3 * Implementation of class for extended truncated power series and
4 * methods for series expansion. */
7 * GiNaC Copyright (C) 1999-2006 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 for (unsigned int i=0; true; ++i) {
88 if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
89 seq.push_back(expair(rest, coeff));
93 n.find_ex("var", var, sym_lst);
94 n.find_ex("point", point, sym_lst);
97 void pseries::archive(archive_node &n) const
99 inherited::archive(n);
100 epvector::const_iterator i = seq.begin(), iend = seq.end();
102 n.add_ex("coeff", i->rest);
103 n.add_ex("power", i->coeff);
106 n.add_ex("var", var);
107 n.add_ex("point", point);
110 DEFAULT_UNARCHIVE(pseries)
113 // functions overriding virtual functions from base classes
116 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
118 if (precedence() <= level)
121 // objects of type pseries must not have any zero entries, so the
122 // trivial (zero) pseries needs a special treatment here:
126 epvector::const_iterator i = seq.begin(), end = seq.end();
129 // print a sign, if needed
130 if (i != seq.begin())
133 if (!is_order_function(i->rest)) {
135 // print 'rest', i.e. the expansion coefficient
136 if (i->rest.info(info_flags::numeric) &&
137 i->rest.info(info_flags::positive)) {
140 c.s << openbrace << '(';
142 c.s << ')' << closebrace;
145 // print 'coeff', something like (x-1)^42
146 if (!i->coeff.is_zero()) {
148 if (!point.is_zero()) {
149 c.s << openbrace << '(';
150 (var-point).print(c);
151 c.s << ')' << closebrace;
154 if (i->coeff.compare(_ex1)) {
157 if (i->coeff.info(info_flags::negative)) {
167 Order(power(var-point,i->coeff)).print(c);
171 if (precedence() <= level)
175 void pseries::do_print(const print_context & c, unsigned level) const
177 print_series(c, "", "", "*", "^", level);
180 void pseries::do_print_latex(const print_latex & c, unsigned level) const
182 print_series(c, "{", "}", " ", "^", level);
185 void pseries::do_print_python(const print_python & c, unsigned level) const
187 print_series(c, "", "", "*", "**", level);
190 void pseries::do_print_tree(const print_tree & c, unsigned level) const
192 c.s << std::string(level, ' ') << class_name() << " @" << this
193 << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
195 size_t num = seq.size();
196 for (size_t i=0; i<num; ++i) {
197 seq[i].rest.print(c, level + c.delta_indent);
198 seq[i].coeff.print(c, level + c.delta_indent);
199 c.s << std::string(level + c.delta_indent, ' ') << "-----" << std::endl;
201 var.print(c, level + c.delta_indent);
202 point.print(c, level + c.delta_indent);
205 void pseries::do_print_python_repr(const print_python_repr & c, unsigned level) const
207 c.s << class_name() << "(relational(";
212 size_t num = seq.size();
213 for (size_t i=0; i<num; ++i) {
217 seq[i].rest.print(c);
219 seq[i].coeff.print(c);
225 int pseries::compare_same_type(const basic & other) const
227 GINAC_ASSERT(is_a<pseries>(other));
228 const pseries &o = static_cast<const pseries &>(other);
230 // first compare the lengths of the series...
231 if (seq.size()>o.seq.size())
233 if (seq.size()<o.seq.size())
236 // ...then the expansion point...
237 int cmpval = var.compare(o.var);
240 cmpval = point.compare(o.point);
244 // ...and if that failed the individual elements
245 epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
246 while (it!=seq.end() && o_it!=o.seq.end()) {
247 cmpval = it->compare(*o_it);
254 // so they are equal.
258 /** Return the number of operands including a possible order term. */
259 size_t pseries::nops() const
264 /** Return the ith term in the series when represented as a sum. */
265 ex pseries::op(size_t i) const
268 throw (std::out_of_range("op() out of range"));
270 if (is_order_function(seq[i].rest))
271 return Order(power(var-point, seq[i].coeff));
272 return seq[i].rest * power(var - point, seq[i].coeff);
275 /** Return degree of highest power of the series. This is usually the exponent
276 * of the Order term. If s is not the expansion variable of the series, the
277 * series is examined termwise. */
278 int pseries::degree(const ex &s) const
280 if (var.is_equal(s)) {
281 // Return last exponent
283 return ex_to<numeric>((seq.end()-1)->coeff).to_int();
287 epvector::const_iterator it = seq.begin(), itend = seq.end();
290 int max_pow = INT_MIN;
291 while (it != itend) {
292 int pow = it->rest.degree(s);
301 /** Return degree of lowest power of the series. This is usually the exponent
302 * of the leading term. If s is not the expansion variable of the series, the
303 * series is examined termwise. If s is the expansion variable but the
304 * expansion point is not zero the series is not expanded to find the degree.
305 * I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
306 int pseries::ldegree(const ex &s) const
308 if (var.is_equal(s)) {
309 // Return first exponent
311 return ex_to<numeric>((seq.begin())->coeff).to_int();
315 epvector::const_iterator it = seq.begin(), itend = seq.end();
318 int min_pow = INT_MAX;
319 while (it != itend) {
320 int pow = it->rest.ldegree(s);
329 /** Return coefficient of degree n in power series if s is the expansion
330 * variable. If the expansion point is nonzero, by definition the n=1
331 * coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
332 * the expansion took place in the s in the first place).
333 * If s is not the expansion variable, an attempt is made to convert the
334 * series to a polynomial and return the corresponding coefficient from
336 ex pseries::coeff(const ex &s, int n) const
338 if (var.is_equal(s)) {
342 // Binary search in sequence for given power
343 numeric looking_for = numeric(n);
344 int lo = 0, hi = seq.size() - 1;
346 int mid = (lo + hi) / 2;
347 GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
348 int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
354 return seq[mid].rest;
359 throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
364 return convert_to_poly().coeff(s, n);
368 ex pseries::collect(const ex &s, bool distributed) const
373 /** Perform coefficient-wise automatic term rewriting rules in this class. */
374 ex pseries::eval(int level) const
379 if (level == -max_recursion_level)
380 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
382 // Construct a new series with evaluated coefficients
384 new_seq.reserve(seq.size());
385 epvector::const_iterator it = seq.begin(), itend = seq.end();
386 while (it != itend) {
387 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
390 return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
393 /** Evaluate coefficients numerically. */
394 ex pseries::evalf(int level) const
399 if (level == -max_recursion_level)
400 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
402 // Construct a new series with evaluated coefficients
404 new_seq.reserve(seq.size());
405 epvector::const_iterator it = seq.begin(), itend = seq.end();
406 while (it != itend) {
407 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
410 return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
413 ex pseries::conjugate() const
415 if(!var.info(info_flags::real))
416 return conjugate_function(*this).hold();
418 epvector * newseq = conjugateepvector(seq);
419 ex newpoint = point.conjugate();
421 if (!newseq && are_ex_trivially_equal(point, newpoint)) {
425 ex result = (new pseries(var==newpoint, newseq ? *newseq : seq))->setflag(status_flags::dynallocated);
432 ex pseries::real_part() const
434 if(!var.info(info_flags::real))
435 return real_part_function(*this).hold();
436 ex newpoint = point.real_part();
437 if(newpoint != point)
438 return real_part_function(*this).hold();
441 v.reserve(seq.size());
442 for(epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i)
443 v.push_back(expair((i->rest).real_part(), i->coeff));
444 return (new pseries(var==point, v))->setflag(status_flags::dynallocated);
447 ex pseries::imag_part() const
449 if(!var.info(info_flags::real))
450 return imag_part_function(*this).hold();
451 ex newpoint = point.real_part();
452 if(newpoint != point)
453 return imag_part_function(*this).hold();
456 v.reserve(seq.size());
457 for(epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i)
458 v.push_back(expair((i->rest).imag_part(), i->coeff));
459 return (new pseries(var==point, v))->setflag(status_flags::dynallocated);
462 ex pseries::eval_integ() const
464 epvector *newseq = NULL;
465 for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
467 newseq->push_back(expair(i->rest.eval_integ(), i->coeff));
470 ex newterm = i->rest.eval_integ();
471 if (!are_ex_trivially_equal(newterm, i->rest)) {
472 newseq = new epvector;
473 newseq->reserve(seq.size());
474 for (epvector::const_iterator j=seq.begin(); j!=i; ++j)
475 newseq->push_back(*j);
476 newseq->push_back(expair(newterm, i->coeff));
480 ex newpoint = point.eval_integ();
481 if (newseq || !are_ex_trivially_equal(newpoint, point))
482 return (new pseries(var==newpoint, *newseq))
483 ->setflag(status_flags::dynallocated);
487 ex pseries::subs(const exmap & m, unsigned options) const
489 // If expansion variable is being substituted, convert the series to a
490 // polynomial and do the substitution there because the result might
491 // no longer be a power series
492 if (m.find(var) != m.end())
493 return convert_to_poly(true).subs(m, options);
495 // Otherwise construct a new series with substituted coefficients and
498 newseq.reserve(seq.size());
499 epvector::const_iterator it = seq.begin(), itend = seq.end();
500 while (it != itend) {
501 newseq.push_back(expair(it->rest.subs(m, options), it->coeff));
504 return (new pseries(relational(var,point.subs(m, options)), newseq))->setflag(status_flags::dynallocated);
507 /** Implementation of ex::expand() for a power series. It expands all the
508 * terms individually and returns the resulting series as a new pseries. */
509 ex pseries::expand(unsigned options) const
512 epvector::const_iterator i = seq.begin(), end = seq.end();
514 ex restexp = i->rest.expand();
515 if (!restexp.is_zero())
516 newseq.push_back(expair(restexp, i->coeff));
519 return (new pseries(relational(var,point), newseq))
520 ->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
523 /** Implementation of ex::diff() for a power series.
525 ex pseries::derivative(const symbol & s) const
528 epvector::const_iterator it = seq.begin(), itend = seq.end();
532 // FIXME: coeff might depend on var
533 while (it != itend) {
534 if (is_order_function(it->rest)) {
535 new_seq.push_back(expair(it->rest, it->coeff - 1));
537 ex c = it->rest * it->coeff;
539 new_seq.push_back(expair(c, it->coeff - 1));
546 while (it != itend) {
547 if (is_order_function(it->rest)) {
548 new_seq.push_back(*it);
550 ex c = it->rest.diff(s);
552 new_seq.push_back(expair(c, it->coeff));
558 return pseries(relational(var,point), new_seq);
561 ex pseries::convert_to_poly(bool no_order) const
564 epvector::const_iterator it = seq.begin(), itend = seq.end();
566 while (it != itend) {
567 if (is_order_function(it->rest)) {
569 e += Order(power(var - point, it->coeff));
571 e += it->rest * power(var - point, it->coeff);
577 bool pseries::is_terminating() const
579 return seq.empty() || !is_order_function((seq.end()-1)->rest);
582 ex pseries::coeffop(size_t i) const
585 throw (std::out_of_range("coeffop() out of range"));
589 ex pseries::exponop(size_t i) const
592 throw (std::out_of_range("exponop() out of range"));
598 * Implementations of series expansion
601 /** Default implementation of ex::series(). This performs Taylor expansion.
603 ex basic::series(const relational & r, int order, unsigned options) const
606 const symbol &s = ex_to<symbol>(r.lhs());
608 // default for order-values that make no sense for Taylor expansion
609 if ((order <= 0) && this->has(s)) {
610 seq.push_back(expair(Order(_ex1), order));
611 return pseries(r, seq);
614 // do Taylor expansion
617 ex coeff = deriv.subs(r, subs_options::no_pattern);
619 if (!coeff.is_zero()) {
620 seq.push_back(expair(coeff, _ex0));
624 for (n=1; n<order; ++n) {
626 // We need to test for zero in order to see if the series terminates.
627 // The problem is that there is no such thing as a perfect test for
628 // zero. Expanding the term occasionally helps a little...
629 deriv = deriv.diff(s).expand();
630 if (deriv.is_zero()) // Series terminates
631 return pseries(r, seq);
633 coeff = deriv.subs(r, subs_options::no_pattern);
634 if (!coeff.is_zero())
635 seq.push_back(expair(fac.inverse() * coeff, n));
638 // Higher-order terms, if present
639 deriv = deriv.diff(s);
640 if (!deriv.expand().is_zero())
641 seq.push_back(expair(Order(_ex1), n));
642 return pseries(r, seq);
646 /** Implementation of ex::series() for symbols.
648 ex symbol::series(const relational & r, int order, unsigned options) const
651 const ex point = r.rhs();
652 GINAC_ASSERT(is_a<symbol>(r.lhs()));
654 if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
655 if (order > 0 && !point.is_zero())
656 seq.push_back(expair(point, _ex0));
658 seq.push_back(expair(_ex1, _ex1));
660 seq.push_back(expair(Order(_ex1), numeric(order)));
662 seq.push_back(expair(*this, _ex0));
663 return pseries(r, seq);
667 /** Add one series object to another, producing a pseries object that
668 * represents the sum.
670 * @param other pseries object to add with
671 * @return the sum as a pseries */
672 ex pseries::add_series(const pseries &other) const
674 // Adding two series with different variables or expansion points
675 // results in an empty (constant) series
676 if (!is_compatible_to(other)) {
678 nul.push_back(expair(Order(_ex1), _ex0));
679 return pseries(relational(var,point), nul);
684 epvector::const_iterator a = seq.begin();
685 epvector::const_iterator b = other.seq.begin();
686 epvector::const_iterator a_end = seq.end();
687 epvector::const_iterator b_end = other.seq.end();
688 int pow_a = INT_MAX, pow_b = INT_MAX;
690 // If a is empty, fill up with elements from b and stop
693 new_seq.push_back(*b);
698 pow_a = ex_to<numeric>((*a).coeff).to_int();
700 // If b is empty, fill up with elements from a and stop
703 new_seq.push_back(*a);
708 pow_b = ex_to<numeric>((*b).coeff).to_int();
710 // a and b are non-empty, compare powers
712 // a has lesser power, get coefficient from a
713 new_seq.push_back(*a);
714 if (is_order_function((*a).rest))
717 } else if (pow_b < pow_a) {
718 // b has lesser power, get coefficient from b
719 new_seq.push_back(*b);
720 if (is_order_function((*b).rest))
724 // Add coefficient of a and b
725 if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
726 new_seq.push_back(expair(Order(_ex1), (*a).coeff));
727 break; // Order term ends the sequence
729 ex sum = (*a).rest + (*b).rest;
730 if (!(sum.is_zero()))
731 new_seq.push_back(expair(sum, numeric(pow_a)));
737 return pseries(relational(var,point), new_seq);
741 /** Implementation of ex::series() for sums. This performs series addition when
742 * adding pseries objects.
744 ex add::series(const relational & r, int order, unsigned options) const
746 ex acc; // Series accumulator
748 // Get first term from overall_coeff
749 acc = overall_coeff.series(r, order, options);
751 // Add remaining terms
752 epvector::const_iterator it = seq.begin();
753 epvector::const_iterator itend = seq.end();
754 for (; it!=itend; ++it) {
756 if (is_exactly_a<pseries>(it->rest))
759 op = it->rest.series(r, order, options);
760 if (!it->coeff.is_equal(_ex1))
761 op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it->coeff));
764 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
770 /** Multiply a pseries object with a numeric constant, producing a pseries
771 * object that represents the product.
773 * @param other constant to multiply with
774 * @return the product as a pseries */
775 ex pseries::mul_const(const numeric &other) const
778 new_seq.reserve(seq.size());
780 epvector::const_iterator it = seq.begin(), itend = seq.end();
781 while (it != itend) {
782 if (!is_order_function(it->rest))
783 new_seq.push_back(expair(it->rest * other, it->coeff));
785 new_seq.push_back(*it);
788 return pseries(relational(var,point), new_seq);
792 /** Multiply one pseries object to another, producing a pseries object that
793 * represents the product.
795 * @param other pseries object to multiply with
796 * @return the product as a pseries */
797 ex pseries::mul_series(const pseries &other) const
799 // Multiplying two series with different variables or expansion points
800 // results in an empty (constant) series
801 if (!is_compatible_to(other)) {
803 nul.push_back(expair(Order(_ex1), _ex0));
804 return pseries(relational(var,point), nul);
807 if (seq.empty() || other.seq.empty()) {
808 return (new pseries(var==point, epvector()))
809 ->setflag(status_flags::dynallocated);
812 // Series multiplication
814 int a_max = degree(var);
815 int b_max = other.degree(var);
816 int a_min = ldegree(var);
817 int b_min = other.ldegree(var);
818 int cdeg_min = a_min + b_min;
819 int cdeg_max = a_max + b_max;
821 int higher_order_a = INT_MAX;
822 int higher_order_b = INT_MAX;
823 if (is_order_function(coeff(var, a_max)))
824 higher_order_a = a_max + b_min;
825 if (is_order_function(other.coeff(var, b_max)))
826 higher_order_b = b_max + a_min;
827 int higher_order_c = std::min(higher_order_a, higher_order_b);
828 if (cdeg_max >= higher_order_c)
829 cdeg_max = higher_order_c - 1;
831 for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
833 // c(i)=a(0)b(i)+...+a(i)b(0)
834 for (int i=a_min; cdeg-i>=b_min; ++i) {
835 ex a_coeff = coeff(var, i);
836 ex b_coeff = other.coeff(var, cdeg-i);
837 if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
838 co += a_coeff * b_coeff;
841 new_seq.push_back(expair(co, numeric(cdeg)));
843 if (higher_order_c < INT_MAX)
844 new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c)));
845 return pseries(relational(var, point), new_seq);
849 /** Implementation of ex::series() for product. This performs series
850 * multiplication when multiplying series.
852 ex mul::series(const relational & r, int order, unsigned options) const
854 pseries acc; // Series accumulator
856 GINAC_ASSERT(is_a<symbol>(r.lhs()));
857 const ex& sym = r.lhs();
859 // holds ldegrees of the series of individual factors
860 std::vector<int> ldegrees;
862 // find minimal degrees
863 const epvector::const_iterator itbeg = seq.begin();
864 const epvector::const_iterator itend = seq.end();
865 for (epvector::const_iterator it=itbeg; it!=itend; ++it) {
867 ex expon = it->coeff;
870 if (expon.info(info_flags::integer)) {
872 factor = ex_to<numeric>(expon).to_int();
874 buf = recombine_pair_to_ex(*it);
877 int real_ldegree = 0;
879 real_ldegree = buf.expand().ldegree(sym-r.rhs());
880 } catch (std::runtime_error) {}
882 if (real_ldegree == 0) {
886 real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
887 } while (real_ldegree == orderloop);
890 ldegrees.push_back(factor * real_ldegree);
893 int degsum = std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
895 if (degsum >= order) {
897 epv.push_back(expair(Order(_ex1), order));
898 return (new pseries(r, epv))->setflag(status_flags::dynallocated);
901 // Multiply with remaining terms
902 std::vector<int>::const_iterator itd = ldegrees.begin();
903 for (epvector::const_iterator it=itbeg; it!=itend; ++it, ++itd) {
905 // do series expansion with adjusted order
906 ex op = recombine_pair_to_ex(*it).series(r, order-degsum+(*itd), options);
908 // Series multiplication
910 acc = ex_to<pseries>(op);
912 acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
915 return acc.mul_const(ex_to<numeric>(overall_coeff));
919 /** Compute the p-th power of a series.
921 * @param p power to compute
922 * @param deg truncation order of series calculation */
923 ex pseries::power_const(const numeric &p, int deg) const
926 // (due to Leonhard Euler)
927 // let A(x) be this series and for the time being let it start with a
928 // constant (later we'll generalize):
929 // A(x) = a_0 + a_1*x + a_2*x^2 + ...
930 // We want to compute
932 // C(x) = c_0 + c_1*x + c_2*x^2 + ...
933 // Taking the derivative on both sides and multiplying with A(x) one
934 // immediately arrives at
935 // C'(x)*A(x) = p*C(x)*A'(x)
936 // Multiplying this out and comparing coefficients we get the recurrence
938 // c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
939 // ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
940 // which can easily be solved given the starting value c_0 = (a_0)^p.
941 // For the more general case where the leading coefficient of A(x) is not
942 // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
943 // repeat the above derivation. The leading power of C2(x) = A2(x)^2 is
944 // then of course x^(p*m) but the recurrence formula still holds.
947 // as a special case, handle the empty (zero) series honoring the
948 // usual power laws such as implemented in power::eval()
949 if (p.real().is_zero())
950 throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
951 else if (p.real().is_negative())
952 throw pole_error("pseries::power_const(): division by zero",1);
957 const int ldeg = ldegree(var);
958 if (!(p*ldeg).is_integer())
959 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
961 // adjust number of coefficients
962 int numcoeff = deg - (p*ldeg).to_int();
966 epv.push_back(expair(Order(_ex1), deg));
967 return (new pseries(relational(var,point), epv))
968 ->setflag(status_flags::dynallocated);
971 // O(x^n)^(-m) is undefined
972 if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
973 throw pole_error("pseries::power_const(): division by zero",1);
975 // Compute coefficients of the powered series
977 co.reserve(numcoeff);
978 co.push_back(power(coeff(var, ldeg), p));
979 for (int i=1; i<numcoeff; ++i) {
981 for (int j=1; j<=i; ++j) {
982 ex c = coeff(var, j + ldeg);
983 if (is_order_function(c)) {
984 co.push_back(Order(_ex1));
987 sum += (p * j - (i - j)) * co[i - j] * c;
989 co.push_back(sum / coeff(var, ldeg) / i);
992 // Construct new series (of non-zero coefficients)
994 bool higher_order = false;
995 for (int i=0; i<numcoeff; ++i) {
996 if (!co[i].is_zero())
997 new_seq.push_back(expair(co[i], p * ldeg + i));
998 if (is_order_function(co[i])) {
1004 new_seq.push_back(expair(Order(_ex1), p * ldeg + numcoeff));
1006 return pseries(relational(var,point), new_seq);
1010 /** Return a new pseries object with the powers shifted by deg. */
1011 pseries pseries::shift_exponents(int deg) const
1013 epvector newseq = seq;
1014 epvector::iterator i = newseq.begin(), end = newseq.end();
1019 return pseries(relational(var, point), newseq);
1023 /** Implementation of ex::series() for powers. This performs Laurent expansion
1024 * of reciprocals of series at singularities.
1025 * @see ex::series */
1026 ex power::series(const relational & r, int order, unsigned options) const
1028 // If basis is already a series, just power it
1029 if (is_exactly_a<pseries>(basis))
1030 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
1032 // Basis is not a series, may there be a singularity?
1033 bool must_expand_basis = false;
1035 basis.subs(r, subs_options::no_pattern);
1036 } catch (pole_error) {
1037 must_expand_basis = true;
1040 // Is the expression of type something^(-int)?
1041 if (!must_expand_basis && !exponent.info(info_flags::negint)
1042 && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1043 return basic::series(r, order, options);
1045 // Is the expression of type 0^something?
1046 if (!must_expand_basis && !basis.subs(r, subs_options::no_pattern).is_zero()
1047 && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1048 return basic::series(r, order, options);
1050 // Singularity encountered, is the basis equal to (var - point)?
1051 if (basis.is_equal(r.lhs() - r.rhs())) {
1053 if (ex_to<numeric>(exponent).to_int() < order)
1054 new_seq.push_back(expair(_ex1, exponent));
1056 new_seq.push_back(expair(Order(_ex1), exponent));
1057 return pseries(r, new_seq);
1060 // No, expand basis into series
1063 if (is_a<numeric>(exponent)) {
1064 numexp = ex_to<numeric>(exponent);
1068 const ex& sym = r.lhs();
1069 // find existing minimal degree
1070 int real_ldegree = basis.expand().ldegree(sym-r.rhs());
1071 if (real_ldegree == 0) {
1075 real_ldegree = basis.series(r, orderloop, options).ldegree(sym);
1076 } while (real_ldegree == orderloop);
1079 if (!(real_ldegree*numexp).is_integer())
1080 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
1081 ex e = basis.series(r, (order + real_ldegree*(1-numexp)).to_int(), options);
1085 result = ex_to<pseries>(e).power_const(numexp, order);
1086 } catch (pole_error) {
1088 ser.push_back(expair(Order(_ex1), order));
1089 result = pseries(r, ser);
1096 /** Re-expansion of a pseries object. */
1097 ex pseries::series(const relational & r, int order, unsigned options) const
1099 const ex p = r.rhs();
1100 GINAC_ASSERT(is_a<symbol>(r.lhs()));
1101 const symbol &s = ex_to<symbol>(r.lhs());
1103 if (var.is_equal(s) && point.is_equal(p)) {
1104 if (order > degree(s))
1108 epvector::const_iterator it = seq.begin(), itend = seq.end();
1109 while (it != itend) {
1110 int o = ex_to<numeric>(it->coeff).to_int();
1112 new_seq.push_back(expair(Order(_ex1), o));
1115 new_seq.push_back(*it);
1118 return pseries(r, new_seq);
1121 return convert_to_poly().series(r, order, options);
1124 ex integral::series(const relational & r, int order, unsigned options) const
1127 throw std::logic_error("Cannot series expand wrt dummy variable");
1129 // Expanding integrant with r substituted taken in boundaries.
1130 ex fseries = f.series(r, order, options);
1131 epvector fexpansion;
1132 fexpansion.reserve(fseries.nops());
1133 for (size_t i=0; i<fseries.nops(); ++i) {
1134 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1135 currcoeff = (currcoeff == Order(_ex1))
1137 : integral(x, a.subs(r), b.subs(r), currcoeff);
1139 fexpansion.push_back(
1140 expair(currcoeff, ex_to<pseries>(fseries).exponop(i)));
1143 // Expanding lower boundary
1144 ex result = (new pseries(r, fexpansion))->setflag(status_flags::dynallocated);
1145 ex aseries = (a-a.subs(r)).series(r, order, options);
1146 fseries = f.series(x == (a.subs(r)), order, options);
1147 for (size_t i=0; i<fseries.nops(); ++i) {
1148 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1149 if (is_order_function(currcoeff))
1151 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1152 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1153 currcoeff = currcoeff.series(r, orderforf);
1154 ex term = ex_to<pseries>(aseries).power_const(ex_to<numeric>(currexpon+1),order);
1155 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(-1/(currexpon+1)));
1156 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1157 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1160 // Expanding upper boundary
1161 ex bseries = (b-b.subs(r)).series(r, order, options);
1162 fseries = f.series(x == (b.subs(r)), order, options);
1163 for (size_t i=0; i<fseries.nops(); ++i) {
1164 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1165 if (is_order_function(currcoeff))
1167 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1168 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1169 currcoeff = currcoeff.series(r, orderforf);
1170 ex term = ex_to<pseries>(bseries).power_const(ex_to<numeric>(currexpon+1),order);
1171 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(1/(currexpon+1)));
1172 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1173 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1180 /** Compute the truncated series expansion of an expression.
1181 * This function returns an expression containing an object of class pseries
1182 * to represent the series. If the series does not terminate within the given
1183 * truncation order, the last term of the series will be an order term.
1185 * @param r expansion relation, lhs holds variable and rhs holds point
1186 * @param order truncation order of series calculations
1187 * @param options of class series_options
1188 * @return an expression holding a pseries object */
1189 ex ex::series(const ex & r, int order, unsigned options) const
1194 if (is_a<relational>(r))
1195 rel_ = ex_to<relational>(r);
1196 else if (is_a<symbol>(r))
1197 rel_ = relational(r,_ex0);
1199 throw (std::logic_error("ex::series(): expansion point has unknown type"));
1202 e = bp->series(rel_, order, options);
1203 } catch (std::exception &x) {
1204 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));
1209 } // namespace GiNaC