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)
54 void matrix::copy(const matrix & other)
56 inherited::copy(other);
59 m = other.m; // STL's vector copying invoked here
62 DEFAULT_DESTROY(matrix)
70 /** Very common ctor. Initializes to r x c-dimensional zero-matrix.
72 * @param r number of rows
73 * @param c number of cols */
74 matrix::matrix(unsigned r, unsigned c)
75 : inherited(TINFO_matrix), row(r), col(c)
82 /** Ctor from representation, for internal use only. */
83 matrix::matrix(unsigned r, unsigned c, const exvector & m2)
84 : inherited(TINFO_matrix), row(r), col(c), m(m2) {}
86 /** Construct matrix from (flat) list of elements. If the list has fewer
87 * elements than the matrix, the remaining matrix elements are set to zero.
88 * If the list has more elements than the matrix, the excessive elements are
90 matrix::matrix(unsigned r, unsigned c, const lst & l)
91 : inherited(TINFO_matrix), row(r), col(c)
95 for (unsigned i=0; i<l.nops(); i++) {
99 break; // matrix smaller than list: throw away excessive elements
108 matrix::matrix(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
110 if (!(n.find_unsigned("row", row)) || !(n.find_unsigned("col", col)))
111 throw (std::runtime_error("unknown matrix dimensions in archive"));
112 m.reserve(row * col);
113 for (unsigned int i=0; true; i++) {
115 if (n.find_ex("m", e, sym_lst, i))
122 void matrix::archive(archive_node &n) const
124 inherited::archive(n);
125 n.add_unsigned("row", row);
126 n.add_unsigned("col", col);
127 exvector::const_iterator i = m.begin(), iend = m.end();
134 DEFAULT_UNARCHIVE(matrix)
137 // functions overriding virtual functions from base classes
142 void matrix::print(const print_context & c, unsigned level) const
144 if (is_a<print_tree>(c)) {
146 inherited::print(c, level);
151 for (unsigned y=0; y<row-1; ++y) {
153 for (unsigned x=0; x<col-1; ++x) {
157 m[col*(y+1)-1].print(c);
161 for (unsigned x=0; x<col-1; ++x) {
162 m[(row-1)*col+x].print(c);
165 m[row*col-1].print(c);
171 /** nops is defined to be rows x columns. */
172 unsigned matrix::nops() const
177 /** returns matrix entry at position (i/col, i%col). */
178 ex matrix::op(int i) const
183 /** returns matrix entry at position (i/col, i%col). */
184 ex & matrix::let_op(int i)
187 GINAC_ASSERT(i<nops());
192 /** Evaluate matrix entry by entry. */
193 ex matrix::eval(int level) const
195 // check if we have to do anything at all
196 if ((level==1)&&(flags & status_flags::evaluated))
200 if (level == -max_recursion_level)
201 throw (std::runtime_error("matrix::eval(): recursion limit exceeded"));
203 // eval() entry by entry
204 exvector m2(row*col);
206 for (unsigned r=0; r<row; ++r)
207 for (unsigned c=0; c<col; ++c)
208 m2[r*col+c] = m[r*col+c].eval(level);
210 return (new matrix(row, col, m2))->setflag(status_flags::dynallocated |
211 status_flags::evaluated );
214 ex matrix::subs(const lst & ls, const lst & lr, bool no_pattern) const
216 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].subs(ls, lr, no_pattern);
221 return matrix(row, col, m2).basic::subs(ls, lr, no_pattern);
226 int matrix::compare_same_type(const basic & other) const
228 GINAC_ASSERT(is_exactly_a<matrix>(other));
229 const matrix &o = static_cast<const matrix &>(other);
231 // compare number of rows
233 return row < o.rows() ? -1 : 1;
235 // compare number of columns
237 return col < o.cols() ? -1 : 1;
239 // equal number of rows and columns, compare individual elements
241 for (unsigned r=0; r<row; ++r) {
242 for (unsigned c=0; c<col; ++c) {
243 cmpval = ((*this)(r,c)).compare(o(r,c));
244 if (cmpval!=0) return cmpval;
247 // all elements are equal => matrices are equal;
251 bool matrix::match_same_type(const basic & other) const
253 GINAC_ASSERT(is_exactly_a<matrix>(other));
254 const matrix & o = static_cast<const matrix &>(other);
256 // The number of rows and columns must be the same. This is necessary to
257 // prevent a 2x3 matrix from matching a 3x2 one.
258 return row == o.rows() && col == o.cols();
261 /** Automatic symbolic evaluation of an indexed matrix. */
262 ex matrix::eval_indexed(const basic & i) const
264 GINAC_ASSERT(is_a<indexed>(i));
265 GINAC_ASSERT(is_a<matrix>(i.op(0)));
267 bool all_indices_unsigned = static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint);
272 // One index, must be one-dimensional vector
273 if (row != 1 && col != 1)
274 throw (std::runtime_error("matrix::eval_indexed(): vector must have exactly 1 index"));
276 const idx & i1 = ex_to<idx>(i.op(1));
281 if (!i1.get_dim().is_equal(row))
282 throw (std::runtime_error("matrix::eval_indexed(): dimension of index must match number of vector elements"));
284 // Index numeric -> return vector element
285 if (all_indices_unsigned) {
286 unsigned n1 = ex_to<numeric>(i1.get_value()).to_int();
288 throw (std::runtime_error("matrix::eval_indexed(): value of index exceeds number of vector elements"));
289 return (*this)(n1, 0);
295 if (!i1.get_dim().is_equal(col))
296 throw (std::runtime_error("matrix::eval_indexed(): dimension of index must match number of vector elements"));
298 // Index numeric -> return vector element
299 if (all_indices_unsigned) {
300 unsigned n1 = ex_to<numeric>(i1.get_value()).to_int();
302 throw (std::runtime_error("matrix::eval_indexed(): value of index exceeds number of vector elements"));
303 return (*this)(0, n1);
307 } else if (i.nops() == 3) {
310 const idx & i1 = ex_to<idx>(i.op(1));
311 const idx & i2 = ex_to<idx>(i.op(2));
313 if (!i1.get_dim().is_equal(row))
314 throw (std::runtime_error("matrix::eval_indexed(): dimension of first index must match number of rows"));
315 if (!i2.get_dim().is_equal(col))
316 throw (std::runtime_error("matrix::eval_indexed(): dimension of second index must match number of columns"));
318 // Pair of dummy indices -> compute trace
319 if (is_dummy_pair(i1, i2))
322 // Both indices numeric -> return matrix element
323 if (all_indices_unsigned) {
324 unsigned n1 = ex_to<numeric>(i1.get_value()).to_int(), n2 = ex_to<numeric>(i2.get_value()).to_int();
326 throw (std::runtime_error("matrix::eval_indexed(): value of first index exceeds number of rows"));
328 throw (std::runtime_error("matrix::eval_indexed(): value of second index exceeds number of columns"));
329 return (*this)(n1, n2);
333 throw (std::runtime_error("matrix::eval_indexed(): matrix must have exactly 2 indices"));
338 /** Sum of two indexed matrices. */
339 ex matrix::add_indexed(const ex & self, const ex & other) const
341 GINAC_ASSERT(is_a<indexed>(self));
342 GINAC_ASSERT(is_a<matrix>(self.op(0)));
343 GINAC_ASSERT(is_a<indexed>(other));
344 GINAC_ASSERT(self.nops() == 2 || self.nops() == 3);
346 // Only add two matrices
347 if (is_ex_of_type(other.op(0), matrix)) {
348 GINAC_ASSERT(other.nops() == 2 || other.nops() == 3);
350 const matrix &self_matrix = ex_to<matrix>(self.op(0));
351 const matrix &other_matrix = ex_to<matrix>(other.op(0));
353 if (self.nops() == 2 && other.nops() == 2) { // vector + vector
355 if (self_matrix.row == other_matrix.row)
356 return indexed(self_matrix.add(other_matrix), self.op(1));
357 else if (self_matrix.row == other_matrix.col)
358 return indexed(self_matrix.add(other_matrix.transpose()), self.op(1));
360 } else if (self.nops() == 3 && other.nops() == 3) { // matrix + matrix
362 if (self.op(1).is_equal(other.op(1)) && self.op(2).is_equal(other.op(2)))
363 return indexed(self_matrix.add(other_matrix), self.op(1), self.op(2));
364 else if (self.op(1).is_equal(other.op(2)) && self.op(2).is_equal(other.op(1)))
365 return indexed(self_matrix.add(other_matrix.transpose()), self.op(1), self.op(2));
370 // Don't know what to do, return unevaluated sum
374 /** Product of an indexed matrix with a number. */
375 ex matrix::scalar_mul_indexed(const ex & self, const numeric & other) const
377 GINAC_ASSERT(is_a<indexed>(self));
378 GINAC_ASSERT(is_a<matrix>(self.op(0)));
379 GINAC_ASSERT(self.nops() == 2 || self.nops() == 3);
381 const matrix &self_matrix = ex_to<matrix>(self.op(0));
383 if (self.nops() == 2)
384 return indexed(self_matrix.mul(other), self.op(1));
385 else // self.nops() == 3
386 return indexed(self_matrix.mul(other), self.op(1), self.op(2));
389 /** Contraction of an indexed matrix with something else. */
390 bool matrix::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
392 GINAC_ASSERT(is_a<indexed>(*self));
393 GINAC_ASSERT(is_a<indexed>(*other));
394 GINAC_ASSERT(self->nops() == 2 || self->nops() == 3);
395 GINAC_ASSERT(is_a<matrix>(self->op(0)));
397 // Only contract with other matrices
398 if (!is_ex_of_type(other->op(0), matrix))
401 GINAC_ASSERT(other->nops() == 2 || other->nops() == 3);
403 const matrix &self_matrix = ex_to<matrix>(self->op(0));
404 const matrix &other_matrix = ex_to<matrix>(other->op(0));
406 if (self->nops() == 2) {
408 if (other->nops() == 2) { // vector * vector (scalar product)
410 if (self_matrix.col == 1) {
411 if (other_matrix.col == 1) {
412 // Column vector * column vector, transpose first vector
413 *self = self_matrix.transpose().mul(other_matrix)(0, 0);
415 // Column vector * row vector, swap factors
416 *self = other_matrix.mul(self_matrix)(0, 0);
419 if (other_matrix.col == 1) {
420 // Row vector * column vector, perfect
421 *self = self_matrix.mul(other_matrix)(0, 0);
423 // Row vector * row vector, transpose second vector
424 *self = self_matrix.mul(other_matrix.transpose())(0, 0);
430 } else { // vector * matrix
432 // B_i * A_ij = (B*A)_j (B is row vector)
433 if (is_dummy_pair(self->op(1), other->op(1))) {
434 if (self_matrix.row == 1)
435 *self = indexed(self_matrix.mul(other_matrix), other->op(2));
437 *self = indexed(self_matrix.transpose().mul(other_matrix), other->op(2));
442 // B_j * A_ij = (A*B)_i (B is column vector)
443 if (is_dummy_pair(self->op(1), other->op(2))) {
444 if (self_matrix.col == 1)
445 *self = indexed(other_matrix.mul(self_matrix), other->op(1));
447 *self = indexed(other_matrix.mul(self_matrix.transpose()), other->op(1));
453 } else if (other->nops() == 3) { // matrix * matrix
455 // A_ij * B_jk = (A*B)_ik
456 if (is_dummy_pair(self->op(2), other->op(1))) {
457 *self = indexed(self_matrix.mul(other_matrix), self->op(1), other->op(2));
462 // A_ij * B_kj = (A*Btrans)_ik
463 if (is_dummy_pair(self->op(2), other->op(2))) {
464 *self = indexed(self_matrix.mul(other_matrix.transpose()), self->op(1), other->op(1));
469 // A_ji * B_jk = (Atrans*B)_ik
470 if (is_dummy_pair(self->op(1), other->op(1))) {
471 *self = indexed(self_matrix.transpose().mul(other_matrix), self->op(2), other->op(2));
476 // A_ji * B_kj = (B*A)_ki
477 if (is_dummy_pair(self->op(1), other->op(2))) {
478 *self = indexed(other_matrix.mul(self_matrix), other->op(1), self->op(2));
489 // non-virtual functions in this class
496 * @exception logic_error (incompatible matrices) */
497 matrix matrix::add(const matrix & other) const
499 if (col != other.col || row != other.row)
500 throw std::logic_error("matrix::add(): incompatible matrices");
502 exvector sum(this->m);
503 exvector::iterator i = sum.begin(), end = sum.end();
504 exvector::const_iterator ci = other.m.begin();
508 return matrix(row,col,sum);
512 /** Difference of matrices.
514 * @exception logic_error (incompatible matrices) */
515 matrix matrix::sub(const matrix & other) const
517 if (col != other.col || row != other.row)
518 throw std::logic_error("matrix::sub(): incompatible matrices");
520 exvector dif(this->m);
521 exvector::iterator i = dif.begin(), end = dif.end();
522 exvector::const_iterator ci = other.m.begin();
526 return matrix(row,col,dif);
530 /** Product of matrices.
532 * @exception logic_error (incompatible matrices) */
533 matrix matrix::mul(const matrix & other) const
535 if (this->cols() != other.rows())
536 throw std::logic_error("matrix::mul(): incompatible matrices");
538 exvector prod(this->rows()*other.cols());
540 for (unsigned r1=0; r1<this->rows(); ++r1) {
541 for (unsigned c=0; c<this->cols(); ++c) {
542 if (m[r1*col+c].is_zero())
544 for (unsigned r2=0; r2<other.cols(); ++r2)
545 prod[r1*other.col+r2] += (m[r1*col+c] * other.m[c*other.col+r2]).expand();
548 return matrix(row, other.col, prod);
552 /** Product of matrix and scalar. */
553 matrix matrix::mul(const numeric & other) const
555 exvector prod(row * col);
557 for (unsigned r=0; r<row; ++r)
558 for (unsigned c=0; c<col; ++c)
559 prod[r*col+c] = m[r*col+c] * other;
561 return matrix(row, col, prod);
565 /** Product of matrix and scalar expression. */
566 matrix matrix::mul_scalar(const ex & other) const
568 if (other.return_type() != return_types::commutative)
569 throw std::runtime_error("matrix::mul_scalar(): non-commutative scalar");
571 exvector prod(row * col);
573 for (unsigned r=0; r<row; ++r)
574 for (unsigned c=0; c<col; ++c)
575 prod[r*col+c] = m[r*col+c] * other;
577 return matrix(row, col, prod);
581 /** Power of a matrix. Currently handles integer exponents only. */
582 matrix matrix::pow(const ex & expn) const
585 throw (std::logic_error("matrix::pow(): matrix not square"));
587 if (is_ex_exactly_of_type(expn, numeric)) {
588 // Integer cases are computed by successive multiplication, using the
589 // obvious shortcut of storing temporaries, like A^4 == (A*A)*(A*A).
590 if (expn.info(info_flags::integer)) {
591 numeric b = ex_to<numeric>(expn);
593 if (expn.info(info_flags::negative)) {
600 for (unsigned r=0; r<row; ++r)
602 // This loop computes the representation of b in base 2 from right
603 // to left and multiplies the factors whenever needed. Note
604 // that this is not entirely optimal but close to optimal and
605 // "better" algorithms are much harder to implement. (See Knuth,
606 // TAoCP2, section "Evaluation of Powers" for a good discussion.)
612 b *= _num1_2; // b /= 2, still integer.
618 throw (std::runtime_error("matrix::pow(): don't know how to handle exponent"));
622 /** operator() to access elements for reading.
624 * @param ro row of element
625 * @param co column of element
626 * @exception range_error (index out of range) */
627 const ex & matrix::operator() (unsigned ro, unsigned co) const
629 if (ro>=row || co>=col)
630 throw (std::range_error("matrix::operator(): index out of range"));
636 /** operator() to access elements for writing.
638 * @param ro row of element
639 * @param co column of element
640 * @exception range_error (index out of range) */
641 ex & matrix::operator() (unsigned ro, unsigned co)
643 if (ro>=row || co>=col)
644 throw (std::range_error("matrix::operator(): index out of range"));
646 ensure_if_modifiable();
651 /** Transposed of an m x n matrix, producing a new n x m matrix object that
652 * represents the transposed. */
653 matrix matrix::transpose(void) const
655 exvector trans(this->cols()*this->rows());
657 for (unsigned r=0; r<this->cols(); ++r)
658 for (unsigned c=0; c<this->rows(); ++c)
659 trans[r*this->rows()+c] = m[c*this->cols()+r];
661 return matrix(this->cols(),this->rows(),trans);
664 /** Determinant of square matrix. This routine doesn't actually calculate the
665 * determinant, it only implements some heuristics about which algorithm to
666 * run. If all the elements of the matrix are elements of an integral domain
667 * the determinant is also in that integral domain and the result is expanded
668 * only. If one or more elements are from a quotient field the determinant is
669 * usually also in that quotient field and the result is normalized before it
670 * is returned. This implies that the determinant of the symbolic 2x2 matrix
671 * [[a/(a-b),1],[b/(a-b),1]] is returned as unity. (In this respect, it
672 * behaves like MapleV and unlike Mathematica.)
674 * @param algo allows to chose an algorithm
675 * @return the determinant as a new expression
676 * @exception logic_error (matrix not square)
677 * @see determinant_algo */
678 ex matrix::determinant(unsigned algo) const
681 throw (std::logic_error("matrix::determinant(): matrix not square"));
682 GINAC_ASSERT(row*col==m.capacity());
684 // Gather some statistical information about this matrix:
685 bool numeric_flag = true;
686 bool normal_flag = false;
687 unsigned sparse_count = 0; // counts non-zero elements
688 exvector::const_iterator r = m.begin(), rend = m.end();
690 lst srl; // symbol replacement list
691 ex rtest = r->to_rational(srl);
692 if (!rtest.is_zero())
694 if (!rtest.info(info_flags::numeric))
695 numeric_flag = false;
696 if (!rtest.info(info_flags::crational_polynomial) &&
697 rtest.info(info_flags::rational_function))
702 // Here is the heuristics in case this routine has to decide:
703 if (algo == determinant_algo::automatic) {
704 // Minor expansion is generally a good guess:
705 algo = determinant_algo::laplace;
706 // Does anybody know when a matrix is really sparse?
707 // Maybe <~row/2.236 nonzero elements average in a row?
708 if (row>3 && 5*sparse_count<=row*col)
709 algo = determinant_algo::bareiss;
710 // Purely numeric matrix can be handled by Gauss elimination.
711 // This overrides any prior decisions.
713 algo = determinant_algo::gauss;
716 // Trap the trivial case here, since some algorithms don't like it
718 // for consistency with non-trivial determinants...
720 return m[0].normal();
722 return m[0].expand();
725 // Compute the determinant
727 case determinant_algo::gauss: {
730 int sign = tmp.gauss_elimination(true);
731 for (unsigned d=0; d<row; ++d)
732 det *= tmp.m[d*col+d];
734 return (sign*det).normal();
736 return (sign*det).normal().expand();
738 case determinant_algo::bareiss: {
741 sign = tmp.fraction_free_elimination(true);
743 return (sign*tmp.m[row*col-1]).normal();
745 return (sign*tmp.m[row*col-1]).expand();
747 case determinant_algo::divfree: {
750 sign = tmp.division_free_elimination(true);
753 ex det = tmp.m[row*col-1];
754 // factor out accumulated bogus slag
755 for (unsigned d=0; d<row-2; ++d)
756 for (unsigned j=0; j<row-d-2; ++j)
757 det = (det/tmp.m[d*col+d]).normal();
760 case determinant_algo::laplace:
762 // This is the minor expansion scheme. We always develop such
763 // that the smallest minors (i.e, the trivial 1x1 ones) are on the
764 // rightmost column. For this to be efficient, empirical tests
765 // have shown that the emptiest columns (i.e. the ones with most
766 // zeros) should be the ones on the right hand side -- although
767 // this might seem counter-intuitive (and in contradiction to some
768 // literature like the FORM manual). Please go ahead and test it
769 // if you don't believe me! Therefore we presort the columns of
771 typedef std::pair<unsigned,unsigned> uintpair;
772 std::vector<uintpair> c_zeros; // number of zeros in column
773 for (unsigned c=0; c<col; ++c) {
775 for (unsigned r=0; r<row; ++r)
776 if (m[r*col+c].is_zero())
778 c_zeros.push_back(uintpair(acc,c));
780 std::sort(c_zeros.begin(),c_zeros.end());
781 std::vector<unsigned> pre_sort;
782 for (std::vector<uintpair>::const_iterator i=c_zeros.begin(); i!=c_zeros.end(); ++i)
783 pre_sort.push_back(i->second);
784 std::vector<unsigned> pre_sort_test(pre_sort); // permutation_sign() modifies the vector so we make a copy here
785 int sign = permutation_sign(pre_sort_test.begin(), pre_sort_test.end());
786 exvector result(row*col); // represents sorted matrix
788 for (std::vector<unsigned>::const_iterator i=pre_sort.begin();
791 for (unsigned r=0; r<row; ++r)
792 result[r*col+c] = m[r*col+(*i)];
796 return (sign*matrix(row,col,result).determinant_minor()).normal();
798 return sign*matrix(row,col,result).determinant_minor();
804 /** Trace of a matrix. The result is normalized if it is in some quotient
805 * field and expanded only otherwise. This implies that the trace of the
806 * symbolic 2x2 matrix [[a/(a-b),x],[y,b/(b-a)]] is recognized to be unity.
808 * @return the sum of diagonal elements
809 * @exception logic_error (matrix not square) */
810 ex matrix::trace(void) const
813 throw (std::logic_error("matrix::trace(): matrix not square"));
816 for (unsigned r=0; r<col; ++r)
819 if (tr.info(info_flags::rational_function) &&
820 !tr.info(info_flags::crational_polynomial))
827 /** Characteristic Polynomial. Following mathematica notation the
828 * characteristic polynomial of a matrix M is defined as the determiant of
829 * (M - lambda * 1) where 1 stands for the unit matrix of the same dimension
830 * as M. Note that some CASs define it with a sign inside the determinant
831 * which gives rise to an overall sign if the dimension is odd. This method
832 * returns the characteristic polynomial collected in powers of lambda as a
835 * @return characteristic polynomial as new expression
836 * @exception logic_error (matrix not square)
837 * @see matrix::determinant() */
838 ex matrix::charpoly(const symbol & lambda) const
841 throw (std::logic_error("matrix::charpoly(): matrix not square"));
843 bool numeric_flag = true;
844 exvector::const_iterator r = m.begin(), rend = m.end();
845 while (r!=rend && numeric_flag==true) {
846 if (!r->info(info_flags::numeric))
847 numeric_flag = false;
851 // The pure numeric case is traditionally rather common. Hence, it is
852 // trapped and we use Leverrier's algorithm which goes as row^3 for
853 // every coefficient. The expensive part is the matrix multiplication.
857 ex poly = power(lambda,row)-c*power(lambda,row-1);
858 for (unsigned i=1; i<row; ++i) {
859 for (unsigned j=0; j<row; ++j)
862 c = B.trace()/ex(i+1);
863 poly -= c*power(lambda,row-i-1);
872 for (unsigned r=0; r<col; ++r)
873 M.m[r*col+r] -= lambda;
875 return M.determinant().collect(lambda);
879 /** Inverse of this matrix.
881 * @return the inverted matrix
882 * @exception logic_error (matrix not square)
883 * @exception runtime_error (singular matrix) */
884 matrix matrix::inverse(void) const
887 throw (std::logic_error("matrix::inverse(): matrix not square"));
889 // This routine actually doesn't do anything fancy at all. We compute the
890 // inverse of the matrix A by solving the system A * A^{-1} == Id.
892 // First populate the identity matrix supposed to become the right hand side.
893 matrix identity(row,col);
894 for (unsigned i=0; i<row; ++i)
895 identity(i,i) = _ex1;
897 // Populate a dummy matrix of variables, just because of compatibility with
898 // matrix::solve() which wants this (for compatibility with under-determined
899 // systems of equations).
900 matrix vars(row,col);
901 for (unsigned r=0; r<row; ++r)
902 for (unsigned c=0; c<col; ++c)
903 vars(r,c) = symbol();
907 sol = this->solve(vars,identity);
908 } catch (const std::runtime_error & e) {
909 if (e.what()==std::string("matrix::solve(): inconsistent linear system"))
910 throw (std::runtime_error("matrix::inverse(): singular matrix"));
918 /** Solve a linear system consisting of a m x n matrix and a m x p right hand
919 * side by applying an elimination scheme to the augmented matrix.
921 * @param vars n x p matrix, all elements must be symbols
922 * @param rhs m x p matrix
923 * @return n x p solution matrix
924 * @exception logic_error (incompatible matrices)
925 * @exception invalid_argument (1st argument must be matrix of symbols)
926 * @exception runtime_error (inconsistent linear system)
928 matrix matrix::solve(const matrix & vars,
932 const unsigned m = this->rows();
933 const unsigned n = this->cols();
934 const unsigned p = rhs.cols();
937 if ((rhs.rows() != m) || (vars.rows() != n) || (vars.col != p))
938 throw (std::logic_error("matrix::solve(): incompatible matrices"));
939 for (unsigned ro=0; ro<n; ++ro)
940 for (unsigned co=0; co<p; ++co)
941 if (!vars(ro,co).info(info_flags::symbol))
942 throw (std::invalid_argument("matrix::solve(): 1st argument must be matrix of symbols"));
944 // build the augmented matrix of *this with rhs attached to the right
946 for (unsigned r=0; r<m; ++r) {
947 for (unsigned c=0; c<n; ++c)
948 aug.m[r*(n+p)+c] = this->m[r*n+c];
949 for (unsigned c=0; c<p; ++c)
950 aug.m[r*(n+p)+c+n] = rhs.m[r*p+c];
953 // Gather some statistical information about the augmented matrix:
954 bool numeric_flag = true;
955 exvector::const_iterator r = aug.m.begin(), rend = aug.m.end();
956 while (r!=rend && numeric_flag==true) {
957 if (!r->info(info_flags::numeric))
958 numeric_flag = false;
962 // Here is the heuristics in case this routine has to decide:
963 if (algo == solve_algo::automatic) {
964 // Bareiss (fraction-free) elimination is generally a good guess:
965 algo = solve_algo::bareiss;
966 // For m<3, Bareiss elimination is equivalent to division free
967 // elimination but has more logistic overhead
969 algo = solve_algo::divfree;
970 // This overrides any prior decisions.
972 algo = solve_algo::gauss;
975 // Eliminate the augmented matrix:
977 case solve_algo::gauss:
978 aug.gauss_elimination();
980 case solve_algo::divfree:
981 aug.division_free_elimination();
983 case solve_algo::bareiss:
985 aug.fraction_free_elimination();
988 // assemble the solution matrix:
990 for (unsigned co=0; co<p; ++co) {
991 unsigned last_assigned_sol = n+1;
992 for (int r=m-1; r>=0; --r) {
993 unsigned fnz = 1; // first non-zero in row
994 while ((fnz<=n) && (aug.m[r*(n+p)+(fnz-1)].is_zero()))
997 // row consists only of zeros, corresponding rhs must be 0, too
998 if (!aug.m[r*(n+p)+n+co].is_zero()) {
999 throw (std::runtime_error("matrix::solve(): inconsistent linear system"));
1002 // assign solutions for vars between fnz+1 and
1003 // last_assigned_sol-1: free parameters
1004 for (unsigned c=fnz; c<last_assigned_sol-1; ++c)
1005 sol(c,co) = vars.m[c*p+co];
1006 ex e = aug.m[r*(n+p)+n+co];
1007 for (unsigned c=fnz; c<n; ++c)
1008 e -= aug.m[r*(n+p)+c]*sol.m[c*p+co];
1009 sol(fnz-1,co) = (e/(aug.m[r*(n+p)+(fnz-1)])).normal();
1010 last_assigned_sol = fnz;
1013 // assign solutions for vars between 1 and
1014 // last_assigned_sol-1: free parameters
1015 for (unsigned ro=0; ro<last_assigned_sol-1; ++ro)
1016 sol(ro,co) = vars(ro,co);
1025 /** Recursive determinant for small matrices having at least one symbolic
1026 * entry. The basic algorithm, known as Laplace-expansion, is enhanced by
1027 * some bookkeeping to avoid calculation of the same submatrices ("minors")
1028 * more than once. According to W.M.Gentleman and S.C.Johnson this algorithm
1029 * is better than elimination schemes for matrices of sparse multivariate
1030 * polynomials and also for matrices of dense univariate polynomials if the
1031 * matrix' dimesion is larger than 7.
1033 * @return the determinant as a new expression (in expanded form)
1034 * @see matrix::determinant() */
1035 ex matrix::determinant_minor(void) const
1037 // for small matrices the algorithm does not make any sense:
1038 const unsigned n = this->cols();
1040 return m[0].expand();
1042 return (m[0]*m[3]-m[2]*m[1]).expand();
1044 return (m[0]*m[4]*m[8]-m[0]*m[5]*m[7]-
1045 m[1]*m[3]*m[8]+m[2]*m[3]*m[7]+
1046 m[1]*m[5]*m[6]-m[2]*m[4]*m[6]).expand();
1048 // This algorithm can best be understood by looking at a naive
1049 // implementation of Laplace-expansion, like this one:
1051 // matrix minorM(this->rows()-1,this->cols()-1);
1052 // for (unsigned r1=0; r1<this->rows(); ++r1) {
1053 // // shortcut if element(r1,0) vanishes
1054 // if (m[r1*col].is_zero())
1056 // // assemble the minor matrix
1057 // for (unsigned r=0; r<minorM.rows(); ++r) {
1058 // for (unsigned c=0; c<minorM.cols(); ++c) {
1060 // minorM(r,c) = m[r*col+c+1];
1062 // minorM(r,c) = m[(r+1)*col+c+1];
1065 // // recurse down and care for sign:
1067 // det -= m[r1*col] * minorM.determinant_minor();
1069 // det += m[r1*col] * minorM.determinant_minor();
1071 // return det.expand();
1072 // What happens is that while proceeding down many of the minors are
1073 // computed more than once. In particular, there are binomial(n,k)
1074 // kxk minors and each one is computed factorial(n-k) times. Therefore
1075 // it is reasonable to store the results of the minors. We proceed from
1076 // right to left. At each column c we only need to retrieve the minors
1077 // calculated in step c-1. We therefore only have to store at most
1078 // 2*binomial(n,n/2) minors.
1080 // Unique flipper counter for partitioning into minors
1081 std::vector<unsigned> Pkey;
1083 // key for minor determinant (a subpartition of Pkey)
1084 std::vector<unsigned> Mkey;
1086 // we store our subminors in maps, keys being the rows they arise from
1087 typedef std::map<std::vector<unsigned>,class ex> Rmap;
1088 typedef std::map<std::vector<unsigned>,class ex>::value_type Rmap_value;
1092 // initialize A with last column:
1093 for (unsigned r=0; r<n; ++r) {
1094 Pkey.erase(Pkey.begin(),Pkey.end());
1096 A.insert(Rmap_value(Pkey,m[n*(r+1)-1]));
1098 // proceed from right to left through matrix
1099 for (int c=n-2; c>=0; --c) {
1100 Pkey.erase(Pkey.begin(),Pkey.end()); // don't change capacity
1101 Mkey.erase(Mkey.begin(),Mkey.end());
1102 for (unsigned i=0; i<n-c; ++i)
1104 unsigned fc = 0; // controls logic for our strange flipper counter
1107 for (unsigned r=0; r<n-c; ++r) {
1108 // maybe there is nothing to do?
1109 if (m[Pkey[r]*n+c].is_zero())
1111 // create the sorted key for all possible minors
1112 Mkey.erase(Mkey.begin(),Mkey.end());
1113 for (unsigned i=0; i<n-c; ++i)
1115 Mkey.push_back(Pkey[i]);
1116 // Fetch the minors and compute the new determinant
1118 det -= m[Pkey[r]*n+c]*A[Mkey];
1120 det += m[Pkey[r]*n+c]*A[Mkey];
1122 // prevent build-up of deep nesting of expressions saves time:
1124 // store the new determinant at its place in B:
1126 B.insert(Rmap_value(Pkey,det));
1127 // increment our strange flipper counter
1128 for (fc=n-c; fc>0; --fc) {
1130 if (Pkey[fc-1]<fc+c)
1134 for (unsigned j=fc; j<n-c; ++j)
1135 Pkey[j] = Pkey[j-1]+1;
1137 // next column, so change the role of A and B:
1146 /** Perform the steps of an ordinary Gaussian elimination to bring the m x n
1147 * matrix into an upper echelon form. The algorithm is ok for matrices
1148 * with numeric coefficients but quite unsuited for symbolic matrices.
1150 * @param det may be set to true to save a lot of space if one is only
1151 * interested in the diagonal elements (i.e. for calculating determinants).
1152 * The others are set to zero in this case.
1153 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1154 * number of rows was swapped and 0 if the matrix is singular. */
1155 int matrix::gauss_elimination(const bool det)
1157 ensure_if_modifiable();
1158 const unsigned m = this->rows();
1159 const unsigned n = this->cols();
1160 GINAC_ASSERT(!det || n==m);
1164 for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
1165 int indx = pivot(r0, r1, true);
1169 return 0; // leaves *this in a messy state
1174 for (unsigned r2=r0+1; r2<m; ++r2) {
1175 if (!this->m[r2*n+r1].is_zero()) {
1176 // yes, there is something to do in this row
1177 ex piv = this->m[r2*n+r1] / this->m[r0*n+r1];
1178 for (unsigned c=r1+1; c<n; ++c) {
1179 this->m[r2*n+c] -= piv * this->m[r0*n+c];
1180 if (!this->m[r2*n+c].info(info_flags::numeric))
1181 this->m[r2*n+c] = this->m[r2*n+c].normal();
1184 // fill up left hand side with zeros
1185 for (unsigned c=0; c<=r1; ++c)
1186 this->m[r2*n+c] = _ex0;
1189 // save space by deleting no longer needed elements
1190 for (unsigned c=r0+1; c<n; ++c)
1191 this->m[r0*n+c] = _ex0;
1201 /** Perform the steps of division free elimination to bring the m x n matrix
1202 * into an upper echelon form.
1204 * @param det may be set to true to save a lot of space if one is only
1205 * interested in the diagonal elements (i.e. for calculating determinants).
1206 * The others are set to zero in this case.
1207 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1208 * number of rows was swapped and 0 if the matrix is singular. */
1209 int matrix::division_free_elimination(const bool det)
1211 ensure_if_modifiable();
1212 const unsigned m = this->rows();
1213 const unsigned n = this->cols();
1214 GINAC_ASSERT(!det || n==m);
1218 for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
1219 int indx = pivot(r0, r1, true);
1223 return 0; // leaves *this in a messy state
1228 for (unsigned r2=r0+1; r2<m; ++r2) {
1229 for (unsigned c=r1+1; c<n; ++c)
1230 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();
1231 // fill up left hand side with zeros
1232 for (unsigned c=0; c<=r1; ++c)
1233 this->m[r2*n+c] = _ex0;
1236 // save space by deleting no longer needed elements
1237 for (unsigned c=r0+1; c<n; ++c)
1238 this->m[r0*n+c] = _ex0;
1248 /** Perform the steps of Bareiss' one-step fraction free elimination to bring
1249 * the matrix into an upper echelon form. Fraction free elimination means
1250 * that divide is used straightforwardly, without computing GCDs first. This
1251 * is possible, since we know the divisor at each step.
1253 * @param det may be set to true to save a lot of space if one is only
1254 * interested in the last element (i.e. for calculating determinants). The
1255 * others are set to zero in this case.
1256 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1257 * number of rows was swapped and 0 if the matrix is singular. */
1258 int matrix::fraction_free_elimination(const bool det)
1261 // (single-step fraction free elimination scheme, already known to Jordan)
1263 // Usual division-free elimination sets m[0](r,c) = m(r,c) and then sets
1264 // m[k+1](r,c) = m[k](k,k) * m[k](r,c) - m[k](r,k) * m[k](k,c).
1266 // Bareiss (fraction-free) elimination in addition divides that element
1267 // by m[k-1](k-1,k-1) for k>1, where it can be shown by means of the
1268 // Sylvester determinant that this really divides m[k+1](r,c).
1270 // We also allow rational functions where the original prove still holds.
1271 // However, we must care for numerator and denominator separately and
1272 // "manually" work in the integral domains because of subtle cancellations
1273 // (see below). This blows up the bookkeeping a bit and the formula has
1274 // to be modified to expand like this (N{x} stands for numerator of x,
1275 // D{x} for denominator of x):
1276 // 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)}
1277 // -N{m[k](r,k)}*N{m[k](k,c)}*D{m[k](k,k)}*D{m[k](r,c)}
1278 // 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)}
1279 // where for k>1 we now divide N{m[k+1](r,c)} by
1280 // N{m[k-1](k-1,k-1)}
1281 // and D{m[k+1](r,c)} by
1282 // D{m[k-1](k-1,k-1)}.
1284 ensure_if_modifiable();
1285 const unsigned m = this->rows();
1286 const unsigned n = this->cols();
1287 GINAC_ASSERT(!det || n==m);
1296 // We populate temporary matrices to subsequently operate on. There is
1297 // one holding numerators and another holding denominators of entries.
1298 // This is a must since the evaluator (or even earlier mul's constructor)
1299 // might cancel some trivial element which causes divide() to fail. The
1300 // elements are normalized first (yes, even though this algorithm doesn't
1301 // need GCDs) since the elements of *this might be unnormalized, which
1302 // makes things more complicated than they need to be.
1303 matrix tmp_n(*this);
1304 matrix tmp_d(m,n); // for denominators, if needed
1305 lst srl; // symbol replacement list
1306 exvector::const_iterator cit = this->m.begin(), citend = this->m.end();
1307 exvector::iterator tmp_n_it = tmp_n.m.begin(), tmp_d_it = tmp_d.m.begin();
1308 while (cit != citend) {
1309 ex nd = cit->normal().to_rational(srl).numer_denom();
1311 *tmp_n_it++ = nd.op(0);
1312 *tmp_d_it++ = nd.op(1);
1316 for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
1317 int indx = tmp_n.pivot(r0, r1, true);
1326 // tmp_n's rows r0 and indx were swapped, do the same in tmp_d:
1327 for (unsigned c=r1; c<n; ++c)
1328 tmp_d.m[n*indx+c].swap(tmp_d.m[n*r0+c]);
1330 for (unsigned r2=r0+1; r2<m; ++r2) {
1331 for (unsigned c=r1+1; c<n; ++c) {
1332 dividend_n = (tmp_n.m[r0*n+r1]*tmp_n.m[r2*n+c]*
1333 tmp_d.m[r2*n+r1]*tmp_d.m[r0*n+c]
1334 -tmp_n.m[r2*n+r1]*tmp_n.m[r0*n+c]*
1335 tmp_d.m[r0*n+r1]*tmp_d.m[r2*n+c]).expand();
1336 dividend_d = (tmp_d.m[r2*n+r1]*tmp_d.m[r0*n+c]*
1337 tmp_d.m[r0*n+r1]*tmp_d.m[r2*n+c]).expand();
1338 bool check = divide(dividend_n, divisor_n,
1339 tmp_n.m[r2*n+c], true);
1340 check &= divide(dividend_d, divisor_d,
1341 tmp_d.m[r2*n+c], true);
1342 GINAC_ASSERT(check);
1344 // fill up left hand side with zeros
1345 for (unsigned c=0; c<=r1; ++c)
1346 tmp_n.m[r2*n+c] = _ex0;
1348 if ((r1<n-1)&&(r0<m-1)) {
1349 // compute next iteration's divisor
1350 divisor_n = tmp_n.m[r0*n+r1].expand();
1351 divisor_d = tmp_d.m[r0*n+r1].expand();
1353 // save space by deleting no longer needed elements
1354 for (unsigned c=0; c<n; ++c) {
1355 tmp_n.m[r0*n+c] = _ex0;
1356 tmp_d.m[r0*n+c] = _ex1;
1363 // repopulate *this matrix:
1364 exvector::iterator it = this->m.begin(), itend = this->m.end();
1365 tmp_n_it = tmp_n.m.begin();
1366 tmp_d_it = tmp_d.m.begin();
1368 *it++ = ((*tmp_n_it++)/(*tmp_d_it++)).subs(srl);
1374 /** Partial pivoting method for matrix elimination schemes.
1375 * Usual pivoting (symbolic==false) returns the index to the element with the
1376 * largest absolute value in column ro and swaps the current row with the one
1377 * where the element was found. With (symbolic==true) it does the same thing
1378 * with the first non-zero element.
1380 * @param ro is the row from where to begin
1381 * @param co is the column to be inspected
1382 * @param symbolic signal if we want the first non-zero element to be pivoted
1383 * (true) or the one with the largest absolute value (false).
1384 * @return 0 if no interchange occured, -1 if all are zero (usually signaling
1385 * a degeneracy) and positive integer k means that rows ro and k were swapped.
1387 int matrix::pivot(unsigned ro, unsigned co, bool symbolic)
1391 // search first non-zero element in column co beginning at row ro
1392 while ((k<row) && (this->m[k*col+co].expand().is_zero()))
1395 // search largest element in column co beginning at row ro
1396 GINAC_ASSERT(is_a<numeric>(this->m[k*col+co]));
1397 unsigned kmax = k+1;
1398 numeric mmax = abs(ex_to<numeric>(m[kmax*col+co]));
1400 GINAC_ASSERT(is_a<numeric>(this->m[kmax*col+co]));
1401 numeric tmp = ex_to<numeric>(this->m[kmax*col+co]);
1402 if (abs(tmp) > mmax) {
1408 if (!mmax.is_zero())
1412 // all elements in column co below row ro vanish
1415 // matrix needs no pivoting
1417 // matrix needs pivoting, so swap rows k and ro
1418 ensure_if_modifiable();
1419 for (unsigned c=0; c<col; ++c)
1420 this->m[k*col+c].swap(this->m[ro*col+c]);
1425 ex lst_to_matrix(const lst & l)
1427 // Find number of rows and columns
1428 unsigned rows = l.nops(), cols = 0, i, j;
1429 for (i=0; i<rows; i++)
1430 if (l.op(i).nops() > cols)
1431 cols = l.op(i).nops();
1433 // Allocate and fill matrix
1434 matrix &m = *new matrix(rows, cols);
1435 m.setflag(status_flags::dynallocated);
1436 for (i=0; i<rows; i++)
1437 for (j=0; j<cols; j++)
1438 if (l.op(i).nops() > j)
1439 m(i, j) = l.op(i).op(j);
1445 ex diag_matrix(const lst & l)
1447 unsigned dim = l.nops();
1449 matrix &m = *new matrix(dim, dim);
1450 m.setflag(status_flags::dynallocated);
1451 for (unsigned i=0; i<dim; i++)
1457 } // namespace GiNaC