3 * Implementation of class for extended truncated power series and
4 * methods for series expansion. */
7 * GiNaC Copyright (C) 1999-2019 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 range = n.find_property_range("coeff", "power");
123 seq.reserve((range.end-range.begin)/2);
125 for (auto loc = range.begin; loc < range.end;) {
128 n.find_ex_by_loc(loc++, rest, sym_lst);
129 n.find_ex_by_loc(loc++, coeff, sym_lst);
130 seq.emplace_back(expair(rest, coeff));
133 n.find_ex("var", var, sym_lst);
134 n.find_ex("point", point, sym_lst);
137 void pseries::archive(archive_node &n) const
139 inherited::archive(n);
140 for (auto & it : seq) {
141 n.add_ex("coeff", it.rest);
142 n.add_ex("power", it.coeff);
144 n.add_ex("var", var);
145 n.add_ex("point", point);
150 // functions overriding virtual functions from base classes
153 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
155 if (precedence() <= level)
158 // objects of type pseries must not have any zero entries, so the
159 // trivial (zero) pseries needs a special treatment here:
163 auto i = seq.begin(), end = seq.end();
166 // print a sign, if needed
167 if (i != seq.begin())
170 if (!is_order_function(i->rest)) {
172 // print 'rest', i.e. the expansion coefficient
173 if (i->rest.info(info_flags::numeric) &&
174 i->rest.info(info_flags::positive)) {
177 c.s << openbrace << '(';
179 c.s << ')' << closebrace;
182 // print 'coeff', something like (x-1)^42
183 if (!i->coeff.is_zero()) {
185 if (!point.is_zero()) {
186 c.s << openbrace << '(';
187 (var-point).print(c);
188 c.s << ')' << closebrace;
191 if (i->coeff.compare(_ex1)) {
194 if (i->coeff.info(info_flags::negative)) {
204 Order(pow(var - point, i->coeff)).print(c);
208 if (precedence() <= level)
212 void pseries::do_print(const print_context & c, unsigned level) const
214 print_series(c, "", "", "*", "^", level);
217 void pseries::do_print_latex(const print_latex & c, unsigned level) const
219 print_series(c, "{", "}", " ", "^", level);
222 void pseries::do_print_python(const print_python & c, unsigned level) const
224 print_series(c, "", "", "*", "**", level);
227 void pseries::do_print_tree(const print_tree & c, unsigned level) const
229 c.s << std::string(level, ' ') << class_name() << " @" << this
230 << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
232 size_t num = seq.size();
233 for (size_t i=0; i<num; ++i) {
234 seq[i].rest.print(c, level + c.delta_indent);
235 seq[i].coeff.print(c, level + c.delta_indent);
236 c.s << std::string(level + c.delta_indent, ' ') << "-----" << std::endl;
238 var.print(c, level + c.delta_indent);
239 point.print(c, level + c.delta_indent);
242 void pseries::do_print_python_repr(const print_python_repr & c, unsigned level) const
244 c.s << class_name() << "(relational(";
249 size_t num = seq.size();
250 for (size_t i=0; i<num; ++i) {
254 seq[i].rest.print(c);
256 seq[i].coeff.print(c);
262 int pseries::compare_same_type(const basic & other) const
264 GINAC_ASSERT(is_a<pseries>(other));
265 const pseries &o = static_cast<const pseries &>(other);
267 // first compare the lengths of the series...
268 if (seq.size()>o.seq.size())
270 if (seq.size()<o.seq.size())
273 // ...then the expansion point...
274 int cmpval = var.compare(o.var);
277 cmpval = point.compare(o.point);
281 // ...and if that failed the individual elements
282 auto it = seq.begin(), o_it = o.seq.begin();
283 while (it!=seq.end() && o_it!=o.seq.end()) {
284 cmpval = it->compare(*o_it);
291 // so they are equal.
295 /** Return the number of operands including a possible order term. */
296 size_t pseries::nops() const
301 /** Return the ith term in the series when represented as a sum. */
302 ex pseries::op(size_t i) const
305 throw (std::out_of_range("op() out of range"));
307 if (is_order_function(seq[i].rest))
308 return Order(pow(var-point, seq[i].coeff));
309 return seq[i].rest * pow(var - point, seq[i].coeff);
312 /** Return degree of highest power of the series. This is usually the exponent
313 * of the Order term. If s is not the expansion variable of the series, the
314 * series is examined termwise. */
315 int pseries::degree(const ex &s) const
321 // Return last/greatest exponent
322 return ex_to<numeric>((seq.end()-1)->coeff).to_int();
324 int max_pow = std::numeric_limits<int>::min();
325 for (auto & it : seq)
326 max_pow = std::max(max_pow, it.rest.degree(s));
330 /** Return degree of lowest power of the series. This is usually the exponent
331 * of the leading term. If s is not the expansion variable of the series, the
332 * series is examined termwise. If s is the expansion variable but the
333 * expansion point is not zero the series is not expanded to find the degree.
334 * I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
335 int pseries::ldegree(const ex &s) const
341 // Return first/smallest exponent
342 return ex_to<numeric>((seq.begin())->coeff).to_int();
344 int min_pow = std::numeric_limits<int>::max();
345 for (auto & it : seq)
346 min_pow = std::min(min_pow, it.rest.degree(s));
350 /** Return coefficient of degree n in power series if s is the expansion
351 * variable. If the expansion point is nonzero, by definition the n=1
352 * coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
353 * the expansion took place in the s in the first place).
354 * If s is not the expansion variable, an attempt is made to convert the
355 * series to a polynomial and return the corresponding coefficient from
357 ex pseries::coeff(const ex &s, int n) const
359 if (var.is_equal(s)) {
363 // Binary search in sequence for given power
364 numeric looking_for = numeric(n);
365 int lo = 0, hi = seq.size() - 1;
367 int mid = (lo + hi) / 2;
368 GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
369 int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
375 return seq[mid].rest;
380 throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
385 return convert_to_poly().coeff(s, n);
389 ex pseries::collect(const ex &s, bool distributed) const
394 /** Perform coefficient-wise automatic term rewriting rules in this class. */
395 ex pseries::eval() const
397 if (flags & status_flags::evaluated) {
401 // Construct a new series with evaluated coefficients
403 new_seq.reserve(seq.size());
404 for (auto & it : seq)
405 new_seq.emplace_back(expair(it.rest, it.coeff));
407 return dynallocate<pseries>(relational(var,point), std::move(new_seq)).setflag(status_flags::evaluated);
410 /** Evaluate coefficients numerically. */
411 ex pseries::evalf() const
413 // Construct a new series with evaluated coefficients
415 new_seq.reserve(seq.size());
416 for (auto & it : seq)
417 new_seq.emplace_back(expair(it.rest, it.coeff));
419 return dynallocate<pseries>(relational(var,point), std::move(new_seq)).setflag(status_flags::evaluated);
422 ex pseries::conjugate() const
424 if(!var.info(info_flags::real))
425 return conjugate_function(*this).hold();
427 std::unique_ptr<epvector> newseq(conjugateepvector(seq));
428 ex newpoint = point.conjugate();
430 if (!newseq && are_ex_trivially_equal(point, newpoint)) {
434 return dynallocate<pseries>(var==newpoint, newseq ? std::move(*newseq) : seq);
437 ex pseries::real_part() const
439 if(!var.info(info_flags::real))
440 return real_part_function(*this).hold();
441 ex newpoint = point.real_part();
442 if(newpoint != point)
443 return real_part_function(*this).hold();
446 v.reserve(seq.size());
447 for (auto & it : seq)
448 v.emplace_back(expair(it.rest.real_part(), it.coeff));
449 return dynallocate<pseries>(var==point, std::move(v));
452 ex pseries::imag_part() const
454 if(!var.info(info_flags::real))
455 return imag_part_function(*this).hold();
456 ex newpoint = point.real_part();
457 if(newpoint != point)
458 return imag_part_function(*this).hold();
461 v.reserve(seq.size());
462 for (auto & it : seq)
463 v.emplace_back(expair(it.rest.imag_part(), it.coeff));
464 return dynallocate<pseries>(var==point, std::move(v));
467 ex pseries::eval_integ() const
469 std::unique_ptr<epvector> newseq(nullptr);
470 for (auto i=seq.begin(); i!=seq.end(); ++i) {
472 newseq->emplace_back(expair(i->rest.eval_integ(), i->coeff));
475 ex newterm = i->rest.eval_integ();
476 if (!are_ex_trivially_equal(newterm, i->rest)) {
477 newseq.reset(new epvector);
478 newseq->reserve(seq.size());
479 for (auto j=seq.begin(); j!=i; ++j)
480 newseq->push_back(*j);
481 newseq->emplace_back(expair(newterm, i->coeff));
485 ex newpoint = point.eval_integ();
486 if (newseq || !are_ex_trivially_equal(newpoint, point))
487 return dynallocate<pseries>(var==newpoint, std::move(*newseq));
491 ex pseries::evalm() const
493 // evalm each coefficient
495 bool something_changed = false;
496 for (auto i=seq.begin(); i!=seq.end(); ++i) {
497 if (something_changed) {
498 ex newcoeff = i->rest.evalm();
499 if (!newcoeff.is_zero())
500 newseq.emplace_back(expair(newcoeff, i->coeff));
502 ex newcoeff = i->rest.evalm();
503 if (!are_ex_trivially_equal(newcoeff, i->rest)) {
504 something_changed = true;
505 newseq.reserve(seq.size());
506 std::copy(seq.begin(), i, std::back_inserter<epvector>(newseq));
507 if (!newcoeff.is_zero())
508 newseq.emplace_back(expair(newcoeff, i->coeff));
512 if (something_changed)
513 return dynallocate<pseries>(var==point, std::move(newseq));
518 ex pseries::subs(const exmap & m, unsigned options) const
520 // If expansion variable is being substituted, convert the series to a
521 // polynomial and do the substitution there because the result might
522 // no longer be a power series
523 if (m.find(var) != m.end())
524 return convert_to_poly(true).subs(m, options);
526 // Otherwise construct a new series with substituted coefficients and
529 newseq.reserve(seq.size());
530 for (auto & it : seq)
531 newseq.emplace_back(expair(it.rest.subs(m, options), it.coeff));
532 return dynallocate<pseries>(relational(var,point.subs(m, options)), std::move(newseq));
535 /** Implementation of ex::expand() for a power series. It expands all the
536 * terms individually and returns the resulting series as a new pseries. */
537 ex pseries::expand(unsigned options) const
540 for (auto & it : seq) {
541 ex restexp = it.rest.expand();
542 if (!restexp.is_zero())
543 newseq.emplace_back(expair(restexp, it.coeff));
545 return dynallocate<pseries>(relational(var,point), std::move(newseq)).setflag(options == 0 ? status_flags::expanded : 0);
548 /** Implementation of ex::diff() for a power series.
550 ex pseries::derivative(const symbol & s) const
556 // FIXME: coeff might depend on var
557 for (auto & it : seq) {
558 if (is_order_function(it.rest)) {
559 new_seq.emplace_back(expair(it.rest, it.coeff - 1));
561 ex c = it.rest * it.coeff;
563 new_seq.emplace_back(expair(c, it.coeff - 1));
569 for (auto & it : seq) {
570 if (is_order_function(it.rest)) {
571 new_seq.push_back(it);
573 ex c = it.rest.diff(s);
575 new_seq.emplace_back(expair(c, it.coeff));
580 return pseries(relational(var,point), std::move(new_seq));
583 ex pseries::convert_to_poly(bool no_order) const
586 for (auto & it : seq) {
587 if (is_order_function(it.rest)) {
589 e += Order(pow(var - point, it.coeff));
591 e += it.rest * pow(var - point, it.coeff);
596 bool pseries::is_terminating() const
598 return seq.empty() || !is_order_function((seq.end()-1)->rest);
601 ex pseries::coeffop(size_t i) const
604 throw (std::out_of_range("coeffop() out of range"));
608 ex pseries::exponop(size_t i) const
611 throw (std::out_of_range("exponop() out of range"));
617 * Implementations of series expansion
620 /** Default implementation of ex::series(). This performs Taylor expansion.
622 ex basic::series(const relational & r, int order, unsigned options) const
625 const symbol &s = ex_to<symbol>(r.lhs());
627 // default for order-values that make no sense for Taylor expansion
628 if ((order <= 0) && this->has(s)) {
629 seq.emplace_back(expair(Order(_ex1), order));
630 return pseries(r, std::move(seq));
633 // do Taylor expansion
636 ex coeff = deriv.subs(r, subs_options::no_pattern);
638 if (!coeff.is_zero()) {
639 seq.emplace_back(expair(coeff, _ex0));
643 for (n=1; n<order; ++n) {
645 // We need to test for zero in order to see if the series terminates.
646 // The problem is that there is no such thing as a perfect test for
647 // zero. Expanding the term occasionally helps a little...
648 deriv = deriv.diff(s).expand();
649 if (deriv.is_zero()) // Series terminates
650 return pseries(r, std::move(seq));
652 coeff = deriv.subs(r, subs_options::no_pattern);
653 if (!coeff.is_zero())
654 seq.emplace_back(expair(fac * coeff, n));
657 // Higher-order terms, if present
658 deriv = deriv.diff(s);
659 if (!deriv.expand().is_zero())
660 seq.emplace_back(expair(Order(_ex1), n));
661 return pseries(r, std::move(seq));
665 /** Implementation of ex::series() for symbols.
667 ex symbol::series(const relational & r, int order, unsigned options) const
670 const ex point = r.rhs();
671 GINAC_ASSERT(is_a<symbol>(r.lhs()));
673 if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
674 if (order > 0 && !point.is_zero())
675 seq.emplace_back(expair(point, _ex0));
677 seq.emplace_back(expair(_ex1, _ex1));
679 seq.emplace_back(expair(Order(_ex1), numeric(order)));
681 seq.emplace_back(expair(*this, _ex0));
682 return pseries(r, std::move(seq));
686 /** Add one series object to another, producing a pseries object that
687 * represents the sum.
689 * @param other pseries object to add with
690 * @return the sum as a pseries */
691 ex pseries::add_series(const pseries &other) const
693 // Adding two series with different variables or expansion points
694 // results in an empty (constant) series
695 if (!is_compatible_to(other)) {
696 epvector nul { expair(Order(_ex1), _ex0) };
697 return pseries(relational(var,point), std::move(nul));
702 auto a = seq.begin(), a_end = seq.end();
703 auto b = other.seq.begin(), b_end = other.seq.end();
704 int pow_a = std::numeric_limits<int>::max(), pow_b = std::numeric_limits<int>::max();
706 // If a is empty, fill up with elements from b and stop
709 new_seq.push_back(*b);
714 pow_a = ex_to<numeric>((*a).coeff).to_int();
716 // If b is empty, fill up with elements from a and stop
719 new_seq.push_back(*a);
724 pow_b = ex_to<numeric>((*b).coeff).to_int();
726 // a and b are non-empty, compare powers
728 // a has lesser power, get coefficient from a
729 new_seq.push_back(*a);
730 if (is_order_function((*a).rest))
733 } else if (pow_b < pow_a) {
734 // b has lesser power, get coefficient from b
735 new_seq.push_back(*b);
736 if (is_order_function((*b).rest))
740 // Add coefficient of a and b
741 if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
742 new_seq.emplace_back(expair(Order(_ex1), (*a).coeff));
743 break; // Order term ends the sequence
745 ex sum = (*a).rest + (*b).rest;
746 if (!(sum.is_zero()))
747 new_seq.emplace_back(expair(sum, numeric(pow_a)));
753 return pseries(relational(var,point), std::move(new_seq));
757 /** Implementation of ex::series() for sums. This performs series addition when
758 * adding pseries objects.
760 ex add::series(const relational & r, int order, unsigned options) const
762 ex acc; // Series accumulator
764 // Get first term from overall_coeff
765 acc = overall_coeff.series(r, order, options);
767 // Add remaining terms
768 for (auto & it : seq) {
770 if (is_exactly_a<pseries>(it.rest))
773 op = it.rest.series(r, order, options);
774 if (!it.coeff.is_equal(_ex1))
775 op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it.coeff));
778 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
784 /** Multiply a pseries object with a numeric constant, producing a pseries
785 * object that represents the product.
787 * @param other constant to multiply with
788 * @return the product as a pseries */
789 ex pseries::mul_const(const numeric &other) const
792 new_seq.reserve(seq.size());
794 for (auto & it : seq) {
795 if (!is_order_function(it.rest))
796 new_seq.emplace_back(expair(it.rest * other, it.coeff));
798 new_seq.push_back(it);
800 return pseries(relational(var,point), std::move(new_seq));
804 /** Multiply one pseries object to another, producing a pseries object that
805 * represents the product.
807 * @param other pseries object to multiply with
808 * @return the product as a pseries */
809 ex pseries::mul_series(const pseries &other) const
811 // Multiplying two series with different variables or expansion points
812 // results in an empty (constant) series
813 if (!is_compatible_to(other)) {
814 epvector nul { expair(Order(_ex1), _ex0) };
815 return pseries(relational(var,point), std::move(nul));
818 if (seq.empty() || other.seq.empty()) {
819 return dynallocate<pseries>(var==point, epvector());
822 // Series multiplication
824 const int a_max = degree(var);
825 const int b_max = other.degree(var);
826 const int a_min = ldegree(var);
827 const int b_min = other.ldegree(var);
828 const int cdeg_min = a_min + b_min;
829 int cdeg_max = a_max + b_max;
831 int higher_order_a = std::numeric_limits<int>::max();
832 int higher_order_b = std::numeric_limits<int>::max();
833 if (is_order_function(coeff(var, a_max)))
834 higher_order_a = a_max + b_min;
835 if (is_order_function(other.coeff(var, b_max)))
836 higher_order_b = b_max + a_min;
837 const int higher_order_c = std::min(higher_order_a, higher_order_b);
838 if (cdeg_max >= higher_order_c)
839 cdeg_max = higher_order_c - 1;
841 std::map<int, ex> rest_map_a, rest_map_b;
842 for (const auto& it : seq)
843 rest_map_a[ex_to<numeric>(it.coeff).to_int()] = it.rest;
845 if (other.var.is_equal(var))
846 for (const auto& it : other.seq)
847 rest_map_b[ex_to<numeric>(it.coeff).to_int()] = it.rest;
849 for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
851 // c(i)=a(0)b(i)+...+a(i)b(0)
852 for (int i=a_min; cdeg-i>=b_min; ++i) {
853 const auto& ita = rest_map_a.find(i);
854 if (ita == rest_map_a.end())
856 const auto& itb = rest_map_b.find(cdeg-i);
857 if (itb == rest_map_b.end())
859 if (!is_order_function(ita->second) && !is_order_function(itb->second))
860 co += ita->second * itb->second;
863 new_seq.emplace_back(expair(co, numeric(cdeg)));
865 if (higher_order_c < std::numeric_limits<int>::max())
866 new_seq.emplace_back(expair(Order(_ex1), numeric(higher_order_c)));
867 return pseries(relational(var, point), std::move(new_seq));
871 /** Implementation of ex::series() for product. This performs series
872 * multiplication when multiplying series.
874 ex mul::series(const relational & r, int order, unsigned options) const
876 pseries acc; // Series accumulator
878 GINAC_ASSERT(is_a<symbol>(r.lhs()));
879 const ex& sym = r.lhs();
881 // holds ldegrees of the series of individual factors
882 std::vector<int> ldegrees;
883 std::vector<bool> ldegree_redo;
885 // find minimal degrees
886 // first round: obtain a bound up to which minimal degrees have to be
888 for (auto & it : seq) {
893 if (expon.info(info_flags::integer)) {
895 factor = ex_to<numeric>(expon).to_int();
897 buf = recombine_pair_to_ex(it);
900 int real_ldegree = 0;
901 bool flag_redo = false;
903 real_ldegree = buf.expand().ldegree(sym-r.rhs());
904 } catch (std::runtime_error) {}
906 if (real_ldegree == 0) {
908 // This case must terminate, otherwise we would have division by
913 real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
914 } while (real_ldegree == orderloop);
916 // Here it is possible that buf does not have a ldegree, therefore
917 // check only if ldegree is negative, otherwise reconsider the case
918 // in the second round.
919 real_ldegree = buf.series(r, 0, options).ldegree(sym);
920 if (real_ldegree == 0)
925 ldegrees.push_back(factor * real_ldegree);
926 ldegree_redo.push_back(flag_redo);
929 int degbound = order-std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
930 // Second round: determine the remaining positive ldegrees by the series
932 // here we can ignore ldegrees larger than degbound
934 for (auto & it : seq) {
935 if ( ldegree_redo[j] ) {
939 if (expon.info(info_flags::integer)) {
941 factor = ex_to<numeric>(expon).to_int();
943 buf = recombine_pair_to_ex(it);
945 int real_ldegree = 0;
949 real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
950 } while ((real_ldegree == orderloop)
951 && (factor*real_ldegree < degbound));
952 ldegrees[j] = factor * real_ldegree;
953 degbound -= factor * real_ldegree;
958 int degsum = std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
960 if (degsum > order) {
961 return dynallocate<pseries>(r, epvector{{Order(_ex1), order}});
964 // Multiply with remaining terms
965 auto itd = ldegrees.begin();
966 for (auto it=seq.begin(), itend=seq.end(); it!=itend; ++it, ++itd) {
968 // do series expansion with adjusted order
969 ex op = recombine_pair_to_ex(*it).series(r, order-degsum+(*itd), options);
971 // Series multiplication
972 if (it == seq.begin())
973 acc = ex_to<pseries>(op);
975 acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
978 return acc.mul_const(ex_to<numeric>(overall_coeff));
982 /** Compute the p-th power of a series.
984 * @param p power to compute
985 * @param deg truncation order of series calculation */
986 ex pseries::power_const(const numeric &p, int deg) const
989 // (due to Leonhard Euler)
990 // let A(x) be this series and for the time being let it start with a
991 // constant (later we'll generalize):
992 // A(x) = a_0 + a_1*x + a_2*x^2 + ...
993 // We want to compute
995 // C(x) = c_0 + c_1*x + c_2*x^2 + ...
996 // Taking the derivative on both sides and multiplying with A(x) one
997 // immediately arrives at
998 // C'(x)*A(x) = p*C(x)*A'(x)
999 // Multiplying this out and comparing coefficients we get the recurrence
1001 // c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
1002 // ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
1003 // which can easily be solved given the starting value c_0 = (a_0)^p.
1004 // For the more general case where the leading coefficient of A(x) is not
1005 // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
1006 // repeat the above derivation. The leading power of C2(x) = A2(x)^2 is
1007 // then of course x^(p*m) but the recurrence formula still holds.
1010 // as a special case, handle the empty (zero) series honoring the
1011 // usual power laws such as implemented in power::eval()
1012 if (p.real().is_zero())
1013 throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
1014 else if (p.real().is_negative())
1015 throw pole_error("pseries::power_const(): division by zero",1);
1020 const int ldeg = ldegree(var);
1021 if (!(p*ldeg).is_integer())
1022 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
1024 // adjust number of coefficients
1025 int numcoeff = deg - (p*ldeg).to_int();
1026 if (numcoeff <= 0) {
1027 epvector epv { expair(Order(_ex1), deg) };
1028 return dynallocate<pseries>(relational(var,point), std::move(epv));
1031 // O(x^n)^(-m) is undefined
1032 if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
1033 throw pole_error("pseries::power_const(): division by zero",1);
1035 // Compute coefficients of the powered series
1037 co.reserve(numcoeff);
1038 co.push_back(pow(coeff(var, ldeg), p));
1039 for (int i=1; i<numcoeff; ++i) {
1041 for (int j=1; j<=i; ++j) {
1042 ex c = coeff(var, j + ldeg);
1043 if (is_order_function(c)) {
1044 co.push_back(Order(_ex1));
1047 sum += (p * j - (i - j)) * co[i - j] * c;
1049 co.push_back(sum / coeff(var, ldeg) / i);
1052 // Construct new series (of non-zero coefficients)
1054 bool higher_order = false;
1055 for (int i=0; i<numcoeff; ++i) {
1056 if (!co[i].is_zero())
1057 new_seq.emplace_back(expair(co[i], p * ldeg + i));
1058 if (is_order_function(co[i])) {
1059 higher_order = true;
1064 new_seq.emplace_back(expair(Order(_ex1), p * ldeg + numcoeff));
1066 return pseries(relational(var,point), std::move(new_seq));
1070 /** Return a new pseries object with the powers shifted by deg. */
1071 pseries pseries::shift_exponents(int deg) const
1073 epvector newseq = seq;
1074 for (auto & it : newseq)
1076 return pseries(relational(var, point), std::move(newseq));
1080 /** Implementation of ex::series() for powers. This performs Laurent expansion
1081 * of reciprocals of series at singularities.
1082 * @see ex::series */
1083 ex power::series(const relational & r, int order, unsigned options) const
1085 // If basis is already a series, just power it
1086 if (is_exactly_a<pseries>(basis))
1087 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
1089 // Basis is not a series, may there be a singularity?
1090 bool must_expand_basis = false;
1092 basis.subs(r, subs_options::no_pattern);
1093 } catch (pole_error) {
1094 must_expand_basis = true;
1097 bool exponent_is_regular = true;
1099 exponent.subs(r, subs_options::no_pattern);
1100 } catch (pole_error) {
1101 exponent_is_regular = false;
1104 if (!exponent_is_regular) {
1105 ex l = exponent*log(basis);
1107 ex le = l.series(r, order, options);
1108 // Note: expanding exp(l) won't help, since that will attempt
1109 // Taylor expansion, and fail (because exponent is "singular")
1110 // Still l itself might be expanded in Taylor series.
1112 // sin(x)/x*log(cos(x))
1114 return exp(le).series(r, order, options);
1115 // Note: if l happens to have a Laurent expansion (with
1116 // negative powers of (var - point)), expanding exp(le)
1117 // will barf (which is The Right Thing).
1120 // Is the expression of type something^(-int)?
1121 if (!must_expand_basis && !exponent.info(info_flags::negint)
1122 && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1123 return basic::series(r, order, options);
1125 // Is the expression of type 0^something?
1126 if (!must_expand_basis && !basis.subs(r, subs_options::no_pattern).is_zero()
1127 && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1128 return basic::series(r, order, options);
1130 // Singularity encountered, is the basis equal to (var - point)?
1131 if (basis.is_equal(r.lhs() - r.rhs())) {
1133 if (ex_to<numeric>(exponent).to_int() < order)
1134 new_seq.emplace_back(expair(_ex1, exponent));
1136 new_seq.emplace_back(expair(Order(_ex1), exponent));
1137 return pseries(r, std::move(new_seq));
1140 // No, expand basis into series
1143 if (is_a<numeric>(exponent)) {
1144 numexp = ex_to<numeric>(exponent);
1148 const ex& sym = r.lhs();
1149 // find existing minimal degree
1150 ex eb = basis.expand();
1151 int real_ldegree = 0;
1152 if (eb.info(info_flags::rational_function))
1153 real_ldegree = eb.ldegree(sym-r.rhs());
1154 if (real_ldegree == 0) {
1158 real_ldegree = basis.series(r, orderloop, options).ldegree(sym);
1159 } while (real_ldegree == orderloop);
1162 if (!(real_ldegree*numexp).is_integer())
1163 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
1164 ex e = basis.series(r, (order + real_ldegree*(1-numexp)).to_int(), options);
1168 result = ex_to<pseries>(e).power_const(numexp, order);
1169 } catch (pole_error) {
1170 epvector ser { expair(Order(_ex1), order) };
1171 result = pseries(r, std::move(ser));
1178 /** Re-expansion of a pseries object. */
1179 ex pseries::series(const relational & r, int order, unsigned options) const
1181 const ex p = r.rhs();
1182 GINAC_ASSERT(is_a<symbol>(r.lhs()));
1183 const symbol &s = ex_to<symbol>(r.lhs());
1185 if (var.is_equal(s) && point.is_equal(p)) {
1186 if (order > degree(s))
1190 for (auto & it : seq) {
1191 int o = ex_to<numeric>(it.coeff).to_int();
1193 new_seq.emplace_back(expair(Order(_ex1), o));
1196 new_seq.push_back(it);
1198 return pseries(r, std::move(new_seq));
1201 return convert_to_poly().series(r, order, options);
1204 ex integral::series(const relational & r, int order, unsigned options) const
1207 throw std::logic_error("Cannot series expand wrt dummy variable");
1209 // Expanding integrand with r substituted taken in boundaries.
1210 ex fseries = f.series(r, order, options);
1211 epvector fexpansion;
1212 fexpansion.reserve(fseries.nops());
1213 for (size_t i=0; i<fseries.nops(); ++i) {
1214 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1215 currcoeff = (currcoeff == Order(_ex1))
1217 : integral(x, a.subs(r), b.subs(r), currcoeff);
1219 fexpansion.emplace_back(
1220 expair(currcoeff, ex_to<pseries>(fseries).exponop(i)));
1223 // Expanding lower boundary
1224 ex result = dynallocate<pseries>(r, std::move(fexpansion));
1225 ex aseries = (a-a.subs(r)).series(r, order, options);
1226 fseries = f.series(x == (a.subs(r)), order, options);
1227 for (size_t i=0; i<fseries.nops(); ++i) {
1228 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1229 if (is_order_function(currcoeff))
1231 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1232 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1233 currcoeff = currcoeff.series(r, orderforf);
1234 ex term = ex_to<pseries>(aseries).power_const(ex_to<numeric>(currexpon+1),order);
1235 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(-1/(currexpon+1)));
1236 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1237 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1240 // Expanding upper boundary
1241 ex bseries = (b-b.subs(r)).series(r, order, options);
1242 fseries = f.series(x == (b.subs(r)), order, options);
1243 for (size_t i=0; i<fseries.nops(); ++i) {
1244 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1245 if (is_order_function(currcoeff))
1247 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1248 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1249 currcoeff = currcoeff.series(r, orderforf);
1250 ex term = ex_to<pseries>(bseries).power_const(ex_to<numeric>(currexpon+1),order);
1251 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(1/(currexpon+1)));
1252 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1253 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1260 /** Compute the truncated series expansion of an expression.
1261 * This function returns an expression containing an object of class pseries
1262 * to represent the series. If the series does not terminate within the given
1263 * truncation order, the last term of the series will be an order term.
1265 * @param r expansion relation, lhs holds variable and rhs holds point
1266 * @param order truncation order of series calculations
1267 * @param options of class series_options
1268 * @return an expression holding a pseries object */
1269 ex ex::series(const ex & r, int order, unsigned options) const
1274 if (is_a<relational>(r))
1275 rel_ = ex_to<relational>(r);
1276 else if (is_a<symbol>(r))
1277 rel_ = relational(r,_ex0);
1279 throw (std::logic_error("ex::series(): expansion point has unknown type"));
1281 e = bp->series(rel_, order, options);
1285 GINAC_BIND_UNARCHIVER(pseries);
1287 } // namespace GiNaC