3 * Implementation of class for extended truncated power series and
4 * methods for series expansion. */
7 * GiNaC Copyright (C) 1999-2015 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_) : seq(ops_)
73 GINAC_ASSERT(is_a<relational>(rel_));
74 GINAC_ASSERT(is_a<symbol>(rel_.lhs()));
84 void pseries::read_archive(const archive_node &n, lst &sym_lst)
86 inherited::read_archive(n, sym_lst);
87 archive_node::archive_node_cit first = n.find_first("coeff");
88 archive_node::archive_node_cit last = n.find_last("power");
90 seq.reserve((last-first)/2);
92 for (archive_node::archive_node_cit loc = first; loc < last;) {
95 n.find_ex_by_loc(loc++, rest, sym_lst);
96 n.find_ex_by_loc(loc++, coeff, sym_lst);
97 seq.push_back(expair(rest, coeff));
100 n.find_ex("var", var, sym_lst);
101 n.find_ex("point", point, sym_lst);
104 void pseries::archive(archive_node &n) const
106 inherited::archive(n);
107 epvector::const_iterator i = seq.begin(), iend = seq.end();
109 n.add_ex("coeff", i->rest);
110 n.add_ex("power", i->coeff);
113 n.add_ex("var", var);
114 n.add_ex("point", point);
119 // functions overriding virtual functions from base classes
122 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
124 if (precedence() <= level)
127 // objects of type pseries must not have any zero entries, so the
128 // trivial (zero) pseries needs a special treatment here:
132 epvector::const_iterator i = seq.begin(), end = seq.end();
135 // print a sign, if needed
136 if (i != seq.begin())
139 if (!is_order_function(i->rest)) {
141 // print 'rest', i.e. the expansion coefficient
142 if (i->rest.info(info_flags::numeric) &&
143 i->rest.info(info_flags::positive)) {
146 c.s << openbrace << '(';
148 c.s << ')' << closebrace;
151 // print 'coeff', something like (x-1)^42
152 if (!i->coeff.is_zero()) {
154 if (!point.is_zero()) {
155 c.s << openbrace << '(';
156 (var-point).print(c);
157 c.s << ')' << closebrace;
160 if (i->coeff.compare(_ex1)) {
163 if (i->coeff.info(info_flags::negative)) {
173 Order(power(var-point,i->coeff)).print(c);
177 if (precedence() <= level)
181 void pseries::do_print(const print_context & c, unsigned level) const
183 print_series(c, "", "", "*", "^", level);
186 void pseries::do_print_latex(const print_latex & c, unsigned level) const
188 print_series(c, "{", "}", " ", "^", level);
191 void pseries::do_print_python(const print_python & c, unsigned level) const
193 print_series(c, "", "", "*", "**", level);
196 void pseries::do_print_tree(const print_tree & c, unsigned level) const
198 c.s << std::string(level, ' ') << class_name() << " @" << this
199 << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
201 size_t num = seq.size();
202 for (size_t i=0; i<num; ++i) {
203 seq[i].rest.print(c, level + c.delta_indent);
204 seq[i].coeff.print(c, level + c.delta_indent);
205 c.s << std::string(level + c.delta_indent, ' ') << "-----" << std::endl;
207 var.print(c, level + c.delta_indent);
208 point.print(c, level + c.delta_indent);
211 void pseries::do_print_python_repr(const print_python_repr & c, unsigned level) const
213 c.s << class_name() << "(relational(";
218 size_t num = seq.size();
219 for (size_t i=0; i<num; ++i) {
223 seq[i].rest.print(c);
225 seq[i].coeff.print(c);
231 int pseries::compare_same_type(const basic & other) const
233 GINAC_ASSERT(is_a<pseries>(other));
234 const pseries &o = static_cast<const pseries &>(other);
236 // first compare the lengths of the series...
237 if (seq.size()>o.seq.size())
239 if (seq.size()<o.seq.size())
242 // ...then the expansion point...
243 int cmpval = var.compare(o.var);
246 cmpval = point.compare(o.point);
250 // ...and if that failed the individual elements
251 epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
252 while (it!=seq.end() && o_it!=o.seq.end()) {
253 cmpval = it->compare(*o_it);
260 // so they are equal.
264 /** Return the number of operands including a possible order term. */
265 size_t pseries::nops() const
270 /** Return the ith term in the series when represented as a sum. */
271 ex pseries::op(size_t i) const
274 throw (std::out_of_range("op() out of range"));
276 if (is_order_function(seq[i].rest))
277 return Order(power(var-point, seq[i].coeff));
278 return seq[i].rest * power(var - point, seq[i].coeff);
281 /** Return degree of highest power of the series. This is usually the exponent
282 * of the Order term. If s is not the expansion variable of the series, the
283 * series is examined termwise. */
284 int pseries::degree(const ex &s) const
286 if (var.is_equal(s)) {
287 // Return last exponent
289 return ex_to<numeric>((seq.end()-1)->coeff).to_int();
293 epvector::const_iterator it = seq.begin(), itend = seq.end();
296 int max_pow = std::numeric_limits<int>::min();
297 while (it != itend) {
298 int pow = it->rest.degree(s);
307 /** Return degree of lowest power of the series. This is usually the exponent
308 * of the leading term. If s is not the expansion variable of the series, the
309 * series is examined termwise. If s is the expansion variable but the
310 * expansion point is not zero the series is not expanded to find the degree.
311 * I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
312 int pseries::ldegree(const ex &s) const
314 if (var.is_equal(s)) {
315 // Return first exponent
317 return ex_to<numeric>((seq.begin())->coeff).to_int();
321 epvector::const_iterator it = seq.begin(), itend = seq.end();
324 int min_pow = std::numeric_limits<int>::max();
325 while (it != itend) {
326 int pow = it->rest.ldegree(s);
335 /** Return coefficient of degree n in power series if s is the expansion
336 * variable. If the expansion point is nonzero, by definition the n=1
337 * coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
338 * the expansion took place in the s in the first place).
339 * If s is not the expansion variable, an attempt is made to convert the
340 * series to a polynomial and return the corresponding coefficient from
342 ex pseries::coeff(const ex &s, int n) const
344 if (var.is_equal(s)) {
348 // Binary search in sequence for given power
349 numeric looking_for = numeric(n);
350 int lo = 0, hi = seq.size() - 1;
352 int mid = (lo + hi) / 2;
353 GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
354 int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
360 return seq[mid].rest;
365 throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
370 return convert_to_poly().coeff(s, n);
374 ex pseries::collect(const ex &s, bool distributed) const
379 /** Perform coefficient-wise automatic term rewriting rules in this class. */
380 ex pseries::eval(int level) const
385 if (level == -max_recursion_level)
386 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
388 // Construct a new series with evaluated coefficients
390 new_seq.reserve(seq.size());
391 epvector::const_iterator it = seq.begin(), itend = seq.end();
392 while (it != itend) {
393 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
396 return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
399 /** Evaluate coefficients numerically. */
400 ex pseries::evalf(int level) const
405 if (level == -max_recursion_level)
406 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
408 // Construct a new series with evaluated coefficients
410 new_seq.reserve(seq.size());
411 epvector::const_iterator it = seq.begin(), itend = seq.end();
412 while (it != itend) {
413 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
416 return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
419 ex pseries::conjugate() const
421 if(!var.info(info_flags::real))
422 return conjugate_function(*this).hold();
424 epvector * newseq = conjugateepvector(seq);
425 ex newpoint = point.conjugate();
427 if (!newseq && are_ex_trivially_equal(point, newpoint)) {
431 ex result = (new pseries(var==newpoint, newseq ? *newseq : seq))->setflag(status_flags::dynallocated);
436 ex pseries::real_part() const
438 if(!var.info(info_flags::real))
439 return real_part_function(*this).hold();
440 ex newpoint = point.real_part();
441 if(newpoint != point)
442 return real_part_function(*this).hold();
445 v.reserve(seq.size());
446 for(epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i)
447 v.push_back(expair((i->rest).real_part(), i->coeff));
448 return (new pseries(var==point, v))->setflag(status_flags::dynallocated);
451 ex pseries::imag_part() const
453 if(!var.info(info_flags::real))
454 return imag_part_function(*this).hold();
455 ex newpoint = point.real_part();
456 if(newpoint != point)
457 return imag_part_function(*this).hold();
460 v.reserve(seq.size());
461 for(epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i)
462 v.push_back(expair((i->rest).imag_part(), i->coeff));
463 return (new pseries(var==point, v))->setflag(status_flags::dynallocated);
466 ex pseries::eval_integ() const
468 epvector *newseq = NULL;
469 for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
471 newseq->push_back(expair(i->rest.eval_integ(), i->coeff));
474 ex newterm = i->rest.eval_integ();
475 if (!are_ex_trivially_equal(newterm, i->rest)) {
476 newseq = new epvector;
477 newseq->reserve(seq.size());
478 for (epvector::const_iterator j=seq.begin(); j!=i; ++j)
479 newseq->push_back(*j);
480 newseq->push_back(expair(newterm, i->coeff));
484 ex newpoint = point.eval_integ();
485 if (newseq || !are_ex_trivially_equal(newpoint, point))
486 return (new pseries(var==newpoint, *newseq))
487 ->setflag(status_flags::dynallocated);
491 ex pseries::evalm() const
493 // evalm each coefficient
495 bool something_changed = false;
496 for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
497 if (something_changed) {
498 ex newcoeff = i->rest.evalm();
499 if (!newcoeff.is_zero())
500 newseq.push_back(expair(newcoeff, i->coeff));
503 ex newcoeff = i->rest.evalm();
504 if (!are_ex_trivially_equal(newcoeff, i->rest)) {
505 something_changed = true;
506 newseq.reserve(seq.size());
507 std::copy(seq.begin(), i, std::back_inserter<epvector>(newseq));
508 if (!newcoeff.is_zero())
509 newseq.push_back(expair(newcoeff, i->coeff));
513 if (something_changed)
514 return (new pseries(var==point, newseq))->setflag(status_flags::dynallocated);
519 ex pseries::subs(const exmap & m, unsigned options) const
521 // If expansion variable is being substituted, convert the series to a
522 // polynomial and do the substitution there because the result might
523 // no longer be a power series
524 if (m.find(var) != m.end())
525 return convert_to_poly(true).subs(m, options);
527 // Otherwise construct a new series with substituted coefficients and
530 newseq.reserve(seq.size());
531 epvector::const_iterator it = seq.begin(), itend = seq.end();
532 while (it != itend) {
533 newseq.push_back(expair(it->rest.subs(m, options), it->coeff));
536 return (new pseries(relational(var,point.subs(m, options)), newseq))->setflag(status_flags::dynallocated);
539 /** Implementation of ex::expand() for a power series. It expands all the
540 * terms individually and returns the resulting series as a new pseries. */
541 ex pseries::expand(unsigned options) const
544 epvector::const_iterator i = seq.begin(), end = seq.end();
546 ex restexp = i->rest.expand();
547 if (!restexp.is_zero())
548 newseq.push_back(expair(restexp, i->coeff));
551 return (new pseries(relational(var,point), newseq))
552 ->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
555 /** Implementation of ex::diff() for a power series.
557 ex pseries::derivative(const symbol & s) const
560 epvector::const_iterator it = seq.begin(), itend = seq.end();
564 // FIXME: coeff might depend on var
565 while (it != itend) {
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));
578 while (it != itend) {
579 if (is_order_function(it->rest)) {
580 new_seq.push_back(*it);
582 ex c = it->rest.diff(s);
584 new_seq.push_back(expair(c, it->coeff));
590 return pseries(relational(var,point), new_seq);
593 ex pseries::convert_to_poly(bool no_order) const
596 epvector::const_iterator it = seq.begin(), itend = seq.end();
598 while (it != itend) {
599 if (is_order_function(it->rest)) {
601 e += Order(power(var - point, it->coeff));
603 e += it->rest * power(var - point, it->coeff);
609 bool pseries::is_terminating() const
611 return seq.empty() || !is_order_function((seq.end()-1)->rest);
614 ex pseries::coeffop(size_t i) const
617 throw (std::out_of_range("coeffop() out of range"));
621 ex pseries::exponop(size_t i) const
624 throw (std::out_of_range("exponop() out of range"));
630 * Implementations of series expansion
633 /** Default implementation of ex::series(). This performs Taylor expansion.
635 ex basic::series(const relational & r, int order, unsigned options) const
638 const symbol &s = ex_to<symbol>(r.lhs());
640 // default for order-values that make no sense for Taylor expansion
641 if ((order <= 0) && this->has(s)) {
642 seq.push_back(expair(Order(_ex1), order));
643 return pseries(r, seq);
646 // do Taylor expansion
649 ex coeff = deriv.subs(r, subs_options::no_pattern);
651 if (!coeff.is_zero()) {
652 seq.push_back(expair(coeff, _ex0));
656 for (n=1; n<order; ++n) {
658 // We need to test for zero in order to see if the series terminates.
659 // The problem is that there is no such thing as a perfect test for
660 // zero. Expanding the term occasionally helps a little...
661 deriv = deriv.diff(s).expand();
662 if (deriv.is_zero()) // Series terminates
663 return pseries(r, seq);
665 coeff = deriv.subs(r, subs_options::no_pattern);
666 if (!coeff.is_zero())
667 seq.push_back(expair(fac.inverse() * coeff, n));
670 // Higher-order terms, if present
671 deriv = deriv.diff(s);
672 if (!deriv.expand().is_zero())
673 seq.push_back(expair(Order(_ex1), n));
674 return pseries(r, seq);
678 /** Implementation of ex::series() for symbols.
680 ex symbol::series(const relational & r, int order, unsigned options) const
683 const ex point = r.rhs();
684 GINAC_ASSERT(is_a<symbol>(r.lhs()));
686 if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
687 if (order > 0 && !point.is_zero())
688 seq.push_back(expair(point, _ex0));
690 seq.push_back(expair(_ex1, _ex1));
692 seq.push_back(expair(Order(_ex1), numeric(order)));
694 seq.push_back(expair(*this, _ex0));
695 return pseries(r, seq);
699 /** Add one series object to another, producing a pseries object that
700 * represents the sum.
702 * @param other pseries object to add with
703 * @return the sum as a pseries */
704 ex pseries::add_series(const pseries &other) const
706 // Adding two series with different variables or expansion points
707 // results in an empty (constant) series
708 if (!is_compatible_to(other)) {
710 nul.push_back(expair(Order(_ex1), _ex0));
711 return pseries(relational(var,point), nul);
716 epvector::const_iterator a = seq.begin();
717 epvector::const_iterator b = other.seq.begin();
718 epvector::const_iterator a_end = seq.end();
719 epvector::const_iterator b_end = other.seq.end();
720 int pow_a = std::numeric_limits<int>::max(), pow_b = std::numeric_limits<int>::max();
722 // If a is empty, fill up with elements from b and stop
725 new_seq.push_back(*b);
730 pow_a = ex_to<numeric>((*a).coeff).to_int();
732 // If b is empty, fill up with elements from a and stop
735 new_seq.push_back(*a);
740 pow_b = ex_to<numeric>((*b).coeff).to_int();
742 // a and b are non-empty, compare powers
744 // a has lesser power, get coefficient from a
745 new_seq.push_back(*a);
746 if (is_order_function((*a).rest))
749 } else if (pow_b < pow_a) {
750 // b has lesser power, get coefficient from b
751 new_seq.push_back(*b);
752 if (is_order_function((*b).rest))
756 // Add coefficient of a and b
757 if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
758 new_seq.push_back(expair(Order(_ex1), (*a).coeff));
759 break; // Order term ends the sequence
761 ex sum = (*a).rest + (*b).rest;
762 if (!(sum.is_zero()))
763 new_seq.push_back(expair(sum, numeric(pow_a)));
769 return pseries(relational(var,point), new_seq);
773 /** Implementation of ex::series() for sums. This performs series addition when
774 * adding pseries objects.
776 ex add::series(const relational & r, int order, unsigned options) const
778 ex acc; // Series accumulator
780 // Get first term from overall_coeff
781 acc = overall_coeff.series(r, order, options);
783 // Add remaining terms
784 epvector::const_iterator it = seq.begin();
785 epvector::const_iterator itend = seq.end();
786 for (; it!=itend; ++it) {
788 if (is_exactly_a<pseries>(it->rest))
791 op = it->rest.series(r, order, options);
792 if (!it->coeff.is_equal(_ex1))
793 op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it->coeff));
796 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
802 /** Multiply a pseries object with a numeric constant, producing a pseries
803 * object that represents the product.
805 * @param other constant to multiply with
806 * @return the product as a pseries */
807 ex pseries::mul_const(const numeric &other) const
810 new_seq.reserve(seq.size());
812 epvector::const_iterator it = seq.begin(), itend = seq.end();
813 while (it != itend) {
814 if (!is_order_function(it->rest))
815 new_seq.push_back(expair(it->rest * other, it->coeff));
817 new_seq.push_back(*it);
820 return pseries(relational(var,point), new_seq);
824 /** Multiply one pseries object to another, producing a pseries object that
825 * represents the product.
827 * @param other pseries object to multiply with
828 * @return the product as a pseries */
829 ex pseries::mul_series(const pseries &other) const
831 // Multiplying two series with different variables or expansion points
832 // results in an empty (constant) series
833 if (!is_compatible_to(other)) {
835 nul.push_back(expair(Order(_ex1), _ex0));
836 return pseries(relational(var,point), nul);
839 if (seq.empty() || other.seq.empty()) {
840 return (new pseries(var==point, epvector()))
841 ->setflag(status_flags::dynallocated);
844 // Series multiplication
846 int a_max = degree(var);
847 int b_max = other.degree(var);
848 int a_min = ldegree(var);
849 int b_min = other.ldegree(var);
850 int cdeg_min = a_min + b_min;
851 int cdeg_max = a_max + b_max;
853 int higher_order_a = std::numeric_limits<int>::max();
854 int higher_order_b = std::numeric_limits<int>::max();
855 if (is_order_function(coeff(var, a_max)))
856 higher_order_a = a_max + b_min;
857 if (is_order_function(other.coeff(var, b_max)))
858 higher_order_b = b_max + a_min;
859 int higher_order_c = std::min(higher_order_a, higher_order_b);
860 if (cdeg_max >= higher_order_c)
861 cdeg_max = higher_order_c - 1;
863 for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
865 // c(i)=a(0)b(i)+...+a(i)b(0)
866 for (int i=a_min; cdeg-i>=b_min; ++i) {
867 ex a_coeff = coeff(var, i);
868 ex b_coeff = other.coeff(var, cdeg-i);
869 if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
870 co += a_coeff * b_coeff;
873 new_seq.push_back(expair(co, numeric(cdeg)));
875 if (higher_order_c < std::numeric_limits<int>::max())
876 new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c)));
877 return pseries(relational(var, point), new_seq);
881 /** Implementation of ex::series() for product. This performs series
882 * multiplication when multiplying series.
884 ex mul::series(const relational & r, int order, unsigned options) const
886 pseries acc; // Series accumulator
888 GINAC_ASSERT(is_a<symbol>(r.lhs()));
889 const ex& sym = r.lhs();
891 // holds ldegrees of the series of individual factors
892 std::vector<int> ldegrees;
893 std::vector<bool> ldegree_redo;
895 // find minimal degrees
896 const epvector::const_iterator itbeg = seq.begin();
897 const epvector::const_iterator itend = seq.end();
898 // first round: obtain a bound up to which minimal degrees have to be
900 for (epvector::const_iterator it=itbeg; it!=itend; ++it) {
902 ex expon = it->coeff;
905 if (expon.info(info_flags::integer)) {
907 factor = ex_to<numeric>(expon).to_int();
909 buf = recombine_pair_to_ex(*it);
912 int real_ldegree = 0;
913 bool flag_redo = false;
915 real_ldegree = buf.expand().ldegree(sym-r.rhs());
916 } catch (std::runtime_error) {}
918 if (real_ldegree == 0) {
920 // This case must terminate, otherwise we would have division by
925 real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
926 } while (real_ldegree == orderloop);
928 // Here it is possible that buf does not have a ldegree, therefore
929 // check only if ldegree is negative, otherwise reconsider the case
930 // in the second round.
931 real_ldegree = buf.series(r, 0, options).ldegree(sym);
932 if (real_ldegree == 0)
937 ldegrees.push_back(factor * real_ldegree);
938 ldegree_redo.push_back(flag_redo);
941 int degbound = order-std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
942 // Second round: determine the remaining positive ldegrees by the series
944 // here we can ignore ldegrees larger than degbound
946 for (epvector::const_iterator it=itbeg; it!=itend; ++it) {
947 if ( ldegree_redo[j] ) {
948 ex expon = it->coeff;
951 if (expon.info(info_flags::integer)) {
953 factor = ex_to<numeric>(expon).to_int();
955 buf = recombine_pair_to_ex(*it);
957 int real_ldegree = 0;
961 real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
962 } while ((real_ldegree == orderloop)
963 && ( factor*real_ldegree < degbound));
964 ldegrees[j] = factor * real_ldegree;
965 degbound -= factor * real_ldegree;
970 int degsum = std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
972 if (degsum >= order) {
974 epv.push_back(expair(Order(_ex1), order));
975 return (new pseries(r, epv))->setflag(status_flags::dynallocated);
978 // Multiply with remaining terms
979 std::vector<int>::const_iterator itd = ldegrees.begin();
980 for (epvector::const_iterator it=itbeg; it!=itend; ++it, ++itd) {
982 // do series expansion with adjusted order
983 ex op = recombine_pair_to_ex(*it).series(r, order-degsum+(*itd), options);
985 // Series multiplication
987 acc = ex_to<pseries>(op);
989 acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
992 return acc.mul_const(ex_to<numeric>(overall_coeff));
996 /** Compute the p-th power of a series.
998 * @param p power to compute
999 * @param deg truncation order of series calculation */
1000 ex pseries::power_const(const numeric &p, int deg) const
1003 // (due to Leonhard Euler)
1004 // let A(x) be this series and for the time being let it start with a
1005 // constant (later we'll generalize):
1006 // A(x) = a_0 + a_1*x + a_2*x^2 + ...
1007 // We want to compute
1009 // C(x) = c_0 + c_1*x + c_2*x^2 + ...
1010 // Taking the derivative on both sides and multiplying with A(x) one
1011 // immediately arrives at
1012 // C'(x)*A(x) = p*C(x)*A'(x)
1013 // Multiplying this out and comparing coefficients we get the recurrence
1015 // c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
1016 // ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
1017 // which can easily be solved given the starting value c_0 = (a_0)^p.
1018 // For the more general case where the leading coefficient of A(x) is not
1019 // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
1020 // repeat the above derivation. The leading power of C2(x) = A2(x)^2 is
1021 // then of course x^(p*m) but the recurrence formula still holds.
1024 // as a special case, handle the empty (zero) series honoring the
1025 // usual power laws such as implemented in power::eval()
1026 if (p.real().is_zero())
1027 throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
1028 else if (p.real().is_negative())
1029 throw pole_error("pseries::power_const(): division by zero",1);
1034 const int ldeg = ldegree(var);
1035 if (!(p*ldeg).is_integer())
1036 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
1038 // adjust number of coefficients
1039 int numcoeff = deg - (p*ldeg).to_int();
1040 if (numcoeff <= 0) {
1043 epv.push_back(expair(Order(_ex1), deg));
1044 return (new pseries(relational(var,point), epv))
1045 ->setflag(status_flags::dynallocated);
1048 // O(x^n)^(-m) is undefined
1049 if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
1050 throw pole_error("pseries::power_const(): division by zero",1);
1052 // Compute coefficients of the powered series
1054 co.reserve(numcoeff);
1055 co.push_back(power(coeff(var, ldeg), p));
1056 for (int i=1; i<numcoeff; ++i) {
1058 for (int j=1; j<=i; ++j) {
1059 ex c = coeff(var, j + ldeg);
1060 if (is_order_function(c)) {
1061 co.push_back(Order(_ex1));
1064 sum += (p * j - (i - j)) * co[i - j] * c;
1066 co.push_back(sum / coeff(var, ldeg) / i);
1069 // Construct new series (of non-zero coefficients)
1071 bool higher_order = false;
1072 for (int i=0; i<numcoeff; ++i) {
1073 if (!co[i].is_zero())
1074 new_seq.push_back(expair(co[i], p * ldeg + i));
1075 if (is_order_function(co[i])) {
1076 higher_order = true;
1081 new_seq.push_back(expair(Order(_ex1), p * ldeg + numcoeff));
1083 return pseries(relational(var,point), new_seq);
1087 /** Return a new pseries object with the powers shifted by deg. */
1088 pseries pseries::shift_exponents(int deg) const
1090 epvector newseq = seq;
1091 epvector::iterator i = newseq.begin(), end = newseq.end();
1096 return pseries(relational(var, point), newseq);
1100 /** Implementation of ex::series() for powers. This performs Laurent expansion
1101 * of reciprocals of series at singularities.
1102 * @see ex::series */
1103 ex power::series(const relational & r, int order, unsigned options) const
1105 // If basis is already a series, just power it
1106 if (is_exactly_a<pseries>(basis))
1107 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
1109 // Basis is not a series, may there be a singularity?
1110 bool must_expand_basis = false;
1112 basis.subs(r, subs_options::no_pattern);
1113 } catch (pole_error) {
1114 must_expand_basis = true;
1117 bool exponent_is_regular = true;
1119 exponent.subs(r, subs_options::no_pattern);
1120 } catch (pole_error) {
1121 exponent_is_regular = false;
1124 if (!exponent_is_regular) {
1125 ex l = exponent*log(basis);
1127 ex le = l.series(r, order, options);
1128 // Note: expanding exp(l) won't help, since that will attempt
1129 // Taylor expansion, and fail (because exponent is "singular")
1130 // Still l itself might be expanded in Taylor series.
1132 // sin(x)/x*log(cos(x))
1134 return exp(le).series(r, order, options);
1135 // Note: if l happens to have a Laurent expansion (with
1136 // negative powers of (var - point)), expanding exp(le)
1137 // will barf (which is The Right Thing).
1140 // Is the expression of type something^(-int)?
1141 if (!must_expand_basis && !exponent.info(info_flags::negint)
1142 && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1143 return basic::series(r, order, options);
1145 // Is the expression of type 0^something?
1146 if (!must_expand_basis && !basis.subs(r, subs_options::no_pattern).is_zero()
1147 && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1148 return basic::series(r, order, options);
1150 // Singularity encountered, is the basis equal to (var - point)?
1151 if (basis.is_equal(r.lhs() - r.rhs())) {
1153 if (ex_to<numeric>(exponent).to_int() < order)
1154 new_seq.push_back(expair(_ex1, exponent));
1156 new_seq.push_back(expair(Order(_ex1), exponent));
1157 return pseries(r, new_seq);
1160 // No, expand basis into series
1163 if (is_a<numeric>(exponent)) {
1164 numexp = ex_to<numeric>(exponent);
1168 const ex& sym = r.lhs();
1169 // find existing minimal degree
1170 ex eb = basis.expand();
1171 int real_ldegree = 0;
1172 if (eb.info(info_flags::rational_function))
1173 real_ldegree = eb.ldegree(sym-r.rhs());
1174 if (real_ldegree == 0) {
1178 real_ldegree = basis.series(r, orderloop, options).ldegree(sym);
1179 } while (real_ldegree == orderloop);
1182 if (!(real_ldegree*numexp).is_integer())
1183 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
1184 ex e = basis.series(r, (order + real_ldegree*(1-numexp)).to_int(), options);
1188 result = ex_to<pseries>(e).power_const(numexp, order);
1189 } catch (pole_error) {
1191 ser.push_back(expair(Order(_ex1), order));
1192 result = pseries(r, ser);
1199 /** Re-expansion of a pseries object. */
1200 ex pseries::series(const relational & r, int order, unsigned options) const
1202 const ex p = r.rhs();
1203 GINAC_ASSERT(is_a<symbol>(r.lhs()));
1204 const symbol &s = ex_to<symbol>(r.lhs());
1206 if (var.is_equal(s) && point.is_equal(p)) {
1207 if (order > degree(s))
1211 epvector::const_iterator it = seq.begin(), itend = seq.end();
1212 while (it != itend) {
1213 int o = ex_to<numeric>(it->coeff).to_int();
1215 new_seq.push_back(expair(Order(_ex1), o));
1218 new_seq.push_back(*it);
1221 return pseries(r, new_seq);
1224 return convert_to_poly().series(r, order, options);
1227 ex integral::series(const relational & r, int order, unsigned options) const
1230 throw std::logic_error("Cannot series expand wrt dummy variable");
1232 // Expanding integrant with r substituted taken in boundaries.
1233 ex fseries = f.series(r, order, options);
1234 epvector fexpansion;
1235 fexpansion.reserve(fseries.nops());
1236 for (size_t i=0; i<fseries.nops(); ++i) {
1237 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1238 currcoeff = (currcoeff == Order(_ex1))
1240 : integral(x, a.subs(r), b.subs(r), currcoeff);
1242 fexpansion.push_back(
1243 expair(currcoeff, ex_to<pseries>(fseries).exponop(i)));
1246 // Expanding lower boundary
1247 ex result = (new pseries(r, fexpansion))->setflag(status_flags::dynallocated);
1248 ex aseries = (a-a.subs(r)).series(r, order, options);
1249 fseries = f.series(x == (a.subs(r)), order, options);
1250 for (size_t i=0; i<fseries.nops(); ++i) {
1251 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1252 if (is_order_function(currcoeff))
1254 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1255 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1256 currcoeff = currcoeff.series(r, orderforf);
1257 ex term = ex_to<pseries>(aseries).power_const(ex_to<numeric>(currexpon+1),order);
1258 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(-1/(currexpon+1)));
1259 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1260 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1263 // Expanding upper boundary
1264 ex bseries = (b-b.subs(r)).series(r, order, options);
1265 fseries = f.series(x == (b.subs(r)), order, options);
1266 for (size_t i=0; i<fseries.nops(); ++i) {
1267 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1268 if (is_order_function(currcoeff))
1270 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1271 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1272 currcoeff = currcoeff.series(r, orderforf);
1273 ex term = ex_to<pseries>(bseries).power_const(ex_to<numeric>(currexpon+1),order);
1274 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(1/(currexpon+1)));
1275 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1276 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1283 /** Compute the truncated series expansion of an expression.
1284 * This function returns an expression containing an object of class pseries
1285 * to represent the series. If the series does not terminate within the given
1286 * truncation order, the last term of the series will be an order term.
1288 * @param r expansion relation, lhs holds variable and rhs holds point
1289 * @param order truncation order of series calculations
1290 * @param options of class series_options
1291 * @return an expression holding a pseries object */
1292 ex ex::series(const ex & r, int order, unsigned options) const
1297 if (is_a<relational>(r))
1298 rel_ = ex_to<relational>(r);
1299 else if (is_a<symbol>(r))
1300 rel_ = relational(r,_ex0);
1302 throw (std::logic_error("ex::series(): expansion point has unknown type"));
1304 e = bp->series(rel_, order, options);
1308 GINAC_BIND_UNARCHIVER(pseries);
1310 } // namespace GiNaC