3 * Implementation of class for extended truncated power series and
4 * methods for series expansion. */
7 * GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
9 * This program is free software; you can redistribute it and/or modify
10 * it under the terms of the GNU General Public License as published by
11 * the Free Software Foundation; either version 2 of the License, or
12 * (at your option) any later version.
14 * This program is distributed in the hope that it will be useful,
15 * but WITHOUT ANY WARRANTY; without even the implied warranty of
16 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
17 * GNU General Public License for more details.
19 * You should have received a copy of the GNU General Public License
20 * along with this program; if not, write to the Free Software
21 * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
32 #include "relational.h"
38 #ifndef NO_NAMESPACE_GINAC
40 #endif // ndef NO_NAMESPACE_GINAC
42 GINAC_IMPLEMENT_REGISTERED_CLASS(pseries, basic)
45 * Default constructor, destructor, copy constructor, assignment operator and helpers
48 pseries::pseries() : basic(TINFO_pseries)
50 debugmsg("pseries default constructor", LOGLEVEL_CONSTRUCT);
53 void pseries::copy(const pseries &other)
55 inherited::copy(other);
61 void pseries::destroy(bool call_parent)
64 inherited::destroy(call_parent);
72 /** Construct pseries from a vector of coefficients and powers.
73 * expair.rest holds the coefficient, expair.coeff holds the power.
74 * The powers must be integers (positive or negative) and in ascending order;
75 * the last coefficient can be Order(_ex1()) to represent a truncated,
76 * non-terminating series.
78 * @param rel_ expansion variable and point (must hold a relational)
79 * @param ops_ vector of {coefficient, power} pairs (coefficient must not be zero)
80 * @return newly constructed pseries */
81 pseries::pseries(const ex &rel_, const epvector &ops_) : basic(TINFO_pseries), seq(ops_)
83 debugmsg("pseries constructor from ex,epvector", LOGLEVEL_CONSTRUCT);
84 GINAC_ASSERT(is_ex_exactly_of_type(rel_, relational));
85 GINAC_ASSERT(is_ex_exactly_of_type(rel_.lhs(),symbol));
87 var = *static_cast<symbol *>(rel_.lhs().bp);
95 /** Construct object from archive_node. */
96 pseries::pseries(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
98 debugmsg("pseries constructor from archive_node", LOGLEVEL_CONSTRUCT);
99 for (unsigned int i=0; true; ++i) {
102 if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
103 seq.push_back(expair(rest, coeff));
107 n.find_ex("var", var, sym_lst);
108 n.find_ex("point", point, sym_lst);
111 /** Unarchive the object. */
112 ex pseries::unarchive(const archive_node &n, const lst &sym_lst)
114 return (new pseries(n, sym_lst))->setflag(status_flags::dynallocated);
117 /** Archive the object. */
118 void pseries::archive(archive_node &n) const
120 inherited::archive(n);
121 epvector::const_iterator i = seq.begin(), iend = seq.end();
123 n.add_ex("coeff", i->rest);
124 n.add_ex("power", i->coeff);
127 n.add_ex("var", var);
128 n.add_ex("point", point);
132 // functions overriding virtual functions from bases classes
135 void pseries::print(std::ostream &os, unsigned upper_precedence) const
137 debugmsg("pseries print", LOGLEVEL_PRINT);
138 if (precedence<=upper_precedence) os << "(";
139 for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
141 if (i->rest.is_zero())
143 // print a sign, if needed
146 if (!is_order_function(i->rest)) {
147 // print 'rest', i.e. the expansion coefficient
148 if (i->rest.info(info_flags::numeric) &&
149 i->rest.info(info_flags::positive)) {
152 os << "(" << i->rest << ')';
153 // print 'coeff', something like (x-1)^42
154 if (!i->coeff.is_zero()) {
156 if (!point.is_zero())
157 os << '(' << var-point << ')';
160 if (i->coeff.compare(_ex1())) {
162 if (i->coeff.info(info_flags::negative))
163 os << '(' << i->coeff << ')';
169 os << Order(power(var-point,i->coeff));
172 if (precedence<=upper_precedence) os << ")";
176 void pseries::printraw(std::ostream &os) const
178 debugmsg("pseries printraw", LOGLEVEL_PRINT);
179 os << "pseries(" << var << ";" << point << ";";
180 for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
181 os << "(" << (*i).rest << "," << (*i).coeff << "),";
187 void pseries::printtree(std::ostream & os, unsigned indent) const
189 debugmsg("pseries printtree",LOGLEVEL_PRINT);
190 os << std::string(indent,' ') << "pseries "
191 << ", hash=" << hashvalue
192 << " (0x" << std::hex << hashvalue << std::dec << ")"
193 << ", flags=" << flags << std::endl;
194 for (unsigned i=0; i<seq.size(); ++i) {
195 seq[i].rest.printtree(os,indent+delta_indent);
196 seq[i].coeff.printtree(os,indent+delta_indent);
198 os << std::string(indent+delta_indent,' ') << "-----" << std::endl;
200 var.printtree(os, indent+delta_indent);
201 point.printtree(os, indent+delta_indent);
204 int pseries::compare_same_type(const basic & other) const
206 GINAC_ASSERT(is_of_type(other, pseries));
207 const pseries &o = static_cast<const pseries &>(other);
209 int cmpval = var.compare(o.var);
212 cmpval = point.compare(o.point);
216 epvector::const_iterator it1 = seq.begin(), it2 = o.seq.begin(), it1end = seq.end(), it2end = o.seq.end();
217 while ((it1 != it1end) && (it2 != it2end)) {
218 cmpval = it1->compare(*it2);
224 return it2 == it2end ? 0 : -1;
229 /** Return the number of operands including a possible order term. */
230 unsigned pseries::nops(void) const
236 /** Return the ith term in the series when represented as a sum. */
237 ex pseries::op(int i) const
239 if (i < 0 || unsigned(i) >= seq.size())
240 throw (std::out_of_range("op() out of range"));
241 return seq[i].rest * power(var - point, seq[i].coeff);
245 ex &pseries::let_op(int i)
247 throw (std::logic_error("let_op not defined for pseries"));
251 /** Return degree of highest power of the series. This is usually the exponent
252 * of the Order term. If s is not the expansion variable of the series, the
253 * series is examined termwise. */
254 int pseries::degree(const symbol &s) const
256 if (var.is_equal(s)) {
257 // Return last exponent
259 return ex_to_numeric((*(seq.end() - 1)).coeff).to_int();
263 epvector::const_iterator it = seq.begin(), itend = seq.end();
266 int max_pow = INT_MIN;
267 while (it != itend) {
268 int pow = it->rest.degree(s);
277 /** Return degree of lowest power of the series. This is usually the exponent
278 * of the leading term. If s is not the expansion variable of the series, the
279 * series is examined termwise. If s is the expansion variable but the
280 * expansion point is not zero the series is not expanded to find the degree.
281 * I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
282 int pseries::ldegree(const symbol &s) const
284 if (var.is_equal(s)) {
285 // Return first exponent
287 return ex_to_numeric((*(seq.begin())).coeff).to_int();
291 epvector::const_iterator it = seq.begin(), itend = seq.end();
294 int min_pow = INT_MAX;
295 while (it != itend) {
296 int pow = it->rest.ldegree(s);
305 ex pseries::coeff(const symbol &s, int n) const
307 if (var.is_equal(s)) {
311 // Binary search in sequence for given power
312 numeric looking_for = numeric(n);
313 int lo = 0, hi = seq.size() - 1;
315 int mid = (lo + hi) / 2;
316 GINAC_ASSERT(is_ex_exactly_of_type(seq[mid].coeff, numeric));
317 int cmp = ex_to_numeric(seq[mid].coeff).compare(looking_for);
323 return seq[mid].rest;
328 throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
333 return convert_to_poly().coeff(s, n);
337 ex pseries::collect(const symbol &s) const
343 /** Evaluate coefficients. */
344 ex pseries::eval(int level) const
349 if (level == -max_recursion_level)
350 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
352 // Construct a new series with evaluated coefficients
354 new_seq.reserve(seq.size());
355 epvector::const_iterator it = seq.begin(), itend = seq.end();
356 while (it != itend) {
357 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
360 return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
364 /** Evaluate coefficients numerically. */
365 ex pseries::evalf(int level) const
370 if (level == -max_recursion_level)
371 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
373 // Construct a new series with evaluated coefficients
375 new_seq.reserve(seq.size());
376 epvector::const_iterator it = seq.begin(), itend = seq.end();
377 while (it != itend) {
378 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
381 return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
385 ex pseries::subs(const lst & ls, const lst & lr) const
387 // If expansion variable is being substituted, convert the series to a
388 // polynomial and do the substitution there because the result might
389 // no longer be a power series
391 return convert_to_poly(true).subs(ls, lr);
393 // Otherwise construct a new series with substituted coefficients and
396 newseq.reserve(seq.size());
397 epvector::const_iterator it = seq.begin(), itend = seq.end();
398 while (it != itend) {
399 newseq.push_back(expair(it->rest.subs(ls, lr), it->coeff));
402 return (new pseries(relational(var,point.subs(ls, lr)), newseq))->setflag(status_flags::dynallocated);
406 /** Implementation of ex::expand() for a power series. It expands all the
407 * terms individually and returns the resulting series as a new pseries.
409 ex pseries::expand(unsigned options) const
412 newseq.reserve(seq.size());
413 for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i)
414 newseq.push_back(expair(i->rest.expand(), i->coeff));
415 return (new pseries(relational(var,point), newseq))
416 ->setflag(status_flags::dynallocated | status_flags::expanded);
420 /** Implementation of ex::diff() for a power series. It treats the series as a
423 ex pseries::derivative(const symbol & s) const
427 epvector::const_iterator it = seq.begin(), itend = seq.end();
429 // FIXME: coeff might depend on var
430 while (it != itend) {
431 if (is_order_function(it->rest)) {
432 new_seq.push_back(expair(it->rest, it->coeff - 1));
434 ex c = it->rest * it->coeff;
436 new_seq.push_back(expair(c, it->coeff - 1));
440 return pseries(relational(var,point), new_seq);
448 * Construct ordinary polynomial out of series
451 /** Convert a pseries object to an ordinary polynomial.
453 * @param no_order flag: discard higher order terms */
454 ex pseries::convert_to_poly(bool no_order) const
457 epvector::const_iterator it = seq.begin(), itend = seq.end();
459 while (it != itend) {
460 if (is_order_function(it->rest)) {
462 e += Order(power(var - point, it->coeff));
464 e += it->rest * power(var - point, it->coeff);
470 /** Returns true if there is no order term, i.e. the series terminates and
471 * false otherwise. */
472 bool pseries::is_terminating(void) const
474 return seq.size() == 0 || !is_order_function((seq.end()-1)->rest);
479 * Implementation of series expansion
482 /** Default implementation of ex::series(). This performs Taylor expansion.
484 ex basic::series(const relational & r, int order, unsigned options) const
489 ex coeff = deriv.subs(r);
490 const symbol *s = static_cast<symbol *>(r.lhs().bp);
492 if (!coeff.is_zero())
493 seq.push_back(expair(coeff, numeric(0)));
496 for (n=1; n<order; ++n) {
497 fac = fac.mul(numeric(n));
498 deriv = deriv.diff(*s).expand();
499 if (deriv.is_zero()) {
501 return pseries(r, seq);
503 coeff = deriv.subs(r);
504 if (!coeff.is_zero())
505 seq.push_back(expair(fac.inverse() * coeff, numeric(n)));
508 // Higher-order terms, if present
509 deriv = deriv.diff(*s);
510 if (!deriv.expand().is_zero())
511 seq.push_back(expair(Order(_ex1()), numeric(n)));
512 return pseries(r, seq);
516 /** Implementation of ex::series() for symbols.
518 ex symbol::series(const relational & r, int order, unsigned options) const
521 const ex point = r.rhs();
522 GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
523 const symbol *s = static_cast<symbol *>(r.lhs().bp);
525 if (this->is_equal(*s)) {
526 if (order > 0 && !point.is_zero())
527 seq.push_back(expair(point, _ex0()));
529 seq.push_back(expair(_ex1(), _ex1()));
531 seq.push_back(expair(Order(_ex1()), numeric(order)));
533 seq.push_back(expair(*this, _ex0()));
534 return pseries(r, seq);
538 /** Add one series object to another, producing a pseries object that
539 * represents the sum.
541 * @param other pseries object to add with
542 * @return the sum as a pseries */
543 ex pseries::add_series(const pseries &other) const
545 // Adding two series with different variables or expansion points
546 // results in an empty (constant) series
547 if (!is_compatible_to(other)) {
549 nul.push_back(expair(Order(_ex1()), _ex0()));
550 return pseries(relational(var,point), nul);
555 epvector::const_iterator a = seq.begin();
556 epvector::const_iterator b = other.seq.begin();
557 epvector::const_iterator a_end = seq.end();
558 epvector::const_iterator b_end = other.seq.end();
559 int pow_a = INT_MAX, pow_b = INT_MAX;
561 // If a is empty, fill up with elements from b and stop
564 new_seq.push_back(*b);
569 pow_a = ex_to_numeric((*a).coeff).to_int();
571 // If b is empty, fill up with elements from a and stop
574 new_seq.push_back(*a);
579 pow_b = ex_to_numeric((*b).coeff).to_int();
581 // a and b are non-empty, compare powers
583 // a has lesser power, get coefficient from a
584 new_seq.push_back(*a);
585 if (is_order_function((*a).rest))
588 } else if (pow_b < pow_a) {
589 // b has lesser power, get coefficient from b
590 new_seq.push_back(*b);
591 if (is_order_function((*b).rest))
595 // Add coefficient of a and b
596 if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
597 new_seq.push_back(expair(Order(_ex1()), (*a).coeff));
598 break; // Order term ends the sequence
600 ex sum = (*a).rest + (*b).rest;
601 if (!(sum.is_zero()))
602 new_seq.push_back(expair(sum, numeric(pow_a)));
608 return pseries(relational(var,point), new_seq);
612 /** Implementation of ex::series() for sums. This performs series addition when
613 * adding pseries objects.
615 ex add::series(const relational & r, int order, unsigned options) const
617 ex acc; // Series accumulator
619 // Get first term from overall_coeff
620 acc = overall_coeff.series(r, order, options);
622 // Add remaining terms
623 epvector::const_iterator it = seq.begin();
624 epvector::const_iterator itend = seq.end();
625 for (; it!=itend; ++it) {
627 if (is_ex_exactly_of_type(it->rest, pseries))
630 op = it->rest.series(r, order, options);
631 if (!it->coeff.is_equal(_ex1()))
632 op = ex_to_pseries(op).mul_const(ex_to_numeric(it->coeff));
635 acc = ex_to_pseries(acc).add_series(ex_to_pseries(op));
641 /** Multiply a pseries object with a numeric constant, producing a pseries
642 * object that represents the product.
644 * @param other constant to multiply with
645 * @return the product as a pseries */
646 ex pseries::mul_const(const numeric &other) const
649 new_seq.reserve(seq.size());
651 epvector::const_iterator it = seq.begin(), itend = seq.end();
652 while (it != itend) {
653 if (!is_order_function(it->rest))
654 new_seq.push_back(expair(it->rest * other, it->coeff));
656 new_seq.push_back(*it);
659 return pseries(relational(var,point), new_seq);
663 /** Multiply one pseries object to another, producing a pseries object that
664 * represents the product.
666 * @param other pseries object to multiply with
667 * @return the product as a pseries */
668 ex pseries::mul_series(const pseries &other) const
670 // Multiplying two series with different variables or expansion points
671 // results in an empty (constant) series
672 if (!is_compatible_to(other)) {
674 nul.push_back(expair(Order(_ex1()), _ex0()));
675 return pseries(relational(var,point), nul);
678 // Series multiplication
681 const symbol *s = static_cast<symbol *>(var.bp);
682 int a_max = degree(*s);
683 int b_max = other.degree(*s);
684 int a_min = ldegree(*s);
685 int b_min = other.ldegree(*s);
686 int cdeg_min = a_min + b_min;
687 int cdeg_max = a_max + b_max;
689 int higher_order_a = INT_MAX;
690 int higher_order_b = INT_MAX;
691 if (is_order_function(coeff(*s, a_max)))
692 higher_order_a = a_max + b_min;
693 if (is_order_function(other.coeff(*s, b_max)))
694 higher_order_b = b_max + a_min;
695 int higher_order_c = std::min(higher_order_a, higher_order_b);
696 if (cdeg_max >= higher_order_c)
697 cdeg_max = higher_order_c - 1;
699 for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
701 // c(i)=a(0)b(i)+...+a(i)b(0)
702 for (int i=a_min; cdeg-i>=b_min; ++i) {
703 ex a_coeff = coeff(*s, i);
704 ex b_coeff = other.coeff(*s, cdeg-i);
705 if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
706 co += a_coeff * b_coeff;
709 new_seq.push_back(expair(co, numeric(cdeg)));
711 if (higher_order_c < INT_MAX)
712 new_seq.push_back(expair(Order(_ex1()), numeric(higher_order_c)));
713 return pseries(relational(var,point), new_seq);
717 /** Implementation of ex::series() for product. This performs series
718 * multiplication when multiplying series.
720 ex mul::series(const relational & r, int order, unsigned options) const
722 ex acc; // Series accumulator
724 // Get first term from overall_coeff
725 acc = overall_coeff.series(r, order, options);
727 // Multiply with remaining terms
728 epvector::const_iterator it = seq.begin();
729 epvector::const_iterator itend = seq.end();
730 for (; it!=itend; ++it) {
732 if (op.info(info_flags::numeric)) {
733 // series * const (special case, faster)
734 ex f = power(op, it->coeff);
735 acc = ex_to_pseries(acc).mul_const(ex_to_numeric(f));
737 } else if (!is_ex_exactly_of_type(op, pseries))
738 op = op.series(r, order, options);
739 if (!it->coeff.is_equal(_ex1()))
740 op = ex_to_pseries(op).power_const(ex_to_numeric(it->coeff), order);
742 // Series multiplication
743 acc = ex_to_pseries(acc).mul_series(ex_to_pseries(op));
749 /** Compute the p-th power of a series.
751 * @param p power to compute
752 * @param deg truncation order of series calculation */
753 ex pseries::power_const(const numeric &p, int deg) const
756 // let A(x) be this series and for the time being let it start with a
757 // constant (later we'll generalize):
758 // A(x) = a_0 + a_1*x + a_2*x^2 + ...
759 // We want to compute
761 // C(x) = c_0 + c_1*x + c_2*x^2 + ...
762 // Taking the derivative on both sides and multiplying with A(x) one
763 // immediately arrives at
764 // C'(x)*A(x) = p*C(x)*A'(x)
765 // Multiplying this out and comparing coefficients we get the recurrence
767 // c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
768 // ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
769 // which can easily be solved given the starting value c_0 = (a_0)^p.
770 // For the more general case where the leading coefficient of A(x) is not
771 // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
772 // repeat the above derivation. The leading power of C2(x) = A2(x)^2 is
773 // then of course x^(p*m) but the recurrence formula still holds.
774 const symbol *s = static_cast<symbol *>(var.bp);
775 int ldeg = ldegree(*s);
777 // Compute coefficients of the powered series
780 co.push_back(power(coeff(*s, ldeg), p));
781 bool all_sums_zero = true;
782 for (int i=1; i<deg; ++i) {
784 for (int j=1; j<=i; ++j) {
785 ex c = coeff(*s, j + ldeg);
786 if (is_order_function(c)) {
787 co.push_back(Order(_ex1()));
790 sum += (p * j - (i - j)) * co[i - j] * c;
793 all_sums_zero = false;
794 co.push_back(sum / coeff(*s, ldeg) / numeric(i));
797 // Construct new series (of non-zero coefficients)
799 bool higher_order = false;
800 for (int i=0; i<deg; ++i) {
801 if (!co[i].is_zero())
802 new_seq.push_back(expair(co[i], numeric(i) + p * ldeg));
803 if (is_order_function(co[i])) {
808 if (!higher_order && !all_sums_zero)
809 new_seq.push_back(expair(Order(_ex1()), numeric(deg) + p * ldeg));
810 return pseries(relational(var,point), new_seq);
814 /** Return a new pseries object with the powers shifted by deg. */
815 pseries pseries::shift_exponents(int deg) const
817 epvector newseq(seq);
818 for (epvector::iterator i=newseq.begin(); i!=newseq.end(); ++i)
819 i->coeff = i->coeff + deg;
820 return pseries(relational(var, point), newseq);
824 /** Implementation of ex::series() for powers. This performs Laurent expansion
825 * of reciprocals of series at singularities.
827 ex power::series(const relational & r, int order, unsigned options) const
830 if (!is_ex_exactly_of_type(basis, pseries)) {
831 // Basis is not a series, may there be a singularity?
832 bool must_expand_basis = false;
835 } catch (pole_error) {
836 must_expand_basis = true;
839 // Is the expression of type something^(-int)?
840 if (!must_expand_basis && !exponent.info(info_flags::negint))
841 return basic::series(r, order, options);
843 // Is the expression of type 0^something?
844 if (!must_expand_basis && !basis.subs(r).is_zero())
845 return basic::series(r, order, options);
847 // Singularity encountered, expand basis into series
848 e = basis.series(r, order, options);
855 return ex_to_pseries(e).power_const(ex_to_numeric(exponent), order);
859 /** Re-expansion of a pseries object. */
860 ex pseries::series(const relational & r, int order, unsigned options) const
862 const ex p = r.rhs();
863 GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
864 const symbol *s = static_cast<symbol *>(r.lhs().bp);
866 if (var.is_equal(*s) && point.is_equal(p)) {
867 if (order > degree(*s))
871 epvector::const_iterator it = seq.begin(), itend = seq.end();
872 while (it != itend) {
873 int o = ex_to_numeric(it->coeff).to_int();
875 new_seq.push_back(expair(Order(_ex1()), o));
878 new_seq.push_back(*it);
881 return pseries(r, new_seq);
884 return convert_to_poly().series(r, order, options);
888 /** Compute the truncated series expansion of an expression.
889 * This function returns an expression containing an object of class pseries
890 * to represent the series. If the series does not terminate within the given
891 * truncation order, the last term of the series will be an order term.
893 * @param r expansion relation, lhs holds variable and rhs holds point
894 * @param order truncation order of series calculations
895 * @param options of class series_options
896 * @return an expression holding a pseries object */
897 ex ex::series(const ex & r, int order, unsigned options) const
903 if (is_ex_exactly_of_type(r,relational))
904 rel_ = ex_to_relational(r);
905 else if (is_ex_exactly_of_type(r,symbol))
906 rel_ = relational(r,_ex0());
908 throw (std::logic_error("ex::series(): expansion point has unknown type"));
911 e = bp->series(rel_, order, options);
912 } catch (std::exception &x) {
913 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));
919 // static member variables
924 unsigned pseries::precedence = 38; // for clarity just below add::precedence
926 #ifndef NO_NAMESPACE_GINAC
928 #endif // ndef NO_NAMESPACE_GINAC