31a92002958399818659b65e83094b498a66dbc4
[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(power(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(power(var-point, seq[i].coeff));
285         return seq[i].rest * power(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                 }
504                 else {
505                         ex newcoeff = i->rest.evalm();
506                         if (!are_ex_trivially_equal(newcoeff, i->rest)) {
507                                 something_changed = true;
508                                 newseq.reserve(seq.size());
509                                 std::copy(seq.begin(), i, std::back_inserter<epvector>(newseq));
510                                 if (!newcoeff.is_zero())
511                                         newseq.push_back(expair(newcoeff, i->coeff));
512                         }
513                 }
514         }
515         if (something_changed)
516                 return dynallocate<pseries>(var==point, std::move(newseq));
517         else
518                 return *this;
519 }
520
521 ex pseries::subs(const exmap & m, unsigned options) const
522 {
523         // If expansion variable is being substituted, convert the series to a
524         // polynomial and do the substitution there because the result might
525         // no longer be a power series
526         if (m.find(var) != m.end())
527                 return convert_to_poly(true).subs(m, options);
528         
529         // Otherwise construct a new series with substituted coefficients and
530         // expansion point
531         epvector newseq;
532         newseq.reserve(seq.size());
533         for (auto & it : seq)
534                 newseq.push_back(expair(it.rest.subs(m, options), it.coeff));
535         return dynallocate<pseries>(relational(var,point.subs(m, options)), std::move(newseq));
536 }
537
538 /** Implementation of ex::expand() for a power series.  It expands all the
539  *  terms individually and returns the resulting series as a new pseries. */
540 ex pseries::expand(unsigned options) const
541 {
542         epvector newseq;
543         for (auto & it : seq) {
544                 ex restexp = it.rest.expand();
545                 if (!restexp.is_zero())
546                         newseq.push_back(expair(restexp, it.coeff));
547         }
548         return dynallocate<pseries>(relational(var,point), std::move(newseq)).setflag(options == 0 ? status_flags::expanded : 0);
549 }
550
551 /** Implementation of ex::diff() for a power series.
552  *  @see ex::diff */
553 ex pseries::derivative(const symbol & s) const
554 {
555         epvector new_seq;
556
557         if (s == var) {
558                 
559                 // FIXME: coeff might depend on var
560                 for (auto & it : seq) {
561                         if (is_order_function(it.rest)) {
562                                 new_seq.push_back(expair(it.rest, it.coeff - 1));
563                         } else {
564                                 ex c = it.rest * it.coeff;
565                                 if (!c.is_zero())
566                                         new_seq.push_back(expair(c, it.coeff - 1));
567                         }
568                 }
569
570         } else {
571
572                 for (auto & it : seq) {
573                         if (is_order_function(it.rest)) {
574                                 new_seq.push_back(it);
575                         } else {
576                                 ex c = it.rest.diff(s);
577                                 if (!c.is_zero())
578                                         new_seq.push_back(expair(c, it.coeff));
579                         }
580                 }
581         }
582
583         return pseries(relational(var,point), std::move(new_seq));
584 }
585
586 ex pseries::convert_to_poly(bool no_order) const
587 {
588         ex e;
589         for (auto & it : seq) {
590                 if (is_order_function(it.rest)) {
591                         if (!no_order)
592                                 e += Order(power(var - point, it.coeff));
593                 } else
594                         e += it.rest * power(var - point, it.coeff);
595         }
596         return e;
597 }
598
599 bool pseries::is_terminating() const
600 {
601         return seq.empty() || !is_order_function((seq.end()-1)->rest);
602 }
603
604 ex pseries::coeffop(size_t i) const
605 {
606         if (i >= nops())
607                 throw (std::out_of_range("coeffop() out of range"));
608         return seq[i].rest;
609 }
610
611 ex pseries::exponop(size_t i) const
612 {
613         if (i >= nops())
614                 throw (std::out_of_range("exponop() out of range"));
615         return seq[i].coeff;
616 }
617
618
619 /*
620  *  Implementations of series expansion
621  */
622
623 /** Default implementation of ex::series(). This performs Taylor expansion.
624  *  @see ex::series */
625 ex basic::series(const relational & r, int order, unsigned options) const
626 {
627         epvector seq;
628         const symbol &s = ex_to<symbol>(r.lhs());
629
630         // default for order-values that make no sense for Taylor expansion
631         if ((order <= 0) && this->has(s)) {
632                 seq.push_back(expair(Order(_ex1), order));
633                 return pseries(r, std::move(seq));
634         }
635
636         // do Taylor expansion
637         numeric fac = 1;
638         ex deriv = *this;
639         ex coeff = deriv.subs(r, subs_options::no_pattern);
640
641         if (!coeff.is_zero()) {
642                 seq.push_back(expair(coeff, _ex0));
643         }
644
645         int n;
646         for (n=1; n<order; ++n) {
647                 fac = fac.mul(n);
648                 // We need to test for zero in order to see if the series terminates.
649                 // The problem is that there is no such thing as a perfect test for
650                 // zero.  Expanding the term occasionally helps a little...
651                 deriv = deriv.diff(s).expand();
652                 if (deriv.is_zero())  // Series terminates
653                         return pseries(r, std::move(seq));
654
655                 coeff = deriv.subs(r, subs_options::no_pattern);
656                 if (!coeff.is_zero())
657                         seq.push_back(expair(fac.inverse() * coeff, n));
658         }
659         
660         // Higher-order terms, if present
661         deriv = deriv.diff(s);
662         if (!deriv.expand().is_zero())
663                 seq.push_back(expair(Order(_ex1), n));
664         return pseries(r, std::move(seq));
665 }
666
667
668 /** Implementation of ex::series() for symbols.
669  *  @see ex::series */
670 ex symbol::series(const relational & r, int order, unsigned options) const
671 {
672         epvector seq;
673         const ex point = r.rhs();
674         GINAC_ASSERT(is_a<symbol>(r.lhs()));
675
676         if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
677                 if (order > 0 && !point.is_zero())
678                         seq.push_back(expair(point, _ex0));
679                 if (order > 1)
680                         seq.push_back(expair(_ex1, _ex1));
681                 else
682                         seq.push_back(expair(Order(_ex1), numeric(order)));
683         } else
684                 seq.push_back(expair(*this, _ex0));
685         return pseries(r, std::move(seq));
686 }
687
688
689 /** Add one series object to another, producing a pseries object that
690  *  represents the sum.
691  *
692  *  @param other  pseries object to add with
693  *  @return the sum as a pseries */
694 ex pseries::add_series(const pseries &other) const
695 {
696         // Adding two series with different variables or expansion points
697         // results in an empty (constant) series 
698         if (!is_compatible_to(other)) {
699                 epvector nul { expair(Order(_ex1), _ex0) };
700                 return pseries(relational(var,point), std::move(nul));
701         }
702         
703         // Series addition
704         epvector new_seq;
705         auto a = seq.begin(), a_end = seq.end();
706         auto b = other.seq.begin(), b_end = other.seq.end();
707         int pow_a = std::numeric_limits<int>::max(), pow_b = std::numeric_limits<int>::max();
708         for (;;) {
709                 // If a is empty, fill up with elements from b and stop
710                 if (a == a_end) {
711                         while (b != b_end) {
712                                 new_seq.push_back(*b);
713                                 ++b;
714                         }
715                         break;
716                 } else
717                         pow_a = ex_to<numeric>((*a).coeff).to_int();
718                 
719                 // If b is empty, fill up with elements from a and stop
720                 if (b == b_end) {
721                         while (a != a_end) {
722                                 new_seq.push_back(*a);
723                                 ++a;
724                         }
725                         break;
726                 } else
727                         pow_b = ex_to<numeric>((*b).coeff).to_int();
728                 
729                 // a and b are non-empty, compare powers
730                 if (pow_a < pow_b) {
731                         // a has lesser power, get coefficient from a
732                         new_seq.push_back(*a);
733                         if (is_order_function((*a).rest))
734                                 break;
735                         ++a;
736                 } else if (pow_b < pow_a) {
737                         // b has lesser power, get coefficient from b
738                         new_seq.push_back(*b);
739                         if (is_order_function((*b).rest))
740                                 break;
741                         ++b;
742                 } else {
743                         // Add coefficient of a and b
744                         if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
745                                 new_seq.push_back(expair(Order(_ex1), (*a).coeff));
746                                 break;  // Order term ends the sequence
747                         } else {
748                                 ex sum = (*a).rest + (*b).rest;
749                                 if (!(sum.is_zero()))
750                                         new_seq.push_back(expair(sum, numeric(pow_a)));
751                                 ++a;
752                                 ++b;
753                         }
754                 }
755         }
756         return pseries(relational(var,point), std::move(new_seq));
757 }
758
759
760 /** Implementation of ex::series() for sums. This performs series addition when
761  *  adding pseries objects.
762  *  @see ex::series */
763 ex add::series(const relational & r, int order, unsigned options) const
764 {
765         ex acc; // Series accumulator
766         
767         // Get first term from overall_coeff
768         acc = overall_coeff.series(r, order, options);
769         
770         // Add remaining terms
771         for (auto & it : seq) {
772                 ex op;
773                 if (is_exactly_a<pseries>(it.rest))
774                         op = it.rest;
775                 else
776                         op = it.rest.series(r, order, options);
777                 if (!it.coeff.is_equal(_ex1))
778                         op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it.coeff));
779                 
780                 // Series addition
781                 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
782         }
783         return acc;
784 }
785
786
787 /** Multiply a pseries object with a numeric constant, producing a pseries
788  *  object that represents the product.
789  *
790  *  @param other  constant to multiply with
791  *  @return the product as a pseries */
792 ex pseries::mul_const(const numeric &other) const
793 {
794         epvector new_seq;
795         new_seq.reserve(seq.size());
796         
797         for (auto & it : seq) {
798                 if (!is_order_function(it.rest))
799                         new_seq.push_back(expair(it.rest * other, it.coeff));
800                 else
801                         new_seq.push_back(it);
802         }
803         return pseries(relational(var,point), std::move(new_seq));
804 }
805
806
807 /** Multiply one pseries object to another, producing a pseries object that
808  *  represents the product.
809  *
810  *  @param other  pseries object to multiply with
811  *  @return the product as a pseries */
812 ex pseries::mul_series(const pseries &other) const
813 {
814         // Multiplying two series with different variables or expansion points
815         // results in an empty (constant) series 
816         if (!is_compatible_to(other)) {
817                 epvector nul { expair(Order(_ex1), _ex0) };
818                 return pseries(relational(var,point), std::move(nul));
819         }
820
821         if (seq.empty() || other.seq.empty()) {
822                 return dynallocate<pseries>(var==point, epvector());
823         }
824         
825         // Series multiplication
826         epvector new_seq;
827         int a_max = degree(var);
828         int b_max = other.degree(var);
829         int a_min = ldegree(var);
830         int b_min = other.ldegree(var);
831         int cdeg_min = a_min + b_min;
832         int cdeg_max = a_max + b_max;
833         
834         int higher_order_a = std::numeric_limits<int>::max();
835         int higher_order_b = std::numeric_limits<int>::max();
836         if (is_order_function(coeff(var, a_max)))
837                 higher_order_a = a_max + b_min;
838         if (is_order_function(other.coeff(var, b_max)))
839                 higher_order_b = b_max + a_min;
840         int higher_order_c = std::min(higher_order_a, higher_order_b);
841         if (cdeg_max >= higher_order_c)
842                 cdeg_max = higher_order_c - 1;
843         
844         for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
845                 ex co = _ex0;
846                 // c(i)=a(0)b(i)+...+a(i)b(0)
847                 for (int i=a_min; cdeg-i>=b_min; ++i) {
848                         ex a_coeff = coeff(var, i);
849                         ex b_coeff = other.coeff(var, cdeg-i);
850                         if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
851                                 co += a_coeff * b_coeff;
852                 }
853                 if (!co.is_zero())
854                         new_seq.push_back(expair(co, numeric(cdeg)));
855         }
856         if (higher_order_c < std::numeric_limits<int>::max())
857                 new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c)));
858         return pseries(relational(var, point), std::move(new_seq));
859 }
860
861
862 /** Implementation of ex::series() for product. This performs series
863  *  multiplication when multiplying series.
864  *  @see ex::series */
865 ex mul::series(const relational & r, int order, unsigned options) const
866 {
867         pseries acc; // Series accumulator
868
869         GINAC_ASSERT(is_a<symbol>(r.lhs()));
870         const ex& sym = r.lhs();
871                 
872         // holds ldegrees of the series of individual factors
873         std::vector<int> ldegrees;
874         std::vector<bool> ldegree_redo;
875
876         // find minimal degrees
877         // first round: obtain a bound up to which minimal degrees have to be
878         // considered
879         for (auto & it : seq) {
880
881                 ex expon = it.coeff;
882                 int factor = 1;
883                 ex buf;
884                 if (expon.info(info_flags::integer)) {
885                         buf = it.rest;
886                         factor = ex_to<numeric>(expon).to_int();
887                 } else {
888                         buf = recombine_pair_to_ex(it);
889                 }
890
891                 int real_ldegree = 0;
892                 bool flag_redo = false;
893                 try {
894                         real_ldegree = buf.expand().ldegree(sym-r.rhs());
895                 } catch (std::runtime_error) {}
896
897                 if (real_ldegree == 0) {
898                         if ( factor < 0 ) {
899                                 // This case must terminate, otherwise we would have division by
900                                 // zero.
901                                 int orderloop = 0;
902                                 do {
903                                         orderloop++;
904                                         real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
905                                 } while (real_ldegree == orderloop);
906                         } else {
907                                 // Here it is possible that buf does not have a ldegree, therefore
908                                 // check only if ldegree is negative, otherwise reconsider the case
909                                 // in the second round.
910                                 real_ldegree = buf.series(r, 0, options).ldegree(sym);
911                                 if (real_ldegree == 0)
912                                         flag_redo = true;
913                         }
914                 }
915
916                 ldegrees.push_back(factor * real_ldegree);
917                 ldegree_redo.push_back(flag_redo);
918         }
919
920         int degbound = order-std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
921         // Second round: determine the remaining positive ldegrees by the series
922         // method.
923         // here we can ignore ldegrees larger than degbound
924         size_t j = 0;
925         for (auto & it : seq) {
926                 if ( ldegree_redo[j] ) {
927                         ex expon = it.coeff;
928                         int factor = 1;
929                         ex buf;
930                         if (expon.info(info_flags::integer)) {
931                                 buf = it.rest;
932                                 factor = ex_to<numeric>(expon).to_int();
933                         } else {
934                                 buf = recombine_pair_to_ex(it);
935                         }
936                         int real_ldegree = 0;
937                         int orderloop = 0;
938                         do {
939                                 orderloop++;
940                                 real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
941                         } while ((real_ldegree == orderloop)
942                               && (factor*real_ldegree < degbound));
943                         ldegrees[j] = factor * real_ldegree;
944                         degbound -= factor * real_ldegree;
945                 }
946                 j++;
947         }
948
949         int degsum = std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
950
951         if (degsum >= order) {
952                 epvector epv { expair(Order(_ex1), order) };
953                 return dynallocate<pseries>(r, std::move(epv));
954         }
955
956         // Multiply with remaining terms
957         auto itd = ldegrees.begin();
958         for (auto it=seq.begin(), itend=seq.end(); it!=itend; ++it, ++itd) {
959
960                 // do series expansion with adjusted order
961                 ex op = recombine_pair_to_ex(*it).series(r, order-degsum+(*itd), options);
962
963                 // Series multiplication
964                 if (it == seq.begin())
965                         acc = ex_to<pseries>(op);
966                 else
967                         acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
968         }
969
970         return acc.mul_const(ex_to<numeric>(overall_coeff));
971 }
972
973
974 /** Compute the p-th power of a series.
975  *
976  *  @param p  power to compute
977  *  @param deg  truncation order of series calculation */
978 ex pseries::power_const(const numeric &p, int deg) const
979 {
980         // method:
981         // (due to Leonhard Euler)
982         // let A(x) be this series and for the time being let it start with a
983         // constant (later we'll generalize):
984         //     A(x) = a_0 + a_1*x + a_2*x^2 + ...
985         // We want to compute
986         //     C(x) = A(x)^p
987         //     C(x) = c_0 + c_1*x + c_2*x^2 + ...
988         // Taking the derivative on both sides and multiplying with A(x) one
989         // immediately arrives at
990         //     C'(x)*A(x) = p*C(x)*A'(x)
991         // Multiplying this out and comparing coefficients we get the recurrence
992         // formula
993         //     c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
994         //                    ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
995         // which can easily be solved given the starting value c_0 = (a_0)^p.
996         // For the more general case where the leading coefficient of A(x) is not
997         // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
998         // repeat the above derivation.  The leading power of C2(x) = A2(x)^2 is
999         // then of course x^(p*m) but the recurrence formula still holds.
1000         
1001         if (seq.empty()) {
1002                 // as a special case, handle the empty (zero) series honoring the
1003                 // usual power laws such as implemented in power::eval()
1004                 if (p.real().is_zero())
1005                         throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
1006                 else if (p.real().is_negative())
1007                         throw pole_error("pseries::power_const(): division by zero",1);
1008                 else
1009                         return *this;
1010         }
1011         
1012         const int ldeg = ldegree(var);
1013         if (!(p*ldeg).is_integer())
1014                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
1015
1016         // adjust number of coefficients
1017         int numcoeff = deg - (p*ldeg).to_int();
1018         if (numcoeff <= 0) {
1019                 epvector epv { expair(Order(_ex1), deg) };
1020                 return dynallocate<pseries>(relational(var,point), std::move(epv));
1021         }
1022         
1023         // O(x^n)^(-m) is undefined
1024         if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
1025                 throw pole_error("pseries::power_const(): division by zero",1);
1026         
1027         // Compute coefficients of the powered series
1028         exvector co;
1029         co.reserve(numcoeff);
1030         co.push_back(power(coeff(var, ldeg), p));
1031         for (int i=1; i<numcoeff; ++i) {
1032                 ex sum = _ex0;
1033                 for (int j=1; j<=i; ++j) {
1034                         ex c = coeff(var, j + ldeg);
1035                         if (is_order_function(c)) {
1036                                 co.push_back(Order(_ex1));
1037                                 break;
1038                         } else
1039                                 sum += (p * j - (i - j)) * co[i - j] * c;
1040                 }
1041                 co.push_back(sum / coeff(var, ldeg) / i);
1042         }
1043         
1044         // Construct new series (of non-zero coefficients)
1045         epvector new_seq;
1046         bool higher_order = false;
1047         for (int i=0; i<numcoeff; ++i) {
1048                 if (!co[i].is_zero())
1049                         new_seq.push_back(expair(co[i], p * ldeg + i));
1050                 if (is_order_function(co[i])) {
1051                         higher_order = true;
1052                         break;
1053                 }
1054         }
1055         if (!higher_order)
1056                 new_seq.push_back(expair(Order(_ex1), p * ldeg + numcoeff));
1057
1058         return pseries(relational(var,point), std::move(new_seq));
1059 }
1060
1061
1062 /** Return a new pseries object with the powers shifted by deg. */
1063 pseries pseries::shift_exponents(int deg) const
1064 {
1065         epvector newseq = seq;
1066         for (auto & it : newseq)
1067                 it.coeff += deg;
1068         return pseries(relational(var, point), std::move(newseq));
1069 }
1070
1071
1072 /** Implementation of ex::series() for powers. This performs Laurent expansion
1073  *  of reciprocals of series at singularities.
1074  *  @see ex::series */
1075 ex power::series(const relational & r, int order, unsigned options) const
1076 {
1077         // If basis is already a series, just power it
1078         if (is_exactly_a<pseries>(basis))
1079                 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
1080
1081         // Basis is not a series, may there be a singularity?
1082         bool must_expand_basis = false;
1083         try {
1084                 basis.subs(r, subs_options::no_pattern);
1085         } catch (pole_error) {
1086                 must_expand_basis = true;
1087         }
1088
1089         bool exponent_is_regular = true;
1090         try {
1091                 exponent.subs(r, subs_options::no_pattern);
1092         } catch (pole_error) {
1093                 exponent_is_regular = false;
1094         }
1095
1096         if (!exponent_is_regular) {
1097                 ex l = exponent*log(basis);
1098                 // this == exp(l);
1099                 ex le = l.series(r, order, options);
1100                 // Note: expanding exp(l) won't help, since that will attempt
1101                 // Taylor expansion, and fail (because exponent is "singular")
1102                 // Still l itself might be expanded in Taylor series.
1103                 // Examples:
1104                 // sin(x)/x*log(cos(x))
1105                 // 1/x*log(1 + x)
1106                 return exp(le).series(r, order, options);
1107                 // Note: if l happens to have a Laurent expansion (with
1108                 // negative powers of (var - point)), expanding exp(le)
1109                 // will barf (which is The Right Thing).
1110         }
1111
1112         // Is the expression of type something^(-int)?
1113         if (!must_expand_basis && !exponent.info(info_flags::negint)
1114          && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1115                 return basic::series(r, order, options);
1116
1117         // Is the expression of type 0^something?
1118         if (!must_expand_basis && !basis.subs(r, subs_options::no_pattern).is_zero()
1119          && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1120                 return basic::series(r, order, options);
1121
1122         // Singularity encountered, is the basis equal to (var - point)?
1123         if (basis.is_equal(r.lhs() - r.rhs())) {
1124                 epvector new_seq;
1125                 if (ex_to<numeric>(exponent).to_int() < order)
1126                         new_seq.push_back(expair(_ex1, exponent));
1127                 else
1128                         new_seq.push_back(expair(Order(_ex1), exponent));
1129                 return pseries(r, std::move(new_seq));
1130         }
1131
1132         // No, expand basis into series
1133
1134         numeric numexp;
1135         if (is_a<numeric>(exponent)) {
1136                 numexp = ex_to<numeric>(exponent);
1137         } else {
1138                 numexp = 0;
1139         }
1140         const ex& sym = r.lhs();
1141         // find existing minimal degree
1142         ex eb = basis.expand();
1143         int real_ldegree = 0;
1144         if (eb.info(info_flags::rational_function))
1145                 real_ldegree = eb.ldegree(sym-r.rhs());
1146         if (real_ldegree == 0) {
1147                 int orderloop = 0;
1148                 do {
1149                         orderloop++;
1150                         real_ldegree = basis.series(r, orderloop, options).ldegree(sym);
1151                 } while (real_ldegree == orderloop);
1152         }
1153
1154         if (!(real_ldegree*numexp).is_integer())
1155                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
1156         ex e = basis.series(r, (order + real_ldegree*(1-numexp)).to_int(), options);
1157         
1158         ex result;
1159         try {
1160                 result = ex_to<pseries>(e).power_const(numexp, order);
1161         } catch (pole_error) {
1162                 epvector ser { expair(Order(_ex1), order) };
1163                 result = pseries(r, std::move(ser));
1164         }
1165
1166         return result;
1167 }
1168
1169
1170 /** Re-expansion of a pseries object. */
1171 ex pseries::series(const relational & r, int order, unsigned options) const
1172 {
1173         const ex p = r.rhs();
1174         GINAC_ASSERT(is_a<symbol>(r.lhs()));
1175         const symbol &s = ex_to<symbol>(r.lhs());
1176         
1177         if (var.is_equal(s) && point.is_equal(p)) {
1178                 if (order > degree(s))
1179                         return *this;
1180                 else {
1181                         epvector new_seq;
1182                         for (auto & it : seq) {
1183                                 int o = ex_to<numeric>(it.coeff).to_int();
1184                                 if (o >= order) {
1185                                         new_seq.push_back(expair(Order(_ex1), o));
1186                                         break;
1187                                 }
1188                                 new_seq.push_back(it);
1189                         }
1190                         return pseries(r, std::move(new_seq));
1191                 }
1192         } else
1193                 return convert_to_poly().series(r, order, options);
1194 }
1195
1196 ex integral::series(const relational & r, int order, unsigned options) const
1197 {
1198         if (x.subs(r) != x)
1199                 throw std::logic_error("Cannot series expand wrt dummy variable");
1200         
1201         // Expanding integrand with r substituted taken in boundaries.
1202         ex fseries = f.series(r, order, options);
1203         epvector fexpansion;
1204         fexpansion.reserve(fseries.nops());
1205         for (size_t i=0; i<fseries.nops(); ++i) {
1206                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1207                 currcoeff = (currcoeff == Order(_ex1))
1208                         ? currcoeff
1209                         : integral(x, a.subs(r), b.subs(r), currcoeff);
1210                 if (currcoeff != 0)
1211                         fexpansion.push_back(
1212                                 expair(currcoeff, ex_to<pseries>(fseries).exponop(i)));
1213         }
1214
1215         // Expanding lower boundary
1216         ex result = dynallocate<pseries>(r, std::move(fexpansion));
1217         ex aseries = (a-a.subs(r)).series(r, order, options);
1218         fseries = f.series(x == (a.subs(r)), order, options);
1219         for (size_t i=0; i<fseries.nops(); ++i) {
1220                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1221                 if (is_order_function(currcoeff))
1222                         break;
1223                 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1224                 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1225                 currcoeff = currcoeff.series(r, orderforf);
1226                 ex term = ex_to<pseries>(aseries).power_const(ex_to<numeric>(currexpon+1),order);
1227                 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(-1/(currexpon+1)));
1228                 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1229                 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1230         }
1231
1232         // Expanding upper boundary
1233         ex bseries = (b-b.subs(r)).series(r, order, options);
1234         fseries = f.series(x == (b.subs(r)), order, options);
1235         for (size_t i=0; i<fseries.nops(); ++i) {
1236                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1237                 if (is_order_function(currcoeff))
1238                         break;
1239                 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1240                 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1241                 currcoeff = currcoeff.series(r, orderforf);
1242                 ex term = ex_to<pseries>(bseries).power_const(ex_to<numeric>(currexpon+1),order);
1243                 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(1/(currexpon+1)));
1244                 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1245                 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1246         }
1247
1248         return result;
1249 }
1250
1251
1252 /** Compute the truncated series expansion of an expression.
1253  *  This function returns an expression containing an object of class pseries 
1254  *  to represent the series. If the series does not terminate within the given
1255  *  truncation order, the last term of the series will be an order term.
1256  *
1257  *  @param r  expansion relation, lhs holds variable and rhs holds point
1258  *  @param order  truncation order of series calculations
1259  *  @param options  of class series_options
1260  *  @return an expression holding a pseries object */
1261 ex ex::series(const ex & r, int order, unsigned options) const
1262 {
1263         ex e;
1264         relational rel_;
1265         
1266         if (is_a<relational>(r))
1267                 rel_ = ex_to<relational>(r);
1268         else if (is_a<symbol>(r))
1269                 rel_ = relational(r,_ex0);
1270         else
1271                 throw (std::logic_error("ex::series(): expansion point has unknown type"));
1272         
1273         e = bp->series(rel_, order, options);
1274         return e;
1275 }
1276
1277 GINAC_BIND_UNARCHIVER(pseries);
1278
1279 } // namespace GiNaC