Merge some cosmetic patches.
[ginac.git] / ginac / pseries.cpp
1 /** @file pseries.cpp
2  *
3  *  Implementation of class for extended truncated power series and
4  *  methods for series expansion. */
5
6 /*
7  *  GiNaC Copyright (C) 1999-2015 Johannes Gutenberg University Mainz, Germany
8  *
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.
13  *
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.
18  *
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
22  */
23
24 #include "pseries.h"
25 #include "add.h"
26 #include "inifcns.h" // for Order function
27 #include "lst.h"
28 #include "mul.h"
29 #include "power.h"
30 #include "relational.h"
31 #include "operators.h"
32 #include "symbol.h"
33 #include "integral.h"
34 #include "archive.h"
35 #include "utils.h"
36
37 #include <limits>
38 #include <numeric>
39 #include <stdexcept>
40
41 namespace GiNaC {
42
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))
49
50
51 /*
52  *  Default constructor
53  */
54
55 pseries::pseries() { }
56
57
58 /*
59  *  Other ctors
60  */
61
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.
67  *
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_)
72   : seq(ops_)
73 {
74         GINAC_ASSERT(is_a<relational>(rel_));
75         GINAC_ASSERT(is_a<symbol>(rel_.lhs()));
76         point = rel_.rhs();
77         var = rel_.lhs();
78 }
79 pseries::pseries(const ex &rel_, epvector &&ops_)
80   : seq(std::move(ops_))
81 {
82         GINAC_ASSERT(is_a<relational>(rel_));
83         GINAC_ASSERT(is_a<symbol>(rel_.lhs()));
84         point = rel_.rhs();
85         var = rel_.lhs();
86 }
87
88
89 /*
90  *  Archiving
91  */
92
93 void pseries::read_archive(const archive_node &n, lst &sym_lst) 
94 {
95         inherited::read_archive(n, sym_lst);
96         auto first = n.find_first("coeff");
97         auto last = n.find_last("power");
98         ++last;
99         seq.reserve((last-first)/2);
100
101         for (auto loc = first; loc < last;) {
102                 ex rest;
103                 ex coeff;
104                 n.find_ex_by_loc(loc++, rest, sym_lst);
105                 n.find_ex_by_loc(loc++, coeff, sym_lst);
106                 seq.push_back(expair(rest, coeff));
107         }
108
109         n.find_ex("var", var, sym_lst);
110         n.find_ex("point", point, sym_lst);
111 }
112
113 void pseries::archive(archive_node &n) const
114 {
115         inherited::archive(n);
116         for (auto & it : seq) {
117                 n.add_ex("coeff", it.rest);
118                 n.add_ex("power", it.coeff);
119         }
120         n.add_ex("var", var);
121         n.add_ex("point", point);
122 }
123
124
125 //////////
126 // functions overriding virtual functions from base classes
127 //////////
128
129 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
130 {
131         if (precedence() <= level)
132                 c.s << '(';
133                 
134         // objects of type pseries must not have any zero entries, so the
135         // trivial (zero) pseries needs a special treatment here:
136         if (seq.empty())
137                 c.s << '0';
138
139         auto i = seq.begin(), end = seq.end();
140         while (i != end) {
141
142                 // print a sign, if needed
143                 if (i != seq.begin())
144                         c.s << '+';
145
146                 if (!is_order_function(i->rest)) {
147
148                         // print 'rest', i.e. the expansion coefficient
149                         if (i->rest.info(info_flags::numeric) &&
150                                 i->rest.info(info_flags::positive)) {
151                                 i->rest.print(c);
152                         } else {
153                                 c.s << openbrace << '(';
154                                 i->rest.print(c);
155                                 c.s << ')' << closebrace;
156                         }
157
158                         // print 'coeff', something like (x-1)^42
159                         if (!i->coeff.is_zero()) {
160                                 c.s << mul_sym;
161                                 if (!point.is_zero()) {
162                                         c.s << openbrace << '(';
163                                         (var-point).print(c);
164                                         c.s << ')' << closebrace;
165                                 } else
166                                         var.print(c);
167                                 if (i->coeff.compare(_ex1)) {
168                                         c.s << pow_sym;
169                                         c.s << openbrace;
170                                         if (i->coeff.info(info_flags::negative)) {
171                                                 c.s << '(';
172                                                 i->coeff.print(c);
173                                                 c.s << ')';
174                                         } else
175                                                 i->coeff.print(c);
176                                         c.s << closebrace;
177                                 }
178                         }
179                 } else
180                         Order(pow(var - point, i->coeff)).print(c);
181                 ++i;
182         }
183
184         if (precedence() <= level)
185                 c.s << ')';
186 }
187
188 void pseries::do_print(const print_context & c, unsigned level) const
189 {
190         print_series(c, "", "", "*", "^", level);
191 }
192
193 void pseries::do_print_latex(const print_latex & c, unsigned level) const
194 {
195         print_series(c, "{", "}", " ", "^", level);
196 }
197
198 void pseries::do_print_python(const print_python & c, unsigned level) const
199 {
200         print_series(c, "", "", "*", "**", level);
201 }
202
203 void pseries::do_print_tree(const print_tree & c, unsigned level) const
204 {
205         c.s << std::string(level, ' ') << class_name() << " @" << this
206             << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
207             << std::endl;
208         size_t num = seq.size();
209         for (size_t i=0; i<num; ++i) {
210                 seq[i].rest.print(c, level + c.delta_indent);
211                 seq[i].coeff.print(c, level + c.delta_indent);
212                 c.s << std::string(level + c.delta_indent, ' ') << "-----" << std::endl;
213         }
214         var.print(c, level + c.delta_indent);
215         point.print(c, level + c.delta_indent);
216 }
217
218 void pseries::do_print_python_repr(const print_python_repr & c, unsigned level) const
219 {
220         c.s << class_name() << "(relational(";
221         var.print(c);
222         c.s << ',';
223         point.print(c);
224         c.s << "),[";
225         size_t num = seq.size();
226         for (size_t i=0; i<num; ++i) {
227                 if (i)
228                         c.s << ',';
229                 c.s << '(';
230                 seq[i].rest.print(c);
231                 c.s << ',';
232                 seq[i].coeff.print(c);
233                 c.s << ')';
234         }
235         c.s << "])";
236 }
237
238 int pseries::compare_same_type(const basic & other) const
239 {
240         GINAC_ASSERT(is_a<pseries>(other));
241         const pseries &o = static_cast<const pseries &>(other);
242         
243         // first compare the lengths of the series...
244         if (seq.size()>o.seq.size())
245                 return 1;
246         if (seq.size()<o.seq.size())
247                 return -1;
248         
249         // ...then the expansion point...
250         int cmpval = var.compare(o.var);
251         if (cmpval)
252                 return cmpval;
253         cmpval = point.compare(o.point);
254         if (cmpval)
255                 return cmpval;
256         
257         // ...and if that failed the individual elements
258         epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
259         while (it!=seq.end() && o_it!=o.seq.end()) {
260                 cmpval = it->compare(*o_it);
261                 if (cmpval)
262                         return cmpval;
263                 ++it;
264                 ++o_it;
265         }
266
267         // so they are equal.
268         return 0;
269 }
270
271 /** Return the number of operands including a possible order term. */
272 size_t pseries::nops() const
273 {
274         return seq.size();
275 }
276
277 /** Return the ith term in the series when represented as a sum. */
278 ex pseries::op(size_t i) const
279 {
280         if (i >= seq.size())
281                 throw (std::out_of_range("op() out of range"));
282
283         if (is_order_function(seq[i].rest))
284                 return Order(pow(var-point, seq[i].coeff));
285         return seq[i].rest * pow(var - point, seq[i].coeff);
286 }
287
288 /** Return degree of highest power of the series.  This is usually the exponent
289  *  of the Order term.  If s is not the expansion variable of the series, the
290  *  series is examined termwise. */
291 int pseries::degree(const ex &s) const
292 {
293         if (var.is_equal(s)) {
294                 // Return last exponent
295                 if (seq.size())
296                         return ex_to<numeric>((seq.end()-1)->coeff).to_int();
297                 else
298                         return 0;
299         } else {
300                 epvector::const_iterator it = seq.begin(), itend = seq.end();
301                 if (it == itend)
302                         return 0;
303                 int max_pow = std::numeric_limits<int>::min();
304                 while (it != itend) {
305                         int pow = it->rest.degree(s);
306                         if (pow > max_pow)
307                                 max_pow = pow;
308                         ++it;
309                 }
310                 return max_pow;
311         }
312 }
313
314 /** Return degree of lowest power of the series.  This is usually the exponent
315  *  of the leading term.  If s is not the expansion variable of the series, the
316  *  series is examined termwise.  If s is the expansion variable but the
317  *  expansion point is not zero the series is not expanded to find the degree.
318  *  I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
319 int pseries::ldegree(const ex &s) const
320 {
321         if (var.is_equal(s)) {
322                 // Return first exponent
323                 if (seq.size())
324                         return ex_to<numeric>((seq.begin())->coeff).to_int();
325                 else
326                         return 0;
327         } else {
328                 epvector::const_iterator it = seq.begin(), itend = seq.end();
329                 if (it == itend)
330                         return 0;
331                 int min_pow = std::numeric_limits<int>::max();
332                 while (it != itend) {
333                         int pow = it->rest.ldegree(s);
334                         if (pow < min_pow)
335                                 min_pow = pow;
336                         ++it;
337                 }
338                 return min_pow;
339         }
340 }
341
342 /** Return coefficient of degree n in power series if s is the expansion
343  *  variable.  If the expansion point is nonzero, by definition the n=1
344  *  coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
345  *  the expansion took place in the s in the first place).
346  *  If s is not the expansion variable, an attempt is made to convert the
347  *  series to a polynomial and return the corresponding coefficient from
348  *  there. */
349 ex pseries::coeff(const ex &s, int n) const
350 {
351         if (var.is_equal(s)) {
352                 if (seq.empty())
353                         return _ex0;
354                 
355                 // Binary search in sequence for given power
356                 numeric looking_for = numeric(n);
357                 int lo = 0, hi = seq.size() - 1;
358                 while (lo <= hi) {
359                         int mid = (lo + hi) / 2;
360                         GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
361                         int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
362                         switch (cmp) {
363                                 case -1:
364                                         lo = mid + 1;
365                                         break;
366                                 case 0:
367                                         return seq[mid].rest;
368                                 case 1:
369                                         hi = mid - 1;
370                                         break;
371                                 default:
372                                         throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
373                         }
374                 }
375                 return _ex0;
376         } else
377                 return convert_to_poly().coeff(s, n);
378 }
379
380 /** Does nothing. */
381 ex pseries::collect(const ex &s, bool distributed) const
382 {
383         return *this;
384 }
385
386 /** Perform coefficient-wise automatic term rewriting rules in this class. */
387 ex pseries::eval() const
388 {
389         if (flags & status_flags::evaluated) {
390                 return *this;
391         }
392         
393         // Construct a new series with evaluated coefficients
394         epvector new_seq;
395         new_seq.reserve(seq.size());
396         epvector::const_iterator it = seq.begin(), itend = seq.end();
397         while (it != itend) {
398                 new_seq.push_back(expair(it->rest, it->coeff));
399                 ++it;
400         }
401         return dynallocate<pseries>(relational(var,point), std::move(new_seq)).setflag(status_flags::evaluated);
402 }
403
404 /** Evaluate coefficients numerically. */
405 ex pseries::evalf(int level) const
406 {
407         if (level == 1)
408                 return *this;
409         
410         if (level == -max_recursion_level)
411                 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
412         
413         // Construct a new series with evaluated coefficients
414         epvector new_seq;
415         new_seq.reserve(seq.size());
416         epvector::const_iterator it = seq.begin(), itend = seq.end();
417         while (it != itend) {
418                 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
419                 ++it;
420         }
421         return dynallocate<pseries>(relational(var,point), std::move(new_seq)).setflag(status_flags::evaluated);
422 }
423
424 ex pseries::conjugate() const
425 {
426         if(!var.info(info_flags::real))
427                 return conjugate_function(*this).hold();
428
429         std::unique_ptr<epvector> newseq(conjugateepvector(seq));
430         ex newpoint = point.conjugate();
431
432         if (!newseq && are_ex_trivially_equal(point, newpoint)) {
433                 return *this;
434         }
435
436         return dynallocate<pseries>(var==newpoint, newseq ? std::move(*newseq) : seq);
437 }
438
439 ex pseries::real_part() const
440 {
441         if(!var.info(info_flags::real))
442                 return real_part_function(*this).hold();
443         ex newpoint = point.real_part();
444         if(newpoint != point)
445                 return real_part_function(*this).hold();
446
447         epvector v;
448         v.reserve(seq.size());
449         for (auto & it : seq)
450                 v.push_back(expair((it.rest).real_part(), it.coeff));
451         return dynallocate<pseries>(var==point, std::move(v));
452 }
453
454 ex pseries::imag_part() const
455 {
456         if(!var.info(info_flags::real))
457                 return imag_part_function(*this).hold();
458         ex newpoint = point.real_part();
459         if(newpoint != point)
460                 return imag_part_function(*this).hold();
461
462         epvector v;
463         v.reserve(seq.size());
464         for (auto & it : seq)
465                 v.push_back(expair((it.rest).imag_part(), it.coeff));
466         return dynallocate<pseries>(var==point, std::move(v));
467 }
468
469 ex pseries::eval_integ() const
470 {
471         std::unique_ptr<epvector> newseq(nullptr);
472         for (auto i=seq.begin(); i!=seq.end(); ++i) {
473                 if (newseq) {
474                         newseq->push_back(expair(i->rest.eval_integ(), i->coeff));
475                         continue;
476                 }
477                 ex newterm = i->rest.eval_integ();
478                 if (!are_ex_trivially_equal(newterm, i->rest)) {
479                         newseq.reset(new epvector);
480                         newseq->reserve(seq.size());
481                         for (auto j=seq.begin(); j!=i; ++j)
482                                 newseq->push_back(*j);
483                         newseq->push_back(expair(newterm, i->coeff));
484                 }
485         }
486
487         ex newpoint = point.eval_integ();
488         if (newseq || !are_ex_trivially_equal(newpoint, point))
489                 return dynallocate<pseries>(var==newpoint, std::move(*newseq));
490         return *this;
491 }
492
493 ex pseries::evalm() const
494 {
495         // evalm each coefficient
496         epvector newseq;
497         bool something_changed = false;
498         for (auto i=seq.begin(); i!=seq.end(); ++i) {
499                 if (something_changed) {
500                         ex newcoeff = i->rest.evalm();
501                         if (!newcoeff.is_zero())
502                                 newseq.push_back(expair(newcoeff, i->coeff));
503                 } else {
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));
511                         }
512                 }
513         }
514         if (something_changed)
515                 return dynallocate<pseries>(var==point, std::move(newseq));
516         else
517                 return *this;
518 }
519
520 ex pseries::subs(const exmap & m, unsigned options) const
521 {
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);
527         
528         // Otherwise construct a new series with substituted coefficients and
529         // expansion point
530         epvector newseq;
531         newseq.reserve(seq.size());
532         for (auto & it : seq)
533                 newseq.push_back(expair(it.rest.subs(m, options), it.coeff));
534         return dynallocate<pseries>(relational(var,point.subs(m, options)), std::move(newseq));
535 }
536
537 /** Implementation of ex::expand() for a power series.  It expands all the
538  *  terms individually and returns the resulting series as a new pseries. */
539 ex pseries::expand(unsigned options) const
540 {
541         epvector newseq;
542         for (auto & it : seq) {
543                 ex restexp = it.rest.expand();
544                 if (!restexp.is_zero())
545                         newseq.push_back(expair(restexp, it.coeff));
546         }
547         return dynallocate<pseries>(relational(var,point), std::move(newseq)).setflag(options == 0 ? status_flags::expanded : 0);
548 }
549
550 /** Implementation of ex::diff() for a power series.
551  *  @see ex::diff */
552 ex pseries::derivative(const symbol & s) const
553 {
554         epvector new_seq;
555
556         if (s == var) {
557                 
558                 // FIXME: coeff might depend on var
559                 for (auto & it : seq) {
560                         if (is_order_function(it.rest)) {
561                                 new_seq.push_back(expair(it.rest, it.coeff - 1));
562                         } else {
563                                 ex c = it.rest * it.coeff;
564                                 if (!c.is_zero())
565                                         new_seq.push_back(expair(c, it.coeff - 1));
566                         }
567                 }
568
569         } else {
570
571                 for (auto & it : seq) {
572                         if (is_order_function(it.rest)) {
573                                 new_seq.push_back(it);
574                         } else {
575                                 ex c = it.rest.diff(s);
576                                 if (!c.is_zero())
577                                         new_seq.push_back(expair(c, it.coeff));
578                         }
579                 }
580         }
581
582         return pseries(relational(var,point), std::move(new_seq));
583 }
584
585 ex pseries::convert_to_poly(bool no_order) const
586 {
587         ex e;
588         for (auto & it : seq) {
589                 if (is_order_function(it.rest)) {
590                         if (!no_order)
591                                 e += Order(pow(var - point, it.coeff));
592                 } else
593                         e += it.rest * pow(var - point, it.coeff);
594         }
595         return e;
596 }
597
598 bool pseries::is_terminating() const
599 {
600         return seq.empty() || !is_order_function((seq.end()-1)->rest);
601 }
602
603 ex pseries::coeffop(size_t i) const
604 {
605         if (i >= nops())
606                 throw (std::out_of_range("coeffop() out of range"));
607         return seq[i].rest;
608 }
609
610 ex pseries::exponop(size_t i) const
611 {
612         if (i >= nops())
613                 throw (std::out_of_range("exponop() out of range"));
614         return seq[i].coeff;
615 }
616
617
618 /*
619  *  Implementations of series expansion
620  */
621
622 /** Default implementation of ex::series(). This performs Taylor expansion.
623  *  @see ex::series */
624 ex basic::series(const relational & r, int order, unsigned options) const
625 {
626         epvector seq;
627         const symbol &s = ex_to<symbol>(r.lhs());
628
629         // default for order-values that make no sense for Taylor expansion
630         if ((order <= 0) && this->has(s)) {
631                 seq.push_back(expair(Order(_ex1), order));
632                 return pseries(r, std::move(seq));
633         }
634
635         // do Taylor expansion
636         numeric fac = 1;
637         ex deriv = *this;
638         ex coeff = deriv.subs(r, subs_options::no_pattern);
639
640         if (!coeff.is_zero()) {
641                 seq.push_back(expair(coeff, _ex0));
642         }
643
644         int n;
645         for (n=1; n<order; ++n) {
646                 fac = fac.div(n);
647                 // We need to test for zero in order to see if the series terminates.
648                 // The problem is that there is no such thing as a perfect test for
649                 // zero.  Expanding the term occasionally helps a little...
650                 deriv = deriv.diff(s).expand();
651                 if (deriv.is_zero())  // Series terminates
652                         return pseries(r, std::move(seq));
653
654                 coeff = deriv.subs(r, subs_options::no_pattern);
655                 if (!coeff.is_zero())
656                         seq.push_back(expair(fac * coeff, n));
657         }
658         
659         // Higher-order terms, if present
660         deriv = deriv.diff(s);
661         if (!deriv.expand().is_zero())
662                 seq.push_back(expair(Order(_ex1), n));
663         return pseries(r, std::move(seq));
664 }
665
666
667 /** Implementation of ex::series() for symbols.
668  *  @see ex::series */
669 ex symbol::series(const relational & r, int order, unsigned options) const
670 {
671         epvector seq;
672         const ex point = r.rhs();
673         GINAC_ASSERT(is_a<symbol>(r.lhs()));
674
675         if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
676                 if (order > 0 && !point.is_zero())
677                         seq.push_back(expair(point, _ex0));
678                 if (order > 1)
679                         seq.push_back(expair(_ex1, _ex1));
680                 else
681                         seq.push_back(expair(Order(_ex1), numeric(order)));
682         } else
683                 seq.push_back(expair(*this, _ex0));
684         return pseries(r, std::move(seq));
685 }
686
687
688 /** Add one series object to another, producing a pseries object that
689  *  represents the sum.
690  *
691  *  @param other  pseries object to add with
692  *  @return the sum as a pseries */
693 ex pseries::add_series(const pseries &other) const
694 {
695         // Adding two series with different variables or expansion points
696         // results in an empty (constant) series 
697         if (!is_compatible_to(other)) {
698                 epvector nul { expair(Order(_ex1), _ex0) };
699                 return pseries(relational(var,point), std::move(nul));
700         }
701         
702         // Series addition
703         epvector new_seq;
704         auto a = seq.begin(), a_end = seq.end();
705         auto b = other.seq.begin(), b_end = other.seq.end();
706         int pow_a = std::numeric_limits<int>::max(), pow_b = std::numeric_limits<int>::max();
707         for (;;) {
708                 // If a is empty, fill up with elements from b and stop
709                 if (a == a_end) {
710                         while (b != b_end) {
711                                 new_seq.push_back(*b);
712                                 ++b;
713                         }
714                         break;
715                 } else
716                         pow_a = ex_to<numeric>((*a).coeff).to_int();
717                 
718                 // If b is empty, fill up with elements from a and stop
719                 if (b == b_end) {
720                         while (a != a_end) {
721                                 new_seq.push_back(*a);
722                                 ++a;
723                         }
724                         break;
725                 } else
726                         pow_b = ex_to<numeric>((*b).coeff).to_int();
727                 
728                 // a and b are non-empty, compare powers
729                 if (pow_a < pow_b) {
730                         // a has lesser power, get coefficient from a
731                         new_seq.push_back(*a);
732                         if (is_order_function((*a).rest))
733                                 break;
734                         ++a;
735                 } else if (pow_b < pow_a) {
736                         // b has lesser power, get coefficient from b
737                         new_seq.push_back(*b);
738                         if (is_order_function((*b).rest))
739                                 break;
740                         ++b;
741                 } else {
742                         // Add coefficient of a and b
743                         if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
744                                 new_seq.push_back(expair(Order(_ex1), (*a).coeff));
745                                 break;  // Order term ends the sequence
746                         } else {
747                                 ex sum = (*a).rest + (*b).rest;
748                                 if (!(sum.is_zero()))
749                                         new_seq.push_back(expair(sum, numeric(pow_a)));
750                                 ++a;
751                                 ++b;
752                         }
753                 }
754         }
755         return pseries(relational(var,point), std::move(new_seq));
756 }
757
758
759 /** Implementation of ex::series() for sums. This performs series addition when
760  *  adding pseries objects.
761  *  @see ex::series */
762 ex add::series(const relational & r, int order, unsigned options) const
763 {
764         ex acc; // Series accumulator
765         
766         // Get first term from overall_coeff
767         acc = overall_coeff.series(r, order, options);
768         
769         // Add remaining terms
770         for (auto & it : seq) {
771                 ex op;
772                 if (is_exactly_a<pseries>(it.rest))
773                         op = it.rest;
774                 else
775                         op = it.rest.series(r, order, options);
776                 if (!it.coeff.is_equal(_ex1))
777                         op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it.coeff));
778                 
779                 // Series addition
780                 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
781         }
782         return acc;
783 }
784
785
786 /** Multiply a pseries object with a numeric constant, producing a pseries
787  *  object that represents the product.
788  *
789  *  @param other  constant to multiply with
790  *  @return the product as a pseries */
791 ex pseries::mul_const(const numeric &other) const
792 {
793         epvector new_seq;
794         new_seq.reserve(seq.size());
795         
796         for (auto & it : seq) {
797                 if (!is_order_function(it.rest))
798                         new_seq.push_back(expair(it.rest * other, it.coeff));
799                 else
800                         new_seq.push_back(it);
801         }
802         return pseries(relational(var,point), std::move(new_seq));
803 }
804
805
806 /** Multiply one pseries object to another, producing a pseries object that
807  *  represents the product.
808  *
809  *  @param other  pseries object to multiply with
810  *  @return the product as a pseries */
811 ex pseries::mul_series(const pseries &other) const
812 {
813         // Multiplying two series with different variables or expansion points
814         // results in an empty (constant) series 
815         if (!is_compatible_to(other)) {
816                 epvector nul { expair(Order(_ex1), _ex0) };
817                 return pseries(relational(var,point), std::move(nul));
818         }
819
820         if (seq.empty() || other.seq.empty()) {
821                 return dynallocate<pseries>(var==point, epvector());
822         }
823         
824         // Series multiplication
825         epvector new_seq;
826         int a_max = degree(var);
827         int b_max = other.degree(var);
828         int a_min = ldegree(var);
829         int b_min = other.ldegree(var);
830         int cdeg_min = a_min + b_min;
831         int cdeg_max = a_max + b_max;
832         
833         int higher_order_a = std::numeric_limits<int>::max();
834         int higher_order_b = std::numeric_limits<int>::max();
835         if (is_order_function(coeff(var, a_max)))
836                 higher_order_a = a_max + b_min;
837         if (is_order_function(other.coeff(var, b_max)))
838                 higher_order_b = b_max + a_min;
839         int higher_order_c = std::min(higher_order_a, higher_order_b);
840         if (cdeg_max >= higher_order_c)
841                 cdeg_max = higher_order_c - 1;
842         
843         for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
844                 ex co = _ex0;
845                 // c(i)=a(0)b(i)+...+a(i)b(0)
846                 for (int i=a_min; cdeg-i>=b_min; ++i) {
847                         ex a_coeff = coeff(var, i);
848                         ex b_coeff = other.coeff(var, cdeg-i);
849                         if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
850                                 co += a_coeff * b_coeff;
851                 }
852                 if (!co.is_zero())
853                         new_seq.push_back(expair(co, numeric(cdeg)));
854         }
855         if (higher_order_c < std::numeric_limits<int>::max())
856                 new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c)));
857         return pseries(relational(var, point), std::move(new_seq));
858 }
859
860
861 /** Implementation of ex::series() for product. This performs series
862  *  multiplication when multiplying series.
863  *  @see ex::series */
864 ex mul::series(const relational & r, int order, unsigned options) const
865 {
866         pseries acc; // Series accumulator
867
868         GINAC_ASSERT(is_a<symbol>(r.lhs()));
869         const ex& sym = r.lhs();
870                 
871         // holds ldegrees of the series of individual factors
872         std::vector<int> ldegrees;
873         std::vector<bool> ldegree_redo;
874
875         // find minimal degrees
876         // first round: obtain a bound up to which minimal degrees have to be
877         // considered
878         for (auto & it : seq) {
879
880                 ex expon = it.coeff;
881                 int factor = 1;
882                 ex buf;
883                 if (expon.info(info_flags::integer)) {
884                         buf = it.rest;
885                         factor = ex_to<numeric>(expon).to_int();
886                 } else {
887                         buf = recombine_pair_to_ex(it);
888                 }
889
890                 int real_ldegree = 0;
891                 bool flag_redo = false;
892                 try {
893                         real_ldegree = buf.expand().ldegree(sym-r.rhs());
894                 } catch (std::runtime_error) {}
895
896                 if (real_ldegree == 0) {
897                         if ( factor < 0 ) {
898                                 // This case must terminate, otherwise we would have division by
899                                 // zero.
900                                 int orderloop = 0;
901                                 do {
902                                         orderloop++;
903                                         real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
904                                 } while (real_ldegree == orderloop);
905                         } else {
906                                 // Here it is possible that buf does not have a ldegree, therefore
907                                 // check only if ldegree is negative, otherwise reconsider the case
908                                 // in the second round.
909                                 real_ldegree = buf.series(r, 0, options).ldegree(sym);
910                                 if (real_ldegree == 0)
911                                         flag_redo = true;
912                         }
913                 }
914
915                 ldegrees.push_back(factor * real_ldegree);
916                 ldegree_redo.push_back(flag_redo);
917         }
918
919         int degbound = order-std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
920         // Second round: determine the remaining positive ldegrees by the series
921         // method.
922         // here we can ignore ldegrees larger than degbound
923         size_t j = 0;
924         for (auto & it : seq) {
925                 if ( ldegree_redo[j] ) {
926                         ex expon = it.coeff;
927                         int factor = 1;
928                         ex buf;
929                         if (expon.info(info_flags::integer)) {
930                                 buf = it.rest;
931                                 factor = ex_to<numeric>(expon).to_int();
932                         } else {
933                                 buf = recombine_pair_to_ex(it);
934                         }
935                         int real_ldegree = 0;
936                         int orderloop = 0;
937                         do {
938                                 orderloop++;
939                                 real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
940                         } while ((real_ldegree == orderloop)
941                               && (factor*real_ldegree < degbound));
942                         ldegrees[j] = factor * real_ldegree;
943                         degbound -= factor * real_ldegree;
944                 }
945                 j++;
946         }
947
948         int degsum = std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
949
950         if (degsum >= order) {
951                 epvector epv { expair(Order(_ex1), order) };
952                 return dynallocate<pseries>(r, std::move(epv));
953         }
954
955         // Multiply with remaining terms
956         auto itd = ldegrees.begin();
957         for (auto it=seq.begin(), itend=seq.end(); it!=itend; ++it, ++itd) {
958
959                 // do series expansion with adjusted order
960                 ex op = recombine_pair_to_ex(*it).series(r, order-degsum+(*itd), options);
961
962                 // Series multiplication
963                 if (it == seq.begin())
964                         acc = ex_to<pseries>(op);
965                 else
966                         acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
967         }
968
969         return acc.mul_const(ex_to<numeric>(overall_coeff));
970 }
971
972
973 /** Compute the p-th power of a series.
974  *
975  *  @param p  power to compute
976  *  @param deg  truncation order of series calculation */
977 ex pseries::power_const(const numeric &p, int deg) const
978 {
979         // method:
980         // (due to Leonhard Euler)
981         // let A(x) be this series and for the time being let it start with a
982         // constant (later we'll generalize):
983         //     A(x) = a_0 + a_1*x + a_2*x^2 + ...
984         // We want to compute
985         //     C(x) = A(x)^p
986         //     C(x) = c_0 + c_1*x + c_2*x^2 + ...
987         // Taking the derivative on both sides and multiplying with A(x) one
988         // immediately arrives at
989         //     C'(x)*A(x) = p*C(x)*A'(x)
990         // Multiplying this out and comparing coefficients we get the recurrence
991         // formula
992         //     c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
993         //                    ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
994         // which can easily be solved given the starting value c_0 = (a_0)^p.
995         // For the more general case where the leading coefficient of A(x) is not
996         // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
997         // repeat the above derivation.  The leading power of C2(x) = A2(x)^2 is
998         // then of course x^(p*m) but the recurrence formula still holds.
999         
1000         if (seq.empty()) {
1001                 // as a special case, handle the empty (zero) series honoring the
1002                 // usual power laws such as implemented in power::eval()
1003                 if (p.real().is_zero())
1004                         throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
1005                 else if (p.real().is_negative())
1006                         throw pole_error("pseries::power_const(): division by zero",1);
1007                 else
1008                         return *this;
1009         }
1010         
1011         const int ldeg = ldegree(var);
1012         if (!(p*ldeg).is_integer())
1013                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
1014
1015         // adjust number of coefficients
1016         int numcoeff = deg - (p*ldeg).to_int();
1017         if (numcoeff <= 0) {
1018                 epvector epv { expair(Order(_ex1), deg) };
1019                 return dynallocate<pseries>(relational(var,point), std::move(epv));
1020         }
1021         
1022         // O(x^n)^(-m) is undefined
1023         if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
1024                 throw pole_error("pseries::power_const(): division by zero",1);
1025         
1026         // Compute coefficients of the powered series
1027         exvector co;
1028         co.reserve(numcoeff);
1029         co.push_back(pow(coeff(var, ldeg), p));
1030         for (int i=1; i<numcoeff; ++i) {
1031                 ex sum = _ex0;
1032                 for (int j=1; j<=i; ++j) {
1033                         ex c = coeff(var, j + ldeg);
1034                         if (is_order_function(c)) {
1035                                 co.push_back(Order(_ex1));
1036                                 break;
1037                         } else
1038                                 sum += (p * j - (i - j)) * co[i - j] * c;
1039                 }
1040                 co.push_back(sum / coeff(var, ldeg) / i);
1041         }
1042         
1043         // Construct new series (of non-zero coefficients)
1044         epvector new_seq;
1045         bool higher_order = false;
1046         for (int i=0; i<numcoeff; ++i) {
1047                 if (!co[i].is_zero())
1048                         new_seq.push_back(expair(co[i], p * ldeg + i));
1049                 if (is_order_function(co[i])) {
1050                         higher_order = true;
1051                         break;
1052                 }
1053         }
1054         if (!higher_order)
1055                 new_seq.push_back(expair(Order(_ex1), p * ldeg + numcoeff));
1056
1057         return pseries(relational(var,point), std::move(new_seq));
1058 }
1059
1060
1061 /** Return a new pseries object with the powers shifted by deg. */
1062 pseries pseries::shift_exponents(int deg) const
1063 {
1064         epvector newseq = seq;
1065         for (auto & it : newseq)
1066                 it.coeff += deg;
1067         return pseries(relational(var, point), std::move(newseq));
1068 }
1069
1070
1071 /** Implementation of ex::series() for powers. This performs Laurent expansion
1072  *  of reciprocals of series at singularities.
1073  *  @see ex::series */
1074 ex power::series(const relational & r, int order, unsigned options) const
1075 {
1076         // If basis is already a series, just power it
1077         if (is_exactly_a<pseries>(basis))
1078                 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
1079
1080         // Basis is not a series, may there be a singularity?
1081         bool must_expand_basis = false;
1082         try {
1083                 basis.subs(r, subs_options::no_pattern);
1084         } catch (pole_error) {
1085                 must_expand_basis = true;
1086         }
1087
1088         bool exponent_is_regular = true;
1089         try {
1090                 exponent.subs(r, subs_options::no_pattern);
1091         } catch (pole_error) {
1092                 exponent_is_regular = false;
1093         }
1094
1095         if (!exponent_is_regular) {
1096                 ex l = exponent*log(basis);
1097                 // this == exp(l);
1098                 ex le = l.series(r, order, options);
1099                 // Note: expanding exp(l) won't help, since that will attempt
1100                 // Taylor expansion, and fail (because exponent is "singular")
1101                 // Still l itself might be expanded in Taylor series.
1102                 // Examples:
1103                 // sin(x)/x*log(cos(x))
1104                 // 1/x*log(1 + x)
1105                 return exp(le).series(r, order, options);
1106                 // Note: if l happens to have a Laurent expansion (with
1107                 // negative powers of (var - point)), expanding exp(le)
1108                 // will barf (which is The Right Thing).
1109         }
1110
1111         // Is the expression of type something^(-int)?
1112         if (!must_expand_basis && !exponent.info(info_flags::negint)
1113          && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1114                 return basic::series(r, order, options);
1115
1116         // Is the expression of type 0^something?
1117         if (!must_expand_basis && !basis.subs(r, subs_options::no_pattern).is_zero()
1118          && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1119                 return basic::series(r, order, options);
1120
1121         // Singularity encountered, is the basis equal to (var - point)?
1122         if (basis.is_equal(r.lhs() - r.rhs())) {
1123                 epvector new_seq;
1124                 if (ex_to<numeric>(exponent).to_int() < order)
1125                         new_seq.push_back(expair(_ex1, exponent));
1126                 else
1127                         new_seq.push_back(expair(Order(_ex1), exponent));
1128                 return pseries(r, std::move(new_seq));
1129         }
1130
1131         // No, expand basis into series
1132
1133         numeric numexp;
1134         if (is_a<numeric>(exponent)) {
1135                 numexp = ex_to<numeric>(exponent);
1136         } else {
1137                 numexp = 0;
1138         }
1139         const ex& sym = r.lhs();
1140         // find existing minimal degree
1141         ex eb = basis.expand();
1142         int real_ldegree = 0;
1143         if (eb.info(info_flags::rational_function))
1144                 real_ldegree = eb.ldegree(sym-r.rhs());
1145         if (real_ldegree == 0) {
1146                 int orderloop = 0;
1147                 do {
1148                         orderloop++;
1149                         real_ldegree = basis.series(r, orderloop, options).ldegree(sym);
1150                 } while (real_ldegree == orderloop);
1151         }
1152
1153         if (!(real_ldegree*numexp).is_integer())
1154                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
1155         ex e = basis.series(r, (order + real_ldegree*(1-numexp)).to_int(), options);
1156         
1157         ex result;
1158         try {
1159                 result = ex_to<pseries>(e).power_const(numexp, order);
1160         } catch (pole_error) {
1161                 epvector ser { expair(Order(_ex1), order) };
1162                 result = pseries(r, std::move(ser));
1163         }
1164
1165         return result;
1166 }
1167
1168
1169 /** Re-expansion of a pseries object. */
1170 ex pseries::series(const relational & r, int order, unsigned options) const
1171 {
1172         const ex p = r.rhs();
1173         GINAC_ASSERT(is_a<symbol>(r.lhs()));
1174         const symbol &s = ex_to<symbol>(r.lhs());
1175         
1176         if (var.is_equal(s) && point.is_equal(p)) {
1177                 if (order > degree(s))
1178                         return *this;
1179                 else {
1180                         epvector new_seq;
1181                         for (auto & it : seq) {
1182                                 int o = ex_to<numeric>(it.coeff).to_int();
1183                                 if (o >= order) {
1184                                         new_seq.push_back(expair(Order(_ex1), o));
1185                                         break;
1186                                 }
1187                                 new_seq.push_back(it);
1188                         }
1189                         return pseries(r, std::move(new_seq));
1190                 }
1191         } else
1192                 return convert_to_poly().series(r, order, options);
1193 }
1194
1195 ex integral::series(const relational & r, int order, unsigned options) const
1196 {
1197         if (x.subs(r) != x)
1198                 throw std::logic_error("Cannot series expand wrt dummy variable");
1199         
1200         // Expanding integrand with r substituted taken in boundaries.
1201         ex fseries = f.series(r, order, options);
1202         epvector fexpansion;
1203         fexpansion.reserve(fseries.nops());
1204         for (size_t i=0; i<fseries.nops(); ++i) {
1205                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1206                 currcoeff = (currcoeff == Order(_ex1))
1207                         ? currcoeff
1208                         : integral(x, a.subs(r), b.subs(r), currcoeff);
1209                 if (currcoeff != 0)
1210                         fexpansion.push_back(
1211                                 expair(currcoeff, ex_to<pseries>(fseries).exponop(i)));
1212         }
1213
1214         // Expanding lower boundary
1215         ex result = dynallocate<pseries>(r, std::move(fexpansion));
1216         ex aseries = (a-a.subs(r)).series(r, order, options);
1217         fseries = f.series(x == (a.subs(r)), order, options);
1218         for (size_t i=0; i<fseries.nops(); ++i) {
1219                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1220                 if (is_order_function(currcoeff))
1221                         break;
1222                 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1223                 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1224                 currcoeff = currcoeff.series(r, orderforf);
1225                 ex term = ex_to<pseries>(aseries).power_const(ex_to<numeric>(currexpon+1),order);
1226                 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(-1/(currexpon+1)));
1227                 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1228                 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1229         }
1230
1231         // Expanding upper boundary
1232         ex bseries = (b-b.subs(r)).series(r, order, options);
1233         fseries = f.series(x == (b.subs(r)), order, options);
1234         for (size_t i=0; i<fseries.nops(); ++i) {
1235                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1236                 if (is_order_function(currcoeff))
1237                         break;
1238                 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1239                 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1240                 currcoeff = currcoeff.series(r, orderforf);
1241                 ex term = ex_to<pseries>(bseries).power_const(ex_to<numeric>(currexpon+1),order);
1242                 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(1/(currexpon+1)));
1243                 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1244                 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1245         }
1246
1247         return result;
1248 }
1249
1250
1251 /** Compute the truncated series expansion of an expression.
1252  *  This function returns an expression containing an object of class pseries 
1253  *  to represent the series. If the series does not terminate within the given
1254  *  truncation order, the last term of the series will be an order term.
1255  *
1256  *  @param r  expansion relation, lhs holds variable and rhs holds point
1257  *  @param order  truncation order of series calculations
1258  *  @param options  of class series_options
1259  *  @return an expression holding a pseries object */
1260 ex ex::series(const ex & r, int order, unsigned options) const
1261 {
1262         ex e;
1263         relational rel_;
1264         
1265         if (is_a<relational>(r))
1266                 rel_ = ex_to<relational>(r);
1267         else if (is_a<symbol>(r))
1268                 rel_ = relational(r,_ex0);
1269         else
1270                 throw (std::logic_error("ex::series(): expansion point has unknown type"));
1271         
1272         e = bp->series(rel_, order, options);
1273         return e;
1274 }
1275
1276 GINAC_BIND_UNARCHIVER(pseries);
1277
1278 } // namespace GiNaC