3 * Implementation of class for extended truncated power series and
4 * methods for series expansion. */
7 * GiNaC Copyright (C) 1999-2016 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., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
26 #include "inifcns.h" // for Order function
30 #include "relational.h"
31 #include "operators.h"
43 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(pseries, basic,
44 print_func<print_context>(&pseries::do_print).
45 print_func<print_latex>(&pseries::do_print_latex).
46 print_func<print_tree>(&pseries::do_print_tree).
47 print_func<print_python>(&pseries::do_print_python).
48 print_func<print_python_repr>(&pseries::do_print_python_repr))
55 pseries::pseries() { }
62 /** Construct pseries from a vector of coefficients and powers.
63 * expair.rest holds the coefficient, expair.coeff holds the power.
64 * The powers must be integers (positive or negative) and in ascending order;
65 * the last coefficient can be Order(_ex1) to represent a truncated,
66 * non-terminating series.
68 * @param rel_ expansion variable and point (must hold a relational)
69 * @param ops_ vector of {coefficient, power} pairs (coefficient must not be zero)
70 * @return newly constructed pseries */
71 pseries::pseries(const ex &rel_, const epvector &ops_)
74 #ifdef DO_GINAC_ASSERT
76 while (i != seq.end()) {
79 GINAC_ASSERT(!is_order_function(i->rest));
82 GINAC_ASSERT(is_a<numeric>(i->coeff));
83 GINAC_ASSERT(ex_to<numeric>(i->coeff) < ex_to<numeric>(ip1->coeff));
86 #endif // def DO_GINAC_ASSERT
87 GINAC_ASSERT(is_a<relational>(rel_));
88 GINAC_ASSERT(is_a<symbol>(rel_.lhs()));
92 pseries::pseries(const ex &rel_, epvector &&ops_)
93 : seq(std::move(ops_))
95 #ifdef DO_GINAC_ASSERT
97 while (i != seq.end()) {
100 GINAC_ASSERT(!is_order_function(i->rest));
103 GINAC_ASSERT(is_a<numeric>(i->coeff));
104 GINAC_ASSERT(ex_to<numeric>(i->coeff) < ex_to<numeric>(ip1->coeff));
107 #endif // def DO_GINAC_ASSERT
108 GINAC_ASSERT(is_a<relational>(rel_));
109 GINAC_ASSERT(is_a<symbol>(rel_.lhs()));
119 void pseries::read_archive(const archive_node &n, lst &sym_lst)
121 inherited::read_archive(n, sym_lst);
122 auto first = n.find_first("coeff");
123 auto last = n.find_last("power");
125 seq.reserve((last-first)/2);
127 for (auto loc = first; loc < last;) {
130 n.find_ex_by_loc(loc++, rest, sym_lst);
131 n.find_ex_by_loc(loc++, coeff, sym_lst);
132 seq.push_back(expair(rest, coeff));
135 n.find_ex("var", var, sym_lst);
136 n.find_ex("point", point, sym_lst);
139 void pseries::archive(archive_node &n) const
141 inherited::archive(n);
142 for (auto & it : seq) {
143 n.add_ex("coeff", it.rest);
144 n.add_ex("power", it.coeff);
146 n.add_ex("var", var);
147 n.add_ex("point", point);
152 // functions overriding virtual functions from base classes
155 void pseries::print_series(const print_context & c, const char *openbrace, const char *closebrace, const char *mul_sym, const char *pow_sym, unsigned level) const
157 if (precedence() <= level)
160 // objects of type pseries must not have any zero entries, so the
161 // trivial (zero) pseries needs a special treatment here:
165 auto i = seq.begin(), end = seq.end();
168 // print a sign, if needed
169 if (i != seq.begin())
172 if (!is_order_function(i->rest)) {
174 // print 'rest', i.e. the expansion coefficient
175 if (i->rest.info(info_flags::numeric) &&
176 i->rest.info(info_flags::positive)) {
179 c.s << openbrace << '(';
181 c.s << ')' << closebrace;
184 // print 'coeff', something like (x-1)^42
185 if (!i->coeff.is_zero()) {
187 if (!point.is_zero()) {
188 c.s << openbrace << '(';
189 (var-point).print(c);
190 c.s << ')' << closebrace;
193 if (i->coeff.compare(_ex1)) {
196 if (i->coeff.info(info_flags::negative)) {
206 Order(pow(var - point, i->coeff)).print(c);
210 if (precedence() <= level)
214 void pseries::do_print(const print_context & c, unsigned level) const
216 print_series(c, "", "", "*", "^", level);
219 void pseries::do_print_latex(const print_latex & c, unsigned level) const
221 print_series(c, "{", "}", " ", "^", level);
224 void pseries::do_print_python(const print_python & c, unsigned level) const
226 print_series(c, "", "", "*", "**", level);
229 void pseries::do_print_tree(const print_tree & c, unsigned level) const
231 c.s << std::string(level, ' ') << class_name() << " @" << this
232 << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
234 size_t num = seq.size();
235 for (size_t i=0; i<num; ++i) {
236 seq[i].rest.print(c, level + c.delta_indent);
237 seq[i].coeff.print(c, level + c.delta_indent);
238 c.s << std::string(level + c.delta_indent, ' ') << "-----" << std::endl;
240 var.print(c, level + c.delta_indent);
241 point.print(c, level + c.delta_indent);
244 void pseries::do_print_python_repr(const print_python_repr & c, unsigned level) const
246 c.s << class_name() << "(relational(";
251 size_t num = seq.size();
252 for (size_t i=0; i<num; ++i) {
256 seq[i].rest.print(c);
258 seq[i].coeff.print(c);
264 int pseries::compare_same_type(const basic & other) const
266 GINAC_ASSERT(is_a<pseries>(other));
267 const pseries &o = static_cast<const pseries &>(other);
269 // first compare the lengths of the series...
270 if (seq.size()>o.seq.size())
272 if (seq.size()<o.seq.size())
275 // ...then the expansion point...
276 int cmpval = var.compare(o.var);
279 cmpval = point.compare(o.point);
283 // ...and if that failed the individual elements
284 epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
285 while (it!=seq.end() && o_it!=o.seq.end()) {
286 cmpval = it->compare(*o_it);
293 // so they are equal.
297 /** Return the number of operands including a possible order term. */
298 size_t pseries::nops() const
303 /** Return the ith term in the series when represented as a sum. */
304 ex pseries::op(size_t i) const
307 throw (std::out_of_range("op() out of range"));
309 if (is_order_function(seq[i].rest))
310 return Order(pow(var-point, seq[i].coeff));
311 return seq[i].rest * pow(var - point, seq[i].coeff);
314 /** Return degree of highest power of the series. This is usually the exponent
315 * of the Order term. If s is not the expansion variable of the series, the
316 * series is examined termwise. */
317 int pseries::degree(const ex &s) const
323 // Return last/greatest exponent
324 return ex_to<numeric>((seq.end()-1)->coeff).to_int();
326 int max_pow = std::numeric_limits<int>::min();
327 for (auto & it : seq)
328 max_pow = std::max(max_pow, it.rest.degree(s));
332 /** Return degree of lowest power of the series. This is usually the exponent
333 * of the leading term. If s is not the expansion variable of the series, the
334 * series is examined termwise. If s is the expansion variable but the
335 * expansion point is not zero the series is not expanded to find the degree.
336 * I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
337 int pseries::ldegree(const ex &s) const
343 // Return first/smallest exponent
344 return ex_to<numeric>((seq.begin())->coeff).to_int();
346 int min_pow = std::numeric_limits<int>::max();
347 for (auto & it : seq)
348 min_pow = std::min(min_pow, it.rest.degree(s));
352 /** Return coefficient of degree n in power series if s is the expansion
353 * variable. If the expansion point is nonzero, by definition the n=1
354 * coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
355 * the expansion took place in the s in the first place).
356 * If s is not the expansion variable, an attempt is made to convert the
357 * series to a polynomial and return the corresponding coefficient from
359 ex pseries::coeff(const ex &s, int n) const
361 if (var.is_equal(s)) {
365 // Binary search in sequence for given power
366 numeric looking_for = numeric(n);
367 int lo = 0, hi = seq.size() - 1;
369 int mid = (lo + hi) / 2;
370 GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
371 int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
377 return seq[mid].rest;
382 throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
387 return convert_to_poly().coeff(s, n);
391 ex pseries::collect(const ex &s, bool distributed) const
396 /** Perform coefficient-wise automatic term rewriting rules in this class. */
397 ex pseries::eval() const
399 if (flags & status_flags::evaluated) {
403 // Construct a new series with evaluated coefficients
405 new_seq.reserve(seq.size());
406 for (auto & it : seq)
407 new_seq.push_back(expair(it.rest, it.coeff));
409 return dynallocate<pseries>(relational(var,point), std::move(new_seq)).setflag(status_flags::evaluated);
412 /** Evaluate coefficients numerically. */
413 ex pseries::evalf(int level) const
418 if (level == -max_recursion_level)
419 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
421 // Construct a new series with evaluated coefficients
423 new_seq.reserve(seq.size());
424 for (auto & it : seq)
425 new_seq.push_back(expair(it.rest, it.coeff));
427 return dynallocate<pseries>(relational(var,point), std::move(new_seq)).setflag(status_flags::evaluated);
430 ex pseries::conjugate() const
432 if(!var.info(info_flags::real))
433 return conjugate_function(*this).hold();
435 std::unique_ptr<epvector> newseq(conjugateepvector(seq));
436 ex newpoint = point.conjugate();
438 if (!newseq && are_ex_trivially_equal(point, newpoint)) {
442 return dynallocate<pseries>(var==newpoint, newseq ? std::move(*newseq) : seq);
445 ex pseries::real_part() const
447 if(!var.info(info_flags::real))
448 return real_part_function(*this).hold();
449 ex newpoint = point.real_part();
450 if(newpoint != point)
451 return real_part_function(*this).hold();
454 v.reserve(seq.size());
455 for (auto & it : seq)
456 v.push_back(expair((it.rest).real_part(), it.coeff));
457 return dynallocate<pseries>(var==point, std::move(v));
460 ex pseries::imag_part() const
462 if(!var.info(info_flags::real))
463 return imag_part_function(*this).hold();
464 ex newpoint = point.real_part();
465 if(newpoint != point)
466 return imag_part_function(*this).hold();
469 v.reserve(seq.size());
470 for (auto & it : seq)
471 v.push_back(expair((it.rest).imag_part(), it.coeff));
472 return dynallocate<pseries>(var==point, std::move(v));
475 ex pseries::eval_integ() const
477 std::unique_ptr<epvector> newseq(nullptr);
478 for (auto i=seq.begin(); i!=seq.end(); ++i) {
480 newseq->push_back(expair(i->rest.eval_integ(), i->coeff));
483 ex newterm = i->rest.eval_integ();
484 if (!are_ex_trivially_equal(newterm, i->rest)) {
485 newseq.reset(new epvector);
486 newseq->reserve(seq.size());
487 for (auto j=seq.begin(); j!=i; ++j)
488 newseq->push_back(*j);
489 newseq->push_back(expair(newterm, i->coeff));
493 ex newpoint = point.eval_integ();
494 if (newseq || !are_ex_trivially_equal(newpoint, point))
495 return dynallocate<pseries>(var==newpoint, std::move(*newseq));
499 ex pseries::evalm() const
501 // evalm each coefficient
503 bool something_changed = false;
504 for (auto i=seq.begin(); i!=seq.end(); ++i) {
505 if (something_changed) {
506 ex newcoeff = i->rest.evalm();
507 if (!newcoeff.is_zero())
508 newseq.push_back(expair(newcoeff, i->coeff));
510 ex newcoeff = i->rest.evalm();
511 if (!are_ex_trivially_equal(newcoeff, i->rest)) {
512 something_changed = true;
513 newseq.reserve(seq.size());
514 std::copy(seq.begin(), i, std::back_inserter<epvector>(newseq));
515 if (!newcoeff.is_zero())
516 newseq.push_back(expair(newcoeff, i->coeff));
520 if (something_changed)
521 return dynallocate<pseries>(var==point, std::move(newseq));
526 ex pseries::subs(const exmap & m, unsigned options) const
528 // If expansion variable is being substituted, convert the series to a
529 // polynomial and do the substitution there because the result might
530 // no longer be a power series
531 if (m.find(var) != m.end())
532 return convert_to_poly(true).subs(m, options);
534 // Otherwise construct a new series with substituted coefficients and
537 newseq.reserve(seq.size());
538 for (auto & it : seq)
539 newseq.push_back(expair(it.rest.subs(m, options), it.coeff));
540 return dynallocate<pseries>(relational(var,point.subs(m, options)), std::move(newseq));
543 /** Implementation of ex::expand() for a power series. It expands all the
544 * terms individually and returns the resulting series as a new pseries. */
545 ex pseries::expand(unsigned options) const
548 for (auto & it : seq) {
549 ex restexp = it.rest.expand();
550 if (!restexp.is_zero())
551 newseq.push_back(expair(restexp, it.coeff));
553 return dynallocate<pseries>(relational(var,point), std::move(newseq)).setflag(options == 0 ? status_flags::expanded : 0);
556 /** Implementation of ex::diff() for a power series.
558 ex pseries::derivative(const symbol & s) const
564 // FIXME: coeff might depend on var
565 for (auto & it : seq) {
566 if (is_order_function(it.rest)) {
567 new_seq.push_back(expair(it.rest, it.coeff - 1));
569 ex c = it.rest * it.coeff;
571 new_seq.push_back(expair(c, it.coeff - 1));
577 for (auto & it : seq) {
578 if (is_order_function(it.rest)) {
579 new_seq.push_back(it);
581 ex c = it.rest.diff(s);
583 new_seq.push_back(expair(c, it.coeff));
588 return pseries(relational(var,point), std::move(new_seq));
591 ex pseries::convert_to_poly(bool no_order) const
594 for (auto & it : seq) {
595 if (is_order_function(it.rest)) {
597 e += Order(pow(var - point, it.coeff));
599 e += it.rest * pow(var - point, it.coeff);
604 bool pseries::is_terminating() const
606 return seq.empty() || !is_order_function((seq.end()-1)->rest);
609 ex pseries::coeffop(size_t i) const
612 throw (std::out_of_range("coeffop() out of range"));
616 ex pseries::exponop(size_t i) const
619 throw (std::out_of_range("exponop() out of range"));
625 * Implementations of series expansion
628 /** Default implementation of ex::series(). This performs Taylor expansion.
630 ex basic::series(const relational & r, int order, unsigned options) const
633 const symbol &s = ex_to<symbol>(r.lhs());
635 // default for order-values that make no sense for Taylor expansion
636 if ((order <= 0) && this->has(s)) {
637 seq.push_back(expair(Order(_ex1), order));
638 return pseries(r, std::move(seq));
641 // do Taylor expansion
644 ex coeff = deriv.subs(r, subs_options::no_pattern);
646 if (!coeff.is_zero()) {
647 seq.push_back(expair(coeff, _ex0));
651 for (n=1; n<order; ++n) {
653 // We need to test for zero in order to see if the series terminates.
654 // The problem is that there is no such thing as a perfect test for
655 // zero. Expanding the term occasionally helps a little...
656 deriv = deriv.diff(s).expand();
657 if (deriv.is_zero()) // Series terminates
658 return pseries(r, std::move(seq));
660 coeff = deriv.subs(r, subs_options::no_pattern);
661 if (!coeff.is_zero())
662 seq.push_back(expair(fac * coeff, n));
665 // Higher-order terms, if present
666 deriv = deriv.diff(s);
667 if (!deriv.expand().is_zero())
668 seq.push_back(expair(Order(_ex1), n));
669 return pseries(r, std::move(seq));
673 /** Implementation of ex::series() for symbols.
675 ex symbol::series(const relational & r, int order, unsigned options) const
678 const ex point = r.rhs();
679 GINAC_ASSERT(is_a<symbol>(r.lhs()));
681 if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
682 if (order > 0 && !point.is_zero())
683 seq.push_back(expair(point, _ex0));
685 seq.push_back(expair(_ex1, _ex1));
687 seq.push_back(expair(Order(_ex1), numeric(order)));
689 seq.push_back(expair(*this, _ex0));
690 return pseries(r, std::move(seq));
694 /** Add one series object to another, producing a pseries object that
695 * represents the sum.
697 * @param other pseries object to add with
698 * @return the sum as a pseries */
699 ex pseries::add_series(const pseries &other) const
701 // Adding two series with different variables or expansion points
702 // results in an empty (constant) series
703 if (!is_compatible_to(other)) {
704 epvector nul { expair(Order(_ex1), _ex0) };
705 return pseries(relational(var,point), std::move(nul));
710 auto a = seq.begin(), a_end = seq.end();
711 auto b = other.seq.begin(), b_end = other.seq.end();
712 int pow_a = std::numeric_limits<int>::max(), pow_b = std::numeric_limits<int>::max();
714 // If a is empty, fill up with elements from b and stop
717 new_seq.push_back(*b);
722 pow_a = ex_to<numeric>((*a).coeff).to_int();
724 // If b is empty, fill up with elements from a and stop
727 new_seq.push_back(*a);
732 pow_b = ex_to<numeric>((*b).coeff).to_int();
734 // a and b are non-empty, compare powers
736 // a has lesser power, get coefficient from a
737 new_seq.push_back(*a);
738 if (is_order_function((*a).rest))
741 } else if (pow_b < pow_a) {
742 // b has lesser power, get coefficient from b
743 new_seq.push_back(*b);
744 if (is_order_function((*b).rest))
748 // Add coefficient of a and b
749 if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
750 new_seq.push_back(expair(Order(_ex1), (*a).coeff));
751 break; // Order term ends the sequence
753 ex sum = (*a).rest + (*b).rest;
754 if (!(sum.is_zero()))
755 new_seq.push_back(expair(sum, numeric(pow_a)));
761 return pseries(relational(var,point), std::move(new_seq));
765 /** Implementation of ex::series() for sums. This performs series addition when
766 * adding pseries objects.
768 ex add::series(const relational & r, int order, unsigned options) const
770 ex acc; // Series accumulator
772 // Get first term from overall_coeff
773 acc = overall_coeff.series(r, order, options);
775 // Add remaining terms
776 for (auto & it : seq) {
778 if (is_exactly_a<pseries>(it.rest))
781 op = it.rest.series(r, order, options);
782 if (!it.coeff.is_equal(_ex1))
783 op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it.coeff));
786 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
792 /** Multiply a pseries object with a numeric constant, producing a pseries
793 * object that represents the product.
795 * @param other constant to multiply with
796 * @return the product as a pseries */
797 ex pseries::mul_const(const numeric &other) const
800 new_seq.reserve(seq.size());
802 for (auto & it : seq) {
803 if (!is_order_function(it.rest))
804 new_seq.push_back(expair(it.rest * other, it.coeff));
806 new_seq.push_back(it);
808 return pseries(relational(var,point), std::move(new_seq));
812 /** Multiply one pseries object to another, producing a pseries object that
813 * represents the product.
815 * @param other pseries object to multiply with
816 * @return the product as a pseries */
817 ex pseries::mul_series(const pseries &other) const
819 // Multiplying two series with different variables or expansion points
820 // results in an empty (constant) series
821 if (!is_compatible_to(other)) {
822 epvector nul { expair(Order(_ex1), _ex0) };
823 return pseries(relational(var,point), std::move(nul));
826 if (seq.empty() || other.seq.empty()) {
827 return dynallocate<pseries>(var==point, epvector());
830 // Series multiplication
832 const int a_max = degree(var);
833 const int b_max = other.degree(var);
834 const int a_min = ldegree(var);
835 const int b_min = other.ldegree(var);
836 const int cdeg_min = a_min + b_min;
837 int cdeg_max = a_max + b_max;
839 int higher_order_a = std::numeric_limits<int>::max();
840 int higher_order_b = std::numeric_limits<int>::max();
841 if (is_order_function(coeff(var, a_max)))
842 higher_order_a = a_max + b_min;
843 if (is_order_function(other.coeff(var, b_max)))
844 higher_order_b = b_max + a_min;
845 const int higher_order_c = std::min(higher_order_a, higher_order_b);
846 if (cdeg_max >= higher_order_c)
847 cdeg_max = higher_order_c - 1;
849 std::map<int, ex> rest_map_a, rest_map_b;
850 for (const auto& it : seq)
851 rest_map_a[ex_to<numeric>(it.coeff).to_int()] = it.rest;
853 if (other.var.is_equal(var))
854 for (const auto& it : other.seq)
855 rest_map_b[ex_to<numeric>(it.coeff).to_int()] = it.rest;
857 for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
859 // c(i)=a(0)b(i)+...+a(i)b(0)
860 for (int i=a_min; cdeg-i>=b_min; ++i) {
861 const auto& ita = rest_map_a.find(i);
862 if (ita == rest_map_a.end())
864 const auto& itb = rest_map_b.find(cdeg-i);
865 if (itb == rest_map_b.end())
867 if (!is_order_function(ita->second) && !is_order_function(itb->second))
868 co += ita->second * itb->second;
871 new_seq.push_back(expair(co, numeric(cdeg)));
873 if (higher_order_c < std::numeric_limits<int>::max())
874 new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c)));
875 return pseries(relational(var, point), std::move(new_seq));
879 /** Implementation of ex::series() for product. This performs series
880 * multiplication when multiplying series.
882 ex mul::series(const relational & r, int order, unsigned options) const
884 pseries acc; // Series accumulator
886 GINAC_ASSERT(is_a<symbol>(r.lhs()));
887 const ex& sym = r.lhs();
889 // holds ldegrees of the series of individual factors
890 std::vector<int> ldegrees;
891 std::vector<bool> ldegree_redo;
893 // find minimal degrees
894 // first round: obtain a bound up to which minimal degrees have to be
896 for (auto & it : seq) {
901 if (expon.info(info_flags::integer)) {
903 factor = ex_to<numeric>(expon).to_int();
905 buf = recombine_pair_to_ex(it);
908 int real_ldegree = 0;
909 bool flag_redo = false;
911 real_ldegree = buf.expand().ldegree(sym-r.rhs());
912 } catch (std::runtime_error) {}
914 if (real_ldegree == 0) {
916 // This case must terminate, otherwise we would have division by
921 real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
922 } while (real_ldegree == orderloop);
924 // Here it is possible that buf does not have a ldegree, therefore
925 // check only if ldegree is negative, otherwise reconsider the case
926 // in the second round.
927 real_ldegree = buf.series(r, 0, options).ldegree(sym);
928 if (real_ldegree == 0)
933 ldegrees.push_back(factor * real_ldegree);
934 ldegree_redo.push_back(flag_redo);
937 int degbound = order-std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
938 // Second round: determine the remaining positive ldegrees by the series
940 // here we can ignore ldegrees larger than degbound
942 for (auto & it : seq) {
943 if ( ldegree_redo[j] ) {
947 if (expon.info(info_flags::integer)) {
949 factor = ex_to<numeric>(expon).to_int();
951 buf = recombine_pair_to_ex(it);
953 int real_ldegree = 0;
957 real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
958 } while ((real_ldegree == orderloop)
959 && (factor*real_ldegree < degbound));
960 ldegrees[j] = factor * real_ldegree;
961 degbound -= factor * real_ldegree;
966 int degsum = std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
968 if (degsum >= order) {
969 epvector epv { expair(Order(_ex1), order) };
970 return dynallocate<pseries>(r, std::move(epv));
973 // Multiply with remaining terms
974 auto itd = ldegrees.begin();
975 for (auto it=seq.begin(), itend=seq.end(); it!=itend; ++it, ++itd) {
977 // do series expansion with adjusted order
978 ex op = recombine_pair_to_ex(*it).series(r, order-degsum+(*itd), options);
980 // Series multiplication
981 if (it == seq.begin())
982 acc = ex_to<pseries>(op);
984 acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
987 return acc.mul_const(ex_to<numeric>(overall_coeff));
991 /** Compute the p-th power of a series.
993 * @param p power to compute
994 * @param deg truncation order of series calculation */
995 ex pseries::power_const(const numeric &p, int deg) const
998 // (due to Leonhard Euler)
999 // let A(x) be this series and for the time being let it start with a
1000 // constant (later we'll generalize):
1001 // A(x) = a_0 + a_1*x + a_2*x^2 + ...
1002 // We want to compute
1004 // C(x) = c_0 + c_1*x + c_2*x^2 + ...
1005 // Taking the derivative on both sides and multiplying with A(x) one
1006 // immediately arrives at
1007 // C'(x)*A(x) = p*C(x)*A'(x)
1008 // Multiplying this out and comparing coefficients we get the recurrence
1010 // c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
1011 // ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
1012 // which can easily be solved given the starting value c_0 = (a_0)^p.
1013 // For the more general case where the leading coefficient of A(x) is not
1014 // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
1015 // repeat the above derivation. The leading power of C2(x) = A2(x)^2 is
1016 // then of course x^(p*m) but the recurrence formula still holds.
1019 // as a special case, handle the empty (zero) series honoring the
1020 // usual power laws such as implemented in power::eval()
1021 if (p.real().is_zero())
1022 throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
1023 else if (p.real().is_negative())
1024 throw pole_error("pseries::power_const(): division by zero",1);
1029 const int ldeg = ldegree(var);
1030 if (!(p*ldeg).is_integer())
1031 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
1033 // adjust number of coefficients
1034 int numcoeff = deg - (p*ldeg).to_int();
1035 if (numcoeff <= 0) {
1036 epvector epv { expair(Order(_ex1), deg) };
1037 return dynallocate<pseries>(relational(var,point), std::move(epv));
1040 // O(x^n)^(-m) is undefined
1041 if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
1042 throw pole_error("pseries::power_const(): division by zero",1);
1044 // Compute coefficients of the powered series
1046 co.reserve(numcoeff);
1047 co.push_back(pow(coeff(var, ldeg), p));
1048 for (int i=1; i<numcoeff; ++i) {
1050 for (int j=1; j<=i; ++j) {
1051 ex c = coeff(var, j + ldeg);
1052 if (is_order_function(c)) {
1053 co.push_back(Order(_ex1));
1056 sum += (p * j - (i - j)) * co[i - j] * c;
1058 co.push_back(sum / coeff(var, ldeg) / i);
1061 // Construct new series (of non-zero coefficients)
1063 bool higher_order = false;
1064 for (int i=0; i<numcoeff; ++i) {
1065 if (!co[i].is_zero())
1066 new_seq.push_back(expair(co[i], p * ldeg + i));
1067 if (is_order_function(co[i])) {
1068 higher_order = true;
1073 new_seq.push_back(expair(Order(_ex1), p * ldeg + numcoeff));
1075 return pseries(relational(var,point), std::move(new_seq));
1079 /** Return a new pseries object with the powers shifted by deg. */
1080 pseries pseries::shift_exponents(int deg) const
1082 epvector newseq = seq;
1083 for (auto & it : newseq)
1085 return pseries(relational(var, point), std::move(newseq));
1089 /** Implementation of ex::series() for powers. This performs Laurent expansion
1090 * of reciprocals of series at singularities.
1091 * @see ex::series */
1092 ex power::series(const relational & r, int order, unsigned options) const
1094 // If basis is already a series, just power it
1095 if (is_exactly_a<pseries>(basis))
1096 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
1098 // Basis is not a series, may there be a singularity?
1099 bool must_expand_basis = false;
1101 basis.subs(r, subs_options::no_pattern);
1102 } catch (pole_error) {
1103 must_expand_basis = true;
1106 bool exponent_is_regular = true;
1108 exponent.subs(r, subs_options::no_pattern);
1109 } catch (pole_error) {
1110 exponent_is_regular = false;
1113 if (!exponent_is_regular) {
1114 ex l = exponent*log(basis);
1116 ex le = l.series(r, order, options);
1117 // Note: expanding exp(l) won't help, since that will attempt
1118 // Taylor expansion, and fail (because exponent is "singular")
1119 // Still l itself might be expanded in Taylor series.
1121 // sin(x)/x*log(cos(x))
1123 return exp(le).series(r, order, options);
1124 // Note: if l happens to have a Laurent expansion (with
1125 // negative powers of (var - point)), expanding exp(le)
1126 // will barf (which is The Right Thing).
1129 // Is the expression of type something^(-int)?
1130 if (!must_expand_basis && !exponent.info(info_flags::negint)
1131 && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1132 return basic::series(r, order, options);
1134 // Is the expression of type 0^something?
1135 if (!must_expand_basis && !basis.subs(r, subs_options::no_pattern).is_zero()
1136 && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1137 return basic::series(r, order, options);
1139 // Singularity encountered, is the basis equal to (var - point)?
1140 if (basis.is_equal(r.lhs() - r.rhs())) {
1142 if (ex_to<numeric>(exponent).to_int() < order)
1143 new_seq.push_back(expair(_ex1, exponent));
1145 new_seq.push_back(expair(Order(_ex1), exponent));
1146 return pseries(r, std::move(new_seq));
1149 // No, expand basis into series
1152 if (is_a<numeric>(exponent)) {
1153 numexp = ex_to<numeric>(exponent);
1157 const ex& sym = r.lhs();
1158 // find existing minimal degree
1159 ex eb = basis.expand();
1160 int real_ldegree = 0;
1161 if (eb.info(info_flags::rational_function))
1162 real_ldegree = eb.ldegree(sym-r.rhs());
1163 if (real_ldegree == 0) {
1167 real_ldegree = basis.series(r, orderloop, options).ldegree(sym);
1168 } while (real_ldegree == orderloop);
1171 if (!(real_ldegree*numexp).is_integer())
1172 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
1173 ex e = basis.series(r, (order + real_ldegree*(1-numexp)).to_int(), options);
1177 result = ex_to<pseries>(e).power_const(numexp, order);
1178 } catch (pole_error) {
1179 epvector ser { expair(Order(_ex1), order) };
1180 result = pseries(r, std::move(ser));
1187 /** Re-expansion of a pseries object. */
1188 ex pseries::series(const relational & r, int order, unsigned options) const
1190 const ex p = r.rhs();
1191 GINAC_ASSERT(is_a<symbol>(r.lhs()));
1192 const symbol &s = ex_to<symbol>(r.lhs());
1194 if (var.is_equal(s) && point.is_equal(p)) {
1195 if (order > degree(s))
1199 for (auto & it : seq) {
1200 int o = ex_to<numeric>(it.coeff).to_int();
1202 new_seq.push_back(expair(Order(_ex1), o));
1205 new_seq.push_back(it);
1207 return pseries(r, std::move(new_seq));
1210 return convert_to_poly().series(r, order, options);
1213 ex integral::series(const relational & r, int order, unsigned options) const
1216 throw std::logic_error("Cannot series expand wrt dummy variable");
1218 // Expanding integrand with r substituted taken in boundaries.
1219 ex fseries = f.series(r, order, options);
1220 epvector fexpansion;
1221 fexpansion.reserve(fseries.nops());
1222 for (size_t i=0; i<fseries.nops(); ++i) {
1223 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1224 currcoeff = (currcoeff == Order(_ex1))
1226 : integral(x, a.subs(r), b.subs(r), currcoeff);
1228 fexpansion.push_back(
1229 expair(currcoeff, ex_to<pseries>(fseries).exponop(i)));
1232 // Expanding lower boundary
1233 ex result = dynallocate<pseries>(r, std::move(fexpansion));
1234 ex aseries = (a-a.subs(r)).series(r, order, options);
1235 fseries = f.series(x == (a.subs(r)), order, options);
1236 for (size_t i=0; i<fseries.nops(); ++i) {
1237 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1238 if (is_order_function(currcoeff))
1240 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1241 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1242 currcoeff = currcoeff.series(r, orderforf);
1243 ex term = ex_to<pseries>(aseries).power_const(ex_to<numeric>(currexpon+1),order);
1244 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(-1/(currexpon+1)));
1245 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1246 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1249 // Expanding upper boundary
1250 ex bseries = (b-b.subs(r)).series(r, order, options);
1251 fseries = f.series(x == (b.subs(r)), order, options);
1252 for (size_t i=0; i<fseries.nops(); ++i) {
1253 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1254 if (is_order_function(currcoeff))
1256 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1257 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1258 currcoeff = currcoeff.series(r, orderforf);
1259 ex term = ex_to<pseries>(bseries).power_const(ex_to<numeric>(currexpon+1),order);
1260 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(1/(currexpon+1)));
1261 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1262 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1269 /** Compute the truncated series expansion of an expression.
1270 * This function returns an expression containing an object of class pseries
1271 * to represent the series. If the series does not terminate within the given
1272 * truncation order, the last term of the series will be an order term.
1274 * @param r expansion relation, lhs holds variable and rhs holds point
1275 * @param order truncation order of series calculations
1276 * @param options of class series_options
1277 * @return an expression holding a pseries object */
1278 ex ex::series(const ex & r, int order, unsigned options) const
1283 if (is_a<relational>(r))
1284 rel_ = ex_to<relational>(r);
1285 else if (is_a<symbol>(r))
1286 rel_ = relational(r,_ex0);
1288 throw (std::logic_error("ex::series(): expansion point has unknown type"));
1290 e = bp->series(rel_, order, options);
1294 GINAC_BIND_UNARCHIVER(pseries);
1296 } // namespace GiNaC