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
41 GINAC_IMPLEMENT_REGISTERED_CLASS(matrix, basic)
44 // default ctor, dtor, copy ctor, assignment operator and helpers:
47 /** Default ctor. Initializes to 1 x 1-dimensional zero-matrix. */
48 matrix::matrix() : inherited(TINFO_matrix), row(1), col(1)
53 void matrix::copy(const matrix & other)
55 inherited::copy(other);
58 m = other.m; // STL's vector copying invoked here
61 DEFAULT_DESTROY(matrix)
69 /** Very common ctor. Initializes to r x c-dimensional zero-matrix.
71 * @param r number of rows
72 * @param c number of cols */
73 matrix::matrix(unsigned r, unsigned c)
74 : inherited(TINFO_matrix), row(r), col(c)
81 /** Ctor from representation, for internal use only. */
82 matrix::matrix(unsigned r, unsigned c, const exvector & m2)
83 : inherited(TINFO_matrix), row(r), col(c), m(m2) {}
85 /** Construct matrix from (flat) list of elements. If the list has fewer
86 * elements than the matrix, the remaining matrix elements are set to zero.
87 * If the list has more elements than the matrix, the excessive elements are
89 matrix::matrix(unsigned r, unsigned c, const lst & l)
90 : inherited(TINFO_matrix), row(r), col(c)
94 for (unsigned i=0; i<l.nops(); i++) {
98 break; // matrix smaller than list: throw away excessive elements
107 matrix::matrix(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
109 if (!(n.find_unsigned("row", row)) || !(n.find_unsigned("col", col)))
110 throw (std::runtime_error("unknown matrix dimensions in archive"));
111 m.reserve(row * col);
112 for (unsigned int i=0; true; i++) {
114 if (n.find_ex("m", e, sym_lst, i))
121 void matrix::archive(archive_node &n) const
123 inherited::archive(n);
124 n.add_unsigned("row", row);
125 n.add_unsigned("col", col);
126 exvector::const_iterator i = m.begin(), iend = m.end();
133 DEFAULT_UNARCHIVE(matrix)
136 // functions overriding virtual functions from base classes
141 void matrix::print(const print_context & c, unsigned level) const
143 if (is_a<print_tree>(c)) {
145 inherited::print(c, level);
150 for (unsigned y=0; y<row-1; ++y) {
152 for (unsigned x=0; x<col-1; ++x) {
156 m[col*(y+1)-1].print(c);
160 for (unsigned x=0; x<col-1; ++x) {
161 m[(row-1)*col+x].print(c);
164 m[row*col-1].print(c);
170 /** nops is defined to be rows x columns. */
171 unsigned matrix::nops() const
176 /** returns matrix entry at position (i/col, i%col). */
177 ex matrix::op(int i) const
182 /** returns matrix entry at position (i/col, i%col). */
183 ex & matrix::let_op(int i)
186 GINAC_ASSERT(i<nops());
191 /** Evaluate matrix entry by entry. */
192 ex matrix::eval(int level) const
194 // check if we have to do anything at all
195 if ((level==1)&&(flags & status_flags::evaluated))
199 if (level == -max_recursion_level)
200 throw (std::runtime_error("matrix::eval(): recursion limit exceeded"));
202 // eval() entry by entry
203 exvector m2(row*col);
205 for (unsigned r=0; r<row; ++r)
206 for (unsigned c=0; c<col; ++c)
207 m2[r*col+c] = m[r*col+c].eval(level);
209 return (new matrix(row, col, m2))->setflag(status_flags::dynallocated |
210 status_flags::evaluated );
213 ex matrix::subs(const lst & ls, const lst & lr, bool no_pattern) const
215 exvector m2(row * col);
216 for (unsigned r=0; r<row; ++r)
217 for (unsigned c=0; c<col; ++c)
218 m2[r*col+c] = m[r*col+c].subs(ls, lr, no_pattern);
220 return matrix(row, col, m2).basic::subs(ls, lr, no_pattern);
225 int matrix::compare_same_type(const basic & other) const
227 GINAC_ASSERT(is_exactly_a<matrix>(other));
228 const matrix &o = static_cast<const matrix &>(other);
230 // compare number of rows
232 return row < o.rows() ? -1 : 1;
234 // compare number of columns
236 return col < o.cols() ? -1 : 1;
238 // equal number of rows and columns, compare individual elements
240 for (unsigned r=0; r<row; ++r) {
241 for (unsigned c=0; c<col; ++c) {
242 cmpval = ((*this)(r,c)).compare(o(r,c));
243 if (cmpval!=0) return cmpval;
246 // all elements are equal => matrices are equal;
250 bool matrix::match_same_type(const basic & other) const
252 GINAC_ASSERT(is_exactly_a<matrix>(other));
253 const matrix & o = static_cast<const matrix &>(other);
255 // The number of rows and columns must be the same. This is necessary to
256 // prevent a 2x3 matrix from matching a 3x2 one.
257 return row == o.rows() && col == o.cols();
260 /** Automatic symbolic evaluation of an indexed matrix. */
261 ex matrix::eval_indexed(const basic & i) const
263 GINAC_ASSERT(is_a<indexed>(i));
264 GINAC_ASSERT(is_a<matrix>(i.op(0)));
266 bool all_indices_unsigned = static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint);
271 // One index, must be one-dimensional vector
272 if (row != 1 && col != 1)
273 throw (std::runtime_error("matrix::eval_indexed(): vector must have exactly 1 index"));
275 const idx & i1 = ex_to<idx>(i.op(1));
280 if (!i1.get_dim().is_equal(row))
281 throw (std::runtime_error("matrix::eval_indexed(): dimension of index must match number of vector elements"));
283 // Index numeric -> return vector element
284 if (all_indices_unsigned) {
285 unsigned n1 = ex_to<numeric>(i1.get_value()).to_int();
287 throw (std::runtime_error("matrix::eval_indexed(): value of index exceeds number of vector elements"));
288 return (*this)(n1, 0);
294 if (!i1.get_dim().is_equal(col))
295 throw (std::runtime_error("matrix::eval_indexed(): dimension of index must match number of vector elements"));
297 // Index numeric -> return vector element
298 if (all_indices_unsigned) {
299 unsigned n1 = ex_to<numeric>(i1.get_value()).to_int();
301 throw (std::runtime_error("matrix::eval_indexed(): value of index exceeds number of vector elements"));
302 return (*this)(0, n1);
306 } else if (i.nops() == 3) {
309 const idx & i1 = ex_to<idx>(i.op(1));
310 const idx & i2 = ex_to<idx>(i.op(2));
312 if (!i1.get_dim().is_equal(row))
313 throw (std::runtime_error("matrix::eval_indexed(): dimension of first index must match number of rows"));
314 if (!i2.get_dim().is_equal(col))
315 throw (std::runtime_error("matrix::eval_indexed(): dimension of second index must match number of columns"));
317 // Pair of dummy indices -> compute trace
318 if (is_dummy_pair(i1, i2))
321 // Both indices numeric -> return matrix element
322 if (all_indices_unsigned) {
323 unsigned n1 = ex_to<numeric>(i1.get_value()).to_int(), n2 = ex_to<numeric>(i2.get_value()).to_int();
325 throw (std::runtime_error("matrix::eval_indexed(): value of first index exceeds number of rows"));
327 throw (std::runtime_error("matrix::eval_indexed(): value of second index exceeds number of columns"));
328 return (*this)(n1, n2);
332 throw (std::runtime_error("matrix::eval_indexed(): matrix must have exactly 2 indices"));
337 /** Sum of two indexed matrices. */
338 ex matrix::add_indexed(const ex & self, const ex & other) const
340 GINAC_ASSERT(is_a<indexed>(self));
341 GINAC_ASSERT(is_a<matrix>(self.op(0)));
342 GINAC_ASSERT(is_a<indexed>(other));
343 GINAC_ASSERT(self.nops() == 2 || self.nops() == 3);
345 // Only add two matrices
346 if (is_ex_of_type(other.op(0), matrix)) {
347 GINAC_ASSERT(other.nops() == 2 || other.nops() == 3);
349 const matrix &self_matrix = ex_to<matrix>(self.op(0));
350 const matrix &other_matrix = ex_to<matrix>(other.op(0));
352 if (self.nops() == 2 && other.nops() == 2) { // vector + vector
354 if (self_matrix.row == other_matrix.row)
355 return indexed(self_matrix.add(other_matrix), self.op(1));
356 else if (self_matrix.row == other_matrix.col)
357 return indexed(self_matrix.add(other_matrix.transpose()), self.op(1));
359 } else if (self.nops() == 3 && other.nops() == 3) { // matrix + matrix
361 if (self.op(1).is_equal(other.op(1)) && self.op(2).is_equal(other.op(2)))
362 return indexed(self_matrix.add(other_matrix), self.op(1), self.op(2));
363 else if (self.op(1).is_equal(other.op(2)) && self.op(2).is_equal(other.op(1)))
364 return indexed(self_matrix.add(other_matrix.transpose()), self.op(1), self.op(2));
369 // Don't know what to do, return unevaluated sum
373 /** Product of an indexed matrix with a number. */
374 ex matrix::scalar_mul_indexed(const ex & self, const numeric & other) const
376 GINAC_ASSERT(is_a<indexed>(self));
377 GINAC_ASSERT(is_a<matrix>(self.op(0)));
378 GINAC_ASSERT(self.nops() == 2 || self.nops() == 3);
380 const matrix &self_matrix = ex_to<matrix>(self.op(0));
382 if (self.nops() == 2)
383 return indexed(self_matrix.mul(other), self.op(1));
384 else // self.nops() == 3
385 return indexed(self_matrix.mul(other), self.op(1), self.op(2));
388 /** Contraction of an indexed matrix with something else. */
389 bool matrix::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
391 GINAC_ASSERT(is_a<indexed>(*self));
392 GINAC_ASSERT(is_a<indexed>(*other));
393 GINAC_ASSERT(self->nops() == 2 || self->nops() == 3);
394 GINAC_ASSERT(is_a<matrix>(self->op(0)));
396 // Only contract with other matrices
397 if (!is_ex_of_type(other->op(0), matrix))
400 GINAC_ASSERT(other->nops() == 2 || other->nops() == 3);
402 const matrix &self_matrix = ex_to<matrix>(self->op(0));
403 const matrix &other_matrix = ex_to<matrix>(other->op(0));
405 if (self->nops() == 2) {
407 if (other->nops() == 2) { // vector * vector (scalar product)
409 if (self_matrix.col == 1) {
410 if (other_matrix.col == 1) {
411 // Column vector * column vector, transpose first vector
412 *self = self_matrix.transpose().mul(other_matrix)(0, 0);
414 // Column vector * row vector, swap factors
415 *self = other_matrix.mul(self_matrix)(0, 0);
418 if (other_matrix.col == 1) {
419 // Row vector * column vector, perfect
420 *self = self_matrix.mul(other_matrix)(0, 0);
422 // Row vector * row vector, transpose second vector
423 *self = self_matrix.mul(other_matrix.transpose())(0, 0);
429 } else { // vector * matrix
431 // B_i * A_ij = (B*A)_j (B is row vector)
432 if (is_dummy_pair(self->op(1), other->op(1))) {
433 if (self_matrix.row == 1)
434 *self = indexed(self_matrix.mul(other_matrix), other->op(2));
436 *self = indexed(self_matrix.transpose().mul(other_matrix), other->op(2));
441 // B_j * A_ij = (A*B)_i (B is column vector)
442 if (is_dummy_pair(self->op(1), other->op(2))) {
443 if (self_matrix.col == 1)
444 *self = indexed(other_matrix.mul(self_matrix), other->op(1));
446 *self = indexed(other_matrix.mul(self_matrix.transpose()), other->op(1));
452 } else if (other->nops() == 3) { // matrix * matrix
454 // A_ij * B_jk = (A*B)_ik
455 if (is_dummy_pair(self->op(2), other->op(1))) {
456 *self = indexed(self_matrix.mul(other_matrix), self->op(1), other->op(2));
461 // A_ij * B_kj = (A*Btrans)_ik
462 if (is_dummy_pair(self->op(2), other->op(2))) {
463 *self = indexed(self_matrix.mul(other_matrix.transpose()), self->op(1), other->op(1));
468 // A_ji * B_jk = (Atrans*B)_ik
469 if (is_dummy_pair(self->op(1), other->op(1))) {
470 *self = indexed(self_matrix.transpose().mul(other_matrix), self->op(2), other->op(2));
475 // A_ji * B_kj = (B*A)_ki
476 if (is_dummy_pair(self->op(1), other->op(2))) {
477 *self = indexed(other_matrix.mul(self_matrix), other->op(1), self->op(2));
488 // non-virtual functions in this class
495 * @exception logic_error (incompatible matrices) */
496 matrix matrix::add(const matrix & other) const
498 if (col != other.col || row != other.row)
499 throw std::logic_error("matrix::add(): incompatible matrices");
501 exvector sum(this->m);
502 exvector::iterator i = sum.begin(), end = sum.end();
503 exvector::const_iterator ci = other.m.begin();
507 return matrix(row,col,sum);
511 /** Difference of matrices.
513 * @exception logic_error (incompatible matrices) */
514 matrix matrix::sub(const matrix & other) const
516 if (col != other.col || row != other.row)
517 throw std::logic_error("matrix::sub(): incompatible matrices");
519 exvector dif(this->m);
520 exvector::iterator i = dif.begin(), end = dif.end();
521 exvector::const_iterator ci = other.m.begin();
525 return matrix(row,col,dif);
529 /** Product of matrices.
531 * @exception logic_error (incompatible matrices) */
532 matrix matrix::mul(const matrix & other) const
534 if (this->cols() != other.rows())
535 throw std::logic_error("matrix::mul(): incompatible matrices");
537 exvector prod(this->rows()*other.cols());
539 for (unsigned r1=0; r1<this->rows(); ++r1) {
540 for (unsigned c=0; c<this->cols(); ++c) {
541 if (m[r1*col+c].is_zero())
543 for (unsigned r2=0; r2<other.cols(); ++r2)
544 prod[r1*other.col+r2] += (m[r1*col+c] * other.m[c*other.col+r2]).expand();
547 return matrix(row, other.col, prod);
551 /** Product of matrix and scalar. */
552 matrix matrix::mul(const numeric & other) const
554 exvector prod(row * col);
556 for (unsigned r=0; r<row; ++r)
557 for (unsigned c=0; c<col; ++c)
558 prod[r*col+c] = m[r*col+c] * other;
560 return matrix(row, col, prod);
564 /** Product of matrix and scalar expression. */
565 matrix matrix::mul_scalar(const ex & other) const
567 if (other.return_type() != return_types::commutative)
568 throw std::runtime_error("matrix::mul_scalar(): non-commutative scalar");
570 exvector prod(row * col);
572 for (unsigned r=0; r<row; ++r)
573 for (unsigned c=0; c<col; ++c)
574 prod[r*col+c] = m[r*col+c] * other;
576 return matrix(row, col, prod);
580 /** Power of a matrix. Currently handles integer exponents only. */
581 matrix matrix::pow(const ex & expn) const
584 throw (std::logic_error("matrix::pow(): matrix not square"));
586 if (is_ex_exactly_of_type(expn, numeric)) {
587 // Integer cases are computed by successive multiplication, using the
588 // obvious shortcut of storing temporaries, like A^4 == (A*A)*(A*A).
589 if (expn.info(info_flags::integer)) {
590 numeric b = ex_to<numeric>(expn);
592 if (expn.info(info_flags::negative)) {
599 for (unsigned r=0; r<row; ++r)
601 // This loop computes the representation of b in base 2 from right
602 // to left and multiplies the factors whenever needed. Note
603 // that this is not entirely optimal but close to optimal and
604 // "better" algorithms are much harder to implement. (See Knuth,
605 // TAoCP2, section "Evaluation of Powers" for a good discussion.)
611 b *= _num1_2; // b /= 2, still integer.
617 throw (std::runtime_error("matrix::pow(): don't know how to handle exponent"));
621 /** operator() to access elements for reading.
623 * @param ro row of element
624 * @param co column of element
625 * @exception range_error (index out of range) */
626 const ex & matrix::operator() (unsigned ro, unsigned co) const
628 if (ro>=row || co>=col)
629 throw (std::range_error("matrix::operator(): index out of range"));
635 /** operator() to access elements for writing.
637 * @param ro row of element
638 * @param co column of element
639 * @exception range_error (index out of range) */
640 ex & matrix::operator() (unsigned ro, unsigned co)
642 if (ro>=row || co>=col)
643 throw (std::range_error("matrix::operator(): index out of range"));
645 ensure_if_modifiable();
650 /** Transposed of an m x n matrix, producing a new n x m matrix object that
651 * represents the transposed. */
652 matrix matrix::transpose(void) const
654 exvector trans(this->cols()*this->rows());
656 for (unsigned r=0; r<this->cols(); ++r)
657 for (unsigned c=0; c<this->rows(); ++c)
658 trans[r*this->rows()+c] = m[c*this->cols()+r];
660 return matrix(this->cols(),this->rows(),trans);
663 /** Determinant of square matrix. This routine doesn't actually calculate the
664 * determinant, it only implements some heuristics about which algorithm to
665 * run. If all the elements of the matrix are elements of an integral domain
666 * the determinant is also in that integral domain and the result is expanded
667 * only. If one or more elements are from a quotient field the determinant is
668 * usually also in that quotient field and the result is normalized before it
669 * is returned. This implies that the determinant of the symbolic 2x2 matrix
670 * [[a/(a-b),1],[b/(a-b),1]] is returned as unity. (In this respect, it
671 * behaves like MapleV and unlike Mathematica.)
673 * @param algo allows to chose an algorithm
674 * @return the determinant as a new expression
675 * @exception logic_error (matrix not square)
676 * @see determinant_algo */
677 ex matrix::determinant(unsigned algo) const
680 throw (std::logic_error("matrix::determinant(): matrix not square"));
681 GINAC_ASSERT(row*col==m.capacity());
683 // Gather some statistical information about this matrix:
684 bool numeric_flag = true;
685 bool normal_flag = false;
686 unsigned sparse_count = 0; // counts non-zero elements
687 exvector::const_iterator r = m.begin(), rend = m.end();
689 lst srl; // symbol replacement list
690 ex rtest = r->to_rational(srl);
691 if (!rtest.is_zero())
693 if (!rtest.info(info_flags::numeric))
694 numeric_flag = false;
695 if (!rtest.info(info_flags::crational_polynomial) &&
696 rtest.info(info_flags::rational_function))
701 // Here is the heuristics in case this routine has to decide:
702 if (algo == determinant_algo::automatic) {
703 // Minor expansion is generally a good guess:
704 algo = determinant_algo::laplace;
705 // Does anybody know when a matrix is really sparse?
706 // Maybe <~row/2.236 nonzero elements average in a row?
707 if (row>3 && 5*sparse_count<=row*col)
708 algo = determinant_algo::bareiss;
709 // Purely numeric matrix can be handled by Gauss elimination.
710 // This overrides any prior decisions.
712 algo = determinant_algo::gauss;
715 // Trap the trivial case here, since some algorithms don't like it
717 // for consistency with non-trivial determinants...
719 return m[0].normal();
721 return m[0].expand();
724 // Compute the determinant
726 case determinant_algo::gauss: {
729 int sign = tmp.gauss_elimination(true);
730 for (unsigned d=0; d<row; ++d)
731 det *= tmp.m[d*col+d];
733 return (sign*det).normal();
735 return (sign*det).normal().expand();
737 case determinant_algo::bareiss: {
740 sign = tmp.fraction_free_elimination(true);
742 return (sign*tmp.m[row*col-1]).normal();
744 return (sign*tmp.m[row*col-1]).expand();
746 case determinant_algo::divfree: {
749 sign = tmp.division_free_elimination(true);
752 ex det = tmp.m[row*col-1];
753 // factor out accumulated bogus slag
754 for (unsigned d=0; d<row-2; ++d)
755 for (unsigned j=0; j<row-d-2; ++j)
756 det = (det/tmp.m[d*col+d]).normal();
759 case determinant_algo::laplace:
761 // This is the minor expansion scheme. We always develop such
762 // that the smallest minors (i.e, the trivial 1x1 ones) are on the
763 // rightmost column. For this to be efficient, empirical tests
764 // have shown that the emptiest columns (i.e. the ones with most
765 // zeros) should be the ones on the right hand side -- although
766 // this might seem counter-intuitive (and in contradiction to some
767 // literature like the FORM manual). Please go ahead and test it
768 // if you don't believe me! Therefore we presort the columns of
770 typedef std::pair<unsigned,unsigned> uintpair;
771 std::vector<uintpair> c_zeros; // number of zeros in column
772 for (unsigned c=0; c<col; ++c) {
774 for (unsigned r=0; r<row; ++r)
775 if (m[r*col+c].is_zero())
777 c_zeros.push_back(uintpair(acc,c));
779 sort(c_zeros.begin(),c_zeros.end());
780 std::vector<unsigned> pre_sort;
781 for (std::vector<uintpair>::const_iterator i=c_zeros.begin(); i!=c_zeros.end(); ++i)
782 pre_sort.push_back(i->second);
783 std::vector<unsigned> pre_sort_test(pre_sort); // permutation_sign() modifies the vector so we make a copy here
784 int sign = permutation_sign(pre_sort_test.begin(), pre_sort_test.end());
785 exvector result(row*col); // represents sorted matrix
787 for (std::vector<unsigned>::const_iterator i=pre_sort.begin();
790 for (unsigned r=0; r<row; ++r)
791 result[r*col+c] = m[r*col+(*i)];
795 return (sign*matrix(row,col,result).determinant_minor()).normal();
797 return sign*matrix(row,col,result).determinant_minor();
803 /** Trace of a matrix. The result is normalized if it is in some quotient
804 * field and expanded only otherwise. This implies that the trace of the
805 * symbolic 2x2 matrix [[a/(a-b),x],[y,b/(b-a)]] is recognized to be unity.
807 * @return the sum of diagonal elements
808 * @exception logic_error (matrix not square) */
809 ex matrix::trace(void) const
812 throw (std::logic_error("matrix::trace(): matrix not square"));
815 for (unsigned r=0; r<col; ++r)
818 if (tr.info(info_flags::rational_function) &&
819 !tr.info(info_flags::crational_polynomial))
826 /** Characteristic Polynomial. Following mathematica notation the
827 * characteristic polynomial of a matrix M is defined as the determiant of
828 * (M - lambda * 1) where 1 stands for the unit matrix of the same dimension
829 * as M. Note that some CASs define it with a sign inside the determinant
830 * which gives rise to an overall sign if the dimension is odd. This method
831 * returns the characteristic polynomial collected in powers of lambda as a
834 * @return characteristic polynomial as new expression
835 * @exception logic_error (matrix not square)
836 * @see matrix::determinant() */
837 ex matrix::charpoly(const symbol & lambda) const
840 throw (std::logic_error("matrix::charpoly(): matrix not square"));
842 bool numeric_flag = true;
843 exvector::const_iterator r = m.begin(), rend = m.end();
844 while (r!=rend && numeric_flag==true) {
845 if (!r->info(info_flags::numeric))
846 numeric_flag = false;
850 // The pure numeric case is traditionally rather common. Hence, it is
851 // trapped and we use Leverrier's algorithm which goes as row^3 for
852 // every coefficient. The expensive part is the matrix multiplication.
856 ex poly = power(lambda,row)-c*power(lambda,row-1);
857 for (unsigned i=1; i<row; ++i) {
858 for (unsigned j=0; j<row; ++j)
861 c = B.trace()/ex(i+1);
862 poly -= c*power(lambda,row-i-1);
871 for (unsigned r=0; r<col; ++r)
872 M.m[r*col+r] -= lambda;
874 return M.determinant().collect(lambda);
878 /** Inverse of this matrix.
880 * @return the inverted matrix
881 * @exception logic_error (matrix not square)
882 * @exception runtime_error (singular matrix) */
883 matrix matrix::inverse(void) const
886 throw (std::logic_error("matrix::inverse(): matrix not square"));
888 // This routine actually doesn't do anything fancy at all. We compute the
889 // inverse of the matrix A by solving the system A * A^{-1} == Id.
891 // First populate the identity matrix supposed to become the right hand side.
892 matrix identity(row,col);
893 for (unsigned i=0; i<row; ++i)
894 identity(i,i) = _ex1;
896 // Populate a dummy matrix of variables, just because of compatibility with
897 // matrix::solve() which wants this (for compatibility with under-determined
898 // systems of equations).
899 matrix vars(row,col);
900 for (unsigned r=0; r<row; ++r)
901 for (unsigned c=0; c<col; ++c)
902 vars(r,c) = symbol();
906 sol = this->solve(vars,identity);
907 } catch (const std::runtime_error & e) {
908 if (e.what()==std::string("matrix::solve(): inconsistent linear system"))
909 throw (std::runtime_error("matrix::inverse(): singular matrix"));
917 /** Solve a linear system consisting of a m x n matrix and a m x p right hand
918 * side by applying an elimination scheme to the augmented matrix.
920 * @param vars n x p matrix, all elements must be symbols
921 * @param rhs m x p matrix
922 * @return n x p solution matrix
923 * @exception logic_error (incompatible matrices)
924 * @exception invalid_argument (1st argument must be matrix of symbols)
925 * @exception runtime_error (inconsistent linear system)
927 matrix matrix::solve(const matrix & vars,
931 const unsigned m = this->rows();
932 const unsigned n = this->cols();
933 const unsigned p = rhs.cols();
936 if ((rhs.rows() != m) || (vars.rows() != n) || (vars.col != p))
937 throw (std::logic_error("matrix::solve(): incompatible matrices"));
938 for (unsigned ro=0; ro<n; ++ro)
939 for (unsigned co=0; co<p; ++co)
940 if (!vars(ro,co).info(info_flags::symbol))
941 throw (std::invalid_argument("matrix::solve(): 1st argument must be matrix of symbols"));
943 // build the augmented matrix of *this with rhs attached to the right
945 for (unsigned r=0; r<m; ++r) {
946 for (unsigned c=0; c<n; ++c)
947 aug.m[r*(n+p)+c] = this->m[r*n+c];
948 for (unsigned c=0; c<p; ++c)
949 aug.m[r*(n+p)+c+n] = rhs.m[r*p+c];
952 // Gather some statistical information about the augmented matrix:
953 bool numeric_flag = true;
954 exvector::const_iterator r = aug.m.begin(), rend = aug.m.end();
955 while (r!=rend && numeric_flag==true) {
956 if (!r->info(info_flags::numeric))
957 numeric_flag = false;
961 // Here is the heuristics in case this routine has to decide:
962 if (algo == solve_algo::automatic) {
963 // Bareiss (fraction-free) elimination is generally a good guess:
964 algo = solve_algo::bareiss;
965 // For m<3, Bareiss elimination is equivalent to division free
966 // elimination but has more logistic overhead
968 algo = solve_algo::divfree;
969 // This overrides any prior decisions.
971 algo = solve_algo::gauss;
974 // Eliminate the augmented matrix:
976 case solve_algo::gauss:
977 aug.gauss_elimination();
979 case solve_algo::divfree:
980 aug.division_free_elimination();
982 case solve_algo::bareiss:
984 aug.fraction_free_elimination();
987 // assemble the solution matrix:
989 for (unsigned co=0; co<p; ++co) {
990 unsigned last_assigned_sol = n+1;
991 for (int r=m-1; r>=0; --r) {
992 unsigned fnz = 1; // first non-zero in row
993 while ((fnz<=n) && (aug.m[r*(n+p)+(fnz-1)].is_zero()))
996 // row consists only of zeros, corresponding rhs must be 0, too
997 if (!aug.m[r*(n+p)+n+co].is_zero()) {
998 throw (std::runtime_error("matrix::solve(): inconsistent linear system"));
1001 // assign solutions for vars between fnz+1 and
1002 // last_assigned_sol-1: free parameters
1003 for (unsigned c=fnz; c<last_assigned_sol-1; ++c)
1004 sol(c,co) = vars.m[c*p+co];
1005 ex e = aug.m[r*(n+p)+n+co];
1006 for (unsigned c=fnz; c<n; ++c)
1007 e -= aug.m[r*(n+p)+c]*sol.m[c*p+co];
1008 sol(fnz-1,co) = (e/(aug.m[r*(n+p)+(fnz-1)])).normal();
1009 last_assigned_sol = fnz;
1012 // assign solutions for vars between 1 and
1013 // last_assigned_sol-1: free parameters
1014 for (unsigned ro=0; ro<last_assigned_sol-1; ++ro)
1015 sol(ro,co) = vars(ro,co);
1024 /** Recursive determinant for small matrices having at least one symbolic
1025 * entry. The basic algorithm, known as Laplace-expansion, is enhanced by
1026 * some bookkeeping to avoid calculation of the same submatrices ("minors")
1027 * more than once. According to W.M.Gentleman and S.C.Johnson this algorithm
1028 * is better than elimination schemes for matrices of sparse multivariate
1029 * polynomials and also for matrices of dense univariate polynomials if the
1030 * matrix' dimesion is larger than 7.
1032 * @return the determinant as a new expression (in expanded form)
1033 * @see matrix::determinant() */
1034 ex matrix::determinant_minor(void) const
1036 // for small matrices the algorithm does not make any sense:
1037 const unsigned n = this->cols();
1039 return m[0].expand();
1041 return (m[0]*m[3]-m[2]*m[1]).expand();
1043 return (m[0]*m[4]*m[8]-m[0]*m[5]*m[7]-
1044 m[1]*m[3]*m[8]+m[2]*m[3]*m[7]+
1045 m[1]*m[5]*m[6]-m[2]*m[4]*m[6]).expand();
1047 // This algorithm can best be understood by looking at a naive
1048 // implementation of Laplace-expansion, like this one:
1050 // matrix minorM(this->rows()-1,this->cols()-1);
1051 // for (unsigned r1=0; r1<this->rows(); ++r1) {
1052 // // shortcut if element(r1,0) vanishes
1053 // if (m[r1*col].is_zero())
1055 // // assemble the minor matrix
1056 // for (unsigned r=0; r<minorM.rows(); ++r) {
1057 // for (unsigned c=0; c<minorM.cols(); ++c) {
1059 // minorM(r,c) = m[r*col+c+1];
1061 // minorM(r,c) = m[(r+1)*col+c+1];
1064 // // recurse down and care for sign:
1066 // det -= m[r1*col] * minorM.determinant_minor();
1068 // det += m[r1*col] * minorM.determinant_minor();
1070 // return det.expand();
1071 // What happens is that while proceeding down many of the minors are
1072 // computed more than once. In particular, there are binomial(n,k)
1073 // kxk minors and each one is computed factorial(n-k) times. Therefore
1074 // it is reasonable to store the results of the minors. We proceed from
1075 // right to left. At each column c we only need to retrieve the minors
1076 // calculated in step c-1. We therefore only have to store at most
1077 // 2*binomial(n,n/2) minors.
1079 // Unique flipper counter for partitioning into minors
1080 std::vector<unsigned> Pkey;
1082 // key for minor determinant (a subpartition of Pkey)
1083 std::vector<unsigned> Mkey;
1085 // we store our subminors in maps, keys being the rows they arise from
1086 typedef std::map<std::vector<unsigned>,class ex> Rmap;
1087 typedef std::map<std::vector<unsigned>,class ex>::value_type Rmap_value;
1091 // initialize A with last column:
1092 for (unsigned r=0; r<n; ++r) {
1093 Pkey.erase(Pkey.begin(),Pkey.end());
1095 A.insert(Rmap_value(Pkey,m[n*(r+1)-1]));
1097 // proceed from right to left through matrix
1098 for (int c=n-2; c>=0; --c) {
1099 Pkey.erase(Pkey.begin(),Pkey.end()); // don't change capacity
1100 Mkey.erase(Mkey.begin(),Mkey.end());
1101 for (unsigned i=0; i<n-c; ++i)
1103 unsigned fc = 0; // controls logic for our strange flipper counter
1106 for (unsigned r=0; r<n-c; ++r) {
1107 // maybe there is nothing to do?
1108 if (m[Pkey[r]*n+c].is_zero())
1110 // create the sorted key for all possible minors
1111 Mkey.erase(Mkey.begin(),Mkey.end());
1112 for (unsigned i=0; i<n-c; ++i)
1114 Mkey.push_back(Pkey[i]);
1115 // Fetch the minors and compute the new determinant
1117 det -= m[Pkey[r]*n+c]*A[Mkey];
1119 det += m[Pkey[r]*n+c]*A[Mkey];
1121 // prevent build-up of deep nesting of expressions saves time:
1123 // store the new determinant at its place in B:
1125 B.insert(Rmap_value(Pkey,det));
1126 // increment our strange flipper counter
1127 for (fc=n-c; fc>0; --fc) {
1129 if (Pkey[fc-1]<fc+c)
1133 for (unsigned j=fc; j<n-c; ++j)
1134 Pkey[j] = Pkey[j-1]+1;
1136 // next column, so change the role of A and B:
1145 /** Perform the steps of an ordinary Gaussian elimination to bring the m x n
1146 * matrix into an upper echelon form. The algorithm is ok for matrices
1147 * with numeric coefficients but quite unsuited for symbolic matrices.
1149 * @param det may be set to true to save a lot of space if one is only
1150 * interested in the diagonal elements (i.e. for calculating determinants).
1151 * The others are set to zero in this case.
1152 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1153 * number of rows was swapped and 0 if the matrix is singular. */
1154 int matrix::gauss_elimination(const bool det)
1156 ensure_if_modifiable();
1157 const unsigned m = this->rows();
1158 const unsigned n = this->cols();
1159 GINAC_ASSERT(!det || n==m);
1163 for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
1164 int indx = pivot(r0, r1, true);
1168 return 0; // leaves *this in a messy state
1173 for (unsigned r2=r0+1; r2<m; ++r2) {
1174 if (!this->m[r2*n+r1].is_zero()) {
1175 // yes, there is something to do in this row
1176 ex piv = this->m[r2*n+r1] / this->m[r0*n+r1];
1177 for (unsigned c=r1+1; c<n; ++c) {
1178 this->m[r2*n+c] -= piv * this->m[r0*n+c];
1179 if (!this->m[r2*n+c].info(info_flags::numeric))
1180 this->m[r2*n+c] = this->m[r2*n+c].normal();
1183 // fill up left hand side with zeros
1184 for (unsigned c=0; c<=r1; ++c)
1185 this->m[r2*n+c] = _ex0;
1188 // save space by deleting no longer needed elements
1189 for (unsigned c=r0+1; c<n; ++c)
1190 this->m[r0*n+c] = _ex0;
1200 /** Perform the steps of division free elimination to bring the m x n matrix
1201 * into an upper echelon form.
1203 * @param det may be set to true to save a lot of space if one is only
1204 * interested in the diagonal elements (i.e. for calculating determinants).
1205 * The others are set to zero in this case.
1206 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1207 * number of rows was swapped and 0 if the matrix is singular. */
1208 int matrix::division_free_elimination(const bool det)
1210 ensure_if_modifiable();
1211 const unsigned m = this->rows();
1212 const unsigned n = this->cols();
1213 GINAC_ASSERT(!det || n==m);
1217 for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
1218 int indx = pivot(r0, r1, true);
1222 return 0; // leaves *this in a messy state
1227 for (unsigned r2=r0+1; r2<m; ++r2) {
1228 for (unsigned c=r1+1; c<n; ++c)
1229 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();
1230 // fill up left hand side with zeros
1231 for (unsigned c=0; c<=r1; ++c)
1232 this->m[r2*n+c] = _ex0;
1235 // save space by deleting no longer needed elements
1236 for (unsigned c=r0+1; c<n; ++c)
1237 this->m[r0*n+c] = _ex0;
1247 /** Perform the steps of Bareiss' one-step fraction free elimination to bring
1248 * the matrix into an upper echelon form. Fraction free elimination means
1249 * that divide is used straightforwardly, without computing GCDs first. This
1250 * is possible, since we know the divisor at each step.
1252 * @param det may be set to true to save a lot of space if one is only
1253 * interested in the last element (i.e. for calculating determinants). The
1254 * others are set to zero in this case.
1255 * @return sign is 1 if an even number of rows was swapped, -1 if an odd
1256 * number of rows was swapped and 0 if the matrix is singular. */
1257 int matrix::fraction_free_elimination(const bool det)
1260 // (single-step fraction free elimination scheme, already known to Jordan)
1262 // Usual division-free elimination sets m[0](r,c) = m(r,c) and then sets
1263 // m[k+1](r,c) = m[k](k,k) * m[k](r,c) - m[k](r,k) * m[k](k,c).
1265 // Bareiss (fraction-free) elimination in addition divides that element
1266 // by m[k-1](k-1,k-1) for k>1, where it can be shown by means of the
1267 // Sylvester determinant that this really divides m[k+1](r,c).
1269 // We also allow rational functions where the original prove still holds.
1270 // However, we must care for numerator and denominator separately and
1271 // "manually" work in the integral domains because of subtle cancellations
1272 // (see below). This blows up the bookkeeping a bit and the formula has
1273 // to be modified to expand like this (N{x} stands for numerator of x,
1274 // D{x} for denominator of x):
1275 // 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)}
1276 // -N{m[k](r,k)}*N{m[k](k,c)}*D{m[k](k,k)}*D{m[k](r,c)}
1277 // 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)}
1278 // where for k>1 we now divide N{m[k+1](r,c)} by
1279 // N{m[k-1](k-1,k-1)}
1280 // and D{m[k+1](r,c)} by
1281 // D{m[k-1](k-1,k-1)}.
1283 ensure_if_modifiable();
1284 const unsigned m = this->rows();
1285 const unsigned n = this->cols();
1286 GINAC_ASSERT(!det || n==m);
1295 // We populate temporary matrices to subsequently operate on. There is
1296 // one holding numerators and another holding denominators of entries.
1297 // This is a must since the evaluator (or even earlier mul's constructor)
1298 // might cancel some trivial element which causes divide() to fail. The
1299 // elements are normalized first (yes, even though this algorithm doesn't
1300 // need GCDs) since the elements of *this might be unnormalized, which
1301 // makes things more complicated than they need to be.
1302 matrix tmp_n(*this);
1303 matrix tmp_d(m,n); // for denominators, if needed
1304 lst srl; // symbol replacement list
1305 exvector::const_iterator cit = this->m.begin(), citend = this->m.end();
1306 exvector::iterator tmp_n_it = tmp_n.m.begin(), tmp_d_it = tmp_d.m.begin();
1307 while (cit != citend) {
1308 ex nd = cit->normal().to_rational(srl).numer_denom();
1310 *tmp_n_it++ = nd.op(0);
1311 *tmp_d_it++ = nd.op(1);
1315 for (unsigned r1=0; (r1<n-1)&&(r0<m-1); ++r1) {
1316 int indx = tmp_n.pivot(r0, r1, true);
1325 // tmp_n's rows r0 and indx were swapped, do the same in tmp_d:
1326 for (unsigned c=r1; c<n; ++c)
1327 tmp_d.m[n*indx+c].swap(tmp_d.m[n*r0+c]);
1329 for (unsigned r2=r0+1; r2<m; ++r2) {
1330 for (unsigned c=r1+1; c<n; ++c) {
1331 dividend_n = (tmp_n.m[r0*n+r1]*tmp_n.m[r2*n+c]*
1332 tmp_d.m[r2*n+r1]*tmp_d.m[r0*n+c]
1333 -tmp_n.m[r2*n+r1]*tmp_n.m[r0*n+c]*
1334 tmp_d.m[r0*n+r1]*tmp_d.m[r2*n+c]).expand();
1335 dividend_d = (tmp_d.m[r2*n+r1]*tmp_d.m[r0*n+c]*
1336 tmp_d.m[r0*n+r1]*tmp_d.m[r2*n+c]).expand();
1337 bool check = divide(dividend_n, divisor_n,
1338 tmp_n.m[r2*n+c], true);
1339 check &= divide(dividend_d, divisor_d,
1340 tmp_d.m[r2*n+c], true);
1341 GINAC_ASSERT(check);
1343 // fill up left hand side with zeros
1344 for (unsigned c=0; c<=r1; ++c)
1345 tmp_n.m[r2*n+c] = _ex0;
1347 if ((r1<n-1)&&(r0<m-1)) {
1348 // compute next iteration's divisor
1349 divisor_n = tmp_n.m[r0*n+r1].expand();
1350 divisor_d = tmp_d.m[r0*n+r1].expand();
1352 // save space by deleting no longer needed elements
1353 for (unsigned c=0; c<n; ++c) {
1354 tmp_n.m[r0*n+c] = _ex0;
1355 tmp_d.m[r0*n+c] = _ex1;
1362 // repopulate *this matrix:
1363 exvector::iterator it = this->m.begin(), itend = this->m.end();
1364 tmp_n_it = tmp_n.m.begin();
1365 tmp_d_it = tmp_d.m.begin();
1367 *it++ = ((*tmp_n_it++)/(*tmp_d_it++)).subs(srl);
1373 /** Partial pivoting method for matrix elimination schemes.
1374 * Usual pivoting (symbolic==false) returns the index to the element with the
1375 * largest absolute value in column ro and swaps the current row with the one
1376 * where the element was found. With (symbolic==true) it does the same thing
1377 * with the first non-zero element.
1379 * @param ro is the row from where to begin
1380 * @param co is the column to be inspected
1381 * @param symbolic signal if we want the first non-zero element to be pivoted
1382 * (true) or the one with the largest absolute value (false).
1383 * @return 0 if no interchange occured, -1 if all are zero (usually signaling
1384 * a degeneracy) and positive integer k means that rows ro and k were swapped.
1386 int matrix::pivot(unsigned ro, unsigned co, bool symbolic)
1390 // search first non-zero element in column co beginning at row ro
1391 while ((k<row) && (this->m[k*col+co].expand().is_zero()))
1394 // search largest element in column co beginning at row ro
1395 GINAC_ASSERT(is_a<numeric>(this->m[k*col+co]));
1396 unsigned kmax = k+1;
1397 numeric mmax = abs(ex_to<numeric>(m[kmax*col+co]));
1399 GINAC_ASSERT(is_a<numeric>(this->m[kmax*col+co]));
1400 numeric tmp = ex_to<numeric>(this->m[kmax*col+co]);
1401 if (abs(tmp) > mmax) {
1407 if (!mmax.is_zero())
1411 // all elements in column co below row ro vanish
1414 // matrix needs no pivoting
1416 // matrix needs pivoting, so swap rows k and ro
1417 ensure_if_modifiable();
1418 for (unsigned c=0; c<col; ++c)
1419 this->m[k*col+c].swap(this->m[ro*col+c]);
1424 ex lst_to_matrix(const lst & l)
1426 // Find number of rows and columns
1427 unsigned rows = l.nops(), cols = 0, i, j;
1428 for (i=0; i<rows; i++)
1429 if (l.op(i).nops() > cols)
1430 cols = l.op(i).nops();
1432 // Allocate and fill matrix
1433 matrix &m = *new matrix(rows, cols);
1434 m.setflag(status_flags::dynallocated);
1435 for (i=0; i<rows; i++)
1436 for (j=0; j<cols; j++)
1437 if (l.op(i).nops() > j)
1438 m(i, j) = l.op(i).op(j);
1444 ex diag_matrix(const lst & l)
1446 unsigned dim = l.nops();
1448 matrix &m = *new matrix(dim, dim);
1449 m.setflag(status_flags::dynallocated);
1450 for (unsigned i=0; i<dim; i++)
1456 } // namespace GiNaC