divide N{m[k+1](r,c)} by
1220 // N{m[k-1](k-1,k-1)}
1221 // and D{m[k+1](r,c)} by
1222 // D{m[k-1](k-1,k-1)}.
1224 ensure_if_modifiable();
1225 const unsigned m = this->rows();
1226 const unsigned n = this->cols();
1227 GINAC_ASSERT(!det || n==m);
1236 // We populate temporary matrices to subsequently operate on. There is
1237 // one holding numerators and another holding denominators of entries.
1238 // This is a must since the evaluator (or even earlier mul's constructor)
1239 // might cancel some trivial element which causes divide() to fail. The
1240 // elements are normalized first (yes, even though this algorithm doesn't
1241 // need GCDs) since the elements of *this might be unnormalized, which
1242 // makes things more complicated than they need to be.
1243 matrix tmp_n(*this);
1244 matrix tmp_d(m,n); // for denominators, if needed
1245 lst srl; // symbol replacement list
1246 exvector::iterator it = this->m.begin();
1247 exvector::iterator tmp_n_it = tmp_n.m.begin();
1248 exvector::iterator tmp_d_it = tmp_d.m.begin();
1249 for (; it!= this->m.end(); ++it, ++tmp_n_it, ++tmp_d_it) {
1250 (*tmp_n_it) = (*it).normal().to_rational(srl);
1251 (*tmp_d_it) = (*tmp_n_it).denom();
1252 (*tmp_n_it) = (*tmp_n_it).numer();
1256 for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
1257 int indx = tmp_n.pivot(r0, r1, true);
1266 // tmp_n's rows r0 and indx were swapped, do the same in tmp_d:
1267 for (unsigned c=r1; c<n; ++c)
1268 tmp_d.m[n*indx+c].swap(tmp_d.m[n*r0+c]);
1270 for (unsigned r2=r0+1; r2<m; ++r2) {
1271 for (unsigned c=r1+1; c<n; ++c) {
1272 dividend_n = (tmp_n.m[r0*n+r1]*tmp_n.m[r2*n+c]*
1273 tmp_d.m[r2*n+r1]*tmp_d.m[r0*n+c]
1274 -tmp_n.m[r2*n+r1]*tmp_n.m[r0*n+c]*
1275 tmp_d.m[r0*n+r1]*tmp_d.m[r2*n+c]).expand();
1276 dividend_d = (tmp_d.m[r2*n+r1]*tmp_d.m[r0*n+c]*
1277 tmp_d.m[r0*n+r1]*tmp_d.m[r2*n+c]).expand();
1278 bool check = divide(dividend_n, divisor_n,
1279 tmp_n.m[r2*n+c], true);
1280 check &= divide(dividend_d, divisor_d,
1281 tmp_d.m[r2*n+c], true);
1282 GINAC_ASSERT(check);
1284 // fill up left hand side with zeros
1285 for (unsigned c=0; c<=r1; ++c)
1286 tmp_n.m[r2*n+c] = _ex0();
1288 if ((r1<n-1)&&(r0<m-1)) {
1289 // compute next iteration's divisor
1290 divisor_n = tmp_n.m[r0*n+r1].expand();
1291 divisor_d = tmp_d.m[r0*n+r1].expand();
1293 // save space by deleting no longer needed elements
1294 for (unsigned c=0; c<n; ++c) {
1295 tmp_n.m[r0*n+c] = _ex0();
1296 tmp_d.m[r0*n+c] = _ex1();
1303 // repopulate *this matrix:
1304 it = this->m.begin();
1305 tmp_n_it = tmp_n.m.begin();
1306 tmp_d_it = tmp_d.m.begin();
1307 for (; it!= this->m.end(); ++it, ++tmp_n_it, ++tmp_d_it)
1308 (*it) = ((*tmp_n_it)/(*tmp_d_it)).subs(srl);
1314 /** Partial pivoting method for matrix elimination schemes.
1315 * Usual pivoting (symbolic==false) returns the index to the element with the
1316 * largest absolute value in column ro and swaps the current row with the one
1317 * where the element was found. With (symbolic==true) it does the same thing
1318 * with the first non-zero element.
1320 * @param ro is the row from where to begin
1321 * @param co is the column to be inspected
1322 * @param symbolic signal if we want the first non-zero element to be pivoted
1323 * (true) or the one with the largest absolute value (false).
1324 * @return 0 if no interchange occured, -1 if all are zero (usually signaling
1325 * a degeneracy) and positive integer k means that rows ro and k were swapped.
1327 int matrix::pivot(unsigned ro, unsigned co, bool symbolic)
1331 // search first non-zero element in column co beginning at row ro
1332 while ((k<row) && (this->m[k*col+co].expand().is_zero()))
1335 // search largest element in column co beginning at row ro
1336 GINAC_ASSERT(is_ex_of_type(this->m[k*col+co],numeric));
1337 unsigned kmax = k+1;
1338 numeric mmax = abs(ex_to_numeric(m[kmax*col+co]));
1340 GINAC_ASSERT(is_ex_of_type(this->m[kmax*col+co],numeric));
1341 numeric tmp = ex_to_numeric(this->m[kmax*col+co]);
1342 if (abs(tmp) > mmax) {
1348 if (!mmax.is_zero())
1352 // all elements in column co below row ro vanish
1355 // matrix needs no pivoting
1357 // matrix needs pivoting, so swap rows k and ro
1358 ensure_if_modifiable();
1359 for (unsigned c=0; c<col; ++c)
1360 this->m[k*col+c].swap(this->m[ro*col+c]);
1365 /** Convert list of lists to matrix. */
1366 ex lst_to_matrix(const ex &l)
1368 if (!is_ex_of_type(l, lst))
1369 throw(std::invalid_argument("argument to lst_to_matrix() must be a lst"));
1371 // Find number of rows and columns
1372 unsigned rows = l.nops(), cols = 0, i, j;
1373 for (i=0; i<rows; i++)
1374 if (l.op(i).nops() > cols)
1375 cols = l.op(i).nops();
1377 // Allocate and fill matrix
1378 matrix &m = *new matrix(rows, cols);
1379 for (i=0; i<rows; i++)
1380 for (j=0; j<cols; j++)
1381 if (l.op(i).nops() > j)
1382 m.set(i, j, l.op(i).op(j));
1388 } // namespace GiNaC