3 * Implementation of symbolic matrices */
6 * GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
8 * This program is free software; you can redistribute it and/or modify
9 * it under the terms of the GNU General Public License as published by
10 * the Free Software Foundation; either version 2 of the License, or
11 * (at your option) any later version.
13 * This program is distributed in the hope that it will be useful,
14 * but WITHOUT ANY WARRANTY; without even the implied warranty of
15 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
16 * GNU General Public License for more details.
18 * You should have received a copy of the GNU General Public License
19 * along with this program; if not, write to the Free Software
20 * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
42 GINAC_IMPLEMENT_REGISTERED_CLASS(matrix, basic)
45 // default ctor, dtor, copy ctor, assignment operator and helpers:
48 /** Default ctor. Initializes to 1 x 1-dimensional zero-matrix. */
49 matrix::matrix() : inherited(TINFO_matrix), row(1), col(1)
51 debugmsg("matrix default ctor",LOGLEVEL_CONSTRUCT);
55 void matrix::copy(const matrix & other)
57 inherited::copy(other);
60 m = other.m; // STL's vector copying invoked here
63 DEFAULT_DESTROY(matrix)
71 /** Very common ctor. Initializes to r x c-dimensional zero-matrix.
73 * @param r number of rows
74 * @param c number of cols */
75 matrix::matrix(unsigned r, unsigned c)
76 : inherited(TINFO_matrix), row(r), col(c)
78 debugmsg("matrix ctor from unsigned,unsigned",LOGLEVEL_CONSTRUCT);
79 m.resize(r*c, _ex0());
84 /** Ctor from representation, for internal use only. */
85 matrix::matrix(unsigned r, unsigned c, const exvector & m2)
86 : inherited(TINFO_matrix), row(r), col(c), m(m2)
88 debugmsg("matrix ctor from unsigned,unsigned,exvector",LOGLEVEL_CONSTRUCT);
91 /** Construct matrix from (flat) list of elements. If the list has fewer
92 * elements than the matrix, the remaining matrix elements are set to zero.
93 * If the list has more elements than the matrix, the excessive elements are
95 matrix::matrix(unsigned r, unsigned c, const lst & l)
96 : inherited(TINFO_matrix), row(r), col(c)
98 debugmsg("matrix ctor from unsigned,unsigned,lst",LOGLEVEL_CONSTRUCT);
99 m.resize(r*c, _ex0());
101 for (unsigned i=0; i<l.nops(); i++) {
105 break; // matrix smaller than list: throw away excessive elements
114 matrix::matrix(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
116 debugmsg("matrix ctor from archive_node", LOGLEVEL_CONSTRUCT);
117 if (!(n.find_unsigned("row", row)) || !(n.find_unsigned("col", col)))
118 throw (std::runtime_error("unknown matrix dimensions in archive"));
119 m.reserve(row * col);
120 for (unsigned int i=0; true; i++) {
122 if (n.find_ex("m", e, sym_lst, i))
129 void matrix::archive(archive_node &n) const
131 inherited::archive(n);
132 n.add_unsigned("row", row);
133 n.add_unsigned("col", col);
134 exvector::const_iterator i = m.begin(), iend = m.end();
141 DEFAULT_UNARCHIVE(matrix)
144 // functions overriding virtual functions from bases classes
149 void matrix::print(const print_context & c, unsigned level) const
151 debugmsg("matrix print", LOGLEVEL_PRINT);
153 if (is_of_type(c, print_tree)) {
155 inherited::print(c, level);
160 for (unsigned y=0; y<row-1; ++y) {
162 for (unsigned x=0; x<col-1; ++x) {
166 m[col*(y+1)-1].print(c);
170 for (unsigned x=0; x<col-1; ++x) {
171 m[(row-1)*col+x].print(c);
174 m[row*col-1].print(c);
180 /** nops is defined to be rows x columns. */
181 unsigned matrix::nops() const
186 /** returns matrix entry at position (i/col, i%col). */
187 ex matrix::op(int i) const
192 /** returns matrix entry at position (i/col, i%col). */
193 ex & matrix::let_op(int i)
196 GINAC_ASSERT(i<nops());
201 /** Evaluate matrix entry by entry. */
202 ex matrix::eval(int level) const
204 debugmsg("matrix eval",LOGLEVEL_MEMBER_FUNCTION);
206 // check if we have to do anything at all
207 if ((level==1)&&(flags & status_flags::evaluated))
211 if (level == -max_recursion_level)
212 throw (std::runtime_error("matrix::eval(): recursion limit exceeded"));
214 // eval() entry by entry
215 exvector m2(row*col);
217 for (unsigned r=0; r<row; ++r)
218 for (unsigned c=0; c<col; ++c)
219 m2[r*col+c] = m[r*col+c].eval(level);
221 return (new matrix(row, col, m2))->setflag(status_flags::dynallocated |
222 status_flags::evaluated );
225 ex matrix::subs(const lst & ls, const lst & lr, bool no_pattern) const
227 exvector m2(row * col);
228 for (unsigned r=0; r<row; ++r)
229 for (unsigned c=0; c<col; ++c)
230 m2[r*col+c] = m[r*col+c].subs(ls, lr, no_pattern);
232 return ex(matrix(row, col, m2)).bp->basic::subs(ls, lr, no_pattern);
237 int matrix::compare_same_type(const basic & other) const
239 GINAC_ASSERT(is_exactly_of_type(other, matrix));
240 const matrix & o = static_cast<const matrix &>(other);
242 // compare number of rows
244 return row < o.rows() ? -1 : 1;
246 // compare number of columns
248 return col < o.cols() ? -1 : 1;
250 // equal number of rows and columns, compare individual elements
252 for (unsigned r=0; r<row; ++r) {
253 for (unsigned c=0; c<col; ++c) {
254 cmpval = ((*this)(r,c)).compare(o(r,c));
255 if (cmpval!=0) return cmpval;
258 // all elements are equal => matrices are equal;
262 /** 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 = sum.begin(), end = sum.end();
507 exvector::const_iterator ci = other.m.begin();
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 = dif.begin(), end = dif.end();
525 exvector::const_iterator ci = other.m.begin();
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 exvector::const_iterator r = m.begin(), rend = m.end();
694 lst srl; // symbol replacement list
695 ex rtest = r->to_rational(srl);
696 if (!rtest.is_zero())
698 if (!rtest.info(info_flags::numeric))
699 numeric_flag = false;
700 if (!rtest.info(info_flags::crational_polynomial) &&
701 rtest.info(info_flags::rational_function))
706 // Here is the heuristics in case this routine has to decide:
707 if (algo == determinant_algo::automatic) {
708 // Minor expansion is generally a good guess:
709 algo = determinant_algo::laplace;
710 // Does anybody know when a matrix is really sparse?
711 // Maybe <~row/2.236 nonzero elements average in a row?
712 if (row>3 && 5*sparse_count<=row*col)
713 algo = determinant_algo::bareiss;
714 // Purely numeric matrix can be handled by Gauss elimination.
715 // This overrides any prior decisions.
717 algo = determinant_algo::gauss;
720 // Trap the trivial case here, since some algorithms don't like it
722 // for consistency with non-trivial determinants...
724 return m[0].normal();
726 return m[0].expand();
729 // Compute the determinant
731 case determinant_algo::gauss: {
734 int sign = tmp.gauss_elimination(true);
735 for (unsigned d=0; d<row; ++d)
736 det *= tmp.m[d*col+d];
738 return (sign*det).normal();
740 return (sign*det).normal().expand();
742 case determinant_algo::bareiss: {
745 sign = tmp.fraction_free_elimination(true);
747 return (sign*tmp.m[row*col-1]).normal();
749 return (sign*tmp.m[row*col-1]).expand();
751 case determinant_algo::divfree: {
754 sign = tmp.division_free_elimination(true);
757 ex det = tmp.m[row*col-1];
758 // factor out accumulated bogus slag
759 for (unsigned d=0; d<row-2; ++d)
760 for (unsigned j=0; j<row-d-2; ++j)
761 det = (det/tmp.m[d*col+d]).normal();
764 case determinant_algo::laplace:
766 // This is the minor expansion scheme. We always develop such
767 // that the smallest minors (i.e, the trivial 1x1 ones) are on the
768 // rightmost column. For this to be efficient it turns out that
769 // the emptiest columns (i.e. the ones with most zeros) should be
770 // the ones on the right hand side. Therefore we presort the
771 // columns of the matrix:
772 typedef std::pair<unsigned,unsigned> uintpair;
773 std::vector<uintpair> c_zeros; // number of zeros in column
774 for (unsigned c=0; c<col; ++c) {
776 for (unsigned r=0; r<row; ++r)
777 if (m[r*col+c].is_zero())
779 c_zeros.push_back(uintpair(acc,c));
781 sort(c_zeros.begin(),c_zeros.end());
782 std::vector<unsigned> pre_sort;
783 for (std::vector<uintpair>::const_iterator i=c_zeros.begin(); i!=c_zeros.end(); ++i)
784 pre_sort.push_back(i->second);
785 std::vector<unsigned> pre_sort_test(pre_sort); // permutation_sign() modifies the vector so we make a copy here
786 int sign = permutation_sign(pre_sort_test.begin(), pre_sort_test.end());
787 exvector result(row*col); // represents sorted matrix
789 for (std::vector<unsigned>::const_iterator i=pre_sort.begin();
792 for (unsigned r=0; r<row; ++r)
793 result[r*col+c] = m[r*col+(*i)];
797 return (sign*matrix(row,col,result).determinant_minor()).normal();
799 return sign*matrix(row,col,result).determinant_minor();
805 /** Trace of a matrix. The result is normalized if it is in some quotient
806 * field and expanded only otherwise. This implies that the trace of the
807 * symbolic 2x2 matrix [[a/(a-b),x],[y,b/(b-a)]] is recognized to be unity.
809 * @return the sum of diagonal elements
810 * @exception logic_error (matrix not square) */
811 ex matrix::trace(void) const
814 throw (std::logic_error("matrix::trace(): matrix not square"));
817 for (unsigned r=0; r<col; ++r)
820 if (tr.info(info_flags::rational_function) &&
821 !tr.info(info_flags::crational_polynomial))
828 /** Characteristic Polynomial. Following mathematica notation the
829 * characteristic polynomial of a matrix M is defined as the determiant of
830 * (M - lambda * 1) where 1 stands for the unit matrix of the same dimension
831 * as M. Note that some CASs define it with a sign inside the determinant
832 * which gives rise to an overall sign if the dimension is odd. This method
833 * returns the characteristic polynomial collected in powers of lambda as a
836 * @return characteristic polynomial as new expression
837 * @exception logic_error (matrix not square)
838 * @see matrix::determinant() */
839 ex matrix::charpoly(const symbol & lambda) const
842 throw (std::logic_error("matrix::charpoly(): matrix not square"));
844 bool numeric_flag = true;
845 exvector::const_iterator r = m.begin(), rend = m.end();
847 if (!r->info(info_flags::numeric))
848 numeric_flag = false;
852 // The pure numeric case is traditionally rather common. Hence, it is
853 // trapped and we use Leverrier's algorithm which goes as row^3 for
854 // every coefficient. The expensive part is the matrix multiplication.
858 ex poly = power(lambda,row)-c*power(lambda,row-1);
859 for (unsigned i=1; i<row; ++i) {
860 for (unsigned j=0; j<row; ++j)
863 c = B.trace()/ex(i+1);
864 poly -= c*power(lambda,row-i-1);
873 for (unsigned r=0; r<col; ++r)
874 M.m[r*col+r] -= lambda;
876 return M.determinant().collect(lambda);
880 /** Inverse of this matrix.
882 * @return the inverted matrix
883 * @exception logic_error (matrix not square)
884 * @exception runtime_error (singular matrix) */
885 matrix matrix::inverse(void) const
888 throw (std::logic_error("matrix::inverse(): matrix not square"));
890 // This routine actually doesn't do anything fancy at all. We compute the
891 // inverse of the matrix A by solving the system A * A^{-1} == Id.
893 // First populate the identity matrix supposed to become the right hand side.
894 matrix identity(row,col);
895 for (unsigned i=0; i<row; ++i)
896 identity(i,i) = _ex1();
898 // Populate a dummy matrix of variables, just because of compatibility with
899 // matrix::solve() which wants this (for compatibility with under-determined
900 // systems of equations).
901 matrix vars(row,col);
902 for (unsigned r=0; r<row; ++r)
903 for (unsigned c=0; c<col; ++c)
904 vars(r,c) = symbol();
908 sol = this->solve(vars,identity);
909 } catch (const std::runtime_error & e) {
910 if (e.what()==std::string("matrix::solve(): inconsistent linear system"))
911 throw (std::runtime_error("matrix::inverse(): singular matrix"));
919 /** Solve a linear system consisting of a m x n matrix and a m x p right hand
920 * side by applying an elimination scheme to the augmented matrix.
922 * @param vars n x p matrix, all elements must be symbols
923 * @param rhs m x p matrix
924 * @return n x p solution matrix
925 * @exception logic_error (incompatible matrices)
926 * @exception invalid_argument (1st argument must be matrix of symbols)
927 * @exception runtime_error (inconsistent linear system)
929 matrix matrix::solve(const matrix & vars,
933 const unsigned m = this->rows();
934 const unsigned n = this->cols();
935 const unsigned p = rhs.cols();
938 if ((rhs.rows() != m) || (vars.rows() != n) || (vars.col != p))
939 throw (std::logic_error("matrix::solve(): incompatible matrices"));
940 for (unsigned ro=0; ro<n; ++ro)
941 for (unsigned co=0; co<p; ++co)
942 if (!vars(ro,co).info(info_flags::symbol))
943 throw (std::invalid_argument("matrix::solve(): 1st argument must be matrix of symbols"));
945 // build the augmented matrix of *this with rhs attached to the right
947 for (unsigned r=0; r<m; ++r) {
948 for (unsigned c=0; c<n; ++c)
949 aug.m[r*(n+p)+c] = this->m[r*n+c];
950 for (unsigned c=0; c<p; ++c)
951 aug.m[r*(n+p)+c+n] = rhs.m[r*p+c];
954 // Gather some statistical information about the augmented matrix:
955 bool numeric_flag = true;
956 exvector::const_iterator r = aug.m.begin(), rend = aug.m.end();
958 if (!r->info(info_flags::numeric))
959 numeric_flag = false;
963 // Here is the heuristics in case this routine has to decide:
964 if (algo == solve_algo::automatic) {
965 // Bareiss (fraction-free) elimination is generally a good guess:
966 algo = solve_algo::bareiss;
967 // For m<3, Bareiss elimination is equivalent to division free
968 // elimination but has more logistic overhead
970 algo = solve_algo::divfree;
971 // This overrides any prior decisions.
973 algo = solve_algo::gauss;
976 // Eliminate the augmented matrix:
978 case solve_algo::gauss:
979 aug.gauss_elimination();
981 case solve_algo::divfree:
982 aug.division_free_elimination();
984 case solve_algo::bareiss:
986 aug.fraction_free_elimination();
989 // assemble the solution matrix:
991 for (unsigned co=0; co<p; ++co) {
992 unsigned last_assigned_sol = n+1;
993 for (int r=m-1; r>=0; --r) {
994 unsigned fnz = 1; // first non-zero in row
995 while ((fnz<=n) && (aug.m[r*(n+p)+(fnz-1)].is_zero()))
998 // row consists only of zeros, corresponding rhs must be 0, too
999 if (!aug.m[r*(n+p)+n+co].is_zero()) {
1000 throw (std::runtime_error("matrix::solve(): inconsistent linear system"));
1003 // assign solutions for vars between fnz+1 and
1004 // last_assigned_sol-1: free parameters
1005 for (unsigned c=fnz; c<last_assigned_sol-1; ++c)
1006 sol(c,co) = vars.m[c*p+co];
1007 ex e = aug.m[r*(n+p)+n+co];
1008 for (unsigned c=fnz; c<n; ++c)
1009 e -= aug.m[r*(n+p)+c]*sol.m[c*p+co];
1010 sol(fnz-1,co) = (e/(aug.m[r*(n+p)+(fnz-1)])).normal();
1011 last_assigned_sol = fnz;
1014 // assign solutions for vars between 1 and
1015 // last_assigned_sol-1: free parameters
1016 for (unsigned ro=0; ro<last_assigned_sol-1; ++ro)
1017 sol(ro,co) = vars(ro,co);
1026 /** Recursive determinant for small matrices having at least one symbolic
1027 * entry. The basic algorithm, known as Laplace-expansion, is enhanced by
1028 * some bookkeeping to avoid calculation of the same submatrices ("minors")
1029 * more than once. According to W.M.Gentleman and S.C.Johnson this algorithm
1030 * is better than elimination schemes for matrices of sparse multivariate
1031 * polynomials and also for matrices of dense univariate polynomials if the
1032 * matrix' dimesion is larger than 7.
1034 * @return the determinant as a new expression (in expanded form)
1035 * @see matrix::determinant() */
1036 ex matrix::determinant_minor(void) const
1038 // for small matrices the algorithm does not make any sense:
1039 const unsigned n = this->cols();
1041 return m[0].expand();
1043 return (m[0]*m[3]-m[2]*m[1]).expand();
1045 return (m[0]*m[4]*m[8]-m[0]*m[5]*m[7]-
1046 m[1]*m[3]*m[8]+m[2]*m[3]*m[7]+
1047 m[1]*m[5]*m[6]-m[2]*m[4]*m[6]).expand();
1049 // This algorithm can best be understood by looking at a naive
1050 // implementation of Laplace-expansion, like this one:
1052 // matrix minorM(this->rows()-1,this->cols()-1);
1053 // for (unsigned r1=0; r1<this->rows(); ++r1) {
1054 // // shortcut if element(r1,0) vanishes
1055 // if (m[r1*col].is_zero())
1057 // // assemble the minor matrix
1058 // for (unsigned r=0; r<minorM.rows(); ++r) {
1059 // for (unsigned c=0; c<minorM.cols(); ++c) {
1061 // minorM(r,c) = m[r*col+c+1];
1063 // minorM(r,c) = m[(r+1)*col+c+1];
1066 // // recurse down and care for sign:
1068 // det -= m[r1*col] * minorM.determinant_minor();
1070 // det += m[r1*col] * minorM.determinant_minor();
1072 // return det.expand();
1073 // What happens is that while proceeding down many of the minors are
1074 // computed more than once. In particular, there are binomial(n,k)
1075 // kxk minors and each one is computed factorial(n-k) times. Therefore
1076 // it is reasonable to store the results of the minors. We proceed from
1077 // right to left. At each column c we only need to retrieve the minors
1078 // calculated in step c-1. We therefore only have to store at most
1079 // 2*binomial(n,n/2) minors.
1081 // Unique flipper counter for partitioning into minors
1082 std::vector<unsigned> Pkey;
1084 // key for minor determinant (a subpartition of Pkey)
1085 std::vector<unsigned> Mkey;
1087 // we store our subminors in maps, keys being the rows they arise from
1088 typedef std::map<std::vector<unsigned>,class ex> Rmap;
1089 typedef std::map<std::vector<unsigned>,class ex>::value_type Rmap_value;
1093 // initialize A with last column:
1094 for (unsigned r=0; r<n; ++r) {
1095 Pkey.erase(Pkey.begin(),Pkey.end());
1097 A.insert(Rmap_value(Pkey,m[n*(r+1)-1]));
1099 // proceed from right to left through matrix
1100 for (int c=n-2; c>=0; --c) {
1101 Pkey.erase(Pkey.begin(),Pkey.end()); // don't change capacity
1102 Mkey.erase(Mkey.begin(),Mkey.end());
1103 for (unsigned i=0; i<n-c; ++i)
1105 unsigned fc = 0; // controls logic for our strange flipper counter
1108 for (unsigned r=0; r<n-c; ++r) {
1109 // maybe there is nothing to do?
1110 if (m[Pkey[r]*n+c].is_zero())
1112 // create the sorted key for all possible minors
1113 Mkey.erase(Mkey.begin(),Mkey.end());
1114 for (unsigned i=0; i<n-c; ++i)
1116 Mkey.push_back(Pkey[i]);
1117 // Fetch the minors and compute the new determinant
1119 det -= m[Pkey[r]*n+c]*A[Mkey];
1121 det += m[Pkey[r]*n+c]*A[Mkey];
1123 // prevent build-up of deep nesting of expressions saves time:
1125 // store the new determinant at its place in B:
1127 B.insert(Rmap_value(Pkey,det));
1128 // increment our strange flipper counter
1129 for (fc=n-c; fc>0; --fc) {
1131 if (Pkey[fc-1]<fc+c)
1135 for (unsigned j=fc; j<n-c; ++j)
1136 Pkey[j] = Pkey[j-1]+1;
1138 // next column, so change the role of A and B:
1147 /** Perform the steps of an ordinary Gaussian elimination to bring the m x n
1148 * matrix into an upper echelon form. The algorithm is ok for matrices
1149 * with numeric coefficients but quite unsuited for symbolic matrices.
1151 * @param det may be set to true to save a lot of space if one is only
1152 * interested in the diagonal elements (i.e. for calculating determinants).
1153 * The others are set to zero in this case.
1154 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1155 * number of rows was swapped and 0 if the matrix is singular. */
1156 int matrix::gauss_elimination(const bool det)
1158 ensure_if_modifiable();
1159 const unsigned m = this->rows();
1160 const unsigned n = this->cols();
1161 GINAC_ASSERT(!det || n==m);
1165 for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
1166 int indx = pivot(r0, r1, true);
1170 return 0; // leaves *this in a messy state
1175 for (unsigned r2=r0+1; r2<m; ++r2) {
1176 if (!this->m[r2*n+r1].is_zero()) {
1177 // yes, there is something to do in this row
1178 ex piv = this->m[r2*n+r1] / this->m[r0*n+r1];
1179 for (unsigned c=r1+1; c<n; ++c) {
1180 this->m[r2*n+c] -= piv * this->m[r0*n+c];
1181 if (!this->m[r2*n+c].info(info_flags::numeric))
1182 this->m[r2*n+c] = this->m[r2*n+c].normal();
1185 // fill up left hand side with zeros
1186 for (unsigned c=0; c<=r1; ++c)
1187 this->m[r2*n+c] = _ex0();
1190 // save space by deleting no longer needed elements
1191 for (unsigned c=r0+1; c<n; ++c)
1192 this->m[r0*n+c] = _ex0();
1202 /** Perform the steps of division free elimination to bring the m x n matrix
1203 * into an upper echelon form.
1205 * @param det may be set to true to save a lot of space if one is only
1206 * interested in the diagonal elements (i.e. for calculating determinants).
1207 * The others are set to zero in this case.
1208 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1209 * number of rows was swapped and 0 if the matrix is singular. */
1210 int matrix::division_free_elimination(const bool det)
1212 ensure_if_modifiable();
1213 const unsigned m = this->rows();
1214 const unsigned n = this->cols();
1215 GINAC_ASSERT(!det || n==m);
1219 for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
1220 int indx = pivot(r0, r1, true);
1224 return 0; // leaves *this in a messy state
1229 for (unsigned r2=r0+1; r2<m; ++r2) {
1230 for (unsigned c=r1+1; c<n; ++c)
1231 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();
1232 // fill up left hand side with zeros
1233 for (unsigned c=0; c<=r1; ++c)
1234 this->m[r2*n+c] = _ex0();
1237 // save space by deleting no longer needed elements
1238 for (unsigned c=r0+1; c<n; ++c)
1239 this->m[r0*n+c] = _ex0();
1249 /** Perform the steps of Bareiss' one-step fraction free elimination to bring
1250 * the matrix into an upper echelon form. Fraction free elimination means
1251 * that divide is used straightforwardly, without computing GCDs first. This
1252 * is possible, since we know the divisor at each step.
1254 * @param det may be set to true to save a lot of space if one is only
1255 * interested in the last element (i.e. for calculating determinants). The
1256 * others are set to zero in this case.
1257 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1258 * number of rows was swapped and 0 if the matrix is singular. */
1259 int matrix::fraction_free_elimination(const bool det)
1262 // (single-step fraction free elimination scheme, already known to Jordan)
1264 // Usual division-free elimination sets m[0](r,c) = m(r,c) and then sets
1265 // m[k+1](r,c) = m[k](k,k) * m[k](r,c) - m[k](r,k) * m[k](k,c).
1267 // Bareiss (fraction-free) elimination in addition divides that element
1268 // by m[k-1](k-1,k-1) for k>1, where it can be shown by means of the
1269 // Sylvester determinant that this really divides m[k+1](r,c).
1271 // We also allow rational functions where the original prove still holds.
1272 // However, we must care for numerator and denominator separately and
1273 // "manually" work in the integral domains because of subtle cancellations
1274 // (see below). This blows up the bookkeeping a bit and the formula has
1275 // to be modified to expand like this (N{x} stands for numerator of x,
1276 // D{x} for denominator of x):
1277 // 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)}
1278 // -N{m[k](r,k)}*N{m[k](k,c)}*D{m[k](k,k)}*D{m[k](r,c)}
1279 // 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)}
1280 // where for k>1 we now divide N{m[k+1](r,c)} by
1281 // N{m[k-1](k-1,k-1)}
1282 // and D{m[k+1](r,c)} by
1283 // D{m[k-1](k-1,k-1)}.
1285 ensure_if_modifiable();
1286 const unsigned m = this->rows();
1287 const unsigned n = this->cols();
1288 GINAC_ASSERT(!det || n==m);
1297 // We populate temporary matrices to subsequently operate on. There is
1298 // one holding numerators and another holding denominators of entries.
1299 // This is a must since the evaluator (or even earlier mul's constructor)
1300 // might cancel some trivial element which causes divide() to fail. The
1301 // elements are normalized first (yes, even though this algorithm doesn't
1302 // need GCDs) since the elements of *this might be unnormalized, which
1303 // makes things more complicated than they need to be.
1304 matrix tmp_n(*this);
1305 matrix tmp_d(m,n); // for denominators, if needed
1306 lst srl; // symbol replacement list
1307 exvector::const_iterator cit = this->m.begin(), citend = this->m.end();
1308 exvector::iterator tmp_n_it = tmp_n.m.begin(), tmp_d_it = tmp_d.m.begin();
1309 while (cit != citend) {
1310 ex nd = cit->normal().to_rational(srl).numer_denom();
1312 *tmp_n_it++ = nd.op(0);
1313 *tmp_d_it++ = nd.op(1);
1317 for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
1318 int indx = tmp_n.pivot(r0, r1, true);
1327 // tmp_n's rows r0 and indx were swapped, do the same in tmp_d:
1328 for (unsigned c=r1; c<n; ++c)
1329 tmp_d.m[n*indx+c].swap(tmp_d.m[n*r0+c]);
1331 for (unsigned r2=r0+1; r2<m; ++r2) {
1332 for (unsigned c=r1+1; c<n; ++c) {
1333 dividend_n = (tmp_n.m[r0*n+r1]*tmp_n.m[r2*n+c]*
1334 tmp_d.m[r2*n+r1]*tmp_d.m[r0*n+c]
1335 -tmp_n.m[r2*n+r1]*tmp_n.m[r0*n+c]*
1336 tmp_d.m[r0*n+r1]*tmp_d.m[r2*n+c]).expand();
1337 dividend_d = (tmp_d.m[r2*n+r1]*tmp_d.m[r0*n+c]*
1338 tmp_d.m[r0*n+r1]*tmp_d.m[r2*n+c]).expand();
1339 bool check = divide(dividend_n, divisor_n,
1340 tmp_n.m[r2*n+c], true);
1341 check &= divide(dividend_d, divisor_d,
1342 tmp_d.m[r2*n+c], true);
1343 GINAC_ASSERT(check);
1345 // fill up left hand side with zeros
1346 for (unsigned c=0; c<=r1; ++c)
1347 tmp_n.m[r2*n+c] = _ex0();
1349 if ((r1<n-1)&&(r0<m-1)) {
1350 // compute next iteration's divisor
1351 divisor_n = tmp_n.m[r0*n+r1].expand();
1352 divisor_d = tmp_d.m[r0*n+r1].expand();
1354 // save space by deleting no longer needed elements
1355 for (unsigned c=0; c<n; ++c) {
1356 tmp_n.m[r0*n+c] = _ex0();
1357 tmp_d.m[r0*n+c] = _ex1();
1364 // repopulate *this matrix:
1365 exvector::iterator it = this->m.begin(), itend = this->m.end();
1366 tmp_n_it = tmp_n.m.begin();
1367 tmp_d_it = tmp_d.m.begin();
1369 *it++ = ((*tmp_n_it++)/(*tmp_d_it++)).subs(srl);
1375 /** Partial pivoting method for matrix elimination schemes.
1376 * Usual pivoting (symbolic==false) returns the index to the element with the
1377 * largest absolute value in column ro and swaps the current row with the one
1378 * where the element was found. With (symbolic==true) it does the same thing
1379 * with the first non-zero element.
1381 * @param ro is the row from where to begin
1382 * @param co is the column to be inspected
1383 * @param symbolic signal if we want the first non-zero element to be pivoted
1384 * (true) or the one with the largest absolute value (false).
1385 * @return 0 if no interchange occured, -1 if all are zero (usually signaling
1386 * a degeneracy) and positive integer k means that rows ro and k were swapped.
1388 int matrix::pivot(unsigned ro, unsigned co, bool symbolic)
1392 // search first non-zero element in column co beginning at row ro
1393 while ((k<row) && (this->m[k*col+co].expand().is_zero()))
1396 // search largest element in column co beginning at row ro
1397 GINAC_ASSERT(is_ex_of_type(this->m[k*col+co],numeric));
1398 unsigned kmax = k+1;
1399 numeric mmax = abs(ex_to<numeric>(m[kmax*col+co]));
1401 GINAC_ASSERT(is_ex_of_type(this->m[kmax*col+co],numeric));
1402 numeric tmp = ex_to<numeric>(this->m[kmax*col+co]);
1403 if (abs(tmp) > mmax) {
1409 if (!mmax.is_zero())
1413 // all elements in column co below row ro vanish
1416 // matrix needs no pivoting
1418 // matrix needs pivoting, so swap rows k and ro
1419 ensure_if_modifiable();
1420 for (unsigned c=0; c<col; ++c)
1421 this->m[k*col+c].swap(this->m[ro*col+c]);
1426 ex lst_to_matrix(const lst & l)
1428 // Find number of rows and columns
1429 unsigned rows = l.nops(), cols = 0, i, j;
1430 for (i=0; i<rows; i++)
1431 if (l.op(i).nops() > cols)
1432 cols = l.op(i).nops();
1434 // Allocate and fill matrix
1435 matrix &m = *new matrix(rows, cols);
1436 m.setflag(status_flags::dynallocated);
1437 for (i=0; i<rows; i++)
1438 for (j=0; j<cols; j++)
1439 if (l.op(i).nops() > j)
1440 m(i, j) = l.op(i).op(j);
1446 ex diag_matrix(const lst & l)
1448 unsigned dim = l.nops();
1450 matrix &m = *new matrix(dim, dim);
1451 m.setflag(status_flags::dynallocated);
1452 for (unsigned i=0; i<dim; i++)
1458 } // namespace GiNaC