Remove 'level' argument of evalf().
[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) 1999-2016 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 #ifdef DO_GINAC_ASSERT
75         auto i = seq.begin();
76         while (i != seq.end()) {
77                 auto ip1 = i+1;
78                 if (ip1 != seq.end())
79                         GINAC_ASSERT(!is_order_function(i->rest));
80                 else
81                         break;
82                 GINAC_ASSERT(is_a<numeric>(i->coeff));
83                 GINAC_ASSERT(ex_to<numeric>(i->coeff) < ex_to<numeric>(ip1->coeff));
84                 ++i;
85         }
86 #endif // def DO_GINAC_ASSERT
87         GINAC_ASSERT(is_a<relational>(rel_));
88         GINAC_ASSERT(is_a<symbol>(rel_.lhs()));
89         point = rel_.rhs();
90         var = rel_.lhs();
91 }
92 pseries::pseries(const ex &rel_, epvector &&ops_)
93   : seq(std::move(ops_))
94 {
95 #ifdef DO_GINAC_ASSERT
96         auto i = seq.begin();
97         while (i != seq.end()) {
98                 auto ip1 = i+1;
99                 if (ip1 != seq.end())
100                         GINAC_ASSERT(!is_order_function(i->rest));
101                 else
102                         break;
103                 GINAC_ASSERT(is_a<numeric>(i->coeff));
104                 GINAC_ASSERT(ex_to<numeric>(i->coeff) < ex_to<numeric>(ip1->coeff));
105                 ++i;
106         }
107 #endif // def DO_GINAC_ASSERT
108         GINAC_ASSERT(is_a<relational>(rel_));
109         GINAC_ASSERT(is_a<symbol>(rel_.lhs()));
110         point = rel_.rhs();
111         var = rel_.lhs();
112 }
113
114
115 /*
116  *  Archiving
117  */
118
119 void pseries::read_archive(const archive_node &n, lst &sym_lst) 
120 {
121         inherited::read_archive(n, sym_lst);
122         auto first = n.find_first("coeff");
123         auto last = n.find_last("power");
124         ++last;
125         seq.reserve((last-first)/2);
126
127         for (auto loc = first; loc < last;) {
128                 ex rest;
129                 ex coeff;
130                 n.find_ex_by_loc(loc++, rest, sym_lst);
131                 n.find_ex_by_loc(loc++, coeff, sym_lst);
132                 seq.push_back(expair(rest, coeff));
133         }
134
135         n.find_ex("var", var, sym_lst);
136         n.find_ex("point", point, sym_lst);
137 }
138
139 void pseries::archive(archive_node &n) const
140 {
141         inherited::archive(n);
142         for (auto & it : seq) {
143                 n.add_ex("coeff", it.rest);
144                 n.add_ex("power", it.coeff);
145         }
146         n.add_ex("var", var);
147         n.add_ex("point", point);
148 }
149
150
151 //////////
152 // functions overriding virtual functions from base classes
153 //////////
154
155 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
156 {
157         if (precedence() <= level)
158                 c.s << '(';
159                 
160         // objects of type pseries must not have any zero entries, so the
161         // trivial (zero) pseries needs a special treatment here:
162         if (seq.empty())
163                 c.s << '0';
164
165         auto i = seq.begin(), end = seq.end();
166         while (i != end) {
167
168                 // print a sign, if needed
169                 if (i != seq.begin())
170                         c.s << '+';
171
172                 if (!is_order_function(i->rest)) {
173
174                         // print 'rest', i.e. the expansion coefficient
175                         if (i->rest.info(info_flags::numeric) &&
176                                 i->rest.info(info_flags::positive)) {
177                                 i->rest.print(c);
178                         } else {
179                                 c.s << openbrace << '(';
180                                 i->rest.print(c);
181                                 c.s << ')' << closebrace;
182                         }
183
184                         // print 'coeff', something like (x-1)^42
185                         if (!i->coeff.is_zero()) {
186                                 c.s << mul_sym;
187                                 if (!point.is_zero()) {
188                                         c.s << openbrace << '(';
189                                         (var-point).print(c);
190                                         c.s << ')' << closebrace;
191                                 } else
192                                         var.print(c);
193                                 if (i->coeff.compare(_ex1)) {
194                                         c.s << pow_sym;
195                                         c.s << openbrace;
196                                         if (i->coeff.info(info_flags::negative)) {
197                                                 c.s << '(';
198                                                 i->coeff.print(c);
199                                                 c.s << ')';
200                                         } else
201                                                 i->coeff.print(c);
202                                         c.s << closebrace;
203                                 }
204                         }
205                 } else
206                         Order(pow(var - point, i->coeff)).print(c);
207                 ++i;
208         }
209
210         if (precedence() <= level)
211                 c.s << ')';
212 }
213
214 void pseries::do_print(const print_context & c, unsigned level) const
215 {
216         print_series(c, "", "", "*", "^", level);
217 }
218
219 void pseries::do_print_latex(const print_latex & c, unsigned level) const
220 {
221         print_series(c, "{", "}", " ", "^", level);
222 }
223
224 void pseries::do_print_python(const print_python & c, unsigned level) const
225 {
226         print_series(c, "", "", "*", "**", level);
227 }
228
229 void pseries::do_print_tree(const print_tree & c, unsigned level) const
230 {
231         c.s << std::string(level, ' ') << class_name() << " @" << this
232             << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
233             << std::endl;
234         size_t num = seq.size();
235         for (size_t i=0; i<num; ++i) {
236                 seq[i].rest.print(c, level + c.delta_indent);
237                 seq[i].coeff.print(c, level + c.delta_indent);
238                 c.s << std::string(level + c.delta_indent, ' ') << "-----" << std::endl;
239         }
240         var.print(c, level + c.delta_indent);
241         point.print(c, level + c.delta_indent);
242 }
243
244 void pseries::do_print_python_repr(const print_python_repr & c, unsigned level) const
245 {
246         c.s << class_name() << "(relational(";
247         var.print(c);
248         c.s << ',';
249         point.print(c);
250         c.s << "),[";
251         size_t num = seq.size();
252         for (size_t i=0; i<num; ++i) {
253                 if (i)
254                         c.s << ',';
255                 c.s << '(';
256                 seq[i].rest.print(c);
257                 c.s << ',';
258                 seq[i].coeff.print(c);
259                 c.s << ')';
260         }
261         c.s << "])";
262 }
263
264 int pseries::compare_same_type(const basic & other) const
265 {
266         GINAC_ASSERT(is_a<pseries>(other));
267         const pseries &o = static_cast<const pseries &>(other);
268         
269         // first compare the lengths of the series...
270         if (seq.size()>o.seq.size())
271                 return 1;
272         if (seq.size()<o.seq.size())
273                 return -1;
274         
275         // ...then the expansion point...
276         int cmpval = var.compare(o.var);
277         if (cmpval)
278                 return cmpval;
279         cmpval = point.compare(o.point);
280         if (cmpval)
281                 return cmpval;
282         
283         // ...and if that failed the individual elements
284         epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
285         while (it!=seq.end() && o_it!=o.seq.end()) {
286                 cmpval = it->compare(*o_it);
287                 if (cmpval)
288                         return cmpval;
289                 ++it;
290                 ++o_it;
291         }
292
293         // so they are equal.
294         return 0;
295 }
296
297 /** Return the number of operands including a possible order term. */
298 size_t pseries::nops() const
299 {
300         return seq.size();
301 }
302
303 /** Return the ith term in the series when represented as a sum. */
304 ex pseries::op(size_t i) const
305 {
306         if (i >= seq.size())
307                 throw (std::out_of_range("op() out of range"));
308
309         if (is_order_function(seq[i].rest))
310                 return Order(pow(var-point, seq[i].coeff));
311         return seq[i].rest * pow(var - point, seq[i].coeff);
312 }
313
314 /** Return degree of highest power of the series.  This is usually the exponent
315  *  of the Order term.  If s is not the expansion variable of the series, the
316  *  series is examined termwise. */
317 int pseries::degree(const ex &s) const
318 {
319         if (seq.empty())
320                 return 0;
321
322         if (var.is_equal(s))
323                 // Return last/greatest exponent
324                 return ex_to<numeric>((seq.end()-1)->coeff).to_int();
325
326         int max_pow = std::numeric_limits<int>::min();
327         for (auto & it : seq)
328                 max_pow = std::max(max_pow, it.rest.degree(s));
329         return max_pow;
330 }
331
332 /** Return degree of lowest power of the series.  This is usually the exponent
333  *  of the leading term.  If s is not the expansion variable of the series, the
334  *  series is examined termwise.  If s is the expansion variable but the
335  *  expansion point is not zero the series is not expanded to find the degree.
336  *  I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
337 int pseries::ldegree(const ex &s) const
338 {
339         if (seq.empty())
340                 return 0;
341
342         if (var.is_equal(s))
343                 // Return first/smallest exponent
344                 return ex_to<numeric>((seq.begin())->coeff).to_int();
345
346         int min_pow = std::numeric_limits<int>::max();
347         for (auto & it : seq)
348                 min_pow = std::min(min_pow, it.rest.degree(s));
349         return min_pow;
350 }
351
352 /** Return coefficient of degree n in power series if s is the expansion
353  *  variable.  If the expansion point is nonzero, by definition the n=1
354  *  coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
355  *  the expansion took place in the s in the first place).
356  *  If s is not the expansion variable, an attempt is made to convert the
357  *  series to a polynomial and return the corresponding coefficient from
358  *  there. */
359 ex pseries::coeff(const ex &s, int n) const
360 {
361         if (var.is_equal(s)) {
362                 if (seq.empty())
363                         return _ex0;
364                 
365                 // Binary search in sequence for given power
366                 numeric looking_for = numeric(n);
367                 int lo = 0, hi = seq.size() - 1;
368                 while (lo <= hi) {
369                         int mid = (lo + hi) / 2;
370                         GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
371                         int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
372                         switch (cmp) {
373                                 case -1:
374                                         lo = mid + 1;
375                                         break;
376                                 case 0:
377                                         return seq[mid].rest;
378                                 case 1:
379                                         hi = mid - 1;
380                                         break;
381                                 default:
382                                         throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
383                         }
384                 }
385                 return _ex0;
386         } else
387                 return convert_to_poly().coeff(s, n);
388 }
389
390 /** Does nothing. */
391 ex pseries::collect(const ex &s, bool distributed) const
392 {
393         return *this;
394 }
395
396 /** Perform coefficient-wise automatic term rewriting rules in this class. */
397 ex pseries::eval() const
398 {
399         if (flags & status_flags::evaluated) {
400                 return *this;
401         }
402
403         // Construct a new series with evaluated coefficients
404         epvector new_seq;
405         new_seq.reserve(seq.size());
406         for (auto & it : seq)
407                 new_seq.push_back(expair(it.rest, it.coeff));
408
409         return dynallocate<pseries>(relational(var,point), std::move(new_seq)).setflag(status_flags::evaluated);
410 }
411
412 /** Evaluate coefficients numerically. */
413 ex pseries::evalf() const
414 {
415         // Construct a new series with evaluated coefficients
416         epvector new_seq;
417         new_seq.reserve(seq.size());
418         for (auto & it : seq)
419                 new_seq.push_back(expair(it.rest, it.coeff));
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         const int a_max = degree(var);
827         const int b_max = other.degree(var);
828         const int a_min = ldegree(var);
829         const int b_min = other.ldegree(var);
830         const 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         const 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         std::map<int, ex> rest_map_a, rest_map_b;
844         for (const auto& it : seq)
845                 rest_map_a[ex_to<numeric>(it.coeff).to_int()] = it.rest;
846
847         if (other.var.is_equal(var))
848                 for (const auto& it : other.seq)
849                         rest_map_b[ex_to<numeric>(it.coeff).to_int()] = it.rest;
850
851         for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
852                 ex co = _ex0;
853                 // c(i)=a(0)b(i)+...+a(i)b(0)
854                 for (int i=a_min; cdeg-i>=b_min; ++i) {
855                         const auto& ita = rest_map_a.find(i);
856                         if (ita == rest_map_a.end())
857                                 continue;
858                         const auto& itb = rest_map_b.find(cdeg-i);
859                         if (itb == rest_map_b.end())
860                                 continue;
861                         if (!is_order_function(ita->second) && !is_order_function(itb->second))
862                                 co += ita->second * itb->second;
863                 }
864                 if (!co.is_zero())
865                         new_seq.push_back(expair(co, numeric(cdeg)));
866         }
867         if (higher_order_c < std::numeric_limits<int>::max())
868                 new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c)));
869         return pseries(relational(var, point), std::move(new_seq));
870 }
871
872
873 /** Implementation of ex::series() for product. This performs series
874  *  multiplication when multiplying series.
875  *  @see ex::series */
876 ex mul::series(const relational & r, int order, unsigned options) const
877 {
878         pseries acc; // Series accumulator
879
880         GINAC_ASSERT(is_a<symbol>(r.lhs()));
881         const ex& sym = r.lhs();
882                 
883         // holds ldegrees of the series of individual factors
884         std::vector<int> ldegrees;
885         std::vector<bool> ldegree_redo;
886
887         // find minimal degrees
888         // first round: obtain a bound up to which minimal degrees have to be
889         // considered
890         for (auto & it : seq) {
891
892                 ex expon = it.coeff;
893                 int factor = 1;
894                 ex buf;
895                 if (expon.info(info_flags::integer)) {
896                         buf = it.rest;
897                         factor = ex_to<numeric>(expon).to_int();
898                 } else {
899                         buf = recombine_pair_to_ex(it);
900                 }
901
902                 int real_ldegree = 0;
903                 bool flag_redo = false;
904                 try {
905                         real_ldegree = buf.expand().ldegree(sym-r.rhs());
906                 } catch (std::runtime_error) {}
907
908                 if (real_ldegree == 0) {
909                         if ( factor < 0 ) {
910                                 // This case must terminate, otherwise we would have division by
911                                 // zero.
912                                 int orderloop = 0;
913                                 do {
914                                         orderloop++;
915                                         real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
916                                 } while (real_ldegree == orderloop);
917                         } else {
918                                 // Here it is possible that buf does not have a ldegree, therefore
919                                 // check only if ldegree is negative, otherwise reconsider the case
920                                 // in the second round.
921                                 real_ldegree = buf.series(r, 0, options).ldegree(sym);
922                                 if (real_ldegree == 0)
923                                         flag_redo = true;
924                         }
925                 }
926
927                 ldegrees.push_back(factor * real_ldegree);
928                 ldegree_redo.push_back(flag_redo);
929         }
930
931         int degbound = order-std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
932         // Second round: determine the remaining positive ldegrees by the series
933         // method.
934         // here we can ignore ldegrees larger than degbound
935         size_t j = 0;
936         for (auto & it : seq) {
937                 if ( ldegree_redo[j] ) {
938                         ex expon = it.coeff;
939                         int factor = 1;
940                         ex buf;
941                         if (expon.info(info_flags::integer)) {
942                                 buf = it.rest;
943                                 factor = ex_to<numeric>(expon).to_int();
944                         } else {
945                                 buf = recombine_pair_to_ex(it);
946                         }
947                         int real_ldegree = 0;
948                         int orderloop = 0;
949                         do {
950                                 orderloop++;
951                                 real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
952                         } while ((real_ldegree == orderloop)
953                               && (factor*real_ldegree < degbound));
954                         ldegrees[j] = factor * real_ldegree;
955                         degbound -= factor * real_ldegree;
956                 }
957                 j++;
958         }
959
960         int degsum = std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
961
962         if (degsum >= order) {
963                 epvector epv { expair(Order(_ex1), order) };
964                 return dynallocate<pseries>(r, std::move(epv));
965         }
966
967         // Multiply with remaining terms
968         auto itd = ldegrees.begin();
969         for (auto it=seq.begin(), itend=seq.end(); it!=itend; ++it, ++itd) {
970
971                 // do series expansion with adjusted order
972                 ex op = recombine_pair_to_ex(*it).series(r, order-degsum+(*itd), options);
973
974                 // Series multiplication
975                 if (it == seq.begin())
976                         acc = ex_to<pseries>(op);
977                 else
978                         acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
979         }
980
981         return acc.mul_const(ex_to<numeric>(overall_coeff));
982 }
983
984
985 /** Compute the p-th power of a series.
986  *
987  *  @param p  power to compute
988  *  @param deg  truncation order of series calculation */
989 ex pseries::power_const(const numeric &p, int deg) const
990 {
991         // method:
992         // (due to Leonhard Euler)
993         // let A(x) be this series and for the time being let it start with a
994         // constant (later we'll generalize):
995         //     A(x) = a_0 + a_1*x + a_2*x^2 + ...
996         // We want to compute
997         //     C(x) = A(x)^p
998         //     C(x) = c_0 + c_1*x + c_2*x^2 + ...
999         // Taking the derivative on both sides and multiplying with A(x) one
1000         // immediately arrives at
1001         //     C'(x)*A(x) = p*C(x)*A'(x)
1002         // Multiplying this out and comparing coefficients we get the recurrence
1003         // formula
1004         //     c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
1005         //                    ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
1006         // which can easily be solved given the starting value c_0 = (a_0)^p.
1007         // For the more general case where the leading coefficient of A(x) is not
1008         // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
1009         // repeat the above derivation.  The leading power of C2(x) = A2(x)^2 is
1010         // then of course x^(p*m) but the recurrence formula still holds.
1011         
1012         if (seq.empty()) {
1013                 // as a special case, handle the empty (zero) series honoring the
1014                 // usual power laws such as implemented in power::eval()
1015                 if (p.real().is_zero())
1016                         throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
1017                 else if (p.real().is_negative())
1018                         throw pole_error("pseries::power_const(): division by zero",1);
1019                 else
1020                         return *this;
1021         }
1022         
1023         const int ldeg = ldegree(var);
1024         if (!(p*ldeg).is_integer())
1025                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
1026
1027         // adjust number of coefficients
1028         int numcoeff = deg - (p*ldeg).to_int();
1029         if (numcoeff <= 0) {
1030                 epvector epv { expair(Order(_ex1), deg) };
1031                 return dynallocate<pseries>(relational(var,point), std::move(epv));
1032         }
1033         
1034         // O(x^n)^(-m) is undefined
1035         if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
1036                 throw pole_error("pseries::power_const(): division by zero",1);
1037         
1038         // Compute coefficients of the powered series
1039         exvector co;
1040         co.reserve(numcoeff);
1041         co.push_back(pow(coeff(var, ldeg), p));
1042         for (int i=1; i<numcoeff; ++i) {
1043                 ex sum = _ex0;
1044                 for (int j=1; j<=i; ++j) {
1045                         ex c = coeff(var, j + ldeg);
1046                         if (is_order_function(c)) {
1047                                 co.push_back(Order(_ex1));
1048                                 break;
1049                         } else
1050                                 sum += (p * j - (i - j)) * co[i - j] * c;
1051                 }
1052                 co.push_back(sum / coeff(var, ldeg) / i);
1053         }
1054         
1055         // Construct new series (of non-zero coefficients)
1056         epvector new_seq;
1057         bool higher_order = false;
1058         for (int i=0; i<numcoeff; ++i) {
1059                 if (!co[i].is_zero())
1060                         new_seq.push_back(expair(co[i], p * ldeg + i));
1061                 if (is_order_function(co[i])) {
1062                         higher_order = true;
1063                         break;
1064                 }
1065         }
1066         if (!higher_order)
1067                 new_seq.push_back(expair(Order(_ex1), p * ldeg + numcoeff));
1068
1069         return pseries(relational(var,point), std::move(new_seq));
1070 }
1071
1072
1073 /** Return a new pseries object with the powers shifted by deg. */
1074 pseries pseries::shift_exponents(int deg) const
1075 {
1076         epvector newseq = seq;
1077         for (auto & it : newseq)
1078                 it.coeff += deg;
1079         return pseries(relational(var, point), std::move(newseq));
1080 }
1081
1082
1083 /** Implementation of ex::series() for powers. This performs Laurent expansion
1084  *  of reciprocals of series at singularities.
1085  *  @see ex::series */
1086 ex power::series(const relational & r, int order, unsigned options) const
1087 {
1088         // If basis is already a series, just power it
1089         if (is_exactly_a<pseries>(basis))
1090                 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
1091
1092         // Basis is not a series, may there be a singularity?
1093         bool must_expand_basis = false;
1094         try {
1095                 basis.subs(r, subs_options::no_pattern);
1096         } catch (pole_error) {
1097                 must_expand_basis = true;
1098         }
1099
1100         bool exponent_is_regular = true;
1101         try {
1102                 exponent.subs(r, subs_options::no_pattern);
1103         } catch (pole_error) {
1104                 exponent_is_regular = false;
1105         }
1106
1107         if (!exponent_is_regular) {
1108                 ex l = exponent*log(basis);
1109                 // this == exp(l);
1110                 ex le = l.series(r, order, options);
1111                 // Note: expanding exp(l) won't help, since that will attempt
1112                 // Taylor expansion, and fail (because exponent is "singular")
1113                 // Still l itself might be expanded in Taylor series.
1114                 // Examples:
1115                 // sin(x)/x*log(cos(x))
1116                 // 1/x*log(1 + x)
1117                 return exp(le).series(r, order, options);
1118                 // Note: if l happens to have a Laurent expansion (with
1119                 // negative powers of (var - point)), expanding exp(le)
1120                 // will barf (which is The Right Thing).
1121         }
1122
1123         // Is the expression of type something^(-int)?
1124         if (!must_expand_basis && !exponent.info(info_flags::negint)
1125          && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1126                 return basic::series(r, order, options);
1127
1128         // Is the expression of type 0^something?
1129         if (!must_expand_basis && !basis.subs(r, subs_options::no_pattern).is_zero()
1130          && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1131                 return basic::series(r, order, options);
1132
1133         // Singularity encountered, is the basis equal to (var - point)?
1134         if (basis.is_equal(r.lhs() - r.rhs())) {
1135                 epvector new_seq;
1136                 if (ex_to<numeric>(exponent).to_int() < order)
1137                         new_seq.push_back(expair(_ex1, exponent));
1138                 else
1139                         new_seq.push_back(expair(Order(_ex1), exponent));
1140                 return pseries(r, std::move(new_seq));
1141         }
1142
1143         // No, expand basis into series
1144
1145         numeric numexp;
1146         if (is_a<numeric>(exponent)) {
1147                 numexp = ex_to<numeric>(exponent);
1148         } else {
1149                 numexp = 0;
1150         }
1151         const ex& sym = r.lhs();
1152         // find existing minimal degree
1153         ex eb = basis.expand();
1154         int real_ldegree = 0;
1155         if (eb.info(info_flags::rational_function))
1156                 real_ldegree = eb.ldegree(sym-r.rhs());
1157         if (real_ldegree == 0) {
1158                 int orderloop = 0;
1159                 do {
1160                         orderloop++;
1161                         real_ldegree = basis.series(r, orderloop, options).ldegree(sym);
1162                 } while (real_ldegree == orderloop);
1163         }
1164
1165         if (!(real_ldegree*numexp).is_integer())
1166                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
1167         ex e = basis.series(r, (order + real_ldegree*(1-numexp)).to_int(), options);
1168         
1169         ex result;
1170         try {
1171                 result = ex_to<pseries>(e).power_const(numexp, order);
1172         } catch (pole_error) {
1173                 epvector ser { expair(Order(_ex1), order) };
1174                 result = pseries(r, std::move(ser));
1175         }
1176
1177         return result;
1178 }
1179
1180
1181 /** Re-expansion of a pseries object. */
1182 ex pseries::series(const relational & r, int order, unsigned options) const
1183 {
1184         const ex p = r.rhs();
1185         GINAC_ASSERT(is_a<symbol>(r.lhs()));
1186         const symbol &s = ex_to<symbol>(r.lhs());
1187         
1188         if (var.is_equal(s) && point.is_equal(p)) {
1189                 if (order > degree(s))
1190                         return *this;
1191                 else {
1192                         epvector new_seq;
1193                         for (auto & it : seq) {
1194                                 int o = ex_to<numeric>(it.coeff).to_int();
1195                                 if (o >= order) {
1196                                         new_seq.push_back(expair(Order(_ex1), o));
1197                                         break;
1198                                 }
1199                                 new_seq.push_back(it);
1200                         }
1201                         return pseries(r, std::move(new_seq));
1202                 }
1203         } else
1204                 return convert_to_poly().series(r, order, options);
1205 }
1206
1207 ex integral::series(const relational & r, int order, unsigned options) const
1208 {
1209         if (x.subs(r) != x)
1210                 throw std::logic_error("Cannot series expand wrt dummy variable");
1211         
1212         // Expanding integrand with r substituted taken in boundaries.
1213         ex fseries = f.series(r, order, options);
1214         epvector fexpansion;
1215         fexpansion.reserve(fseries.nops());
1216         for (size_t i=0; i<fseries.nops(); ++i) {
1217                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1218                 currcoeff = (currcoeff == Order(_ex1))
1219                         ? currcoeff
1220                         : integral(x, a.subs(r), b.subs(r), currcoeff);
1221                 if (currcoeff != 0)
1222                         fexpansion.push_back(
1223                                 expair(currcoeff, ex_to<pseries>(fseries).exponop(i)));
1224         }
1225
1226         // Expanding lower boundary
1227         ex result = dynallocate<pseries>(r, std::move(fexpansion));
1228         ex aseries = (a-a.subs(r)).series(r, order, options);
1229         fseries = f.series(x == (a.subs(r)), order, options);
1230         for (size_t i=0; i<fseries.nops(); ++i) {
1231                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1232                 if (is_order_function(currcoeff))
1233                         break;
1234                 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1235                 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1236                 currcoeff = currcoeff.series(r, orderforf);
1237                 ex term = ex_to<pseries>(aseries).power_const(ex_to<numeric>(currexpon+1),order);
1238                 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(-1/(currexpon+1)));
1239                 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1240                 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1241         }
1242
1243         // Expanding upper boundary
1244         ex bseries = (b-b.subs(r)).series(r, order, options);
1245         fseries = f.series(x == (b.subs(r)), order, options);
1246         for (size_t i=0; i<fseries.nops(); ++i) {
1247                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1248                 if (is_order_function(currcoeff))
1249                         break;
1250                 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1251                 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1252                 currcoeff = currcoeff.series(r, orderforf);
1253                 ex term = ex_to<pseries>(bseries).power_const(ex_to<numeric>(currexpon+1),order);
1254                 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(1/(currexpon+1)));
1255                 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1256                 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1257         }
1258
1259         return result;
1260 }
1261
1262
1263 /** Compute the truncated series expansion of an expression.
1264  *  This function returns an expression containing an object of class pseries 
1265  *  to represent the series. If the series does not terminate within the given
1266  *  truncation order, the last term of the series will be an order term.
1267  *
1268  *  @param r  expansion relation, lhs holds variable and rhs holds point
1269  *  @param order  truncation order of series calculations
1270  *  @param options  of class series_options
1271  *  @return an expression holding a pseries object */
1272 ex ex::series(const ex & r, int order, unsigned options) const
1273 {
1274         ex e;
1275         relational rel_;
1276         
1277         if (is_a<relational>(r))
1278                 rel_ = ex_to<relational>(r);
1279         else if (is_a<symbol>(r))
1280                 rel_ = relational(r,_ex0);
1281         else
1282                 throw (std::logic_error("ex::series(): expansion point has unknown type"));
1283         
1284         e = bp->series(rel_, order, options);
1285         return e;
1286 }
1287
1288 GINAC_BIND_UNARCHIVER(pseries);
1289
1290 } // namespace GiNaC