3 * Implementation of symbolic matrices */
6 * GiNaC Copyright (C) 1999-2002 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);
150 if (is_a<print_python_repr>(c))
151 c.s << class_name() << '(';
154 for (unsigned y=0; y<row-1; ++y) {
156 for (unsigned x=0; x<col-1; ++x) {
160 m[col*(y+1)-1].print(c);
164 for (unsigned x=0; x<col-1; ++x) {
165 m[(row-1)*col+x].print(c);
168 m[row*col-1].print(c);
171 if (is_a<print_python_repr>(c))
177 /** nops is defined to be rows x columns. */
178 unsigned matrix::nops() const
183 /** returns matrix entry at position (i/col, i%col). */
184 ex matrix::op(int i) const
189 /** returns matrix entry at position (i/col, i%col). */
190 ex & matrix::let_op(int i)
193 GINAC_ASSERT(i<nops());
198 /** Evaluate matrix entry by entry. */
199 ex matrix::eval(int level) const
201 // check if we have to do anything at all
202 if ((level==1)&&(flags & status_flags::evaluated))
206 if (level == -max_recursion_level)
207 throw (std::runtime_error("matrix::eval(): recursion limit exceeded"));
209 // eval() entry by entry
210 exvector m2(row*col);
212 for (unsigned r=0; r<row; ++r)
213 for (unsigned c=0; c<col; ++c)
214 m2[r*col+c] = m[r*col+c].eval(level);
216 return (new matrix(row, col, m2))->setflag(status_flags::dynallocated |
217 status_flags::evaluated );
220 ex matrix::subs(const lst & ls, const lst & lr, bool no_pattern) const
222 exvector m2(row * col);
223 for (unsigned r=0; r<row; ++r)
224 for (unsigned c=0; c<col; ++c)
225 m2[r*col+c] = m[r*col+c].subs(ls, lr, no_pattern);
227 return matrix(row, col, m2).basic::subs(ls, lr, no_pattern);
232 int matrix::compare_same_type(const basic & other) const
234 GINAC_ASSERT(is_exactly_a<matrix>(other));
235 const matrix &o = static_cast<const matrix &>(other);
237 // compare number of rows
239 return row < o.rows() ? -1 : 1;
241 // compare number of columns
243 return col < o.cols() ? -1 : 1;
245 // equal number of rows and columns, compare individual elements
247 for (unsigned r=0; r<row; ++r) {
248 for (unsigned c=0; c<col; ++c) {
249 cmpval = ((*this)(r,c)).compare(o(r,c));
250 if (cmpval!=0) return cmpval;
253 // all elements are equal => matrices are equal;
257 bool matrix::match_same_type(const basic & other) const
259 GINAC_ASSERT(is_exactly_a<matrix>(other));
260 const matrix & o = static_cast<const matrix &>(other);
262 // The number of rows and columns must be the same. This is necessary to
263 // prevent a 2x3 matrix from matching a 3x2 one.
264 return row == o.rows() && col == o.cols();
267 /** Automatic symbolic evaluation of an indexed matrix. */
268 ex matrix::eval_indexed(const basic & i) const
270 GINAC_ASSERT(is_a<indexed>(i));
271 GINAC_ASSERT(is_a<matrix>(i.op(0)));
273 bool all_indices_unsigned = static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint);
278 // One index, must be one-dimensional vector
279 if (row != 1 && col != 1)
280 throw (std::runtime_error("matrix::eval_indexed(): vector must have exactly 1 index"));
282 const idx & i1 = ex_to<idx>(i.op(1));
287 if (!i1.get_dim().is_equal(row))
288 throw (std::runtime_error("matrix::eval_indexed(): dimension of index must match number of vector elements"));
290 // Index numeric -> return vector element
291 if (all_indices_unsigned) {
292 unsigned n1 = ex_to<numeric>(i1.get_value()).to_int();
294 throw (std::runtime_error("matrix::eval_indexed(): value of index exceeds number of vector elements"));
295 return (*this)(n1, 0);
301 if (!i1.get_dim().is_equal(col))
302 throw (std::runtime_error("matrix::eval_indexed(): dimension of index must match number of vector elements"));
304 // Index numeric -> return vector element
305 if (all_indices_unsigned) {
306 unsigned n1 = ex_to<numeric>(i1.get_value()).to_int();
308 throw (std::runtime_error("matrix::eval_indexed(): value of index exceeds number of vector elements"));
309 return (*this)(0, n1);
313 } else if (i.nops() == 3) {
316 const idx & i1 = ex_to<idx>(i.op(1));
317 const idx & i2 = ex_to<idx>(i.op(2));
319 if (!i1.get_dim().is_equal(row))
320 throw (std::runtime_error("matrix::eval_indexed(): dimension of first index must match number of rows"));
321 if (!i2.get_dim().is_equal(col))
322 throw (std::runtime_error("matrix::eval_indexed(): dimension of second index must match number of columns"));
324 // Pair of dummy indices -> compute trace
325 if (is_dummy_pair(i1, i2))
328 // Both indices numeric -> return matrix element
329 if (all_indices_unsigned) {
330 unsigned n1 = ex_to<numeric>(i1.get_value()).to_int(), n2 = ex_to<numeric>(i2.get_value()).to_int();
332 throw (std::runtime_error("matrix::eval_indexed(): value of first index exceeds number of rows"));
334 throw (std::runtime_error("matrix::eval_indexed(): value of second index exceeds number of columns"));
335 return (*this)(n1, n2);
339 throw (std::runtime_error("matrix::eval_indexed(): matrix must have exactly 2 indices"));
344 /** Sum of two indexed matrices. */
345 ex matrix::add_indexed(const ex & self, const ex & other) const
347 GINAC_ASSERT(is_a<indexed>(self));
348 GINAC_ASSERT(is_a<matrix>(self.op(0)));
349 GINAC_ASSERT(is_a<indexed>(other));
350 GINAC_ASSERT(self.nops() == 2 || self.nops() == 3);
352 // Only add two matrices
353 if (is_ex_of_type(other.op(0), matrix)) {
354 GINAC_ASSERT(other.nops() == 2 || other.nops() == 3);
356 const matrix &self_matrix = ex_to<matrix>(self.op(0));
357 const matrix &other_matrix = ex_to<matrix>(other.op(0));
359 if (self.nops() == 2 && other.nops() == 2) { // vector + vector
361 if (self_matrix.row == other_matrix.row)
362 return indexed(self_matrix.add(other_matrix), self.op(1));
363 else if (self_matrix.row == other_matrix.col)
364 return indexed(self_matrix.add(other_matrix.transpose()), self.op(1));
366 } else if (self.nops() == 3 && other.nops() == 3) { // matrix + matrix
368 if (self.op(1).is_equal(other.op(1)) && self.op(2).is_equal(other.op(2)))
369 return indexed(self_matrix.add(other_matrix), self.op(1), self.op(2));
370 else if (self.op(1).is_equal(other.op(2)) && self.op(2).is_equal(other.op(1)))
371 return indexed(self_matrix.add(other_matrix.transpose()), self.op(1), self.op(2));
376 // Don't know what to do, return unevaluated sum
380 /** Product of an indexed matrix with a number. */
381 ex matrix::scalar_mul_indexed(const ex & self, const numeric & other) const
383 GINAC_ASSERT(is_a<indexed>(self));
384 GINAC_ASSERT(is_a<matrix>(self.op(0)));
385 GINAC_ASSERT(self.nops() == 2 || self.nops() == 3);
387 const matrix &self_matrix = ex_to<matrix>(self.op(0));
389 if (self.nops() == 2)
390 return indexed(self_matrix.mul(other), self.op(1));
391 else // self.nops() == 3
392 return indexed(self_matrix.mul(other), self.op(1), self.op(2));
395 /** Contraction of an indexed matrix with something else. */
396 bool matrix::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
398 GINAC_ASSERT(is_a<indexed>(*self));
399 GINAC_ASSERT(is_a<indexed>(*other));
400 GINAC_ASSERT(self->nops() == 2 || self->nops() == 3);
401 GINAC_ASSERT(is_a<matrix>(self->op(0)));
403 // Only contract with other matrices
404 if (!is_ex_of_type(other->op(0), matrix))
407 GINAC_ASSERT(other->nops() == 2 || other->nops() == 3);
409 const matrix &self_matrix = ex_to<matrix>(self->op(0));
410 const matrix &other_matrix = ex_to<matrix>(other->op(0));
412 if (self->nops() == 2) {
414 if (other->nops() == 2) { // vector * vector (scalar product)
416 if (self_matrix.col == 1) {
417 if (other_matrix.col == 1) {
418 // Column vector * column vector, transpose first vector
419 *self = self_matrix.transpose().mul(other_matrix)(0, 0);
421 // Column vector * row vector, swap factors
422 *self = other_matrix.mul(self_matrix)(0, 0);
425 if (other_matrix.col == 1) {
426 // Row vector * column vector, perfect
427 *self = self_matrix.mul(other_matrix)(0, 0);
429 // Row vector * row vector, transpose second vector
430 *self = self_matrix.mul(other_matrix.transpose())(0, 0);
436 } else { // vector * matrix
438 // B_i * A_ij = (B*A)_j (B is row vector)
439 if (is_dummy_pair(self->op(1), other->op(1))) {
440 if (self_matrix.row == 1)
441 *self = indexed(self_matrix.mul(other_matrix), other->op(2));
443 *self = indexed(self_matrix.transpose().mul(other_matrix), other->op(2));
448 // B_j * A_ij = (A*B)_i (B is column vector)
449 if (is_dummy_pair(self->op(1), other->op(2))) {
450 if (self_matrix.col == 1)
451 *self = indexed(other_matrix.mul(self_matrix), other->op(1));
453 *self = indexed(other_matrix.mul(self_matrix.transpose()), other->op(1));
459 } else if (other->nops() == 3) { // matrix * matrix
461 // A_ij * B_jk = (A*B)_ik
462 if (is_dummy_pair(self->op(2), other->op(1))) {
463 *self = indexed(self_matrix.mul(other_matrix), self->op(1), other->op(2));
468 // A_ij * B_kj = (A*Btrans)_ik
469 if (is_dummy_pair(self->op(2), other->op(2))) {
470 *self = indexed(self_matrix.mul(other_matrix.transpose()), self->op(1), other->op(1));
475 // A_ji * B_jk = (Atrans*B)_ik
476 if (is_dummy_pair(self->op(1), other->op(1))) {
477 *self = indexed(self_matrix.transpose().mul(other_matrix), self->op(2), other->op(2));
482 // A_ji * B_kj = (B*A)_ki
483 if (is_dummy_pair(self->op(1), other->op(2))) {
484 *self = indexed(other_matrix.mul(self_matrix), other->op(1), self->op(2));
495 // non-virtual functions in this class
502 * @exception logic_error (incompatible matrices) */
503 matrix matrix::add(const matrix & other) const
505 if (col != other.col || row != other.row)
506 throw std::logic_error("matrix::add(): incompatible matrices");
508 exvector sum(this->m);
509 exvector::iterator i = sum.begin(), end = sum.end();
510 exvector::const_iterator ci = other.m.begin();
514 return matrix(row,col,sum);
518 /** Difference of matrices.
520 * @exception logic_error (incompatible matrices) */
521 matrix matrix::sub(const matrix & other) const
523 if (col != other.col || row != other.row)
524 throw std::logic_error("matrix::sub(): incompatible matrices");
526 exvector dif(this->m);
527 exvector::iterator i = dif.begin(), end = dif.end();
528 exvector::const_iterator ci = other.m.begin();
532 return matrix(row,col,dif);
536 /** Product of matrices.
538 * @exception logic_error (incompatible matrices) */
539 matrix matrix::mul(const matrix & other) const
541 if (this->cols() != other.rows())
542 throw std::logic_error("matrix::mul(): incompatible matrices");
544 exvector prod(this->rows()*other.cols());
546 for (unsigned r1=0; r1<this->rows(); ++r1) {
547 for (unsigned c=0; c<this->cols(); ++c) {
548 if (m[r1*col+c].is_zero())
550 for (unsigned r2=0; r2<other.cols(); ++r2)
551 prod[r1*other.col+r2] += (m[r1*col+c] * other.m[c*other.col+r2]).expand();
554 return matrix(row, other.col, prod);
558 /** Product of matrix and scalar. */
559 matrix matrix::mul(const numeric & other) const
561 exvector prod(row * col);
563 for (unsigned r=0; r<row; ++r)
564 for (unsigned c=0; c<col; ++c)
565 prod[r*col+c] = m[r*col+c] * other;
567 return matrix(row, col, prod);
571 /** Product of matrix and scalar expression. */
572 matrix matrix::mul_scalar(const ex & other) const
574 if (other.return_type() != return_types::commutative)
575 throw std::runtime_error("matrix::mul_scalar(): non-commutative scalar");
577 exvector prod(row * col);
579 for (unsigned r=0; r<row; ++r)
580 for (unsigned c=0; c<col; ++c)
581 prod[r*col+c] = m[r*col+c] * other;
583 return matrix(row, col, prod);
587 /** Power of a matrix. Currently handles integer exponents only. */
588 matrix matrix::pow(const ex & expn) const
591 throw (std::logic_error("matrix::pow(): matrix not square"));
593 if (is_ex_exactly_of_type(expn, numeric)) {
594 // Integer cases are computed by successive multiplication, using the
595 // obvious shortcut of storing temporaries, like A^4 == (A*A)*(A*A).
596 if (expn.info(info_flags::integer)) {
597 numeric b = ex_to<numeric>(expn);
599 if (expn.info(info_flags::negative)) {
606 for (unsigned r=0; r<row; ++r)
608 // This loop computes the representation of b in base 2 from right
609 // to left and multiplies the factors whenever needed. Note
610 // that this is not entirely optimal but close to optimal and
611 // "better" algorithms are much harder to implement. (See Knuth,
612 // TAoCP2, section "Evaluation of Powers" for a good discussion.)
618 b *= _num1_2; // b /= 2, still integer.
624 throw (std::runtime_error("matrix::pow(): don't know how to handle exponent"));
628 /** operator() to access elements for reading.
630 * @param ro row of element
631 * @param co column of element
632 * @exception range_error (index out of range) */
633 const ex & matrix::operator() (unsigned ro, unsigned co) const
635 if (ro>=row || co>=col)
636 throw (std::range_error("matrix::operator(): index out of range"));
642 /** operator() to access elements for writing.
644 * @param ro row of element
645 * @param co column of element
646 * @exception range_error (index out of range) */
647 ex & matrix::operator() (unsigned ro, unsigned co)
649 if (ro>=row || co>=col)
650 throw (std::range_error("matrix::operator(): index out of range"));
652 ensure_if_modifiable();
657 /** Transposed of an m x n matrix, producing a new n x m matrix object that
658 * represents the transposed. */
659 matrix matrix::transpose(void) const
661 exvector trans(this->cols()*this->rows());
663 for (unsigned r=0; r<this->cols(); ++r)
664 for (unsigned c=0; c<this->rows(); ++c)
665 trans[r*this->rows()+c] = m[c*this->cols()+r];
667 return matrix(this->cols(),this->rows(),trans);
670 /** Determinant of square matrix. This routine doesn't actually calculate the
671 * determinant, it only implements some heuristics about which algorithm to
672 * run. If all the elements of the matrix are elements of an integral domain
673 * the determinant is also in that integral domain and the result is expanded
674 * only. If one or more elements are from a quotient field the determinant is
675 * usually also in that quotient field and the result is normalized before it
676 * is returned. This implies that the determinant of the symbolic 2x2 matrix
677 * [[a/(a-b),1],[b/(a-b),1]] is returned as unity. (In this respect, it
678 * behaves like MapleV and unlike Mathematica.)
680 * @param algo allows to chose an algorithm
681 * @return the determinant as a new expression
682 * @exception logic_error (matrix not square)
683 * @see determinant_algo */
684 ex matrix::determinant(unsigned algo) const
687 throw (std::logic_error("matrix::determinant(): matrix not square"));
688 GINAC_ASSERT(row*col==m.capacity());
690 // Gather some statistical information about this matrix:
691 bool numeric_flag = true;
692 bool normal_flag = false;
693 unsigned sparse_count = 0; // counts non-zero elements
694 exvector::const_iterator r = m.begin(), rend = m.end();
696 lst srl; // symbol replacement list
697 ex rtest = r->to_rational(srl);
698 if (!rtest.is_zero())
700 if (!rtest.info(info_flags::numeric))
701 numeric_flag = false;
702 if (!rtest.info(info_flags::crational_polynomial) &&
703 rtest.info(info_flags::rational_function))
708 // Here is the heuristics in case this routine has to decide:
709 if (algo == determinant_algo::automatic) {
710 // Minor expansion is generally a good guess:
711 algo = determinant_algo::laplace;
712 // Does anybody know when a matrix is really sparse?
713 // Maybe <~row/2.236 nonzero elements average in a row?
714 if (row>3 && 5*sparse_count<=row*col)
715 algo = determinant_algo::bareiss;
716 // Purely numeric matrix can be handled by Gauss elimination.
717 // This overrides any prior decisions.
719 algo = determinant_algo::gauss;
722 // Trap the trivial case here, since some algorithms don't like it
724 // for consistency with non-trivial determinants...
726 return m[0].normal();
728 return m[0].expand();
731 // Compute the determinant
733 case determinant_algo::gauss: {
736 int sign = tmp.gauss_elimination(true);
737 for (unsigned d=0; d<row; ++d)
738 det *= tmp.m[d*col+d];
740 return (sign*det).normal();
742 return (sign*det).normal().expand();
744 case determinant_algo::bareiss: {
747 sign = tmp.fraction_free_elimination(true);
749 return (sign*tmp.m[row*col-1]).normal();
751 return (sign*tmp.m[row*col-1]).expand();
753 case determinant_algo::divfree: {
756 sign = tmp.division_free_elimination(true);
759 ex det = tmp.m[row*col-1];
760 // factor out accumulated bogus slag
761 for (unsigned d=0; d<row-2; ++d)
762 for (unsigned j=0; j<row-d-2; ++j)
763 det = (det/tmp.m[d*col+d]).normal();
766 case determinant_algo::laplace:
768 // This is the minor expansion scheme. We always develop such
769 // that the smallest minors (i.e, the trivial 1x1 ones) are on the
770 // rightmost column. For this to be efficient, empirical tests
771 // have shown that the emptiest columns (i.e. the ones with most
772 // zeros) should be the ones on the right hand side -- although
773 // this might seem counter-intuitive (and in contradiction to some
774 // literature like the FORM manual). Please go ahead and test it
775 // if you don't believe me! Therefore we presort the columns of
777 typedef std::pair<unsigned,unsigned> uintpair;
778 std::vector<uintpair> c_zeros; // number of zeros in column
779 for (unsigned c=0; c<col; ++c) {
781 for (unsigned r=0; r<row; ++r)
782 if (m[r*col+c].is_zero())
784 c_zeros.push_back(uintpair(acc,c));
786 std::sort(c_zeros.begin(),c_zeros.end());
787 std::vector<unsigned> pre_sort;
788 for (std::vector<uintpair>::const_iterator i=c_zeros.begin(); i!=c_zeros.end(); ++i)
789 pre_sort.push_back(i->second);
790 std::vector<unsigned> pre_sort_test(pre_sort); // permutation_sign() modifies the vector so we make a copy here
791 int sign = permutation_sign(pre_sort_test.begin(), pre_sort_test.end());
792 exvector result(row*col); // represents sorted matrix
794 for (std::vector<unsigned>::const_iterator i=pre_sort.begin();
797 for (unsigned r=0; r<row; ++r)
798 result[r*col+c] = m[r*col+(*i)];
802 return (sign*matrix(row,col,result).determinant_minor()).normal();
804 return sign*matrix(row,col,result).determinant_minor();
810 /** Trace of a matrix. The result is normalized if it is in some quotient
811 * field and expanded only otherwise. This implies that the trace of the
812 * symbolic 2x2 matrix [[a/(a-b),x],[y,b/(b-a)]] is recognized to be unity.
814 * @return the sum of diagonal elements
815 * @exception logic_error (matrix not square) */
816 ex matrix::trace(void) const
819 throw (std::logic_error("matrix::trace(): matrix not square"));
822 for (unsigned r=0; r<col; ++r)
825 if (tr.info(info_flags::rational_function) &&
826 !tr.info(info_flags::crational_polynomial))
833 /** Characteristic Polynomial. Following mathematica notation the
834 * characteristic polynomial of a matrix M is defined as the determiant of
835 * (M - lambda * 1) where 1 stands for the unit matrix of the same dimension
836 * as M. Note that some CASs define it with a sign inside the determinant
837 * which gives rise to an overall sign if the dimension is odd. This method
838 * returns the characteristic polynomial collected in powers of lambda as a
841 * @return characteristic polynomial as new expression
842 * @exception logic_error (matrix not square)
843 * @see matrix::determinant() */
844 ex matrix::charpoly(const symbol & lambda) const
847 throw (std::logic_error("matrix::charpoly(): matrix not square"));
849 bool numeric_flag = true;
850 exvector::const_iterator r = m.begin(), rend = m.end();
851 while (r!=rend && numeric_flag==true) {
852 if (!r->info(info_flags::numeric))
853 numeric_flag = false;
857 // The pure numeric case is traditionally rather common. Hence, it is
858 // trapped and we use Leverrier's algorithm which goes as row^3 for
859 // every coefficient. The expensive part is the matrix multiplication.
863 ex poly = power(lambda,row)-c*power(lambda,row-1);
864 for (unsigned i=1; i<row; ++i) {
865 for (unsigned j=0; j<row; ++j)
868 c = B.trace()/ex(i+1);
869 poly -= c*power(lambda,row-i-1);
878 for (unsigned r=0; r<col; ++r)
879 M.m[r*col+r] -= lambda;
881 return M.determinant().collect(lambda);
885 /** Inverse of this matrix.
887 * @return the inverted matrix
888 * @exception logic_error (matrix not square)
889 * @exception runtime_error (singular matrix) */
890 matrix matrix::inverse(void) const
893 throw (std::logic_error("matrix::inverse(): matrix not square"));
895 // This routine actually doesn't do anything fancy at all. We compute the
896 // inverse of the matrix A by solving the system A * A^{-1} == Id.
898 // First populate the identity matrix supposed to become the right hand side.
899 matrix identity(row,col);
900 for (unsigned i=0; i<row; ++i)
901 identity(i,i) = _ex1;
903 // Populate a dummy matrix of variables, just because of compatibility with
904 // matrix::solve() which wants this (for compatibility with under-determined
905 // systems of equations).
906 matrix vars(row,col);
907 for (unsigned r=0; r<row; ++r)
908 for (unsigned c=0; c<col; ++c)
909 vars(r,c) = symbol();
913 sol = this->solve(vars,identity);
914 } catch (const std::runtime_error & e) {
915 if (e.what()==std::string("matrix::solve(): inconsistent linear system"))
916 throw (std::runtime_error("matrix::inverse(): singular matrix"));
924 /** Solve a linear system consisting of a m x n matrix and a m x p right hand
925 * side by applying an elimination scheme to the augmented matrix.
927 * @param vars n x p matrix, all elements must be symbols
928 * @param rhs m x p matrix
929 * @return n x p solution matrix
930 * @exception logic_error (incompatible matrices)
931 * @exception invalid_argument (1st argument must be matrix of symbols)
932 * @exception runtime_error (inconsistent linear system)
934 matrix matrix::solve(const matrix & vars,
938 const unsigned m = this->rows();
939 const unsigned n = this->cols();
940 const unsigned p = rhs.cols();
943 if ((rhs.rows() != m) || (vars.rows() != n) || (vars.col != p))
944 throw (std::logic_error("matrix::solve(): incompatible matrices"));
945 for (unsigned ro=0; ro<n; ++ro)
946 for (unsigned co=0; co<p; ++co)
947 if (!vars(ro,co).info(info_flags::symbol))
948 throw (std::invalid_argument("matrix::solve(): 1st argument must be matrix of symbols"));
950 // build the augmented matrix of *this with rhs attached to the right
952 for (unsigned r=0; r<m; ++r) {
953 for (unsigned c=0; c<n; ++c)
954 aug.m[r*(n+p)+c] = this->m[r*n+c];
955 for (unsigned c=0; c<p; ++c)
956 aug.m[r*(n+p)+c+n] = rhs.m[r*p+c];
959 // Gather some statistical information about the augmented matrix:
960 bool numeric_flag = true;
961 exvector::const_iterator r = aug.m.begin(), rend = aug.m.end();
962 while (r!=rend && numeric_flag==true) {
963 if (!r->info(info_flags::numeric))
964 numeric_flag = false;
968 // Here is the heuristics in case this routine has to decide:
969 if (algo == solve_algo::automatic) {
970 // Bareiss (fraction-free) elimination is generally a good guess:
971 algo = solve_algo::bareiss;
972 // For m<3, Bareiss elimination is equivalent to division free
973 // elimination but has more logistic overhead
975 algo = solve_algo::divfree;
976 // This overrides any prior decisions.
978 algo = solve_algo::gauss;
981 // Eliminate the augmented matrix:
983 case solve_algo::gauss:
984 aug.gauss_elimination();
986 case solve_algo::divfree:
987 aug.division_free_elimination();
989 case solve_algo::bareiss:
991 aug.fraction_free_elimination();
994 // assemble the solution matrix:
996 for (unsigned co=0; co<p; ++co) {
997 unsigned last_assigned_sol = n+1;
998 for (int r=m-1; r>=0; --r) {
999 unsigned fnz = 1; // first non-zero in row
1000 while ((fnz<=n) && (aug.m[r*(n+p)+(fnz-1)].is_zero()))
1003 // row consists only of zeros, corresponding rhs must be 0, too
1004 if (!aug.m[r*(n+p)+n+co].is_zero()) {
1005 throw (std::runtime_error("matrix::solve(): inconsistent linear system"));
1008 // assign solutions for vars between fnz+1 and
1009 // last_assigned_sol-1: free parameters
1010 for (unsigned c=fnz; c<last_assigned_sol-1; ++c)
1011 sol(c,co) = vars.m[c*p+co];
1012 ex e = aug.m[r*(n+p)+n+co];
1013 for (unsigned c=fnz; c<n; ++c)
1014 e -= aug.m[r*(n+p)+c]*sol.m[c*p+co];
1015 sol(fnz-1,co) = (e/(aug.m[r*(n+p)+(fnz-1)])).normal();
1016 last_assigned_sol = fnz;
1019 // assign solutions for vars between 1 and
1020 // last_assigned_sol-1: free parameters
1021 for (unsigned ro=0; ro<last_assigned_sol-1; ++ro)
1022 sol(ro,co) = vars(ro,co);
1031 /** Recursive determinant for small matrices having at least one symbolic
1032 * entry. The basic algorithm, known as Laplace-expansion, is enhanced by
1033 * some bookkeeping to avoid calculation of the same submatrices ("minors")
1034 * more than once. According to W.M.Gentleman and S.C.Johnson this algorithm
1035 * is better than elimination schemes for matrices of sparse multivariate
1036 * polynomials and also for matrices of dense univariate polynomials if the
1037 * matrix' dimesion is larger than 7.
1039 * @return the determinant as a new expression (in expanded form)
1040 * @see matrix::determinant() */
1041 ex matrix::determinant_minor(void) const
1043 // for small matrices the algorithm does not make any sense:
1044 const unsigned n = this->cols();
1046 return m[0].expand();
1048 return (m[0]*m[3]-m[2]*m[1]).expand();
1050 return (m[0]*m[4]*m[8]-m[0]*m[5]*m[7]-
1051 m[1]*m[3]*m[8]+m[2]*m[3]*m[7]+
1052 m[1]*m[5]*m[6]-m[2]*m[4]*m[6]).expand();
1054 // This algorithm can best be understood by looking at a naive
1055 // implementation of Laplace-expansion, like this one:
1057 // matrix minorM(this->rows()-1,this->cols()-1);
1058 // for (unsigned r1=0; r1<this->rows(); ++r1) {
1059 // // shortcut if element(r1,0) vanishes
1060 // if (m[r1*col].is_zero())
1062 // // assemble the minor matrix
1063 // for (unsigned r=0; r<minorM.rows(); ++r) {
1064 // for (unsigned c=0; c<minorM.cols(); ++c) {
1066 // minorM(r,c) = m[r*col+c+1];
1068 // minorM(r,c) = m[(r+1)*col+c+1];
1071 // // recurse down and care for sign:
1073 // det -= m[r1*col] * minorM.determinant_minor();
1075 // det += m[r1*col] * minorM.determinant_minor();
1077 // return det.expand();
1078 // What happens is that while proceeding down many of the minors are
1079 // computed more than once. In particular, there are binomial(n,k)
1080 // kxk minors and each one is computed factorial(n-k) times. Therefore
1081 // it is reasonable to store the results of the minors. We proceed from
1082 // right to left. At each column c we only need to retrieve the minors
1083 // calculated in step c-1. We therefore only have to store at most
1084 // 2*binomial(n,n/2) minors.
1086 // Unique flipper counter for partitioning into minors
1087 std::vector<unsigned> Pkey;
1089 // key for minor determinant (a subpartition of Pkey)
1090 std::vector<unsigned> Mkey;
1092 // we store our subminors in maps, keys being the rows they arise from
1093 typedef std::map<std::vector<unsigned>,class ex> Rmap;
1094 typedef std::map<std::vector<unsigned>,class ex>::value_type Rmap_value;
1098 // initialize A with last column:
1099 for (unsigned r=0; r<n; ++r) {
1100 Pkey.erase(Pkey.begin(),Pkey.end());
1102 A.insert(Rmap_value(Pkey,m[n*(r+1)-1]));
1104 // proceed from right to left through matrix
1105 for (int c=n-2; c>=0; --c) {
1106 Pkey.erase(Pkey.begin(),Pkey.end()); // don't change capacity
1107 Mkey.erase(Mkey.begin(),Mkey.end());
1108 for (unsigned i=0; i<n-c; ++i)
1110 unsigned fc = 0; // controls logic for our strange flipper counter
1113 for (unsigned r=0; r<n-c; ++r) {
1114 // maybe there is nothing to do?
1115 if (m[Pkey[r]*n+c].is_zero())
1117 // create the sorted key for all possible minors
1118 Mkey.erase(Mkey.begin(),Mkey.end());
1119 for (unsigned i=0; i<n-c; ++i)
1121 Mkey.push_back(Pkey[i]);
1122 // Fetch the minors and compute the new determinant
1124 det -= m[Pkey[r]*n+c]*A[Mkey];
1126 det += m[Pkey[r]*n+c]*A[Mkey];
1128 // prevent build-up of deep nesting of expressions saves time:
1130 // store the new determinant at its place in B:
1132 B.insert(Rmap_value(Pkey,det));
1133 // increment our strange flipper counter
1134 for (fc=n-c; fc>0; --fc) {
1136 if (Pkey[fc-1]<fc+c)
1140 for (unsigned j=fc; j<n-c; ++j)
1141 Pkey[j] = Pkey[j-1]+1;
1143 // next column, so change the role of A and B:
1152 /** Perform the steps of an ordinary Gaussian elimination to bring the m x n
1153 * matrix into an upper echelon form. The algorithm is ok for matrices
1154 * with numeric coefficients but quite unsuited for symbolic matrices.
1156 * @param det may be set to true to save a lot of space if one is only
1157 * interested in the diagonal elements (i.e. for calculating determinants).
1158 * The others are set to zero in this case.
1159 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1160 * number of rows was swapped and 0 if the matrix is singular. */
1161 int matrix::gauss_elimination(const bool det)
1163 ensure_if_modifiable();
1164 const unsigned m = this->rows();
1165 const unsigned n = this->cols();
1166 GINAC_ASSERT(!det || n==m);
1170 for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
1171 int indx = pivot(r0, r1, true);
1175 return 0; // leaves *this in a messy state
1180 for (unsigned r2=r0+1; r2<m; ++r2) {
1181 if (!this->m[r2*n+r1].is_zero()) {
1182 // yes, there is something to do in this row
1183 ex piv = this->m[r2*n+r1] / this->m[r0*n+r1];
1184 for (unsigned c=r1+1; c<n; ++c) {
1185 this->m[r2*n+c] -= piv * this->m[r0*n+c];
1186 if (!this->m[r2*n+c].info(info_flags::numeric))
1187 this->m[r2*n+c] = this->m[r2*n+c].normal();
1190 // fill up left hand side with zeros
1191 for (unsigned c=0; c<=r1; ++c)
1192 this->m[r2*n+c] = _ex0;
1195 // save space by deleting no longer needed elements
1196 for (unsigned c=r0+1; c<n; ++c)
1197 this->m[r0*n+c] = _ex0;
1207 /** Perform the steps of division free elimination to bring the m x n matrix
1208 * into an upper echelon form.
1210 * @param det may be set to true to save a lot of space if one is only
1211 * interested in the diagonal elements (i.e. for calculating determinants).
1212 * The others are set to zero in this case.
1213 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1214 * number of rows was swapped and 0 if the matrix is singular. */
1215 int matrix::division_free_elimination(const bool det)
1217 ensure_if_modifiable();
1218 const unsigned m = this->rows();
1219 const unsigned n = this->cols();
1220 GINAC_ASSERT(!det || n==m);
1224 for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
1225 int indx = pivot(r0, r1, true);
1229 return 0; // leaves *this in a messy state
1234 for (unsigned r2=r0+1; r2<m; ++r2) {
1235 for (unsigned c=r1+1; c<n; ++c)
1236 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();
1237 // fill up left hand side with zeros
1238 for (unsigned c=0; c<=r1; ++c)
1239 this->m[r2*n+c] = _ex0;
1242 // save space by deleting no longer needed elements
1243 for (unsigned c=r0+1; c<n; ++c)
1244 this->m[r0*n+c] = _ex0;
1254 /** Perform the steps of Bareiss' one-step fraction free elimination to bring
1255 * the matrix into an upper echelon form. Fraction free elimination means
1256 * that divide is used straightforwardly, without computing GCDs first. This
1257 * is possible, since we know the divisor at each step.
1259 * @param det may be set to true to save a lot of space if one is only
1260 * interested in the last element (i.e. for calculating determinants). The
1261 * others are set to zero in this case.
1262 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1263 * number of rows was swapped and 0 if the matrix is singular. */
1264 int matrix::fraction_free_elimination(const bool det)
1267 // (single-step fraction free elimination scheme, already known to Jordan)
1269 // Usual division-free elimination sets m[0](r,c) = m(r,c) and then sets
1270 // m[k+1](r,c) = m[k](k,k) * m[k](r,c) - m[k](r,k) * m[k](k,c).
1272 // Bareiss (fraction-free) elimination in addition divides that element
1273 // by m[k-1](k-1,k-1) for k>1, where it can be shown by means of the
1274 // Sylvester determinant that this really divides m[k+1](r,c).
1276 // We also allow rational functions where the original prove still holds.
1277 // However, we must care for numerator and denominator separately and
1278 // "manually" work in the integral domains because of subtle cancellations
1279 // (see below). This blows up the bookkeeping a bit and the formula has
1280 // to be modified to expand like this (N{x} stands for numerator of x,
1281 // D{x} for denominator of x):
1282 // 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)}
1283 // -N{m[k](r,k)}*N{m[k](k,c)}*D{m[k](k,k)}*D{m[k](r,c)}
1284 // 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)}
1285 // where for k>1 we now divide N{m[k+1](r,c)} by
1286 // N{m[k-1](k-1,k-1)}
1287 // and D{m[k+1](r,c)} by
1288 // D{m[k-1](k-1,k-1)}.
1290 ensure_if_modifiable();
1291 const unsigned m = this->rows();
1292 const unsigned n = this->cols();
1293 GINAC_ASSERT(!det || n==m);
1302 // We populate temporary matrices to subsequently operate on. There is
1303 // one holding numerators and another holding denominators of entries.
1304 // This is a must since the evaluator (or even earlier mul's constructor)
1305 // might cancel some trivial element which causes divide() to fail. The
1306 // elements are normalized first (yes, even though this algorithm doesn't
1307 // need GCDs) since the elements of *this might be unnormalized, which
1308 // makes things more complicated than they need to be.
1309 matrix tmp_n(*this);
1310 matrix tmp_d(m,n); // for denominators, if needed
1311 lst srl; // symbol replacement list
1312 exvector::const_iterator cit = this->m.begin(), citend = this->m.end();
1313 exvector::iterator tmp_n_it = tmp_n.m.begin(), tmp_d_it = tmp_d.m.begin();
1314 while (cit != citend) {
1315 ex nd = cit->normal().to_rational(srl).numer_denom();
1317 *tmp_n_it++ = nd.op(0);
1318 *tmp_d_it++ = nd.op(1);
1322 for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
1323 int indx = tmp_n.pivot(r0, r1, true);
1332 // tmp_n's rows r0 and indx were swapped, do the same in tmp_d:
1333 for (unsigned c=r1; c<n; ++c)
1334 tmp_d.m[n*indx+c].swap(tmp_d.m[n*r0+c]);
1336 for (unsigned r2=r0+1; r2<m; ++r2) {
1337 for (unsigned c=r1+1; c<n; ++c) {
1338 dividend_n = (tmp_n.m[r0*n+r1]*tmp_n.m[r2*n+c]*
1339 tmp_d.m[r2*n+r1]*tmp_d.m[r0*n+c]
1340 -tmp_n.m[r2*n+r1]*tmp_n.m[r0*n+c]*
1341 tmp_d.m[r0*n+r1]*tmp_d.m[r2*n+c]).expand();
1342 dividend_d = (tmp_d.m[r2*n+r1]*tmp_d.m[r0*n+c]*
1343 tmp_d.m[r0*n+r1]*tmp_d.m[r2*n+c]).expand();
1344 bool check = divide(dividend_n, divisor_n,
1345 tmp_n.m[r2*n+c], true);
1346 check &= divide(dividend_d, divisor_d,
1347 tmp_d.m[r2*n+c], true);
1348 GINAC_ASSERT(check);
1350 // fill up left hand side with zeros
1351 for (unsigned c=0; c<=r1; ++c)
1352 tmp_n.m[r2*n+c] = _ex0;
1354 if ((r1<n-1)&&(r0<m-1)) {
1355 // compute next iteration's divisor
1356 divisor_n = tmp_n.m[r0*n+r1].expand();
1357 divisor_d = tmp_d.m[r0*n+r1].expand();
1359 // save space by deleting no longer needed elements
1360 for (unsigned c=0; c<n; ++c) {
1361 tmp_n.m[r0*n+c] = _ex0;
1362 tmp_d.m[r0*n+c] = _ex1;
1369 // repopulate *this matrix:
1370 exvector::iterator it = this->m.begin(), itend = this->m.end();
1371 tmp_n_it = tmp_n.m.begin();
1372 tmp_d_it = tmp_d.m.begin();
1374 *it++ = ((*tmp_n_it++)/(*tmp_d_it++)).subs(srl);
1380 /** Partial pivoting method for matrix elimination schemes.
1381 * Usual pivoting (symbolic==false) returns the index to the element with the
1382 * largest absolute value in column ro and swaps the current row with the one
1383 * where the element was found. With (symbolic==true) it does the same thing
1384 * with the first non-zero element.
1386 * @param ro is the row from where to begin
1387 * @param co is the column to be inspected
1388 * @param symbolic signal if we want the first non-zero element to be pivoted
1389 * (true) or the one with the largest absolute value (false).
1390 * @return 0 if no interchange occured, -1 if all are zero (usually signaling
1391 * a degeneracy) and positive integer k means that rows ro and k were swapped.
1393 int matrix::pivot(unsigned ro, unsigned co, bool symbolic)
1397 // search first non-zero element in column co beginning at row ro
1398 while ((k<row) && (this->m[k*col+co].expand().is_zero()))
1401 // search largest element in column co beginning at row ro
1402 GINAC_ASSERT(is_a<numeric>(this->m[k*col+co]));
1403 unsigned kmax = k+1;
1404 numeric mmax = abs(ex_to<numeric>(m[kmax*col+co]));
1406 GINAC_ASSERT(is_a<numeric>(this->m[kmax*col+co]));
1407 numeric tmp = ex_to<numeric>(this->m[kmax*col+co]);
1408 if (abs(tmp) > mmax) {
1414 if (!mmax.is_zero())
1418 // all elements in column co below row ro vanish
1421 // matrix needs no pivoting
1423 // matrix needs pivoting, so swap rows k and ro
1424 ensure_if_modifiable();
1425 for (unsigned c=0; c<col; ++c)
1426 this->m[k*col+c].swap(this->m[ro*col+c]);
1431 ex lst_to_matrix(const lst & l)
1433 // Find number of rows and columns
1434 unsigned rows = l.nops(), cols = 0, i, j;
1435 for (i=0; i<rows; i++)
1436 if (l.op(i).nops() > cols)
1437 cols = l.op(i).nops();
1439 // Allocate and fill matrix
1440 matrix &m = *new matrix(rows, cols);
1441 m.setflag(status_flags::dynallocated);
1442 for (i=0; i<rows; i++)
1443 for (j=0; j<cols; j++)
1444 if (l.op(i).nops() > j)
1445 m(i, j) = l.op(i).op(j);
1451 ex diag_matrix(const lst & l)
1453 unsigned dim = l.nops();
1455 matrix &m = *new matrix(dim, dim);
1456 m.setflag(status_flags::dynallocated);
1457 for (unsigned i=0; i<dim; i++)
1463 } // namespace GiNaC