3 * Implementation of symbolic matrices */
6 * GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
8 * This program is free software; you can redistribute it and/or modify
9 * it under the terms of the GNU General Public License as published by
10 * the Free Software Foundation; either version 2 of the License, or
11 * (at your option) any later version.
13 * This program is distributed in the hope that it will be useful,
14 * but WITHOUT ANY WARRANTY; without even the implied warranty of
15 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
16 * GNU General Public License for more details.
18 * You should have received a copy of the GNU General Public License
19 * along with this program; if not, write to the Free Software
20 * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
42 GINAC_IMPLEMENT_REGISTERED_CLASS(matrix, basic)
45 // default ctor, dtor, copy ctor, assignment operator and helpers:
48 /** Default ctor. Initializes to 1 x 1-dimensional zero-matrix. */
49 matrix::matrix() : inherited(TINFO_matrix), row(1), col(1)
51 debugmsg("matrix default ctor",LOGLEVEL_CONSTRUCT);
55 void matrix::copy(const matrix & other)
57 inherited::copy(other);
60 m = other.m; // STL's vector copying invoked here
63 DEFAULT_DESTROY(matrix)
71 /** Very common ctor. Initializes to r x c-dimensional zero-matrix.
73 * @param r number of rows
74 * @param c number of cols */
75 matrix::matrix(unsigned r, unsigned c)
76 : inherited(TINFO_matrix), row(r), col(c)
78 debugmsg("matrix ctor from unsigned,unsigned",LOGLEVEL_CONSTRUCT);
79 m.resize(r*c, _ex0());
84 /** Ctor from representation, for internal use only. */
85 matrix::matrix(unsigned r, unsigned c, const exvector & m2)
86 : inherited(TINFO_matrix), row(r), col(c), m(m2)
88 debugmsg("matrix ctor from unsigned,unsigned,exvector",LOGLEVEL_CONSTRUCT);
91 /** Construct matrix from (flat) list of elements. If the list has fewer
92 * elements than the matrix, the remaining matrix elements are set to zero.
93 * If the list has more elements than the matrix, the excessive elements are
95 matrix::matrix(unsigned r, unsigned c, const lst & l)
96 : inherited(TINFO_matrix), row(r), col(c)
98 debugmsg("matrix ctor from unsigned,unsigned,lst",LOGLEVEL_CONSTRUCT);
99 m.resize(r*c, _ex0());
101 for (unsigned i=0; i<l.nops(); i++) {
105 break; // matrix smaller than list: throw away excessive elements
114 matrix::matrix(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
116 debugmsg("matrix ctor from archive_node", LOGLEVEL_CONSTRUCT);
117 if (!(n.find_unsigned("row", row)) || !(n.find_unsigned("col", col)))
118 throw (std::runtime_error("unknown matrix dimensions in archive"));
119 m.reserve(row * col);
120 for (unsigned int i=0; true; i++) {
122 if (n.find_ex("m", e, sym_lst, i))
129 void matrix::archive(archive_node &n) const
131 inherited::archive(n);
132 n.add_unsigned("row", row);
133 n.add_unsigned("col", col);
134 exvector::const_iterator i = m.begin(), iend = m.end();
141 DEFAULT_UNARCHIVE(matrix)
144 // functions overriding virtual functions from bases classes
149 void matrix::print(const print_context & c, unsigned level) const
151 debugmsg("matrix print", LOGLEVEL_PRINT);
153 if (is_of_type(c, print_tree)) {
155 inherited::print(c, level);
160 for (unsigned y=0; y<row-1; ++y) {
162 for (unsigned x=0; x<col-1; ++x) {
166 m[col*(y+1)-1].print(c);
170 for (unsigned x=0; x<col-1; ++x) {
171 m[(row-1)*col+x].print(c);
174 m[row*col-1].print(c);
180 /** nops is defined to be rows x columns. */
181 unsigned matrix::nops() const
186 /** returns matrix entry at position (i/col, i%col). */
187 ex matrix::op(int i) const
192 /** returns matrix entry at position (i/col, i%col). */
193 ex & matrix::let_op(int i)
196 GINAC_ASSERT(i<nops());
201 /** Evaluate matrix entry by entry. */
202 ex matrix::eval(int level) const
204 debugmsg("matrix eval",LOGLEVEL_MEMBER_FUNCTION);
206 // check if we have to do anything at all
207 if ((level==1)&&(flags & status_flags::evaluated))
211 if (level == -max_recursion_level)
212 throw (std::runtime_error("matrix::eval(): recursion limit exceeded"));
214 // eval() entry by entry
215 exvector m2(row*col);
217 for (unsigned r=0; r<row; ++r)
218 for (unsigned c=0; c<col; ++c)
219 m2[r*col+c] = m[r*col+c].eval(level);
221 return (new matrix(row, col, m2))->setflag(status_flags::dynallocated |
222 status_flags::evaluated );
225 ex matrix::subs(const lst & ls, const lst & lr, bool no_pattern) const
227 exvector m2(row * col);
228 for (unsigned r=0; r<row; ++r)
229 for (unsigned c=0; c<col; ++c)
230 m2[r*col+c] = m[r*col+c].subs(ls, lr, no_pattern);
232 return ex(matrix(row, col, m2)).bp->basic::subs(ls, lr, no_pattern);
237 int matrix::compare_same_type(const basic & other) const
239 GINAC_ASSERT(is_exactly_of_type(other, matrix));
240 const matrix & o = static_cast<matrix &>(const_cast<basic &>(other));
242 // compare number of rows
244 return row < o.rows() ? -1 : 1;
246 // compare number of columns
248 return col < o.cols() ? -1 : 1;
250 // equal number of rows and columns, compare individual elements
252 for (unsigned r=0; r<row; ++r) {
253 for (unsigned c=0; c<col; ++c) {
254 cmpval = ((*this)(r,c)).compare(o(r,c));
255 if (cmpval!=0) return cmpval;
258 // all elements are equal => matrices are equal;
262 /** Automatic symbolic evaluation of an indexed matrix. */
263 ex matrix::eval_indexed(const basic & i) const
265 GINAC_ASSERT(is_of_type(i, indexed));
266 GINAC_ASSERT(is_ex_of_type(i.op(0), matrix));
268 bool all_indices_unsigned = static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint);
273 // One index, must be one-dimensional vector
274 if (row != 1 && col != 1)
275 throw (std::runtime_error("matrix::eval_indexed(): vector must have exactly 1 index"));
277 const idx & i1 = ex_to<idx>(i.op(1));
282 if (!i1.get_dim().is_equal(row))
283 throw (std::runtime_error("matrix::eval_indexed(): dimension of index must match number of vector elements"));
285 // Index numeric -> return vector element
286 if (all_indices_unsigned) {
287 unsigned n1 = ex_to<numeric>(i1.get_value()).to_int();
289 throw (std::runtime_error("matrix::eval_indexed(): value of index exceeds number of vector elements"));
290 return (*this)(n1, 0);
296 if (!i1.get_dim().is_equal(col))
297 throw (std::runtime_error("matrix::eval_indexed(): dimension of index must match number of vector elements"));
299 // Index numeric -> return vector element
300 if (all_indices_unsigned) {
301 unsigned n1 = ex_to<numeric>(i1.get_value()).to_int();
303 throw (std::runtime_error("matrix::eval_indexed(): value of index exceeds number of vector elements"));
304 return (*this)(0, n1);
308 } else if (i.nops() == 3) {
311 const idx & i1 = ex_to<idx>(i.op(1));
312 const idx & i2 = ex_to<idx>(i.op(2));
314 if (!i1.get_dim().is_equal(row))
315 throw (std::runtime_error("matrix::eval_indexed(): dimension of first index must match number of rows"));
316 if (!i2.get_dim().is_equal(col))
317 throw (std::runtime_error("matrix::eval_indexed(): dimension of second index must match number of columns"));
319 // Pair of dummy indices -> compute trace
320 if (is_dummy_pair(i1, i2))
323 // Both indices numeric -> return matrix element
324 if (all_indices_unsigned) {
325 unsigned n1 = ex_to<numeric>(i1.get_value()).to_int(), n2 = ex_to<numeric>(i2.get_value()).to_int();
327 throw (std::runtime_error("matrix::eval_indexed(): value of first index exceeds number of rows"));
329 throw (std::runtime_error("matrix::eval_indexed(): value of second index exceeds number of columns"));
330 return (*this)(n1, n2);
334 throw (std::runtime_error("matrix::eval_indexed(): matrix must have exactly 2 indices"));
339 /** Sum of two indexed matrices. */
340 ex matrix::add_indexed(const ex & self, const ex & other) const
342 GINAC_ASSERT(is_ex_of_type(self, indexed));
343 GINAC_ASSERT(is_ex_of_type(self.op(0), matrix));
344 GINAC_ASSERT(is_ex_of_type(other, indexed));
345 GINAC_ASSERT(self.nops() == 2 || self.nops() == 3);
347 // Only add two matrices
348 if (is_ex_of_type(other.op(0), matrix)) {
349 GINAC_ASSERT(other.nops() == 2 || other.nops() == 3);
351 const matrix &self_matrix = ex_to<matrix>(self.op(0));
352 const matrix &other_matrix = ex_to<matrix>(other.op(0));
354 if (self.nops() == 2 && other.nops() == 2) { // vector + vector
356 if (self_matrix.row == other_matrix.row)
357 return indexed(self_matrix.add(other_matrix), self.op(1));
358 else if (self_matrix.row == other_matrix.col)
359 return indexed(self_matrix.add(other_matrix.transpose()), self.op(1));
361 } else if (self.nops() == 3 && other.nops() == 3) { // matrix + matrix
363 if (self.op(1).is_equal(other.op(1)) && self.op(2).is_equal(other.op(2)))
364 return indexed(self_matrix.add(other_matrix), self.op(1), self.op(2));
365 else if (self.op(1).is_equal(other.op(2)) && self.op(2).is_equal(other.op(1)))
366 return indexed(self_matrix.add(other_matrix.transpose()), self.op(1), self.op(2));
371 // Don't know what to do, return unevaluated sum
375 /** Product of an indexed matrix with a number. */
376 ex matrix::scalar_mul_indexed(const ex & self, const numeric & other) const
378 GINAC_ASSERT(is_ex_of_type(self, indexed));
379 GINAC_ASSERT(is_ex_of_type(self.op(0), matrix));
380 GINAC_ASSERT(self.nops() == 2 || self.nops() == 3);
382 const matrix &self_matrix = ex_to<matrix>(self.op(0));
384 if (self.nops() == 2)
385 return indexed(self_matrix.mul(other), self.op(1));
386 else // self.nops() == 3
387 return indexed(self_matrix.mul(other), self.op(1), self.op(2));
390 /** Contraction of an indexed matrix with something else. */
391 bool matrix::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
393 GINAC_ASSERT(is_ex_of_type(*self, indexed));
394 GINAC_ASSERT(is_ex_of_type(*other, indexed));
395 GINAC_ASSERT(self->nops() == 2 || self->nops() == 3);
396 GINAC_ASSERT(is_ex_of_type(self->op(0), matrix));
398 // Only contract with other matrices
399 if (!is_ex_of_type(other->op(0), matrix))
402 GINAC_ASSERT(other->nops() == 2 || other->nops() == 3);
404 const matrix &self_matrix = ex_to<matrix>(self->op(0));
405 const matrix &other_matrix = ex_to<matrix>(other->op(0));
407 if (self->nops() == 2) {
408 unsigned self_dim = (self_matrix.col == 1) ? self_matrix.row : self_matrix.col;
410 if (other->nops() == 2) { // vector * vector (scalar product)
411 unsigned other_dim = (other_matrix.col == 1) ? other_matrix.row : other_matrix.col;
413 if (self_matrix.col == 1) {
414 if (other_matrix.col == 1) {
415 // Column vector * column vector, transpose first vector
416 *self = self_matrix.transpose().mul(other_matrix)(0, 0);
418 // Column vector * row vector, swap factors
419 *self = other_matrix.mul(self_matrix)(0, 0);
422 if (other_matrix.col == 1) {
423 // Row vector * column vector, perfect
424 *self = self_matrix.mul(other_matrix)(0, 0);
426 // Row vector * row vector, transpose second vector
427 *self = self_matrix.mul(other_matrix.transpose())(0, 0);
433 } else { // vector * matrix
435 // B_i * A_ij = (B*A)_j (B is row vector)
436 if (is_dummy_pair(self->op(1), other->op(1))) {
437 if (self_matrix.row == 1)
438 *self = indexed(self_matrix.mul(other_matrix), other->op(2));
440 *self = indexed(self_matrix.transpose().mul(other_matrix), other->op(2));
445 // B_j * A_ij = (A*B)_i (B is column vector)
446 if (is_dummy_pair(self->op(1), other->op(2))) {
447 if (self_matrix.col == 1)
448 *self = indexed(other_matrix.mul(self_matrix), other->op(1));
450 *self = indexed(other_matrix.mul(self_matrix.transpose()), other->op(1));
456 } else if (other->nops() == 3) { // matrix * matrix
458 // A_ij * B_jk = (A*B)_ik
459 if (is_dummy_pair(self->op(2), other->op(1))) {
460 *self = indexed(self_matrix.mul(other_matrix), self->op(1), other->op(2));
465 // A_ij * B_kj = (A*Btrans)_ik
466 if (is_dummy_pair(self->op(2), other->op(2))) {
467 *self = indexed(self_matrix.mul(other_matrix.transpose()), self->op(1), other->op(1));
472 // A_ji * B_jk = (Atrans*B)_ik
473 if (is_dummy_pair(self->op(1), other->op(1))) {
474 *self = indexed(self_matrix.transpose().mul(other_matrix), self->op(2), other->op(2));
479 // A_ji * B_kj = (B*A)_ki
480 if (is_dummy_pair(self->op(1), other->op(2))) {
481 *self = indexed(other_matrix.mul(self_matrix), other->op(1), self->op(2));
492 // non-virtual functions in this class
499 * @exception logic_error (incompatible matrices) */
500 matrix matrix::add(const matrix & other) const
502 if (col != other.col || row != other.row)
503 throw std::logic_error("matrix::add(): incompatible matrices");
505 exvector sum(this->m);
506 exvector::iterator i;
507 exvector::const_iterator ci;
508 for (i=sum.begin(), ci=other.m.begin(); i!=sum.end(); ++i, ++ci)
511 return matrix(row,col,sum);
515 /** Difference of matrices.
517 * @exception logic_error (incompatible matrices) */
518 matrix matrix::sub(const matrix & other) const
520 if (col != other.col || row != other.row)
521 throw std::logic_error("matrix::sub(): incompatible matrices");
523 exvector dif(this->m);
524 exvector::iterator i;
525 exvector::const_iterator ci;
526 for (i=dif.begin(), ci=other.m.begin(); i!=dif.end(); ++i, ++ci)
529 return matrix(row,col,dif);
533 /** Product of matrices.
535 * @exception logic_error (incompatible matrices) */
536 matrix matrix::mul(const matrix & other) const
538 if (this->cols() != other.rows())
539 throw std::logic_error("matrix::mul(): incompatible matrices");
541 exvector prod(this->rows()*other.cols());
543 for (unsigned r1=0; r1<this->rows(); ++r1) {
544 for (unsigned c=0; c<this->cols(); ++c) {
545 if (m[r1*col+c].is_zero())
547 for (unsigned r2=0; r2<other.cols(); ++r2)
548 prod[r1*other.col+r2] += (m[r1*col+c] * other.m[c*other.col+r2]).expand();
551 return matrix(row, other.col, prod);
555 /** Product of matrix and scalar. */
556 matrix matrix::mul(const numeric & other) const
558 exvector prod(row * col);
560 for (unsigned r=0; r<row; ++r)
561 for (unsigned c=0; c<col; ++c)
562 prod[r*col+c] = m[r*col+c] * other;
564 return matrix(row, col, prod);
568 /** Product of matrix and scalar expression. */
569 matrix matrix::mul_scalar(const ex & other) const
571 if (other.return_type() != return_types::commutative)
572 throw std::runtime_error("matrix::mul_scalar(): non-commutative scalar");
574 exvector prod(row * col);
576 for (unsigned r=0; r<row; ++r)
577 for (unsigned c=0; c<col; ++c)
578 prod[r*col+c] = m[r*col+c] * other;
580 return matrix(row, col, prod);
584 /** Power of a matrix. Currently handles integer exponents only. */
585 matrix matrix::pow(const ex & expn) const
588 throw (std::logic_error("matrix::pow(): matrix not square"));
590 if (is_ex_exactly_of_type(expn, numeric)) {
591 // Integer cases are computed by successive multiplication, using the
592 // obvious shortcut of storing temporaries, like A^4 == (A*A)*(A*A).
593 if (expn.info(info_flags::integer)) {
595 matrix prod(row,col);
596 if (expn.info(info_flags::negative)) {
597 k = -ex_to<numeric>(expn);
598 prod = this->inverse();
600 k = ex_to<numeric>(expn);
603 matrix result(row,col);
604 for (unsigned r=0; r<row; ++r)
605 result(r,r) = _ex1();
607 // this loop computes the representation of k in base 2 and
608 // multiplies the factors whenever needed:
609 while (b.compare(k)<=0) {
614 result = result.mul(prod);
617 prod = prod.mul(prod);
622 throw (std::runtime_error("matrix::pow(): don't know how to handle exponent"));
626 /** operator() to access elements for reading.
628 * @param ro row of element
629 * @param co column of element
630 * @exception range_error (index out of range) */
631 const ex & matrix::operator() (unsigned ro, unsigned co) const
633 if (ro>=row || co>=col)
634 throw (std::range_error("matrix::operator(): index out of range"));
640 /** operator() to access elements for writing.
642 * @param ro row of element
643 * @param co column of element
644 * @exception range_error (index out of range) */
645 ex & matrix::operator() (unsigned ro, unsigned co)
647 if (ro>=row || co>=col)
648 throw (std::range_error("matrix::operator(): index out of range"));
650 ensure_if_modifiable();
655 /** Transposed of an m x n matrix, producing a new n x m matrix object that
656 * represents the transposed. */
657 matrix matrix::transpose(void) const
659 exvector trans(this->cols()*this->rows());
661 for (unsigned r=0; r<this->cols(); ++r)
662 for (unsigned c=0; c<this->rows(); ++c)
663 trans[r*this->rows()+c] = m[c*this->cols()+r];
665 return matrix(this->cols(),this->rows(),trans);
668 /** Determinant of square matrix. This routine doesn't actually calculate the
669 * determinant, it only implements some heuristics about which algorithm to
670 * run. If all the elements of the matrix are elements of an integral domain
671 * the determinant is also in that integral domain and the result is expanded
672 * only. If one or more elements are from a quotient field the determinant is
673 * usually also in that quotient field and the result is normalized before it
674 * is returned. This implies that the determinant of the symbolic 2x2 matrix
675 * [[a/(a-b),1],[b/(a-b),1]] is returned as unity. (In this respect, it
676 * behaves like MapleV and unlike Mathematica.)
678 * @param algo allows to chose an algorithm
679 * @return the determinant as a new expression
680 * @exception logic_error (matrix not square)
681 * @see determinant_algo */
682 ex matrix::determinant(unsigned algo) const
685 throw (std::logic_error("matrix::determinant(): matrix not square"));
686 GINAC_ASSERT(row*col==m.capacity());
688 // Gather some statistical information about this matrix:
689 bool numeric_flag = true;
690 bool normal_flag = false;
691 unsigned sparse_count = 0; // counts non-zero elements
692 for (exvector::const_iterator r=m.begin(); r!=m.end(); ++r) {
693 lst srl; // symbol replacement list
694 ex rtest = (*r).to_rational(srl);
695 if (!rtest.is_zero())
697 if (!rtest.info(info_flags::numeric))
698 numeric_flag = false;
699 if (!rtest.info(info_flags::crational_polynomial) &&
700 rtest.info(info_flags::rational_function))
704 // Here is the heuristics in case this routine has to decide:
705 if (algo == determinant_algo::automatic) {
706 // Minor expansion is generally a good guess:
707 algo = determinant_algo::laplace;
708 // Does anybody know when a matrix is really sparse?
709 // Maybe <~row/2.236 nonzero elements average in a row?
710 if (row>3 && 5*sparse_count<=row*col)
711 algo = determinant_algo::bareiss;
712 // Purely numeric matrix can be handled by Gauss elimination.
713 // This overrides any prior decisions.
715 algo = determinant_algo::gauss;
718 // Trap the trivial case here, since some algorithms don't like it
720 // for consistency with non-trivial determinants...
722 return m[0].normal();
724 return m[0].expand();
727 // Compute the determinant
729 case determinant_algo::gauss: {
732 int sign = tmp.gauss_elimination(true);
733 for (unsigned d=0; d<row; ++d)
734 det *= tmp.m[d*col+d];
736 return (sign*det).normal();
738 return (sign*det).normal().expand();
740 case determinant_algo::bareiss: {
743 sign = tmp.fraction_free_elimination(true);
745 return (sign*tmp.m[row*col-1]).normal();
747 return (sign*tmp.m[row*col-1]).expand();
749 case determinant_algo::divfree: {
752 sign = tmp.division_free_elimination(true);
755 ex det = tmp.m[row*col-1];
756 // factor out accumulated bogus slag
757 for (unsigned d=0; d<row-2; ++d)
758 for (unsigned j=0; j<row-d-2; ++j)
759 det = (det/tmp.m[d*col+d]).normal();
762 case determinant_algo::laplace:
764 // This is the minor expansion scheme. We always develop such
765 // that the smallest minors (i.e, the trivial 1x1 ones) are on the
766 // rightmost column. For this to be efficient it turns out that
767 // the emptiest columns (i.e. the ones with most zeros) should be
768 // the ones on the right hand side. Therefore we presort the
769 // columns of the matrix:
770 typedef std::pair<unsigned,unsigned> uintpair;
771 std::vector<uintpair> c_zeros; // number of zeros in column
772 for (unsigned c=0; c<col; ++c) {
774 for (unsigned r=0; r<row; ++r)
775 if (m[r*col+c].is_zero())
777 c_zeros.push_back(uintpair(acc,c));
779 sort(c_zeros.begin(),c_zeros.end());
780 std::vector<unsigned> pre_sort;
781 for (std::vector<uintpair>::iterator i=c_zeros.begin(); i!=c_zeros.end(); ++i)
782 pre_sort.push_back(i->second);
783 std::vector<unsigned> pre_sort_test(pre_sort); // permutation_sign() modifies the vector so we make a copy here
784 int sign = permutation_sign(pre_sort_test.begin(), pre_sort_test.end());
785 exvector result(row*col); // represents sorted matrix
787 for (std::vector<unsigned>::iterator i=pre_sort.begin();
790 for (unsigned r=0; r<row; ++r)
791 result[r*col+c] = m[r*col+(*i)];
795 return (sign*matrix(row,col,result).determinant_minor()).normal();
797 return sign*matrix(row,col,result).determinant_minor();
803 /** Trace of a matrix. The result is normalized if it is in some quotient
804 * field and expanded only otherwise. This implies that the trace of the
805 * symbolic 2x2 matrix [[a/(a-b),x],[y,b/(b-a)]] is recognized to be unity.
807 * @return the sum of diagonal elements
808 * @exception logic_error (matrix not square) */
809 ex matrix::trace(void) const
812 throw (std::logic_error("matrix::trace(): matrix not square"));
815 for (unsigned r=0; r<col; ++r)
818 if (tr.info(info_flags::rational_function) &&
819 !tr.info(info_flags::crational_polynomial))
826 /** Characteristic Polynomial. Following mathematica notation the
827 * characteristic polynomial of a matrix M is defined as the determiant of
828 * (M - lambda * 1) where 1 stands for the unit matrix of the same dimension
829 * as M. Note that some CASs define it with a sign inside the determinant
830 * which gives rise to an overall sign if the dimension is odd. This method
831 * returns the characteristic polynomial collected in powers of lambda as a
834 * @return characteristic polynomial as new expression
835 * @exception logic_error (matrix not square)
836 * @see matrix::determinant() */
837 ex matrix::charpoly(const symbol & lambda) const
840 throw (std::logic_error("matrix::charpoly(): matrix not square"));
842 bool numeric_flag = true;
843 for (exvector::const_iterator r=m.begin(); r!=m.end(); ++r) {
844 if (!(*r).info(info_flags::numeric)) {
845 numeric_flag = false;
849 // The pure numeric case is traditionally rather common. Hence, it is
850 // trapped and we use Leverrier's algorithm which goes as row^3 for
851 // every coefficient. The expensive part is the matrix multiplication.
855 ex poly = power(lambda,row)-c*power(lambda,row-1);
856 for (unsigned i=1; i<row; ++i) {
857 for (unsigned j=0; j<row; ++j)
860 c = B.trace()/ex(i+1);
861 poly -= c*power(lambda,row-i-1);
870 for (unsigned r=0; r<col; ++r)
871 M.m[r*col+r] -= lambda;
873 return M.determinant().collect(lambda);
877 /** Inverse of this matrix.
879 * @return the inverted matrix
880 * @exception logic_error (matrix not square)
881 * @exception runtime_error (singular matrix) */
882 matrix matrix::inverse(void) const
885 throw (std::logic_error("matrix::inverse(): matrix not square"));
887 // This routine actually doesn't do anything fancy at all. We compute the
888 // inverse of the matrix A by solving the system A * A^{-1} == Id.
890 // First populate the identity matrix supposed to become the right hand side.
891 matrix identity(row,col);
892 for (unsigned i=0; i<row; ++i)
893 identity(i,i) = _ex1();
895 // Populate a dummy matrix of variables, just because of compatibility with
896 // matrix::solve() which wants this (for compatibility with under-determined
897 // systems of equations).
898 matrix vars(row,col);
899 for (unsigned r=0; r<row; ++r)
900 for (unsigned c=0; c<col; ++c)
901 vars(r,c) = symbol();
905 sol = this->solve(vars,identity);
906 } catch (const std::runtime_error & e) {
907 if (e.what()==std::string("matrix::solve(): inconsistent linear system"))
908 throw (std::runtime_error("matrix::inverse(): singular matrix"));
916 /** Solve a linear system consisting of a m x n matrix and a m x p right hand
917 * side by applying an elimination scheme to the augmented matrix.
919 * @param vars n x p matrix, all elements must be symbols
920 * @param rhs m x p matrix
921 * @return n x p solution matrix
922 * @exception logic_error (incompatible matrices)
923 * @exception invalid_argument (1st argument must be matrix of symbols)
924 * @exception runtime_error (inconsistent linear system)
926 matrix matrix::solve(const matrix & vars,
930 const unsigned m = this->rows();
931 const unsigned n = this->cols();
932 const unsigned p = rhs.cols();
935 if ((rhs.rows() != m) || (vars.rows() != n) || (vars.col != p))
936 throw (std::logic_error("matrix::solve(): incompatible matrices"));
937 for (unsigned ro=0; ro<n; ++ro)
938 for (unsigned co=0; co<p; ++co)
939 if (!vars(ro,co).info(info_flags::symbol))
940 throw (std::invalid_argument("matrix::solve(): 1st argument must be matrix of symbols"));
942 // build the augmented matrix of *this with rhs attached to the right
944 for (unsigned r=0; r<m; ++r) {
945 for (unsigned c=0; c<n; ++c)
946 aug.m[r*(n+p)+c] = this->m[r*n+c];
947 for (unsigned c=0; c<p; ++c)
948 aug.m[r*(n+p)+c+n] = rhs.m[r*p+c];
951 // Gather some statistical information about the augmented matrix:
952 bool numeric_flag = true;
953 for (exvector::const_iterator r=aug.m.begin(); r!=aug.m.end(); ++r) {
954 if (!(*r).info(info_flags::numeric))
955 numeric_flag = false;
958 // Here is the heuristics in case this routine has to decide:
959 if (algo == solve_algo::automatic) {
960 // Bareiss (fraction-free) elimination is generally a good guess:
961 algo = solve_algo::bareiss;
962 // For m<3, Bareiss elimination is equivalent to division free
963 // elimination but has more logistic overhead
965 algo = solve_algo::divfree;
966 // This overrides any prior decisions.
968 algo = solve_algo::gauss;
971 // Eliminate the augmented matrix:
973 case solve_algo::gauss:
974 aug.gauss_elimination();
976 case solve_algo::divfree:
977 aug.division_free_elimination();
979 case solve_algo::bareiss:
981 aug.fraction_free_elimination();
984 // assemble the solution matrix:
986 for (unsigned co=0; co<p; ++co) {
987 unsigned last_assigned_sol = n+1;
988 for (int r=m-1; r>=0; --r) {
989 unsigned fnz = 1; // first non-zero in row
990 while ((fnz<=n) && (aug.m[r*(n+p)+(fnz-1)].is_zero()))
993 // row consists only of zeros, corresponding rhs must be 0, too
994 if (!aug.m[r*(n+p)+n+co].is_zero()) {
995 throw (std::runtime_error("matrix::solve(): inconsistent linear system"));
998 // assign solutions for vars between fnz+1 and
999 // last_assigned_sol-1: free parameters
1000 for (unsigned c=fnz; c<last_assigned_sol-1; ++c)
1001 sol(c,co) = vars.m[c*p+co];
1002 ex e = aug.m[r*(n+p)+n+co];
1003 for (unsigned c=fnz; c<n; ++c)
1004 e -= aug.m[r*(n+p)+c]*sol.m[c*p+co];
1005 sol(fnz-1,co) = (e/(aug.m[r*(n+p)+(fnz-1)])).normal();
1006 last_assigned_sol = fnz;
1009 // assign solutions for vars between 1 and
1010 // last_assigned_sol-1: free parameters
1011 for (unsigned ro=0; ro<last_assigned_sol-1; ++ro)
1012 sol(ro,co) = vars(ro,co);
1021 /** Recursive determinant for small matrices having at least one symbolic
1022 * entry. The basic algorithm, known as Laplace-expansion, is enhanced by
1023 * some bookkeeping to avoid calculation of the same submatrices ("minors")
1024 * more than once. According to W.M.Gentleman and S.C.Johnson this algorithm
1025 * is better than elimination schemes for matrices of sparse multivariate
1026 * polynomials and also for matrices of dense univariate polynomials if the
1027 * matrix' dimesion is larger than 7.
1029 * @return the determinant as a new expression (in expanded form)
1030 * @see matrix::determinant() */
1031 ex matrix::determinant_minor(void) const
1033 // for small matrices the algorithm does not make any sense:
1034 const unsigned n = this->cols();
1036 return m[0].expand();
1038 return (m[0]*m[3]-m[2]*m[1]).expand();
1040 return (m[0]*m[4]*m[8]-m[0]*m[5]*m[7]-
1041 m[1]*m[3]*m[8]+m[2]*m[3]*m[7]+
1042 m[1]*m[5]*m[6]-m[2]*m[4]*m[6]).expand();
1044 // This algorithm can best be understood by looking at a naive
1045 // implementation of Laplace-expansion, like this one:
1047 // matrix minorM(this->rows()-1,this->cols()-1);
1048 // for (unsigned r1=0; r1<this->rows(); ++r1) {
1049 // // shortcut if element(r1,0) vanishes
1050 // if (m[r1*col].is_zero())
1052 // // assemble the minor matrix
1053 // for (unsigned r=0; r<minorM.rows(); ++r) {
1054 // for (unsigned c=0; c<minorM.cols(); ++c) {
1056 // minorM(r,c) = m[r*col+c+1];
1058 // minorM(r,c) = m[(r+1)*col+c+1];
1061 // // recurse down and care for sign:
1063 // det -= m[r1*col] * minorM.determinant_minor();
1065 // det += m[r1*col] * minorM.determinant_minor();
1067 // return det.expand();
1068 // What happens is that while proceeding down many of the minors are
1069 // computed more than once. In particular, there are binomial(n,k)
1070 // kxk minors and each one is computed factorial(n-k) times. Therefore
1071 // it is reasonable to store the results of the minors. We proceed from
1072 // right to left. At each column c we only need to retrieve the minors
1073 // calculated in step c-1. We therefore only have to store at most
1074 // 2*binomial(n,n/2) minors.
1076 // Unique flipper counter for partitioning into minors
1077 std::vector<unsigned> Pkey;
1079 // key for minor determinant (a subpartition of Pkey)
1080 std::vector<unsigned> Mkey;
1082 // we store our subminors in maps, keys being the rows they arise from
1083 typedef std::map<std::vector<unsigned>,class ex> Rmap;
1084 typedef std::map<std::vector<unsigned>,class ex>::value_type Rmap_value;
1088 // initialize A with last column:
1089 for (unsigned r=0; r<n; ++r) {
1090 Pkey.erase(Pkey.begin(),Pkey.end());
1092 A.insert(Rmap_value(Pkey,m[n*(r+1)-1]));
1094 // proceed from right to left through matrix
1095 for (int c=n-2; c>=0; --c) {
1096 Pkey.erase(Pkey.begin(),Pkey.end()); // don't change capacity
1097 Mkey.erase(Mkey.begin(),Mkey.end());
1098 for (unsigned i=0; i<n-c; ++i)
1100 unsigned fc = 0; // controls logic for our strange flipper counter
1103 for (unsigned r=0; r<n-c; ++r) {
1104 // maybe there is nothing to do?
1105 if (m[Pkey[r]*n+c].is_zero())
1107 // create the sorted key for all possible minors
1108 Mkey.erase(Mkey.begin(),Mkey.end());
1109 for (unsigned i=0; i<n-c; ++i)
1111 Mkey.push_back(Pkey[i]);
1112 // Fetch the minors and compute the new determinant
1114 det -= m[Pkey[r]*n+c]*A[Mkey];
1116 det += m[Pkey[r]*n+c]*A[Mkey];
1118 // prevent build-up of deep nesting of expressions saves time:
1120 // store the new determinant at its place in B:
1122 B.insert(Rmap_value(Pkey,det));
1123 // increment our strange flipper counter
1124 for (fc=n-c; fc>0; --fc) {
1126 if (Pkey[fc-1]<fc+c)
1130 for (unsigned j=fc; j<n-c; ++j)
1131 Pkey[j] = Pkey[j-1]+1;
1133 // next column, so change the role of A and B:
1142 /** Perform the steps of an ordinary Gaussian elimination to bring the m x n
1143 * matrix into an upper echelon form. The algorithm is ok for matrices
1144 * with numeric coefficients but quite unsuited for symbolic matrices.
1146 * @param det may be set to true to save a lot of space if one is only
1147 * interested in the diagonal elements (i.e. for calculating determinants).
1148 * The others are set to zero in this case.
1149 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1150 * number of rows was swapped and 0 if the matrix is singular. */
1151 int matrix::gauss_elimination(const bool det)
1153 ensure_if_modifiable();
1154 const unsigned m = this->rows();
1155 const unsigned n = this->cols();
1156 GINAC_ASSERT(!det || n==m);
1160 for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
1161 int indx = pivot(r0, r1, true);
1165 return 0; // leaves *this in a messy state
1170 for (unsigned r2=r0+1; r2<m; ++r2) {
1171 if (!this->m[r2*n+r1].is_zero()) {
1172 // yes, there is something to do in this row
1173 ex piv = this->m[r2*n+r1] / this->m[r0*n+r1];
1174 for (unsigned c=r1+1; c<n; ++c) {
1175 this->m[r2*n+c] -= piv * this->m[r0*n+c];
1176 if (!this->m[r2*n+c].info(info_flags::numeric))
1177 this->m[r2*n+c] = this->m[r2*n+c].normal();
1180 // fill up left hand side with zeros
1181 for (unsigned c=0; c<=r1; ++c)
1182 this->m[r2*n+c] = _ex0();
1185 // save space by deleting no longer needed elements
1186 for (unsigned c=r0+1; c<n; ++c)
1187 this->m[r0*n+c] = _ex0();
1197 /** Perform the steps of division free elimination to bring the m x n matrix
1198 * into an upper echelon form.
1200 * @param det may be set to true to save a lot of space if one is only
1201 * interested in the diagonal elements (i.e. for calculating determinants).
1202 * The others are set to zero in this case.
1203 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1204 * number of rows was swapped and 0 if the matrix is singular. */
1205 int matrix::division_free_elimination(const bool det)
1207 ensure_if_modifiable();
1208 const unsigned m = this->rows();
1209 const unsigned n = this->cols();
1210 GINAC_ASSERT(!det || n==m);
1214 for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
1215 int indx = pivot(r0, r1, true);
1219 return 0; // leaves *this in a messy state
1224 for (unsigned r2=r0+1; r2<m; ++r2) {
1225 for (unsigned c=r1+1; c<n; ++c)
1226 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();
1227 // fill up left hand side with zeros
1228 for (unsigned c=0; c<=r1; ++c)
1229 this->m[r2*n+c] = _ex0();
1232 // save space by deleting no longer needed elements
1233 for (unsigned c=r0+1; c<n; ++c)
1234 this->m[r0*n+c] = _ex0();
1244 /** Perform the steps of Bareiss' one-step fraction free elimination to bring
1245 * the matrix into an upper echelon form. Fraction free elimination means
1246 * that divide is used straightforwardly, without computing GCDs first. This
1247 * is possible, since we know the divisor at each step.
1249 * @param det may be set to true to save a lot of space if one is only
1250 * interested in the last element (i.e. for calculating determinants). The
1251 * others are set to zero in this case.
1252 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1253 * number of rows was swapped and 0 if the matrix is singular. */
1254 int matrix::fraction_free_elimination(const bool det)
1257 // (single-step fraction free elimination scheme, already known to Jordan)
1259 // Usual division-free elimination sets m[0](r,c) = m(r,c) and then sets
1260 // m[k+1](r,c) = m[k](k,k) * m[k](r,c) - m[k](r,k) * m[k](k,c).
1262 // Bareiss (fraction-free) elimination in addition divides that element
1263 // by m[k-1](k-1,k-1) for k>1, where it can be shown by means of the
1264 // Sylvester determinant that this really divides m[k+1](r,c).
1266 // We also allow rational functions where the original prove still holds.
1267 // However, we must care for numerator and denominator separately and
1268 // "manually" work in the integral domains because of subtle cancellations
1269 // (see below). This blows up the bookkeeping a bit and the formula has
1270 // to be modified to expand like this (N{x} stands for numerator of x,
1271 // D{x} for denominator of x):
1272 // 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)}
1273 // -N{m[k](r,k)}*N{m[k](k,c)}*D{m[k](k,k)}*D{m[k](r,c)}
1274 // 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)}
1275 // where for k>1 we now divide N{m[k+1](r,c)} by
1276 // N{m[k-1](k-1,k-1)}
1277 // and D{m[k+1](r,c)} by
1278 // D{m[k-1](k-1,k-1)}.
1280 ensure_if_modifiable();
1281 const unsigned m = this->rows();
1282 const unsigned n = this->cols();
1283 GINAC_ASSERT(!det || n==m);
1292 // We populate temporary matrices to subsequently operate on. There is
1293 // one holding numerators and another holding denominators of entries.
1294 // This is a must since the evaluator (or even earlier mul's constructor)
1295 // might cancel some trivial element which causes divide() to fail. The
1296 // elements are normalized first (yes, even though this algorithm doesn't
1297 // need GCDs) since the elements of *this might be unnormalized, which
1298 // makes things more complicated than they need to be.
1299 matrix tmp_n(*this);
1300 matrix tmp_d(m,n); // for denominators, if needed
1301 lst srl; // symbol replacement list
1302 exvector::iterator it = this->m.begin();
1303 exvector::iterator tmp_n_it = tmp_n.m.begin();
1304 exvector::iterator tmp_d_it = tmp_d.m.begin();
1305 for (; it!= this->m.end(); ++it, ++tmp_n_it, ++tmp_d_it) {
1306 (*tmp_n_it) = (*it).normal().to_rational(srl);
1307 (*tmp_d_it) = (*tmp_n_it).denom();
1308 (*tmp_n_it) = (*tmp_n_it).numer();
1312 for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
1313 int indx = tmp_n.pivot(r0, r1, true);
1322 // tmp_n's rows r0 and indx were swapped, do the same in tmp_d:
1323 for (unsigned c=r1; c<n; ++c)
1324 tmp_d.m[n*indx+c].swap(tmp_d.m[n*r0+c]);
1326 for (unsigned r2=r0+1; r2<m; ++r2) {
1327 for (unsigned c=r1+1; c<n; ++c) {
1328 dividend_n = (tmp_n.m[r0*n+r1]*tmp_n.m[r2*n+c]*
1329 tmp_d.m[r2*n+r1]*tmp_d.m[r0*n+c]
1330 -tmp_n.m[r2*n+r1]*tmp_n.m[r0*n+c]*
1331 tmp_d.m[r0*n+r1]*tmp_d.m[r2*n+c]).expand();
1332 dividend_d = (tmp_d.m[r2*n+r1]*tmp_d.m[r0*n+c]*
1333 tmp_d.m[r0*n+r1]*tmp_d.m[r2*n+c]).expand();
1334 bool check = divide(dividend_n, divisor_n,
1335 tmp_n.m[r2*n+c], true);
1336 check &= divide(dividend_d, divisor_d,
1337 tmp_d.m[r2*n+c], true);
1338 GINAC_ASSERT(check);
1340 // fill up left hand side with zeros
1341 for (unsigned c=0; c<=r1; ++c)
1342 tmp_n.m[r2*n+c] = _ex0();
1344 if ((r1<n-1)&&(r0<m-1)) {
1345 // compute next iteration's divisor
1346 divisor_n = tmp_n.m[r0*n+r1].expand();
1347 divisor_d = tmp_d.m[r0*n+r1].expand();
1349 // save space by deleting no longer needed elements
1350 for (unsigned c=0; c<n; ++c) {
1351 tmp_n.m[r0*n+c] = _ex0();
1352 tmp_d.m[r0*n+c] = _ex1();
1359 // repopulate *this matrix:
1360 it = this->m.begin();
1361 tmp_n_it = tmp_n.m.begin();
1362 tmp_d_it = tmp_d.m.begin();
1363 for (; it!= this->m.end(); ++it, ++tmp_n_it, ++tmp_d_it)
1364 (*it) = ((*tmp_n_it)/(*tmp_d_it)).subs(srl);
1370 /** Partial pivoting method for matrix elimination schemes.
1371 * Usual pivoting (symbolic==false) returns the index to the element with the
1372 * largest absolute value in column ro and swaps the current row with the one
1373 * where the element was found. With (symbolic==true) it does the same thing
1374 * with the first non-zero element.
1376 * @param ro is the row from where to begin
1377 * @param co is the column to be inspected
1378 * @param symbolic signal if we want the first non-zero element to be pivoted
1379 * (true) or the one with the largest absolute value (false).
1380 * @return 0 if no interchange occured, -1 if all are zero (usually signaling
1381 * a degeneracy) and positive integer k means that rows ro and k were swapped.
1383 int matrix::pivot(unsigned ro, unsigned co, bool symbolic)
1387 // search first non-zero element in column co beginning at row ro
1388 while ((k<row) && (this->m[k*col+co].expand().is_zero()))
1391 // search largest element in column co beginning at row ro
1392 GINAC_ASSERT(is_ex_of_type(this->m[k*col+co],numeric));
1393 unsigned kmax = k+1;
1394 numeric mmax = abs(ex_to<numeric>(m[kmax*col+co]));
1396 GINAC_ASSERT(is_ex_of_type(this->m[kmax*col+co],numeric));
1397 numeric tmp = ex_to<numeric>(this->m[kmax*col+co]);
1398 if (abs(tmp) > mmax) {
1404 if (!mmax.is_zero())
1408 // all elements in column co below row ro vanish
1411 // matrix needs no pivoting
1413 // matrix needs pivoting, so swap rows k and ro
1414 ensure_if_modifiable();
1415 for (unsigned c=0; c<col; ++c)
1416 this->m[k*col+c].swap(this->m[ro*col+c]);
1421 ex lst_to_matrix(const lst & l)
1423 // Find number of rows and columns
1424 unsigned rows = l.nops(), cols = 0, i, j;
1425 for (i=0; i<rows; i++)
1426 if (l.op(i).nops() > cols)
1427 cols = l.op(i).nops();
1429 // Allocate and fill matrix
1430 matrix &m = *new matrix(rows, cols);
1431 m.setflag(status_flags::dynallocated);
1432 for (i=0; i<rows; i++)
1433 for (j=0; j<cols; j++)
1434 if (l.op(i).nops() > j)
1435 m(i, j) = l.op(i).op(j);
1441 ex diag_matrix(const lst & l)
1443 unsigned dim = l.nops();
1445 matrix &m = *new matrix(dim, dim);
1446 m.setflag(status_flags::dynallocated);
1447 for (unsigned i=0; i<dim; i++)
1453 } // namespace GiNaC