3 * Implementation of class for extended truncated power series and
4 * methods for series expansion. */
7 * GiNaC Copyright (C) 1999-2007 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
29 #include "inifcns.h" // for Order function
33 #include "relational.h"
34 #include "operators.h"
42 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(pseries, basic,
43 print_func<print_context>(&pseries::do_print).
44 print_func<print_latex>(&pseries::do_print_latex).
45 print_func<print_tree>(&pseries::do_print_tree).
46 print_func<print_python>(&pseries::do_print_python).
47 print_func<print_python_repr>(&pseries::do_print_python_repr))
54 pseries::pseries() : inherited(&pseries::tinfo_static) { }
61 /** Construct pseries from a vector of coefficients and powers.
62 * expair.rest holds the coefficient, expair.coeff holds the power.
63 * The powers must be integers (positive or negative) and in ascending order;
64 * the last coefficient can be Order(_ex1) to represent a truncated,
65 * non-terminating series.
67 * @param rel_ expansion variable and point (must hold a relational)
68 * @param ops_ vector of {coefficient, power} pairs (coefficient must not be zero)
69 * @return newly constructed pseries */
70 pseries::pseries(const ex &rel_, const epvector &ops_) : basic(&pseries::tinfo_static), seq(ops_)
72 GINAC_ASSERT(is_a<relational>(rel_));
73 GINAC_ASSERT(is_a<symbol>(rel_.lhs()));
83 pseries::pseries(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
85 archive_node::archive_node_cit first = n.find_first("coeff");
86 archive_node::archive_node_cit last = n.find_last("power");
88 seq.reserve((last-first)/2);
90 for (archive_node::archive_node_cit loc = first; loc < last;) {
93 n.find_ex_by_loc(loc++, rest, sym_lst);
94 n.find_ex_by_loc(loc++, coeff, sym_lst);
95 seq.push_back(expair(rest, coeff));
98 n.find_ex("var", var, sym_lst);
99 n.find_ex("point", point, sym_lst);
102 void pseries::archive(archive_node &n) const
104 inherited::archive(n);
105 epvector::const_iterator i = seq.begin(), iend = seq.end();
107 n.add_ex("coeff", i->rest);
108 n.add_ex("power", i->coeff);
111 n.add_ex("var", var);
112 n.add_ex("point", point);
115 DEFAULT_UNARCHIVE(pseries)
118 // functions overriding virtual functions from base classes
121 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
123 if (precedence() <= level)
126 // objects of type pseries must not have any zero entries, so the
127 // trivial (zero) pseries needs a special treatment here:
131 epvector::const_iterator i = seq.begin(), end = seq.end();
134 // print a sign, if needed
135 if (i != seq.begin())
138 if (!is_order_function(i->rest)) {
140 // print 'rest', i.e. the expansion coefficient
141 if (i->rest.info(info_flags::numeric) &&
142 i->rest.info(info_flags::positive)) {
145 c.s << openbrace << '(';
147 c.s << ')' << closebrace;
150 // print 'coeff', something like (x-1)^42
151 if (!i->coeff.is_zero()) {
153 if (!point.is_zero()) {
154 c.s << openbrace << '(';
155 (var-point).print(c);
156 c.s << ')' << closebrace;
159 if (i->coeff.compare(_ex1)) {
162 if (i->coeff.info(info_flags::negative)) {
172 Order(power(var-point,i->coeff)).print(c);
176 if (precedence() <= level)
180 void pseries::do_print(const print_context & c, unsigned level) const
182 print_series(c, "", "", "*", "^", level);
185 void pseries::do_print_latex(const print_latex & c, unsigned level) const
187 print_series(c, "{", "}", " ", "^", level);
190 void pseries::do_print_python(const print_python & c, unsigned level) const
192 print_series(c, "", "", "*", "**", level);
195 void pseries::do_print_tree(const print_tree & c, unsigned level) const
197 c.s << std::string(level, ' ') << class_name() << " @" << this
198 << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
200 size_t num = seq.size();
201 for (size_t i=0; i<num; ++i) {
202 seq[i].rest.print(c, level + c.delta_indent);
203 seq[i].coeff.print(c, level + c.delta_indent);
204 c.s << std::string(level + c.delta_indent, ' ') << "-----" << std::endl;
206 var.print(c, level + c.delta_indent);
207 point.print(c, level + c.delta_indent);
210 void pseries::do_print_python_repr(const print_python_repr & c, unsigned level) const
212 c.s << class_name() << "(relational(";
217 size_t num = seq.size();
218 for (size_t i=0; i<num; ++i) {
222 seq[i].rest.print(c);
224 seq[i].coeff.print(c);
230 int pseries::compare_same_type(const basic & other) const
232 GINAC_ASSERT(is_a<pseries>(other));
233 const pseries &o = static_cast<const pseries &>(other);
235 // first compare the lengths of the series...
236 if (seq.size()>o.seq.size())
238 if (seq.size()<o.seq.size())
241 // ...then the expansion point...
242 int cmpval = var.compare(o.var);
245 cmpval = point.compare(o.point);
249 // ...and if that failed the individual elements
250 epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
251 while (it!=seq.end() && o_it!=o.seq.end()) {
252 cmpval = it->compare(*o_it);
259 // so they are equal.
263 /** Return the number of operands including a possible order term. */
264 size_t pseries::nops() const
269 /** Return the ith term in the series when represented as a sum. */
270 ex pseries::op(size_t i) const
273 throw (std::out_of_range("op() out of range"));
275 if (is_order_function(seq[i].rest))
276 return Order(power(var-point, seq[i].coeff));
277 return seq[i].rest * power(var - point, seq[i].coeff);
280 /** Return degree of highest power of the series. This is usually the exponent
281 * of the Order term. If s is not the expansion variable of the series, the
282 * series is examined termwise. */
283 int pseries::degree(const ex &s) const
285 if (var.is_equal(s)) {
286 // Return last exponent
288 return ex_to<numeric>((seq.end()-1)->coeff).to_int();
292 epvector::const_iterator it = seq.begin(), itend = seq.end();
295 int max_pow = INT_MIN;
296 while (it != itend) {
297 int pow = it->rest.degree(s);
306 /** Return degree of lowest power of the series. This is usually the exponent
307 * of the leading term. If s is not the expansion variable of the series, the
308 * series is examined termwise. If s is the expansion variable but the
309 * expansion point is not zero the series is not expanded to find the degree.
310 * I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
311 int pseries::ldegree(const ex &s) const
313 if (var.is_equal(s)) {
314 // Return first exponent
316 return ex_to<numeric>((seq.begin())->coeff).to_int();
320 epvector::const_iterator it = seq.begin(), itend = seq.end();
323 int min_pow = INT_MAX;
324 while (it != itend) {
325 int pow = it->rest.ldegree(s);
334 /** Return coefficient of degree n in power series if s is the expansion
335 * variable. If the expansion point is nonzero, by definition the n=1
336 * coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
337 * the expansion took place in the s in the first place).
338 * If s is not the expansion variable, an attempt is made to convert the
339 * series to a polynomial and return the corresponding coefficient from
341 ex pseries::coeff(const ex &s, int n) const
343 if (var.is_equal(s)) {
347 // Binary search in sequence for given power
348 numeric looking_for = numeric(n);
349 int lo = 0, hi = seq.size() - 1;
351 int mid = (lo + hi) / 2;
352 GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
353 int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
359 return seq[mid].rest;
364 throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
369 return convert_to_poly().coeff(s, n);
373 ex pseries::collect(const ex &s, bool distributed) const
378 /** Perform coefficient-wise automatic term rewriting rules in this class. */
379 ex pseries::eval(int level) const
384 if (level == -max_recursion_level)
385 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
387 // Construct a new series with evaluated coefficients
389 new_seq.reserve(seq.size());
390 epvector::const_iterator it = seq.begin(), itend = seq.end();
391 while (it != itend) {
392 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
395 return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
398 /** Evaluate coefficients numerically. */
399 ex pseries::evalf(int level) const
404 if (level == -max_recursion_level)
405 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
407 // Construct a new series with evaluated coefficients
409 new_seq.reserve(seq.size());
410 epvector::const_iterator it = seq.begin(), itend = seq.end();
411 while (it != itend) {
412 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
415 return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
418 ex pseries::conjugate() const
420 if(!var.info(info_flags::real))
421 return conjugate_function(*this).hold();
423 epvector * newseq = conjugateepvector(seq);
424 ex newpoint = point.conjugate();
426 if (!newseq && are_ex_trivially_equal(point, newpoint)) {
430 ex result = (new pseries(var==newpoint, newseq ? *newseq : seq))->setflag(status_flags::dynallocated);
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(epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i)
448 v.push_back(expair((i->rest).real_part(), i->coeff));
449 return (new pseries(var==point, v))->setflag(status_flags::dynallocated);
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(epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i)
463 v.push_back(expair((i->rest).imag_part(), i->coeff));
464 return (new pseries(var==point, v))->setflag(status_flags::dynallocated);
467 ex pseries::eval_integ() const
469 epvector *newseq = NULL;
470 for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
472 newseq->push_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 = new epvector;
478 newseq->reserve(seq.size());
479 for (epvector::const_iterator j=seq.begin(); j!=i; ++j)
480 newseq->push_back(*j);
481 newseq->push_back(expair(newterm, i->coeff));
485 ex newpoint = point.eval_integ();
486 if (newseq || !are_ex_trivially_equal(newpoint, point))
487 return (new pseries(var==newpoint, *newseq))
488 ->setflag(status_flags::dynallocated);
492 ex pseries::evalm() const
494 // evalm each coefficient
496 bool something_changed = false;
497 for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
498 if (something_changed) {
499 ex newcoeff = i->rest.evalm();
500 if (!newcoeff.is_zero())
501 newseq.push_back(expair(newcoeff, i->coeff));
504 ex newcoeff = i->rest.evalm();
505 if (!are_ex_trivially_equal(newcoeff, i->rest)) {
506 something_changed = true;
507 newseq.reserve(seq.size());
508 std::copy(seq.begin(), i, std::back_inserter<epvector>(newseq));
509 if (!newcoeff.is_zero())
510 newseq.push_back(expair(newcoeff, i->coeff));
514 if (something_changed)
515 return (new pseries(var==point, newseq))->setflag(status_flags::dynallocated);
520 ex pseries::subs(const exmap & m, unsigned options) const
522 // If expansion variable is being substituted, convert the series to a
523 // polynomial and do the substitution there because the result might
524 // no longer be a power series
525 if (m.find(var) != m.end())
526 return convert_to_poly(true).subs(m, options);
528 // Otherwise construct a new series with substituted coefficients and
531 newseq.reserve(seq.size());
532 epvector::const_iterator it = seq.begin(), itend = seq.end();
533 while (it != itend) {
534 newseq.push_back(expair(it->rest.subs(m, options), it->coeff));
537 return (new pseries(relational(var,point.subs(m, options)), newseq))->setflag(status_flags::dynallocated);
540 /** Implementation of ex::expand() for a power series. It expands all the
541 * terms individually and returns the resulting series as a new pseries. */
542 ex pseries::expand(unsigned options) const
545 epvector::const_iterator i = seq.begin(), end = seq.end();
547 ex restexp = i->rest.expand();
548 if (!restexp.is_zero())
549 newseq.push_back(expair(restexp, i->coeff));
552 return (new pseries(relational(var,point), newseq))
553 ->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
556 /** Implementation of ex::diff() for a power series.
558 ex pseries::derivative(const symbol & s) const
561 epvector::const_iterator it = seq.begin(), itend = seq.end();
565 // FIXME: coeff might depend on var
566 while (it != itend) {
567 if (is_order_function(it->rest)) {
568 new_seq.push_back(expair(it->rest, it->coeff - 1));
570 ex c = it->rest * it->coeff;
572 new_seq.push_back(expair(c, it->coeff - 1));
579 while (it != itend) {
580 if (is_order_function(it->rest)) {
581 new_seq.push_back(*it);
583 ex c = it->rest.diff(s);
585 new_seq.push_back(expair(c, it->coeff));
591 return pseries(relational(var,point), new_seq);
594 ex pseries::convert_to_poly(bool no_order) const
597 epvector::const_iterator it = seq.begin(), itend = seq.end();
599 while (it != itend) {
600 if (is_order_function(it->rest)) {
602 e += Order(power(var - point, it->coeff));
604 e += it->rest * power(var - point, it->coeff);
610 bool pseries::is_terminating() const
612 return seq.empty() || !is_order_function((seq.end()-1)->rest);
615 ex pseries::coeffop(size_t i) const
618 throw (std::out_of_range("coeffop() out of range"));
622 ex pseries::exponop(size_t i) const
625 throw (std::out_of_range("exponop() out of range"));
631 * Implementations of series expansion
634 /** Default implementation of ex::series(). This performs Taylor expansion.
636 ex basic::series(const relational & r, int order, unsigned options) const
639 const symbol &s = ex_to<symbol>(r.lhs());
641 // default for order-values that make no sense for Taylor expansion
642 if ((order <= 0) && this->has(s)) {
643 seq.push_back(expair(Order(_ex1), order));
644 return pseries(r, seq);
647 // do Taylor expansion
650 ex coeff = deriv.subs(r, subs_options::no_pattern);
652 if (!coeff.is_zero()) {
653 seq.push_back(expair(coeff, _ex0));
657 for (n=1; n<order; ++n) {
659 // We need to test for zero in order to see if the series terminates.
660 // The problem is that there is no such thing as a perfect test for
661 // zero. Expanding the term occasionally helps a little...
662 deriv = deriv.diff(s).expand();
663 if (deriv.is_zero()) // Series terminates
664 return pseries(r, seq);
666 coeff = deriv.subs(r, subs_options::no_pattern);
667 if (!coeff.is_zero())
668 seq.push_back(expair(fac.inverse() * coeff, n));
671 // Higher-order terms, if present
672 deriv = deriv.diff(s);
673 if (!deriv.expand().is_zero())
674 seq.push_back(expair(Order(_ex1), n));
675 return pseries(r, seq);
679 /** Implementation of ex::series() for symbols.
681 ex symbol::series(const relational & r, int order, unsigned options) const
684 const ex point = r.rhs();
685 GINAC_ASSERT(is_a<symbol>(r.lhs()));
687 if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
688 if (order > 0 && !point.is_zero())
689 seq.push_back(expair(point, _ex0));
691 seq.push_back(expair(_ex1, _ex1));
693 seq.push_back(expair(Order(_ex1), numeric(order)));
695 seq.push_back(expair(*this, _ex0));
696 return pseries(r, seq);
700 /** Add one series object to another, producing a pseries object that
701 * represents the sum.
703 * @param other pseries object to add with
704 * @return the sum as a pseries */
705 ex pseries::add_series(const pseries &other) const
707 // Adding two series with different variables or expansion points
708 // results in an empty (constant) series
709 if (!is_compatible_to(other)) {
711 nul.push_back(expair(Order(_ex1), _ex0));
712 return pseries(relational(var,point), nul);
717 epvector::const_iterator a = seq.begin();
718 epvector::const_iterator b = other.seq.begin();
719 epvector::const_iterator a_end = seq.end();
720 epvector::const_iterator b_end = other.seq.end();
721 int pow_a = INT_MAX, pow_b = INT_MAX;
723 // If a is empty, fill up with elements from b and stop
726 new_seq.push_back(*b);
731 pow_a = ex_to<numeric>((*a).coeff).to_int();
733 // If b is empty, fill up with elements from a and stop
736 new_seq.push_back(*a);
741 pow_b = ex_to<numeric>((*b).coeff).to_int();
743 // a and b are non-empty, compare powers
745 // a has lesser power, get coefficient from a
746 new_seq.push_back(*a);
747 if (is_order_function((*a).rest))
750 } else if (pow_b < pow_a) {
751 // b has lesser power, get coefficient from b
752 new_seq.push_back(*b);
753 if (is_order_function((*b).rest))
757 // Add coefficient of a and b
758 if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
759 new_seq.push_back(expair(Order(_ex1), (*a).coeff));
760 break; // Order term ends the sequence
762 ex sum = (*a).rest + (*b).rest;
763 if (!(sum.is_zero()))
764 new_seq.push_back(expair(sum, numeric(pow_a)));
770 return pseries(relational(var,point), new_seq);
774 /** Implementation of ex::series() for sums. This performs series addition when
775 * adding pseries objects.
777 ex add::series(const relational & r, int order, unsigned options) const
779 ex acc; // Series accumulator
781 // Get first term from overall_coeff
782 acc = overall_coeff.series(r, order, options);
784 // Add remaining terms
785 epvector::const_iterator it = seq.begin();
786 epvector::const_iterator itend = seq.end();
787 for (; it!=itend; ++it) {
789 if (is_exactly_a<pseries>(it->rest))
792 op = it->rest.series(r, order, options);
793 if (!it->coeff.is_equal(_ex1))
794 op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it->coeff));
797 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
803 /** Multiply a pseries object with a numeric constant, producing a pseries
804 * object that represents the product.
806 * @param other constant to multiply with
807 * @return the product as a pseries */
808 ex pseries::mul_const(const numeric &other) const
811 new_seq.reserve(seq.size());
813 epvector::const_iterator it = seq.begin(), itend = seq.end();
814 while (it != itend) {
815 if (!is_order_function(it->rest))
816 new_seq.push_back(expair(it->rest * other, it->coeff));
818 new_seq.push_back(*it);
821 return pseries(relational(var,point), new_seq);
825 /** Multiply one pseries object to another, producing a pseries object that
826 * represents the product.
828 * @param other pseries object to multiply with
829 * @return the product as a pseries */
830 ex pseries::mul_series(const pseries &other) const
832 // Multiplying two series with different variables or expansion points
833 // results in an empty (constant) series
834 if (!is_compatible_to(other)) {
836 nul.push_back(expair(Order(_ex1), _ex0));
837 return pseries(relational(var,point), nul);
840 if (seq.empty() || other.seq.empty()) {
841 return (new pseries(var==point, epvector()))
842 ->setflag(status_flags::dynallocated);
845 // Series multiplication
847 int a_max = degree(var);
848 int b_max = other.degree(var);
849 int a_min = ldegree(var);
850 int b_min = other.ldegree(var);
851 int cdeg_min = a_min + b_min;
852 int cdeg_max = a_max + b_max;
854 int higher_order_a = INT_MAX;
855 int higher_order_b = INT_MAX;
856 if (is_order_function(coeff(var, a_max)))
857 higher_order_a = a_max + b_min;
858 if (is_order_function(other.coeff(var, b_max)))
859 higher_order_b = b_max + a_min;
860 int higher_order_c = std::min(higher_order_a, higher_order_b);
861 if (cdeg_max >= higher_order_c)
862 cdeg_max = higher_order_c - 1;
864 for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
866 // c(i)=a(0)b(i)+...+a(i)b(0)
867 for (int i=a_min; cdeg-i>=b_min; ++i) {
868 ex a_coeff = coeff(var, i);
869 ex b_coeff = other.coeff(var, cdeg-i);
870 if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
871 co += a_coeff * b_coeff;
874 new_seq.push_back(expair(co, numeric(cdeg)));
876 if (higher_order_c < INT_MAX)
877 new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c)));
878 return pseries(relational(var, point), new_seq);
882 /** Implementation of ex::series() for product. This performs series
883 * multiplication when multiplying series.
885 ex mul::series(const relational & r, int order, unsigned options) const
887 pseries acc; // Series accumulator
889 GINAC_ASSERT(is_a<symbol>(r.lhs()));
890 const ex& sym = r.lhs();
892 // holds ldegrees of the series of individual factors
893 std::vector<int> ldegrees;
895 // find minimal degrees
896 const epvector::const_iterator itbeg = seq.begin();
897 const epvector::const_iterator itend = seq.end();
898 for (epvector::const_iterator it=itbeg; it!=itend; ++it) {
900 ex expon = it->coeff;
903 if (expon.info(info_flags::integer)) {
905 factor = ex_to<numeric>(expon).to_int();
907 buf = recombine_pair_to_ex(*it);
910 int real_ldegree = 0;
912 real_ldegree = buf.expand().ldegree(sym-r.rhs());
913 } catch (std::runtime_error) {}
915 if (real_ldegree == 0) {
919 real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
920 } while (real_ldegree == orderloop);
923 ldegrees.push_back(factor * real_ldegree);
926 int degsum = std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
928 if (degsum >= order) {
930 epv.push_back(expair(Order(_ex1), order));
931 return (new pseries(r, epv))->setflag(status_flags::dynallocated);
934 // Multiply with remaining terms
935 std::vector<int>::const_iterator itd = ldegrees.begin();
936 for (epvector::const_iterator it=itbeg; it!=itend; ++it, ++itd) {
938 // do series expansion with adjusted order
939 ex op = recombine_pair_to_ex(*it).series(r, order-degsum+(*itd), options);
941 // Series multiplication
943 acc = ex_to<pseries>(op);
945 acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
948 return acc.mul_const(ex_to<numeric>(overall_coeff));
952 /** Compute the p-th power of a series.
954 * @param p power to compute
955 * @param deg truncation order of series calculation */
956 ex pseries::power_const(const numeric &p, int deg) const
959 // (due to Leonhard Euler)
960 // let A(x) be this series and for the time being let it start with a
961 // constant (later we'll generalize):
962 // A(x) = a_0 + a_1*x + a_2*x^2 + ...
963 // We want to compute
965 // C(x) = c_0 + c_1*x + c_2*x^2 + ...
966 // Taking the derivative on both sides and multiplying with A(x) one
967 // immediately arrives at
968 // C'(x)*A(x) = p*C(x)*A'(x)
969 // Multiplying this out and comparing coefficients we get the recurrence
971 // c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
972 // ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
973 // which can easily be solved given the starting value c_0 = (a_0)^p.
974 // For the more general case where the leading coefficient of A(x) is not
975 // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
976 // repeat the above derivation. The leading power of C2(x) = A2(x)^2 is
977 // then of course x^(p*m) but the recurrence formula still holds.
980 // as a special case, handle the empty (zero) series honoring the
981 // usual power laws such as implemented in power::eval()
982 if (p.real().is_zero())
983 throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
984 else if (p.real().is_negative())
985 throw pole_error("pseries::power_const(): division by zero",1);
990 const int ldeg = ldegree(var);
991 if (!(p*ldeg).is_integer())
992 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
994 // adjust number of coefficients
995 int numcoeff = deg - (p*ldeg).to_int();
999 epv.push_back(expair(Order(_ex1), deg));
1000 return (new pseries(relational(var,point), epv))
1001 ->setflag(status_flags::dynallocated);
1004 // O(x^n)^(-m) is undefined
1005 if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
1006 throw pole_error("pseries::power_const(): division by zero",1);
1008 // Compute coefficients of the powered series
1010 co.reserve(numcoeff);
1011 co.push_back(power(coeff(var, ldeg), p));
1012 for (int i=1; i<numcoeff; ++i) {
1014 for (int j=1; j<=i; ++j) {
1015 ex c = coeff(var, j + ldeg);
1016 if (is_order_function(c)) {
1017 co.push_back(Order(_ex1));
1020 sum += (p * j - (i - j)) * co[i - j] * c;
1022 co.push_back(sum / coeff(var, ldeg) / i);
1025 // Construct new series (of non-zero coefficients)
1027 bool higher_order = false;
1028 for (int i=0; i<numcoeff; ++i) {
1029 if (!co[i].is_zero())
1030 new_seq.push_back(expair(co[i], p * ldeg + i));
1031 if (is_order_function(co[i])) {
1032 higher_order = true;
1037 new_seq.push_back(expair(Order(_ex1), p * ldeg + numcoeff));
1039 return pseries(relational(var,point), new_seq);
1043 /** Return a new pseries object with the powers shifted by deg. */
1044 pseries pseries::shift_exponents(int deg) const
1046 epvector newseq = seq;
1047 epvector::iterator i = newseq.begin(), end = newseq.end();
1052 return pseries(relational(var, point), newseq);
1056 /** Implementation of ex::series() for powers. This performs Laurent expansion
1057 * of reciprocals of series at singularities.
1058 * @see ex::series */
1059 ex power::series(const relational & r, int order, unsigned options) const
1061 // If basis is already a series, just power it
1062 if (is_exactly_a<pseries>(basis))
1063 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
1065 // Basis is not a series, may there be a singularity?
1066 bool must_expand_basis = false;
1068 basis.subs(r, subs_options::no_pattern);
1069 } catch (pole_error) {
1070 must_expand_basis = true;
1073 // Is the expression of type something^(-int)?
1074 if (!must_expand_basis && !exponent.info(info_flags::negint)
1075 && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1076 return basic::series(r, order, options);
1078 // Is the expression of type 0^something?
1079 if (!must_expand_basis && !basis.subs(r, subs_options::no_pattern).is_zero()
1080 && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1081 return basic::series(r, order, options);
1083 // Singularity encountered, is the basis equal to (var - point)?
1084 if (basis.is_equal(r.lhs() - r.rhs())) {
1086 if (ex_to<numeric>(exponent).to_int() < order)
1087 new_seq.push_back(expair(_ex1, exponent));
1089 new_seq.push_back(expair(Order(_ex1), exponent));
1090 return pseries(r, new_seq);
1093 // No, expand basis into series
1096 if (is_a<numeric>(exponent)) {
1097 numexp = ex_to<numeric>(exponent);
1101 const ex& sym = r.lhs();
1102 // find existing minimal degree
1103 ex eb = basis.expand();
1104 int real_ldegree = 0;
1105 if (eb.info(info_flags::rational_function))
1106 real_ldegree = eb.ldegree(sym-r.rhs());
1107 if (real_ldegree == 0) {
1111 real_ldegree = basis.series(r, orderloop, options).ldegree(sym);
1112 } while (real_ldegree == orderloop);
1115 if (!(real_ldegree*numexp).is_integer())
1116 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
1117 ex e = basis.series(r, (order + real_ldegree*(1-numexp)).to_int(), options);
1121 result = ex_to<pseries>(e).power_const(numexp, order);
1122 } catch (pole_error) {
1124 ser.push_back(expair(Order(_ex1), order));
1125 result = pseries(r, ser);
1132 /** Re-expansion of a pseries object. */
1133 ex pseries::series(const relational & r, int order, unsigned options) const
1135 const ex p = r.rhs();
1136 GINAC_ASSERT(is_a<symbol>(r.lhs()));
1137 const symbol &s = ex_to<symbol>(r.lhs());
1139 if (var.is_equal(s) && point.is_equal(p)) {
1140 if (order > degree(s))
1144 epvector::const_iterator it = seq.begin(), itend = seq.end();
1145 while (it != itend) {
1146 int o = ex_to<numeric>(it->coeff).to_int();
1148 new_seq.push_back(expair(Order(_ex1), o));
1151 new_seq.push_back(*it);
1154 return pseries(r, new_seq);
1157 return convert_to_poly().series(r, order, options);
1160 ex integral::series(const relational & r, int order, unsigned options) const
1163 throw std::logic_error("Cannot series expand wrt dummy variable");
1165 // Expanding integrant with r substituted taken in boundaries.
1166 ex fseries = f.series(r, order, options);
1167 epvector fexpansion;
1168 fexpansion.reserve(fseries.nops());
1169 for (size_t i=0; i<fseries.nops(); ++i) {
1170 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1171 currcoeff = (currcoeff == Order(_ex1))
1173 : integral(x, a.subs(r), b.subs(r), currcoeff);
1175 fexpansion.push_back(
1176 expair(currcoeff, ex_to<pseries>(fseries).exponop(i)));
1179 // Expanding lower boundary
1180 ex result = (new pseries(r, fexpansion))->setflag(status_flags::dynallocated);
1181 ex aseries = (a-a.subs(r)).series(r, order, options);
1182 fseries = f.series(x == (a.subs(r)), order, options);
1183 for (size_t i=0; i<fseries.nops(); ++i) {
1184 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1185 if (is_order_function(currcoeff))
1187 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1188 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1189 currcoeff = currcoeff.series(r, orderforf);
1190 ex term = ex_to<pseries>(aseries).power_const(ex_to<numeric>(currexpon+1),order);
1191 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(-1/(currexpon+1)));
1192 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1193 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1196 // Expanding upper boundary
1197 ex bseries = (b-b.subs(r)).series(r, order, options);
1198 fseries = f.series(x == (b.subs(r)), order, options);
1199 for (size_t i=0; i<fseries.nops(); ++i) {
1200 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1201 if (is_order_function(currcoeff))
1203 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1204 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1205 currcoeff = currcoeff.series(r, orderforf);
1206 ex term = ex_to<pseries>(bseries).power_const(ex_to<numeric>(currexpon+1),order);
1207 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(1/(currexpon+1)));
1208 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1209 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1216 /** Compute the truncated series expansion of an expression.
1217 * This function returns an expression containing an object of class pseries
1218 * to represent the series. If the series does not terminate within the given
1219 * truncation order, the last term of the series will be an order term.
1221 * @param r expansion relation, lhs holds variable and rhs holds point
1222 * @param order truncation order of series calculations
1223 * @param options of class series_options
1224 * @return an expression holding a pseries object */
1225 ex ex::series(const ex & r, int order, unsigned options) const
1230 if (is_a<relational>(r))
1231 rel_ = ex_to<relational>(r);
1232 else if (is_a<symbol>(r))
1233 rel_ = relational(r,_ex0);
1235 throw (std::logic_error("ex::series(): expansion point has unknown type"));
1238 e = bp->series(rel_, order, options);
1239 } catch (std::exception &x) {
1240 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));
1245 } // namespace GiNaC