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 /** Evaluate matrix entry by entry. */
212 ex matrix::eval(int level) const
214 debugmsg("matrix eval",LOGLEVEL_MEMBER_FUNCTION);
216 // check if we have to do anything at all
217 if ((level==1)&&(flags & status_flags::evaluated))
221 if (level == -max_recursion_level)
222 throw (std::runtime_error("matrix::eval(): recursion limit exceeded"));
224 // eval() entry by entry
225 exvector m2(row*col);
227 for (unsigned r=0; r<row; ++r)
228 for (unsigned c=0; c<col; ++c)
229 m2[r*col+c] = m[r*col+c].eval(level);
231 return (new matrix(row, col, m2))->setflag(status_flags::dynallocated |
232 status_flags::evaluated );
235 /** Evaluate matrix numerically entry by entry. */
236 ex matrix::evalf(int level) const
238 debugmsg("matrix evalf",LOGLEVEL_MEMBER_FUNCTION);
240 // check if we have to do anything at all
245 if (level == -max_recursion_level) {
246 throw (std::runtime_error("matrix::evalf(): recursion limit exceeded"));
249 // evalf() entry by entry
250 exvector m2(row*col);
252 for (unsigned r=0; r<row; ++r)
253 for (unsigned c=0; c<col; ++c)
254 m2[r*col+c] = m[r*col+c].evalf(level);
256 return matrix(row, col, m2);
259 ex matrix::subs(const lst & ls, const lst & lr, bool no_pattern) const
261 exvector m2(row * col);
262 for (unsigned r=0; r<row; ++r)
263 for (unsigned c=0; c<col; ++c)
264 m2[r*col+c] = m[r*col+c].subs(ls, lr, no_pattern);
266 return ex(matrix(row, col, m2)).bp->basic::subs(ls, lr, no_pattern);
271 int matrix::compare_same_type(const basic & other) const
273 GINAC_ASSERT(is_exactly_of_type(other, matrix));
274 const matrix & o = static_cast<matrix &>(const_cast<basic &>(other));
276 // compare number of rows
278 return row < o.rows() ? -1 : 1;
280 // compare number of columns
282 return col < o.cols() ? -1 : 1;
284 // equal number of rows and columns, compare individual elements
286 for (unsigned r=0; r<row; ++r) {
287 for (unsigned c=0; c<col; ++c) {
288 cmpval = ((*this)(r,c)).compare(o(r,c));
289 if (cmpval!=0) return cmpval;
292 // all elements are equal => matrices are equal;
296 /** Automatic symbolic evaluation of an indexed matrix. */
297 ex matrix::eval_indexed(const basic & i) const
299 GINAC_ASSERT(is_of_type(i, indexed));
300 GINAC_ASSERT(is_ex_of_type(i.op(0), matrix));
302 bool all_indices_unsigned = static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint);
307 // One index, must be one-dimensional vector
308 if (row != 1 && col != 1)
309 throw (std::runtime_error("matrix::eval_indexed(): vector must have exactly 1 index"));
311 const idx & i1 = ex_to_idx(i.op(1));
316 if (!i1.get_dim().is_equal(row))
317 throw (std::runtime_error("matrix::eval_indexed(): dimension of index must match number of vector elements"));
319 // Index numeric -> return vector element
320 if (all_indices_unsigned) {
321 unsigned n1 = ex_to_numeric(i1.get_value()).to_int();
323 throw (std::runtime_error("matrix::eval_indexed(): value of index exceeds number of vector elements"));
324 return (*this)(n1, 0);
330 if (!i1.get_dim().is_equal(col))
331 throw (std::runtime_error("matrix::eval_indexed(): dimension of index must match number of vector elements"));
333 // Index numeric -> return vector element
334 if (all_indices_unsigned) {
335 unsigned n1 = ex_to_numeric(i1.get_value()).to_int();
337 throw (std::runtime_error("matrix::eval_indexed(): value of index exceeds number of vector elements"));
338 return (*this)(0, n1);
342 } else if (i.nops() == 3) {
345 const idx & i1 = ex_to_idx(i.op(1));
346 const idx & i2 = ex_to_idx(i.op(2));
348 if (!i1.get_dim().is_equal(row))
349 throw (std::runtime_error("matrix::eval_indexed(): dimension of first index must match number of rows"));
350 if (!i2.get_dim().is_equal(col))
351 throw (std::runtime_error("matrix::eval_indexed(): dimension of second index must match number of columns"));
353 // Pair of dummy indices -> compute trace
354 if (is_dummy_pair(i1, i2))
357 // Both indices numeric -> return matrix element
358 if (all_indices_unsigned) {
359 unsigned n1 = ex_to_numeric(i1.get_value()).to_int(), n2 = ex_to_numeric(i2.get_value()).to_int();
361 throw (std::runtime_error("matrix::eval_indexed(): value of first index exceeds number of rows"));
363 throw (std::runtime_error("matrix::eval_indexed(): value of second index exceeds number of columns"));
364 return (*this)(n1, n2);
368 throw (std::runtime_error("matrix::eval_indexed(): matrix must have exactly 2 indices"));
373 /** Sum of two indexed matrices. */
374 ex matrix::add_indexed(const ex & self, const ex & other) const
376 GINAC_ASSERT(is_ex_of_type(self, indexed));
377 GINAC_ASSERT(is_ex_of_type(self.op(0), matrix));
378 GINAC_ASSERT(is_ex_of_type(other, indexed));
379 GINAC_ASSERT(self.nops() == 2 || self.nops() == 3);
381 // Only add two matrices
382 if (is_ex_of_type(other.op(0), matrix)) {
383 GINAC_ASSERT(other.nops() == 2 || other.nops() == 3);
385 const matrix &self_matrix = ex_to_matrix(self.op(0));
386 const matrix &other_matrix = ex_to_matrix(other.op(0));
388 if (self.nops() == 2 && other.nops() == 2) { // vector + vector
390 if (self_matrix.row == other_matrix.row)
391 return indexed(self_matrix.add(other_matrix), self.op(1));
392 else if (self_matrix.row == other_matrix.col)
393 return indexed(self_matrix.add(other_matrix.transpose()), self.op(1));
395 } else if (self.nops() == 3 && other.nops() == 3) { // matrix + matrix
397 if (self.op(1).is_equal(other.op(1)) && self.op(2).is_equal(other.op(2)))
398 return indexed(self_matrix.add(other_matrix), self.op(1), self.op(2));
399 else if (self.op(1).is_equal(other.op(2)) && self.op(2).is_equal(other.op(1)))
400 return indexed(self_matrix.add(other_matrix.transpose()), self.op(1), self.op(2));
405 // Don't know what to do, return unevaluated sum
409 /** Product of an indexed matrix with a number. */
410 ex matrix::scalar_mul_indexed(const ex & self, const numeric & other) const
412 GINAC_ASSERT(is_ex_of_type(self, indexed));
413 GINAC_ASSERT(is_ex_of_type(self.op(0), matrix));
414 GINAC_ASSERT(self.nops() == 2 || self.nops() == 3);
416 const matrix &self_matrix = ex_to_matrix(self.op(0));
418 if (self.nops() == 2)
419 return indexed(self_matrix.mul(other), self.op(1));
420 else // self.nops() == 3
421 return indexed(self_matrix.mul(other), self.op(1), self.op(2));
424 /** Contraction of an indexed matrix with something else. */
425 bool matrix::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
427 GINAC_ASSERT(is_ex_of_type(*self, indexed));
428 GINAC_ASSERT(is_ex_of_type(*other, indexed));
429 GINAC_ASSERT(self->nops() == 2 || self->nops() == 3);
430 GINAC_ASSERT(is_ex_of_type(self->op(0), matrix));
432 // Only contract with other matrices
433 if (!is_ex_of_type(other->op(0), matrix))
436 GINAC_ASSERT(other->nops() == 2 || other->nops() == 3);
438 const matrix &self_matrix = ex_to_matrix(self->op(0));
439 const matrix &other_matrix = ex_to_matrix(other->op(0));
441 if (self->nops() == 2) {
442 unsigned self_dim = (self_matrix.col == 1) ? self_matrix.row : self_matrix.col;
444 if (other->nops() == 2) { // vector * vector (scalar product)
445 unsigned other_dim = (other_matrix.col == 1) ? other_matrix.row : other_matrix.col;
447 if (self_matrix.col == 1) {
448 if (other_matrix.col == 1) {
449 // Column vector * column vector, transpose first vector
450 *self = self_matrix.transpose().mul(other_matrix)(0, 0);
452 // Column vector * row vector, swap factors
453 *self = other_matrix.mul(self_matrix)(0, 0);
456 if (other_matrix.col == 1) {
457 // Row vector * column vector, perfect
458 *self = self_matrix.mul(other_matrix)(0, 0);
460 // Row vector * row vector, transpose second vector
461 *self = self_matrix.mul(other_matrix.transpose())(0, 0);
467 } else { // vector * matrix
469 // B_i * A_ij = (B*A)_j (B is row vector)
470 if (is_dummy_pair(self->op(1), other->op(1))) {
471 if (self_matrix.row == 1)
472 *self = indexed(self_matrix.mul(other_matrix), other->op(2));
474 *self = indexed(self_matrix.transpose().mul(other_matrix), other->op(2));
479 // B_j * A_ij = (A*B)_i (B is column vector)
480 if (is_dummy_pair(self->op(1), other->op(2))) {
481 if (self_matrix.col == 1)
482 *self = indexed(other_matrix.mul(self_matrix), other->op(1));
484 *self = indexed(other_matrix.mul(self_matrix.transpose()), other->op(1));
490 } else if (other->nops() == 3) { // matrix * matrix
492 // A_ij * B_jk = (A*B)_ik
493 if (is_dummy_pair(self->op(2), other->op(1))) {
494 *self = indexed(self_matrix.mul(other_matrix), self->op(1), other->op(2));
499 // A_ij * B_kj = (A*Btrans)_ik
500 if (is_dummy_pair(self->op(2), other->op(2))) {
501 *self = indexed(self_matrix.mul(other_matrix.transpose()), self->op(1), other->op(1));
506 // A_ji * B_jk = (Atrans*B)_ik
507 if (is_dummy_pair(self->op(1), other->op(1))) {
508 *self = indexed(self_matrix.transpose().mul(other_matrix), self->op(2), other->op(2));
513 // A_ji * B_kj = (B*A)_ki
514 if (is_dummy_pair(self->op(1), other->op(2))) {
515 *self = indexed(other_matrix.mul(self_matrix), other->op(1), self->op(2));
526 // non-virtual functions in this class
533 * @exception logic_error (incompatible matrices) */
534 matrix matrix::add(const matrix & other) const
536 if (col != other.col || row != other.row)
537 throw std::logic_error("matrix::add(): incompatible matrices");
539 exvector sum(this->m);
540 exvector::iterator i;
541 exvector::const_iterator ci;
542 for (i=sum.begin(), ci=other.m.begin(); i!=sum.end(); ++i, ++ci)
545 return matrix(row,col,sum);
549 /** Difference of matrices.
551 * @exception logic_error (incompatible matrices) */
552 matrix matrix::sub(const matrix & other) const
554 if (col != other.col || row != other.row)
555 throw std::logic_error("matrix::sub(): incompatible matrices");
557 exvector dif(this->m);
558 exvector::iterator i;
559 exvector::const_iterator ci;
560 for (i=dif.begin(), ci=other.m.begin(); i!=dif.end(); ++i, ++ci)
563 return matrix(row,col,dif);
567 /** Product of matrices.
569 * @exception logic_error (incompatible matrices) */
570 matrix matrix::mul(const matrix & other) const
572 if (this->cols() != other.rows())
573 throw std::logic_error("matrix::mul(): incompatible matrices");
575 exvector prod(this->rows()*other.cols());
577 for (unsigned r1=0; r1<this->rows(); ++r1) {
578 for (unsigned c=0; c<this->cols(); ++c) {
579 if (m[r1*col+c].is_zero())
581 for (unsigned r2=0; r2<other.cols(); ++r2)
582 prod[r1*other.col+r2] += (m[r1*col+c] * other.m[c*other.col+r2]).expand();
585 return matrix(row, other.col, prod);
589 /** Product of matrix and scalar. */
590 matrix matrix::mul(const numeric & other) const
592 exvector prod(row * col);
594 for (unsigned r=0; r<row; ++r)
595 for (unsigned c=0; c<col; ++c)
596 prod[r*col+c] = m[r*col+c] * other;
598 return matrix(row, col, prod);
602 /** Product of matrix and scalar expression. */
603 matrix matrix::mul_scalar(const ex & other) const
605 if (other.return_type() != return_types::commutative)
606 throw std::runtime_error("matrix::mul_scalar(): non-commutative scalar");
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 /** Power of a matrix. Currently handles integer exponents only. */
619 matrix matrix::pow(const ex & expn) const
622 throw (std::logic_error("matrix::pow(): matrix not square"));
624 if (is_ex_exactly_of_type(expn, numeric)) {
625 // Integer cases are computed by successive multiplication, using the
626 // obvious shortcut of storing temporaries, like A^4 == (A*A)*(A*A).
627 if (expn.info(info_flags::integer)) {
629 matrix prod(row,col);
630 if (expn.info(info_flags::negative)) {
631 k = -ex_to_numeric(expn);
632 prod = this->inverse();
634 k = ex_to_numeric(expn);
637 matrix result(row,col);
638 for (unsigned r=0; r<row; ++r)
639 result.set(r,r,_ex1());
641 // this loop computes the representation of k in base 2 and multiplies
642 // the factors whenever needed:
643 while (b.compare(k)<=0) {
648 result = result.mul(prod);
650 prod = prod.mul(prod);
655 throw (std::runtime_error("matrix::pow(): don't know how to handle exponent"));
659 /** operator() to access elements.
661 * @param ro row of element
662 * @param co column of element
663 * @exception range_error (index out of range) */
664 const ex & matrix::operator() (unsigned ro, unsigned co) const
666 if (ro>=row || co>=col)
667 throw (std::range_error("matrix::operator(): index out of range"));
673 /** Set individual elements manually.
675 * @exception range_error (index out of range) */
676 matrix & matrix::set(unsigned ro, unsigned co, ex value)
678 if (ro>=row || co>=col)
679 throw (std::range_error("matrix::set(): index out of range"));
680 if (value.return_type() != return_types::commutative)
681 throw std::runtime_error("matrix::set(): non-commutative argument");
683 ensure_if_modifiable();
684 m[ro*col+co] = value;
689 /** Transposed of an m x n matrix, producing a new n x m matrix object that
690 * represents the transposed. */
691 matrix matrix::transpose(void) const
693 exvector trans(this->cols()*this->rows());
695 for (unsigned r=0; r<this->cols(); ++r)
696 for (unsigned c=0; c<this->rows(); ++c)
697 trans[r*this->rows()+c] = m[c*this->cols()+r];
699 return matrix(this->cols(),this->rows(),trans);
702 /** Determinant of square matrix. This routine doesn't actually calculate the
703 * determinant, it only implements some heuristics about which algorithm to
704 * run. If all the elements of the matrix are elements of an integral domain
705 * the determinant is also in that integral domain and the result is expanded
706 * only. If one or more elements are from a quotient field the determinant is
707 * usually also in that quotient field and the result is normalized before it
708 * is returned. This implies that the determinant of the symbolic 2x2 matrix
709 * [[a/(a-b),1],[b/(a-b),1]] is returned as unity. (In this respect, it
710 * behaves like MapleV and unlike Mathematica.)
712 * @param algo allows to chose an algorithm
713 * @return the determinant as a new expression
714 * @exception logic_error (matrix not square)
715 * @see determinant_algo */
716 ex matrix::determinant(unsigned algo) const
719 throw (std::logic_error("matrix::determinant(): matrix not square"));
720 GINAC_ASSERT(row*col==m.capacity());
722 // Gather some statistical information about this matrix:
723 bool numeric_flag = true;
724 bool normal_flag = false;
725 unsigned sparse_count = 0; // counts non-zero elements
726 for (exvector::const_iterator r=m.begin(); r!=m.end(); ++r) {
727 lst srl; // symbol replacement list
728 ex rtest = (*r).to_rational(srl);
729 if (!rtest.is_zero())
731 if (!rtest.info(info_flags::numeric))
732 numeric_flag = false;
733 if (!rtest.info(info_flags::crational_polynomial) &&
734 rtest.info(info_flags::rational_function))
738 // Here is the heuristics in case this routine has to decide:
739 if (algo == determinant_algo::automatic) {
740 // Minor expansion is generally a good guess:
741 algo = determinant_algo::laplace;
742 // Does anybody know when a matrix is really sparse?
743 // Maybe <~row/2.236 nonzero elements average in a row?
744 if (row>3 && 5*sparse_count<=row*col)
745 algo = determinant_algo::bareiss;
746 // Purely numeric matrix can be handled by Gauss elimination.
747 // This overrides any prior decisions.
749 algo = determinant_algo::gauss;
752 // Trap the trivial case here, since some algorithms don't like it
754 // for consistency with non-trivial determinants...
756 return m[0].normal();
758 return m[0].expand();
761 // Compute the determinant
763 case determinant_algo::gauss: {
766 int sign = tmp.gauss_elimination(true);
767 for (unsigned d=0; d<row; ++d)
768 det *= tmp.m[d*col+d];
770 return (sign*det).normal();
772 return (sign*det).normal().expand();
774 case determinant_algo::bareiss: {
777 sign = tmp.fraction_free_elimination(true);
779 return (sign*tmp.m[row*col-1]).normal();
781 return (sign*tmp.m[row*col-1]).expand();
783 case determinant_algo::divfree: {
786 sign = tmp.division_free_elimination(true);
789 ex det = tmp.m[row*col-1];
790 // factor out accumulated bogus slag
791 for (unsigned d=0; d<row-2; ++d)
792 for (unsigned j=0; j<row-d-2; ++j)
793 det = (det/tmp.m[d*col+d]).normal();
796 case determinant_algo::laplace:
798 // This is the minor expansion scheme. We always develop such
799 // that the smallest minors (i.e, the trivial 1x1 ones) are on the
800 // rightmost column. For this to be efficient it turns out that
801 // the emptiest columns (i.e. the ones with most zeros) should be
802 // the ones on the right hand side. Therefore we presort the
803 // columns of the matrix:
804 typedef std::pair<unsigned,unsigned> uintpair;
805 std::vector<uintpair> c_zeros; // number of zeros in column
806 for (unsigned c=0; c<col; ++c) {
808 for (unsigned r=0; r<row; ++r)
809 if (m[r*col+c].is_zero())
811 c_zeros.push_back(uintpair(acc,c));
813 sort(c_zeros.begin(),c_zeros.end());
814 std::vector<unsigned> pre_sort;
815 for (std::vector<uintpair>::iterator i=c_zeros.begin(); i!=c_zeros.end(); ++i)
816 pre_sort.push_back(i->second);
817 std::vector<unsigned> pre_sort_test(pre_sort); // permutation_sign() modifies the vector so we make a copy here
818 int sign = permutation_sign(pre_sort_test.begin(), pre_sort_test.end());
819 exvector result(row*col); // represents sorted matrix
821 for (std::vector<unsigned>::iterator i=pre_sort.begin();
824 for (unsigned r=0; r<row; ++r)
825 result[r*col+c] = m[r*col+(*i)];
829 return (sign*matrix(row,col,result).determinant_minor()).normal();
831 return sign*matrix(row,col,result).determinant_minor();
837 /** Trace of a matrix. The result is normalized if it is in some quotient
838 * field and expanded only otherwise. This implies that the trace of the
839 * symbolic 2x2 matrix [[a/(a-b),x],[y,b/(b-a)]] is recognized to be unity.
841 * @return the sum of diagonal elements
842 * @exception logic_error (matrix not square) */
843 ex matrix::trace(void) const
846 throw (std::logic_error("matrix::trace(): matrix not square"));
849 for (unsigned r=0; r<col; ++r)
852 if (tr.info(info_flags::rational_function) &&
853 !tr.info(info_flags::crational_polynomial))
860 /** Characteristic Polynomial. Following mathematica notation the
861 * characteristic polynomial of a matrix M is defined as the determiant of
862 * (M - lambda * 1) where 1 stands for the unit matrix of the same dimension
863 * as M. Note that some CASs define it with a sign inside the determinant
864 * which gives rise to an overall sign if the dimension is odd. This method
865 * returns the characteristic polynomial collected in powers of lambda as a
868 * @return characteristic polynomial as new expression
869 * @exception logic_error (matrix not square)
870 * @see matrix::determinant() */
871 ex matrix::charpoly(const symbol & lambda) const
874 throw (std::logic_error("matrix::charpoly(): matrix not square"));
876 bool numeric_flag = true;
877 for (exvector::const_iterator r=m.begin(); r!=m.end(); ++r) {
878 if (!(*r).info(info_flags::numeric)) {
879 numeric_flag = false;
883 // The pure numeric case is traditionally rather common. Hence, it is
884 // trapped and we use Leverrier's algorithm which goes as row^3 for
885 // every coefficient. The expensive part is the matrix multiplication.
889 ex poly = power(lambda,row)-c*power(lambda,row-1);
890 for (unsigned i=1; i<row; ++i) {
891 for (unsigned j=0; j<row; ++j)
894 c = B.trace()/ex(i+1);
895 poly -= c*power(lambda,row-i-1);
904 for (unsigned r=0; r<col; ++r)
905 M.m[r*col+r] -= lambda;
907 return M.determinant().collect(lambda);
911 /** Inverse of this matrix.
913 * @return the inverted matrix
914 * @exception logic_error (matrix not square)
915 * @exception runtime_error (singular matrix) */
916 matrix matrix::inverse(void) const
919 throw (std::logic_error("matrix::inverse(): matrix not square"));
921 // This routine actually doesn't do anything fancy at all. We compute the
922 // inverse of the matrix A by solving the system A * A^{-1} == Id.
924 // First populate the identity matrix supposed to become the right hand side.
925 matrix identity(row,col);
926 for (unsigned i=0; i<row; ++i)
927 identity.set(i,i,_ex1());
929 // Populate a dummy matrix of variables, just because of compatibility with
930 // matrix::solve() which wants this (for compatibility with under-determined
931 // systems of equations).
932 matrix vars(row,col);
933 for (unsigned r=0; r<row; ++r)
934 for (unsigned c=0; c<col; ++c)
935 vars.set(r,c,symbol());
939 sol = this->solve(vars,identity);
940 } catch (const std::runtime_error & e) {
941 if (e.what()==std::string("matrix::solve(): inconsistent linear system"))
942 throw (std::runtime_error("matrix::inverse(): singular matrix"));
950 /** Solve a linear system consisting of a m x n matrix and a m x p right hand
951 * side by applying an elimination scheme to the augmented matrix.
953 * @param vars n x p matrix, all elements must be symbols
954 * @param rhs m x p matrix
955 * @return n x p solution matrix
956 * @exception logic_error (incompatible matrices)
957 * @exception invalid_argument (1st argument must be matrix of symbols)
958 * @exception runtime_error (inconsistent linear system)
960 matrix matrix::solve(const matrix & vars,
964 const unsigned m = this->rows();
965 const unsigned n = this->cols();
966 const unsigned p = rhs.cols();
969 if ((rhs.rows() != m) || (vars.rows() != n) || (vars.col != p))
970 throw (std::logic_error("matrix::solve(): incompatible matrices"));
971 for (unsigned ro=0; ro<n; ++ro)
972 for (unsigned co=0; co<p; ++co)
973 if (!vars(ro,co).info(info_flags::symbol))
974 throw (std::invalid_argument("matrix::solve(): 1st argument must be matrix of symbols"));
976 // build the augmented matrix of *this with rhs attached to the right
978 for (unsigned r=0; r<m; ++r) {
979 for (unsigned c=0; c<n; ++c)
980 aug.m[r*(n+p)+c] = this->m[r*n+c];
981 for (unsigned c=0; c<p; ++c)
982 aug.m[r*(n+p)+c+n] = rhs.m[r*p+c];
985 // Gather some statistical information about the augmented matrix:
986 bool numeric_flag = true;
987 for (exvector::const_iterator r=aug.m.begin(); r!=aug.m.end(); ++r) {
988 if (!(*r).info(info_flags::numeric))
989 numeric_flag = false;
992 // Here is the heuristics in case this routine has to decide:
993 if (algo == solve_algo::automatic) {
994 // Bareiss (fraction-free) elimination is generally a good guess:
995 algo = solve_algo::bareiss;
996 // For m<3, Bareiss elimination is equivalent to division free
997 // elimination but has more logistic overhead
999 algo = solve_algo::divfree;
1000 // This overrides any prior decisions.
1002 algo = solve_algo::gauss;
1005 // Eliminate the augmented matrix:
1007 case solve_algo::gauss:
1008 aug.gauss_elimination();
1010 case solve_algo::divfree:
1011 aug.division_free_elimination();
1013 case solve_algo::bareiss:
1015 aug.fraction_free_elimination();
1018 // assemble the solution matrix:
1020 for (unsigned co=0; co<p; ++co) {
1021 unsigned last_assigned_sol = n+1;
1022 for (int r=m-1; r>=0; --r) {
1023 unsigned fnz = 1; // first non-zero in row
1024 while ((fnz<=n) && (aug.m[r*(n+p)+(fnz-1)].is_zero()))
1027 // row consists only of zeros, corresponding rhs must be 0, too
1028 if (!aug.m[r*(n+p)+n+co].is_zero()) {
1029 throw (std::runtime_error("matrix::solve(): inconsistent linear system"));
1032 // assign solutions for vars between fnz+1 and
1033 // last_assigned_sol-1: free parameters
1034 for (unsigned c=fnz; c<last_assigned_sol-1; ++c)
1035 sol.set(c,co,vars.m[c*p+co]);
1036 ex e = aug.m[r*(n+p)+n+co];
1037 for (unsigned c=fnz; c<n; ++c)
1038 e -= aug.m[r*(n+p)+c]*sol.m[c*p+co];
1040 (e/(aug.m[r*(n+p)+(fnz-1)])).normal());
1041 last_assigned_sol = fnz;
1044 // assign solutions for vars between 1 and
1045 // last_assigned_sol-1: free parameters
1046 for (unsigned ro=0; ro<last_assigned_sol-1; ++ro)
1047 sol.set(ro,co,vars(ro,co));
1056 /** Recursive determinant for small matrices having at least one symbolic
1057 * entry. The basic algorithm, known as Laplace-expansion, is enhanced by
1058 * some bookkeeping to avoid calculation of the same submatrices ("minors")
1059 * more than once. According to W.M.Gentleman and S.C.Johnson this algorithm
1060 * is better than elimination schemes for matrices of sparse multivariate
1061 * polynomials and also for matrices of dense univariate polynomials if the
1062 * matrix' dimesion is larger than 7.
1064 * @return the determinant as a new expression (in expanded form)
1065 * @see matrix::determinant() */
1066 ex matrix::determinant_minor(void) const
1068 // for small matrices the algorithm does not make any sense:
1069 const unsigned n = this->cols();
1071 return m[0].expand();
1073 return (m[0]*m[3]-m[2]*m[1]).expand();
1075 return (m[0]*m[4]*m[8]-m[0]*m[5]*m[7]-
1076 m[1]*m[3]*m[8]+m[2]*m[3]*m[7]+
1077 m[1]*m[5]*m[6]-m[2]*m[4]*m[6]).expand();
1079 // This algorithm can best be understood by looking at a naive
1080 // implementation of Laplace-expansion, like this one:
1082 // matrix minorM(this->rows()-1,this->cols()-1);
1083 // for (unsigned r1=0; r1<this->rows(); ++r1) {
1084 // // shortcut if element(r1,0) vanishes
1085 // if (m[r1*col].is_zero())
1087 // // assemble the minor matrix
1088 // for (unsigned r=0; r<minorM.rows(); ++r) {
1089 // for (unsigned c=0; c<minorM.cols(); ++c) {
1091 // minorM.set(r,c,m[r*col+c+1]);
1093 // minorM.set(r,c,m[(r+1)*col+c+1]);
1096 // // recurse down and care for sign:
1098 // det -= m[r1*col] * minorM.determinant_minor();
1100 // det += m[r1*col] * minorM.determinant_minor();
1102 // return det.expand();
1103 // What happens is that while proceeding down many of the minors are
1104 // computed more than once. In particular, there are binomial(n,k)
1105 // kxk minors and each one is computed factorial(n-k) times. Therefore
1106 // it is reasonable to store the results of the minors. We proceed from
1107 // right to left. At each column c we only need to retrieve the minors
1108 // calculated in step c-1. We therefore only have to store at most
1109 // 2*binomial(n,n/2) minors.
1111 // Unique flipper counter for partitioning into minors
1112 std::vector<unsigned> Pkey;
1114 // key for minor determinant (a subpartition of Pkey)
1115 std::vector<unsigned> Mkey;
1117 // we store our subminors in maps, keys being the rows they arise from
1118 typedef std::map<std::vector<unsigned>,class ex> Rmap;
1119 typedef std::map<std::vector<unsigned>,class ex>::value_type Rmap_value;
1123 // initialize A with last column:
1124 for (unsigned r=0; r<n; ++r) {
1125 Pkey.erase(Pkey.begin(),Pkey.end());
1127 A.insert(Rmap_value(Pkey,m[n*(r+1)-1]));
1129 // proceed from right to left through matrix
1130 for (int c=n-2; c>=0; --c) {
1131 Pkey.erase(Pkey.begin(),Pkey.end()); // don't change capacity
1132 Mkey.erase(Mkey.begin(),Mkey.end());
1133 for (unsigned i=0; i<n-c; ++i)
1135 unsigned fc = 0; // controls logic for our strange flipper counter
1138 for (unsigned r=0; r<n-c; ++r) {
1139 // maybe there is nothing to do?
1140 if (m[Pkey[r]*n+c].is_zero())
1142 // create the sorted key for all possible minors
1143 Mkey.erase(Mkey.begin(),Mkey.end());
1144 for (unsigned i=0; i<n-c; ++i)
1146 Mkey.push_back(Pkey[i]);
1147 // Fetch the minors and compute the new determinant
1149 det -= m[Pkey[r]*n+c]*A[Mkey];
1151 det += m[Pkey[r]*n+c]*A[Mkey];
1153 // prevent build-up of deep nesting of expressions saves time:
1155 // store the new determinant at its place in B:
1157 B.insert(Rmap_value(Pkey,det));
1158 // increment our strange flipper counter
1159 for (fc=n-c; fc>0; --fc) {
1161 if (Pkey[fc-1]<fc+c)
1165 for (unsigned j=fc; j<n-c; ++j)
1166 Pkey[j] = Pkey[j-1]+1;
1168 // next column, so change the role of A and B:
1177 /** Perform the steps of an ordinary Gaussian elimination to bring the m x n
1178 * matrix into an upper echelon form. The algorithm is ok for matrices
1179 * with numeric coefficients but quite unsuited for symbolic matrices.
1181 * @param det may be set to true to save a lot of space if one is only
1182 * interested in the diagonal elements (i.e. for calculating determinants).
1183 * The others are set to zero in this case.
1184 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1185 * number of rows was swapped and 0 if the matrix is singular. */
1186 int matrix::gauss_elimination(const bool det)
1188 ensure_if_modifiable();
1189 const unsigned m = this->rows();
1190 const unsigned n = this->cols();
1191 GINAC_ASSERT(!det || n==m);
1195 for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
1196 int indx = pivot(r0, r1, true);
1200 return 0; // leaves *this in a messy state
1205 for (unsigned r2=r0+1; r2<m; ++r2) {
1206 if (!this->m[r2*n+r1].is_zero()) {
1207 // yes, there is something to do in this row
1208 ex piv = this->m[r2*n+r1] / this->m[r0*n+r1];
1209 for (unsigned c=r1+1; c<n; ++c) {
1210 this->m[r2*n+c] -= piv * this->m[r0*n+c];
1211 if (!this->m[r2*n+c].info(info_flags::numeric))
1212 this->m[r2*n+c] = this->m[r2*n+c].normal();
1215 // fill up left hand side with zeros
1216 for (unsigned c=0; c<=r1; ++c)
1217 this->m[r2*n+c] = _ex0();
1220 // save space by deleting no longer needed elements
1221 for (unsigned c=r0+1; c<n; ++c)
1222 this->m[r0*n+c] = _ex0();
1232 /** Perform the steps of division free elimination to bring the m x n matrix
1233 * into an upper echelon form.
1235 * @param det may be set to true to save a lot of space if one is only
1236 * interested in the diagonal elements (i.e. for calculating determinants).
1237 * The others are set to zero in this case.
1238 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1239 * number of rows was swapped and 0 if the matrix is singular. */
1240 int matrix::division_free_elimination(const bool det)
1242 ensure_if_modifiable();
1243 const unsigned m = this->rows();
1244 const unsigned n = this->cols();
1245 GINAC_ASSERT(!det || n==m);
1249 for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
1250 int indx = pivot(r0, r1, true);
1254 return 0; // leaves *this in a messy state
1259 for (unsigned r2=r0+1; r2<m; ++r2) {
1260 for (unsigned c=r1+1; c<n; ++c)
1261 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();
1262 // fill up left hand side with zeros
1263 for (unsigned c=0; c<=r1; ++c)
1264 this->m[r2*n+c] = _ex0();
1267 // save space by deleting no longer needed elements
1268 for (unsigned c=r0+1; c<n; ++c)
1269 this->m[r0*n+c] = _ex0();
1279 /** Perform the steps of Bareiss' one-step fraction free elimination to bring
1280 * the matrix into an upper echelon form. Fraction free elimination means
1281 * that divide is used straightforwardly, without computing GCDs first. This
1282 * is possible, since we know the divisor at each step.
1284 * @param det may be set to true to save a lot of space if one is only
1285 * interested in the last element (i.e. for calculating determinants). The
1286 * others are set to zero in this case.
1287 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1288 * number of rows was swapped and 0 if the matrix is singular. */
1289 int matrix::fraction_free_elimination(const bool det)
1292 // (single-step fraction free elimination scheme, already known to Jordan)
1294 // Usual division-free elimination sets m[0](r,c) = m(r,c) and then sets
1295 // m[k+1](r,c) = m[k](k,k) * m[k](r,c) - m[k](r,k) * m[k](k,c).
1297 // Bareiss (fraction-free) elimination in addition divides that element
1298 // by m[k-1](k-1,k-1) for k>1, where it can be shown by means of the
1299 // Sylvester determinant that this really divides m[k+1](r,c).
1301 // We also allow rational functions where the original prove still holds.
1302 // However, we must care for numerator and denominator separately and
1303 // "manually" work in the integral domains because of subtle cancellations
1304 // (see below). This blows up the bookkeeping a bit and the formula has
1305 // to be modified to expand like this (N{x} stands for numerator of x,
1306 // D{x} for denominator of x):
1307 // 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)}
1308 // -N{m[k](r,k)}*N{m[k](k,c)}*D{m[k](k,k)}*D{m[k](r,c)}
1309 // 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)}
1310 // where for k>1 we now divide N{m[k+1](r,c)} by
1311 // N{m[k-1](k-1,k-1)}
1312 // and D{m[k+1](r,c)} by
1313 // D{m[k-1](k-1,k-1)}.
1315 ensure_if_modifiable();
1316 const unsigned m = this->rows();
1317 const unsigned n = this->cols();
1318 GINAC_ASSERT(!det || n==m);
1327 // We populate temporary matrices to subsequently operate on. There is
1328 // one holding numerators and another holding denominators of entries.
1329 // This is a must since the evaluator (or even earlier mul's constructor)
1330 // might cancel some trivial element which causes divide() to fail. The
1331 // elements are normalized first (yes, even though this algorithm doesn't
1332 // need GCDs) since the elements of *this might be unnormalized, which
1333 // makes things more complicated than they need to be.
1334 matrix tmp_n(*this);
1335 matrix tmp_d(m,n); // for denominators, if needed
1336 lst srl; // symbol replacement list
1337 exvector::iterator it = this->m.begin();
1338 exvector::iterator tmp_n_it = tmp_n.m.begin();
1339 exvector::iterator tmp_d_it = tmp_d.m.begin();
1340 for (; it!= this->m.end(); ++it, ++tmp_n_it, ++tmp_d_it) {
1341 (*tmp_n_it) = (*it).normal().to_rational(srl);
1342 (*tmp_d_it) = (*tmp_n_it).denom();
1343 (*tmp_n_it) = (*tmp_n_it).numer();
1347 for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
1348 int indx = tmp_n.pivot(r0, r1, true);
1357 // tmp_n's rows r0 and indx were swapped, do the same in tmp_d:
1358 for (unsigned c=r1; c<n; ++c)
1359 tmp_d.m[n*indx+c].swap(tmp_d.m[n*r0+c]);
1361 for (unsigned r2=r0+1; r2<m; ++r2) {
1362 for (unsigned c=r1+1; c<n; ++c) {
1363 dividend_n = (tmp_n.m[r0*n+r1]*tmp_n.m[r2*n+c]*
1364 tmp_d.m[r2*n+r1]*tmp_d.m[r0*n+c]
1365 -tmp_n.m[r2*n+r1]*tmp_n.m[r0*n+c]*
1366 tmp_d.m[r0*n+r1]*tmp_d.m[r2*n+c]).expand();
1367 dividend_d = (tmp_d.m[r2*n+r1]*tmp_d.m[r0*n+c]*
1368 tmp_d.m[r0*n+r1]*tmp_d.m[r2*n+c]).expand();
1369 bool check = divide(dividend_n, divisor_n,
1370 tmp_n.m[r2*n+c], true);
1371 check &= divide(dividend_d, divisor_d,
1372 tmp_d.m[r2*n+c], true);
1373 GINAC_ASSERT(check);
1375 // fill up left hand side with zeros
1376 for (unsigned c=0; c<=r1; ++c)
1377 tmp_n.m[r2*n+c] = _ex0();
1379 if ((r1<n-1)&&(r0<m-1)) {
1380 // compute next iteration's divisor
1381 divisor_n = tmp_n.m[r0*n+r1].expand();
1382 divisor_d = tmp_d.m[r0*n+r1].expand();
1384 // save space by deleting no longer needed elements
1385 for (unsigned c=0; c<n; ++c) {
1386 tmp_n.m[r0*n+c] = _ex0();
1387 tmp_d.m[r0*n+c] = _ex1();
1394 // repopulate *this matrix:
1395 it = this->m.begin();
1396 tmp_n_it = tmp_n.m.begin();
1397 tmp_d_it = tmp_d.m.begin();
1398 for (; it!= this->m.end(); ++it, ++tmp_n_it, ++tmp_d_it)
1399 (*it) = ((*tmp_n_it)/(*tmp_d_it)).subs(srl);
1405 /** Partial pivoting method for matrix elimination schemes.
1406 * Usual pivoting (symbolic==false) returns the index to the element with the
1407 * largest absolute value in column ro and swaps the current row with the one
1408 * where the element was found. With (symbolic==true) it does the same thing
1409 * with the first non-zero element.
1411 * @param ro is the row from where to begin
1412 * @param co is the column to be inspected
1413 * @param symbolic signal if we want the first non-zero element to be pivoted
1414 * (true) or the one with the largest absolute value (false).
1415 * @return 0 if no interchange occured, -1 if all are zero (usually signaling
1416 * a degeneracy) and positive integer k means that rows ro and k were swapped.
1418 int matrix::pivot(unsigned ro, unsigned co, bool symbolic)
1422 // search first non-zero element in column co beginning at row ro
1423 while ((k<row) && (this->m[k*col+co].expand().is_zero()))
1426 // search largest element in column co beginning at row ro
1427 GINAC_ASSERT(is_ex_of_type(this->m[k*col+co],numeric));
1428 unsigned kmax = k+1;
1429 numeric mmax = abs(ex_to_numeric(m[kmax*col+co]));
1431 GINAC_ASSERT(is_ex_of_type(this->m[kmax*col+co],numeric));
1432 numeric tmp = ex_to_numeric(this->m[kmax*col+co]);
1433 if (abs(tmp) > mmax) {
1439 if (!mmax.is_zero())
1443 // all elements in column co below row ro vanish
1446 // matrix needs no pivoting
1448 // matrix needs pivoting, so swap rows k and ro
1449 ensure_if_modifiable();
1450 for (unsigned c=0; c<col; ++c)
1451 this->m[k*col+c].swap(this->m[ro*col+c]);
1456 ex lst_to_matrix(const lst & l)
1458 // Find number of rows and columns
1459 unsigned rows = l.nops(), cols = 0, i, j;
1460 for (i=0; i<rows; i++)
1461 if (l.op(i).nops() > cols)
1462 cols = l.op(i).nops();
1464 // Allocate and fill matrix
1465 matrix &m = *new matrix(rows, cols);
1466 m.setflag(status_flags::dynallocated);
1467 for (i=0; i<rows; i++)
1468 for (j=0; j<cols; j++)
1469 if (l.op(i).nops() > j)
1470 m.set(i, j, l.op(i).op(j));
1476 ex diag_matrix(const lst & l)
1478 unsigned dim = l.nops();
1480 matrix &m = *new matrix(dim, dim);
1481 m.setflag(status_flags::dynallocated);
1482 for (unsigned i=0; i<dim; i++)
1483 m.set(i, i, l.op(i));
1488 } // namespace GiNaC