Minor bug fix for the class modular_form_kernel: Ensure that the series-method includ...
[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) 1999-2021 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 range = n.find_property_range("coeff", "power");
123         seq.reserve((range.end-range.begin)/2);
124
125         for (auto loc = range.begin; loc < range.end;) {
126                 ex rest;
127                 ex coeff;
128                 n.find_ex_by_loc(loc++, rest, sym_lst);
129                 n.find_ex_by_loc(loc++, coeff, sym_lst);
130                 seq.emplace_back(expair(rest, coeff));
131         }
132
133         n.find_ex("var", var, sym_lst);
134         n.find_ex("point", point, sym_lst);
135 }
136
137 void pseries::archive(archive_node &n) const
138 {
139         inherited::archive(n);
140         for (auto & it : seq) {
141                 n.add_ex("coeff", it.rest);
142                 n.add_ex("power", it.coeff);
143         }
144         n.add_ex("var", var);
145         n.add_ex("point", point);
146 }
147
148
149 //////////
150 // functions overriding virtual functions from base classes
151 //////////
152
153 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
154 {
155         if (precedence() <= level)
156                 c.s << '(';
157                 
158         // objects of type pseries must not have any zero entries, so the
159         // trivial (zero) pseries needs a special treatment here:
160         if (seq.empty())
161                 c.s << '0';
162
163         auto i = seq.begin(), end = seq.end();
164         while (i != end) {
165
166                 // print a sign, if needed
167                 if (i != seq.begin())
168                         c.s << '+';
169
170                 if (!is_order_function(i->rest)) {
171
172                         // print 'rest', i.e. the expansion coefficient
173                         if (i->rest.info(info_flags::numeric) &&
174                                 i->rest.info(info_flags::positive)) {
175                                 i->rest.print(c);
176                         } else {
177                                 c.s << openbrace << '(';
178                                 i->rest.print(c);
179                                 c.s << ')' << closebrace;
180                         }
181
182                         // print 'coeff', something like (x-1)^42
183                         if (!i->coeff.is_zero()) {
184                                 c.s << mul_sym;
185                                 if (!point.is_zero()) {
186                                         c.s << openbrace << '(';
187                                         (var-point).print(c);
188                                         c.s << ')' << closebrace;
189                                 } else
190                                         var.print(c);
191                                 if (i->coeff.compare(_ex1)) {
192                                         c.s << pow_sym;
193                                         c.s << openbrace;
194                                         if (i->coeff.info(info_flags::negative)) {
195                                                 c.s << '(';
196                                                 i->coeff.print(c);
197                                                 c.s << ')';
198                                         } else
199                                                 i->coeff.print(c);
200                                         c.s << closebrace;
201                                 }
202                         }
203                 } else
204                         Order(pow(var - point, i->coeff)).print(c);
205                 ++i;
206         }
207
208         if (precedence() <= level)
209                 c.s << ')';
210 }
211
212 void pseries::do_print(const print_context & c, unsigned level) const
213 {
214         print_series(c, "", "", "*", "^", level);
215 }
216
217 void pseries::do_print_latex(const print_latex & c, unsigned level) const
218 {
219         print_series(c, "{", "}", " ", "^", level);
220 }
221
222 void pseries::do_print_python(const print_python & c, unsigned level) const
223 {
224         print_series(c, "", "", "*", "**", level);
225 }
226
227 void pseries::do_print_tree(const print_tree & c, unsigned level) const
228 {
229         c.s << std::string(level, ' ') << class_name() << " @" << this
230             << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
231             << std::endl;
232         size_t num = seq.size();
233         for (size_t i=0; i<num; ++i) {
234                 seq[i].rest.print(c, level + c.delta_indent);
235                 seq[i].coeff.print(c, level + c.delta_indent);
236                 c.s << std::string(level + c.delta_indent, ' ') << "-----" << std::endl;
237         }
238         var.print(c, level + c.delta_indent);
239         point.print(c, level + c.delta_indent);
240 }
241
242 void pseries::do_print_python_repr(const print_python_repr & c, unsigned level) const
243 {
244         c.s << class_name() << "(relational(";
245         var.print(c);
246         c.s << ',';
247         point.print(c);
248         c.s << "),[";
249         size_t num = seq.size();
250         for (size_t i=0; i<num; ++i) {
251                 if (i)
252                         c.s << ',';
253                 c.s << '(';
254                 seq[i].rest.print(c);
255                 c.s << ',';
256                 seq[i].coeff.print(c);
257                 c.s << ')';
258         }
259         c.s << "])";
260 }
261
262 int pseries::compare_same_type(const basic & other) const
263 {
264         GINAC_ASSERT(is_a<pseries>(other));
265         const pseries &o = static_cast<const pseries &>(other);
266         
267         // first compare the lengths of the series...
268         if (seq.size()>o.seq.size())
269                 return 1;
270         if (seq.size()<o.seq.size())
271                 return -1;
272         
273         // ...then the expansion point...
274         int cmpval = var.compare(o.var);
275         if (cmpval)
276                 return cmpval;
277         cmpval = point.compare(o.point);
278         if (cmpval)
279                 return cmpval;
280         
281         // ...and if that failed the individual elements
282         auto it = seq.begin(), o_it = o.seq.begin();
283         while (it!=seq.end() && o_it!=o.seq.end()) {
284                 cmpval = it->compare(*o_it);
285                 if (cmpval)
286                         return cmpval;
287                 ++it;
288                 ++o_it;
289         }
290
291         // so they are equal.
292         return 0;
293 }
294
295 /** Return the number of operands including a possible order term. */
296 size_t pseries::nops() const
297 {
298         return seq.size();
299 }
300
301 /** Return the ith term in the series when represented as a sum. */
302 ex pseries::op(size_t i) const
303 {
304         if (i >= seq.size())
305                 throw (std::out_of_range("op() out of range"));
306
307         if (is_order_function(seq[i].rest))
308                 return Order(pow(var-point, seq[i].coeff));
309         return seq[i].rest * pow(var - point, seq[i].coeff);
310 }
311
312 /** Return degree of highest power of the series.  This is usually the exponent
313  *  of the Order term.  If s is not the expansion variable of the series, the
314  *  series is examined termwise. */
315 int pseries::degree(const ex &s) const
316 {
317         if (seq.empty())
318                 return 0;
319
320         if (var.is_equal(s))
321                 // Return last/greatest exponent
322                 return ex_to<numeric>((seq.end()-1)->coeff).to_int();
323
324         int max_pow = std::numeric_limits<int>::min();
325         for (auto & it : seq)
326                 max_pow = std::max(max_pow, it.rest.degree(s));
327         return max_pow;
328 }
329
330 /** Return degree of lowest power of the series.  This is usually the exponent
331  *  of the leading term.  If s is not the expansion variable of the series, the
332  *  series is examined termwise.  If s is the expansion variable but the
333  *  expansion point is not zero the series is not expanded to find the degree.
334  *  I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
335 int pseries::ldegree(const ex &s) const
336 {
337         if (seq.empty())
338                 return 0;
339
340         if (var.is_equal(s))
341                 // Return first/smallest exponent
342                 return ex_to<numeric>((seq.begin())->coeff).to_int();
343
344         int min_pow = std::numeric_limits<int>::max();
345         for (auto & it : seq)
346                 min_pow = std::min(min_pow, it.rest.degree(s));
347         return min_pow;
348 }
349
350 /** Return coefficient of degree n in power series if s is the expansion
351  *  variable.  If the expansion point is nonzero, by definition the n=1
352  *  coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
353  *  the expansion took place in the s in the first place).
354  *  If s is not the expansion variable, an attempt is made to convert the
355  *  series to a polynomial and return the corresponding coefficient from
356  *  there. */
357 ex pseries::coeff(const ex &s, int n) const
358 {
359         if (var.is_equal(s)) {
360                 if (seq.empty())
361                         return _ex0;
362                 
363                 // Binary search in sequence for given power
364                 numeric looking_for = numeric(n);
365                 int lo = 0, hi = seq.size() - 1;
366                 while (lo <= hi) {
367                         int mid = (lo + hi) / 2;
368                         GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
369                         int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
370                         switch (cmp) {
371                                 case -1:
372                                         lo = mid + 1;
373                                         break;
374                                 case 0:
375                                         return seq[mid].rest;
376                                 case 1:
377                                         hi = mid - 1;
378                                         break;
379                                 default:
380                                         throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
381                         }
382                 }
383                 return _ex0;
384         } else
385                 return convert_to_poly().coeff(s, n);
386 }
387
388 /** Does nothing. */
389 ex pseries::collect(const ex &s, bool distributed) const
390 {
391         return *this;
392 }
393
394 /** Perform coefficient-wise automatic term rewriting rules in this class. */
395 ex pseries::eval() const
396 {
397         return hold();
398 }
399
400 /** Evaluate coefficients numerically. */
401 ex pseries::evalf() const
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.emplace_back(expair(it.rest.evalf(), it.coeff));
408
409         return dynallocate<pseries>(relational(var,point), std::move(new_seq)).setflag(status_flags::evaluated);
410 }
411
412 ex pseries::conjugate() const
413 {
414         if(!var.info(info_flags::real))
415                 return conjugate_function(*this).hold();
416
417         std::unique_ptr<epvector> newseq(conjugateepvector(seq));
418         ex newpoint = point.conjugate();
419
420         if (!newseq && are_ex_trivially_equal(point, newpoint)) {
421                 return *this;
422         }
423
424         return dynallocate<pseries>(var==newpoint, newseq ? std::move(*newseq) : seq);
425 }
426
427 ex pseries::real_part() const
428 {
429         if(!var.info(info_flags::real))
430                 return real_part_function(*this).hold();
431         ex newpoint = point.real_part();
432         if(newpoint != point)
433                 return real_part_function(*this).hold();
434
435         epvector v;
436         v.reserve(seq.size());
437         for (auto & it : seq)
438                 v.emplace_back(expair(it.rest.real_part(), it.coeff));
439         return dynallocate<pseries>(var==point, std::move(v));
440 }
441
442 ex pseries::imag_part() const
443 {
444         if(!var.info(info_flags::real))
445                 return imag_part_function(*this).hold();
446         ex newpoint = point.real_part();
447         if(newpoint != point)
448                 return imag_part_function(*this).hold();
449
450         epvector v;
451         v.reserve(seq.size());
452         for (auto & it : seq)
453                 v.emplace_back(expair(it.rest.imag_part(), it.coeff));
454         return dynallocate<pseries>(var==point, std::move(v));
455 }
456
457 ex pseries::eval_integ() const
458 {
459         std::unique_ptr<epvector> newseq(nullptr);
460         for (auto i=seq.begin(); i!=seq.end(); ++i) {
461                 if (newseq) {
462                         newseq->emplace_back(expair(i->rest.eval_integ(), i->coeff));
463                         continue;
464                 }
465                 ex newterm = i->rest.eval_integ();
466                 if (!are_ex_trivially_equal(newterm, i->rest)) {
467                         newseq.reset(new epvector);
468                         newseq->reserve(seq.size());
469                         for (auto j=seq.begin(); j!=i; ++j)
470                                 newseq->push_back(*j);
471                         newseq->emplace_back(expair(newterm, i->coeff));
472                 }
473         }
474
475         ex newpoint = point.eval_integ();
476         if (newseq || !are_ex_trivially_equal(newpoint, point))
477                 return dynallocate<pseries>(var==newpoint, std::move(*newseq));
478         return *this;
479 }
480
481 ex pseries::evalm() const
482 {
483         // evalm each coefficient
484         epvector newseq;
485         bool something_changed = false;
486         for (auto i=seq.begin(); i!=seq.end(); ++i) {
487                 if (something_changed) {
488                         ex newcoeff = i->rest.evalm();
489                         if (!newcoeff.is_zero())
490                                 newseq.emplace_back(expair(newcoeff, i->coeff));
491                 } else {
492                         ex newcoeff = i->rest.evalm();
493                         if (!are_ex_trivially_equal(newcoeff, i->rest)) {
494                                 something_changed = true;
495                                 newseq.reserve(seq.size());
496                                 std::copy(seq.begin(), i, std::back_inserter<epvector>(newseq));
497                                 if (!newcoeff.is_zero())
498                                         newseq.emplace_back(expair(newcoeff, i->coeff));
499                         }
500                 }
501         }
502         if (something_changed)
503                 return dynallocate<pseries>(var==point, std::move(newseq));
504         else
505                 return *this;
506 }
507
508 ex pseries::subs(const exmap & m, unsigned options) const
509 {
510         // If expansion variable is being substituted, convert the series to a
511         // polynomial and do the substitution there because the result might
512         // no longer be a power series
513         if (m.find(var) != m.end())
514                 return convert_to_poly(true).subs(m, options);
515         
516         // Otherwise construct a new series with substituted coefficients and
517         // expansion point
518         epvector newseq;
519         newseq.reserve(seq.size());
520         for (auto & it : seq)
521                 newseq.emplace_back(expair(it.rest.subs(m, options), it.coeff));
522         return dynallocate<pseries>(relational(var,point.subs(m, options)), std::move(newseq));
523 }
524
525 /** Implementation of ex::expand() for a power series.  It expands all the
526  *  terms individually and returns the resulting series as a new pseries. */
527 ex pseries::expand(unsigned options) const
528 {
529         epvector newseq;
530         for (auto & it : seq) {
531                 ex restexp = it.rest.expand();
532                 if (!restexp.is_zero())
533                         newseq.emplace_back(expair(restexp, it.coeff));
534         }
535         return dynallocate<pseries>(relational(var,point), std::move(newseq)).setflag(options == 0 ? status_flags::expanded : 0);
536 }
537
538 /** Implementation of ex::diff() for a power series.
539  *  @see ex::diff */
540 ex pseries::derivative(const symbol & s) const
541 {
542         epvector new_seq;
543
544         if (s == var) {
545                 
546                 // FIXME: coeff might depend on var
547                 for (auto & it : seq) {
548                         if (is_order_function(it.rest)) {
549                                 new_seq.emplace_back(expair(it.rest, it.coeff - 1));
550                         } else {
551                                 ex c = it.rest * it.coeff;
552                                 if (!c.is_zero())
553                                         new_seq.emplace_back(expair(c, it.coeff - 1));
554                         }
555                 }
556
557         } else {
558
559                 for (auto & it : seq) {
560                         if (is_order_function(it.rest)) {
561                                 new_seq.push_back(it);
562                         } else {
563                                 ex c = it.rest.diff(s);
564                                 if (!c.is_zero())
565                                         new_seq.emplace_back(expair(c, it.coeff));
566                         }
567                 }
568         }
569
570         return pseries(relational(var,point), std::move(new_seq));
571 }
572
573 ex pseries::convert_to_poly(bool no_order) const
574 {
575         ex e;
576         for (auto & it : seq) {
577                 if (is_order_function(it.rest)) {
578                         if (!no_order)
579                                 e += Order(pow(var - point, it.coeff));
580                 } else
581                         e += it.rest * pow(var - point, it.coeff);
582         }
583         return e;
584 }
585
586 bool pseries::is_terminating() const
587 {
588         return seq.empty() || !is_order_function((seq.end()-1)->rest);
589 }
590
591 ex pseries::coeffop(size_t i) const
592 {
593         if (i >= nops())
594                 throw (std::out_of_range("coeffop() out of range"));
595         return seq[i].rest;
596 }
597
598 ex pseries::exponop(size_t i) const
599 {
600         if (i >= nops())
601                 throw (std::out_of_range("exponop() out of range"));
602         return seq[i].coeff;
603 }
604
605
606 /*
607  *  Implementations of series expansion
608  */
609
610 /** Default implementation of ex::series(). This performs Taylor expansion.
611  *  @see ex::series */
612 ex basic::series(const relational & r, int order, unsigned options) const
613 {
614         epvector seq;
615         const symbol &s = ex_to<symbol>(r.lhs());
616
617         // default for order-values that make no sense for Taylor expansion
618         if ((order <= 0) && this->has(s)) {
619                 seq.emplace_back(expair(Order(_ex1), order));
620                 return pseries(r, std::move(seq));
621         }
622
623         // do Taylor expansion
624         numeric fac = 1;
625         ex deriv = *this;
626         ex coeff = deriv.subs(r, subs_options::no_pattern);
627
628         if (!coeff.is_zero()) {
629                 seq.emplace_back(expair(coeff, _ex0));
630         }
631
632         int n;
633         for (n=1; n<order; ++n) {
634                 fac = fac.div(n);
635                 // We need to test for zero in order to see if the series terminates.
636                 // The problem is that there is no such thing as a perfect test for
637                 // zero.  Expanding the term occasionally helps a little...
638                 deriv = deriv.diff(s).expand();
639                 if (deriv.is_zero())  // Series terminates
640                         return pseries(r, std::move(seq));
641
642                 coeff = deriv.subs(r, subs_options::no_pattern);
643                 if (!coeff.is_zero())
644                         seq.emplace_back(expair(fac * coeff, n));
645         }
646         
647         // Higher-order terms, if present
648         deriv = deriv.diff(s);
649         if (!deriv.expand().is_zero())
650                 seq.emplace_back(expair(Order(_ex1), n));
651         return pseries(r, std::move(seq));
652 }
653
654
655 /** Implementation of ex::series() for symbols.
656  *  @see ex::series */
657 ex symbol::series(const relational & r, int order, unsigned options) const
658 {
659         epvector seq;
660         const ex point = r.rhs();
661         GINAC_ASSERT(is_a<symbol>(r.lhs()));
662
663         if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
664                 if (order > 0 && !point.is_zero())
665                         seq.emplace_back(expair(point, _ex0));
666                 if (order > 1)
667                         seq.emplace_back(expair(_ex1, _ex1));
668                 else
669                         seq.emplace_back(expair(Order(_ex1), numeric(order)));
670         } else
671                 seq.emplace_back(expair(*this, _ex0));
672         return pseries(r, std::move(seq));
673 }
674
675
676 /** Add one series object to another, producing a pseries object that
677  *  represents the sum.
678  *
679  *  @param other  pseries object to add with
680  *  @return the sum as a pseries */
681 ex pseries::add_series(const pseries &other) const
682 {
683         // Adding two series with different variables or expansion points
684         // results in an empty (constant) series 
685         if (!is_compatible_to(other)) {
686                 epvector nul { expair(Order(_ex1), _ex0) };
687                 return pseries(relational(var,point), std::move(nul));
688         }
689         
690         // Series addition
691         epvector new_seq;
692         auto a = seq.begin(), a_end = seq.end();
693         auto b = other.seq.begin(), b_end = other.seq.end();
694         int pow_a = std::numeric_limits<int>::max(), pow_b = std::numeric_limits<int>::max();
695         for (;;) {
696                 // If a is empty, fill up with elements from b and stop
697                 if (a == a_end) {
698                         while (b != b_end) {
699                                 new_seq.push_back(*b);
700                                 ++b;
701                         }
702                         break;
703                 } else
704                         pow_a = ex_to<numeric>((*a).coeff).to_int();
705                 
706                 // If b is empty, fill up with elements from a and stop
707                 if (b == b_end) {
708                         while (a != a_end) {
709                                 new_seq.push_back(*a);
710                                 ++a;
711                         }
712                         break;
713                 } else
714                         pow_b = ex_to<numeric>((*b).coeff).to_int();
715                 
716                 // a and b are non-empty, compare powers
717                 if (pow_a < pow_b) {
718                         // a has lesser power, get coefficient from a
719                         new_seq.push_back(*a);
720                         if (is_order_function((*a).rest))
721                                 break;
722                         ++a;
723                 } else if (pow_b < pow_a) {
724                         // b has lesser power, get coefficient from b
725                         new_seq.push_back(*b);
726                         if (is_order_function((*b).rest))
727                                 break;
728                         ++b;
729                 } else {
730                         // Add coefficient of a and b
731                         if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
732                                 new_seq.emplace_back(expair(Order(_ex1), (*a).coeff));
733                                 break;  // Order term ends the sequence
734                         } else {
735                                 ex sum = (*a).rest + (*b).rest;
736                                 if (!(sum.is_zero()))
737                                         new_seq.emplace_back(expair(sum, numeric(pow_a)));
738                                 ++a;
739                                 ++b;
740                         }
741                 }
742         }
743         return pseries(relational(var,point), std::move(new_seq));
744 }
745
746
747 /** Implementation of ex::series() for sums. This performs series addition when
748  *  adding pseries objects.
749  *  @see ex::series */
750 ex add::series(const relational & r, int order, unsigned options) const
751 {
752         ex acc; // Series accumulator
753         
754         // Get first term from overall_coeff
755         acc = overall_coeff.series(r, order, options);
756         
757         // Add remaining terms
758         for (auto & it : seq) {
759                 ex op;
760                 if (is_exactly_a<pseries>(it.rest))
761                         op = it.rest;
762                 else
763                         op = it.rest.series(r, order, options);
764                 if (!it.coeff.is_equal(_ex1))
765                         op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it.coeff));
766                 
767                 // Series addition
768                 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
769         }
770         return acc;
771 }
772
773
774 /** Multiply a pseries object with a numeric constant, producing a pseries
775  *  object that represents the product.
776  *
777  *  @param other  constant to multiply with
778  *  @return the product as a pseries */
779 ex pseries::mul_const(const numeric &other) const
780 {
781         epvector new_seq;
782         new_seq.reserve(seq.size());
783         
784         for (auto & it : seq) {
785                 if (!is_order_function(it.rest))
786                         new_seq.emplace_back(expair(it.rest * other, it.coeff));
787                 else
788                         new_seq.push_back(it);
789         }
790         return pseries(relational(var,point), std::move(new_seq));
791 }
792
793
794 /** Multiply one pseries object to another, producing a pseries object that
795  *  represents the product.
796  *
797  *  @param other  pseries object to multiply with
798  *  @return the product as a pseries */
799 ex pseries::mul_series(const pseries &other) const
800 {
801         // Multiplying two series with different variables or expansion points
802         // results in an empty (constant) series 
803         if (!is_compatible_to(other)) {
804                 epvector nul { expair(Order(_ex1), _ex0) };
805                 return pseries(relational(var,point), std::move(nul));
806         }
807
808         if (seq.empty() || other.seq.empty()) {
809                 return dynallocate<pseries>(var==point, epvector());
810         }
811         
812         // Series multiplication
813         epvector new_seq;
814         const int a_max = degree(var);
815         const int b_max = other.degree(var);
816         const int a_min = ldegree(var);
817         const int b_min = other.ldegree(var);
818         const int cdeg_min = a_min + b_min;
819         int cdeg_max = a_max + b_max;
820         
821         int higher_order_a = std::numeric_limits<int>::max();
822         int higher_order_b = std::numeric_limits<int>::max();
823         if (is_order_function(coeff(var, a_max)))
824                 higher_order_a = a_max + b_min;
825         if (is_order_function(other.coeff(var, b_max)))
826                 higher_order_b = b_max + a_min;
827         const int higher_order_c = std::min(higher_order_a, higher_order_b);
828         if (cdeg_max >= higher_order_c)
829                 cdeg_max = higher_order_c - 1;
830
831         std::map<int, ex> rest_map_a, rest_map_b;
832         for (const auto& it : seq)
833                 rest_map_a[ex_to<numeric>(it.coeff).to_int()] = it.rest;
834
835         if (other.var.is_equal(var))
836                 for (const auto& it : other.seq)
837                         rest_map_b[ex_to<numeric>(it.coeff).to_int()] = it.rest;
838
839         for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
840                 ex co = _ex0;
841                 // c(i)=a(0)b(i)+...+a(i)b(0)
842                 for (int i=a_min; cdeg-i>=b_min; ++i) {
843                         const auto& ita = rest_map_a.find(i);
844                         if (ita == rest_map_a.end())
845                                 continue;
846                         const auto& itb = rest_map_b.find(cdeg-i);
847                         if (itb == rest_map_b.end())
848                                 continue;
849                         if (!is_order_function(ita->second) && !is_order_function(itb->second))
850                                 co += ita->second * itb->second;
851                 }
852                 if (!co.is_zero())
853                         new_seq.emplace_back(expair(co, numeric(cdeg)));
854         }
855         if (higher_order_c < std::numeric_limits<int>::max())
856                 new_seq.emplace_back(expair(Order(_ex1), numeric(higher_order_c)));
857         return pseries(relational(var, point), std::move(new_seq));
858 }
859
860
861 /** Implementation of ex::series() for product. This performs series
862  *  multiplication when multiplying series.
863  *  @see ex::series */
864 ex mul::series(const relational & r, int order, unsigned options) const
865 {
866         pseries acc; // Series accumulator
867
868         GINAC_ASSERT(is_a<symbol>(r.lhs()));
869         const ex& sym = r.lhs();
870                 
871         // holds ldegrees of the series of individual factors
872         std::vector<int> ldegrees;
873         std::vector<bool> ldegree_redo;
874
875         // find minimal degrees
876         // first round: obtain a bound up to which minimal degrees have to be
877         // considered
878         for (auto & it : seq) {
879
880                 ex expon = it.coeff;
881                 int factor = 1;
882                 ex buf;
883                 if (expon.info(info_flags::integer)) {
884                         buf = it.rest;
885                         factor = ex_to<numeric>(expon).to_int();
886                 } else {
887                         buf = recombine_pair_to_ex(it);
888                 }
889
890                 int real_ldegree = 0;
891                 bool flag_redo = false;
892                 try {
893                         real_ldegree = buf.expand().ldegree(sym-r.rhs());
894                 } catch (std::runtime_error &) {}
895
896                 if (real_ldegree == 0) {
897                         if ( factor < 0 ) {
898                                 // This case must terminate, otherwise we would have division by
899                                 // zero.
900                                 int orderloop = 0;
901                                 do {
902                                         orderloop++;
903                                         real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
904                                 } while (real_ldegree == orderloop);
905                         } else {
906                                 // Here it is possible that buf does not have a ldegree, therefore
907                                 // check only if ldegree is negative, otherwise reconsider the case
908                                 // in the second round.
909                                 real_ldegree = buf.series(r, 0, options).ldegree(sym);
910                                 if (real_ldegree == 0)
911                                         flag_redo = true;
912                         }
913                 }
914
915                 ldegrees.push_back(factor * real_ldegree);
916                 ldegree_redo.push_back(flag_redo);
917         }
918
919         int degbound = order-std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
920         // Second round: determine the remaining positive ldegrees by the series
921         // method.
922         // here we can ignore ldegrees larger than degbound
923         size_t j = 0;
924         for (auto & it : seq) {
925                 if ( ldegree_redo[j] ) {
926                         ex expon = it.coeff;
927                         int factor = 1;
928                         ex buf;
929                         if (expon.info(info_flags::integer)) {
930                                 buf = it.rest;
931                                 factor = ex_to<numeric>(expon).to_int();
932                         } else {
933                                 buf = recombine_pair_to_ex(it);
934                         }
935                         int real_ldegree = 0;
936                         int orderloop = 0;
937                         do {
938                                 orderloop++;
939                                 real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
940                         } while ((real_ldegree == orderloop)
941                               && (factor*real_ldegree < degbound));
942                         ldegrees[j] = factor * real_ldegree;
943                         degbound -= factor * real_ldegree;
944                 }
945                 j++;
946         }
947
948         int degsum = std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
949
950         if (degsum > order) {
951                 return dynallocate<pseries>(r, epvector{{Order(_ex1), order}});
952         }
953
954         // Multiply with remaining terms
955         auto itd = ldegrees.begin();
956         for (auto it=seq.begin(), itend=seq.end(); it!=itend; ++it, ++itd) {
957
958                 // do series expansion with adjusted order
959                 ex op = recombine_pair_to_ex(*it).series(r, order-degsum+(*itd), options);
960
961                 // Series multiplication
962                 if (it == seq.begin())
963                         acc = ex_to<pseries>(op);
964                 else
965                         acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
966         }
967
968         return acc.mul_const(ex_to<numeric>(overall_coeff));
969 }
970
971
972 /** Compute the p-th power of a series.
973  *
974  *  @param p  power to compute
975  *  @param deg  truncation order of series calculation */
976 ex pseries::power_const(const numeric &p, int deg) const
977 {
978         // method:
979         // (due to Leonhard Euler)
980         // let A(x) be this series and for the time being let it start with a
981         // constant (later we'll generalize):
982         //     A(x) = a_0 + a_1*x + a_2*x^2 + ...
983         // We want to compute
984         //     C(x) = A(x)^p
985         //     C(x) = c_0 + c_1*x + c_2*x^2 + ...
986         // Taking the derivative on both sides and multiplying with A(x) one
987         // immediately arrives at
988         //     C'(x)*A(x) = p*C(x)*A'(x)
989         // Multiplying this out and comparing coefficients we get the recurrence
990         // formula
991         //     c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
992         //                    ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
993         // which can easily be solved given the starting value c_0 = (a_0)^p.
994         // For the more general case where the leading coefficient of A(x) is not
995         // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
996         // repeat the above derivation.  The leading power of C2(x) = A2(x)^2 is
997         // then of course x^(p*m) but the recurrence formula still holds.
998         
999         if (seq.empty()) {
1000                 // as a special case, handle the empty (zero) series honoring the
1001                 // usual power laws such as implemented in power::eval()
1002                 if (p.real().is_zero())
1003                         throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
1004                 else if (p.real().is_negative())
1005                         throw pole_error("pseries::power_const(): division by zero",1);
1006                 else
1007                         return *this;
1008         }
1009         
1010         const int ldeg = ldegree(var);
1011         if (!(p*ldeg).is_integer())
1012                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
1013
1014         // adjust number of coefficients
1015         int numcoeff = deg - (p*ldeg).to_int();
1016         if (numcoeff <= 0) {
1017                 epvector epv { expair(Order(_ex1), deg) };
1018                 return dynallocate<pseries>(relational(var,point), std::move(epv));
1019         }
1020         
1021         // O(x^n)^(-m) is undefined
1022         if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
1023                 throw pole_error("pseries::power_const(): division by zero",1);
1024         
1025         // Compute coefficients of the powered series
1026         exvector co;
1027         co.reserve(numcoeff);
1028         co.push_back(pow(coeff(var, ldeg), p));
1029         for (int i=1; i<numcoeff; ++i) {
1030                 ex sum = _ex0;
1031                 for (int j=1; j<=i; ++j) {
1032                         ex c = coeff(var, j + ldeg);
1033                         if (is_order_function(c)) {
1034                                 co.push_back(Order(_ex1));
1035                                 break;
1036                         } else
1037                                 sum += (p * j - (i - j)) * co[i - j] * c;
1038                 }
1039                 co.push_back(sum / coeff(var, ldeg) / i);
1040         }
1041         
1042         // Construct new series (of non-zero coefficients)
1043         epvector new_seq;
1044         bool higher_order = false;
1045         for (int i=0; i<numcoeff; ++i) {
1046                 if (!co[i].is_zero())
1047                         new_seq.emplace_back(expair(co[i], p * ldeg + i));
1048                 if (is_order_function(co[i])) {
1049                         higher_order = true;
1050                         break;
1051                 }
1052         }
1053         if (!higher_order)
1054                 new_seq.emplace_back(expair(Order(_ex1), p * ldeg + numcoeff));
1055
1056         return pseries(relational(var,point), std::move(new_seq));
1057 }
1058
1059
1060 /** Return a new pseries object with the powers shifted by deg. */
1061 pseries pseries::shift_exponents(int deg) const
1062 {
1063         epvector newseq = seq;
1064         for (auto & it : newseq)
1065                 it.coeff += deg;
1066         return pseries(relational(var, point), std::move(newseq));
1067 }
1068
1069
1070 /** Implementation of ex::series() for powers. This performs Laurent expansion
1071  *  of reciprocals of series at singularities.
1072  *  @see ex::series */
1073 ex power::series(const relational & r, int order, unsigned options) const
1074 {
1075         // If basis is already a series, just power it
1076         if (is_exactly_a<pseries>(basis))
1077                 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
1078
1079         // Basis is not a series, may there be a singularity?
1080         bool must_expand_basis = false;
1081         try {
1082                 basis.subs(r, subs_options::no_pattern);
1083         } catch (pole_error &) {
1084                 must_expand_basis = true;
1085         }
1086
1087         bool exponent_is_regular = true;
1088         try {
1089                 exponent.subs(r, subs_options::no_pattern);
1090         } catch (pole_error &) {
1091                 exponent_is_regular = false;
1092         }
1093
1094         if (!exponent_is_regular) {
1095                 ex l = exponent*log(basis);
1096                 // this == exp(l);
1097                 ex le = l.series(r, order, options);
1098                 // Note: expanding exp(l) won't help, since that will attempt
1099                 // Taylor expansion, and fail (because exponent is "singular")
1100                 // Still l itself might be expanded in Taylor series.
1101                 // Examples:
1102                 // sin(x)/x*log(cos(x))
1103                 // 1/x*log(1 + x)
1104                 return exp(le).series(r, order, options);
1105                 // Note: if l happens to have a Laurent expansion (with
1106                 // negative powers of (var - point)), expanding exp(le)
1107                 // will barf (which is The Right Thing).
1108         }
1109
1110         // Is the expression of type something^(-int)?
1111         if (!must_expand_basis && !exponent.info(info_flags::negint)
1112          && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1113                 return basic::series(r, order, options);
1114
1115         // Is the expression of type 0^something?
1116         if (!must_expand_basis && !basis.subs(r, subs_options::no_pattern).is_zero()
1117          && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1118                 return basic::series(r, order, options);
1119
1120         // Singularity encountered, is the basis equal to (var - point)?
1121         if (basis.is_equal(r.lhs() - r.rhs())) {
1122                 epvector new_seq;
1123                 if (ex_to<numeric>(exponent).to_int() < order)
1124                         new_seq.emplace_back(expair(_ex1, exponent));
1125                 else
1126                         new_seq.emplace_back(expair(Order(_ex1), exponent));
1127                 return pseries(r, std::move(new_seq));
1128         }
1129
1130         // No, expand basis into series
1131
1132         numeric numexp;
1133         if (is_a<numeric>(exponent)) {
1134                 numexp = ex_to<numeric>(exponent);
1135         } else {
1136                 numexp = 0;
1137         }
1138         const ex& sym = r.lhs();
1139         // find existing minimal degree
1140         ex eb = basis.expand();
1141         int real_ldegree = 0;
1142         if (eb.info(info_flags::rational_function))
1143                 real_ldegree = eb.ldegree(sym-r.rhs());
1144         if (real_ldegree == 0) {
1145                 int orderloop = 0;
1146                 do {
1147                         orderloop++;
1148                         real_ldegree = basis.series(r, orderloop, options).ldegree(sym);
1149                 } while (real_ldegree == orderloop);
1150         }
1151
1152         if (!(real_ldegree*numexp).is_integer())
1153                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
1154         ex e = basis.series(r, (order + real_ldegree*(1-numexp)).to_int(), options);
1155         
1156         ex result;
1157         try {
1158                 result = ex_to<pseries>(e).power_const(numexp, order);
1159         } catch (pole_error &) {
1160                 epvector ser { expair(Order(_ex1), order) };
1161                 result = pseries(r, std::move(ser));
1162         }
1163
1164         return result;
1165 }
1166
1167
1168 /** Re-expansion of a pseries object. */
1169 ex pseries::series(const relational & r, int order, unsigned options) const
1170 {
1171         const ex p = r.rhs();
1172         GINAC_ASSERT(is_a<symbol>(r.lhs()));
1173         const symbol &s = ex_to<symbol>(r.lhs());
1174         
1175         if (var.is_equal(s) && point.is_equal(p)) {
1176                 if (order > degree(s))
1177                         return *this;
1178                 else {
1179                         epvector new_seq;
1180                         for (auto & it : seq) {
1181                                 int o = ex_to<numeric>(it.coeff).to_int();
1182                                 if (o >= order) {
1183                                         new_seq.emplace_back(expair(Order(_ex1), o));
1184                                         break;
1185                                 }
1186                                 new_seq.push_back(it);
1187                         }
1188                         return pseries(r, std::move(new_seq));
1189                 }
1190         } else
1191                 return convert_to_poly().series(r, order, options);
1192 }
1193
1194 ex integral::series(const relational & r, int order, unsigned options) const
1195 {
1196         if (x.subs(r) != x)
1197                 throw std::logic_error("Cannot series expand wrt dummy variable");
1198         
1199         // Expanding integrand with r substituted taken in boundaries.
1200         ex fseries = f.series(r, order, options);
1201         epvector fexpansion;
1202         fexpansion.reserve(fseries.nops());
1203         for (size_t i=0; i<fseries.nops(); ++i) {
1204                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1205                 currcoeff = (currcoeff == Order(_ex1))
1206                         ? currcoeff
1207                         : integral(x, a.subs(r), b.subs(r), currcoeff);
1208                 if (currcoeff != 0)
1209                         fexpansion.emplace_back(
1210                                 expair(currcoeff, ex_to<pseries>(fseries).exponop(i)));
1211         }
1212
1213         // Expanding lower boundary
1214         ex result = dynallocate<pseries>(r, std::move(fexpansion));
1215         ex aseries = (a-a.subs(r)).series(r, order, options);
1216         fseries = f.series(x == (a.subs(r)), order, options);
1217         for (size_t i=0; i<fseries.nops(); ++i) {
1218                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1219                 if (is_order_function(currcoeff))
1220                         break;
1221                 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1222                 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1223                 currcoeff = currcoeff.series(r, orderforf);
1224                 ex term = ex_to<pseries>(aseries).power_const(ex_to<numeric>(currexpon+1),order);
1225                 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(-1/(currexpon+1)));
1226                 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1227                 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1228         }
1229
1230         // Expanding upper boundary
1231         ex bseries = (b-b.subs(r)).series(r, order, options);
1232         fseries = f.series(x == (b.subs(r)), order, options);
1233         for (size_t i=0; i<fseries.nops(); ++i) {
1234                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1235                 if (is_order_function(currcoeff))
1236                         break;
1237                 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1238                 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1239                 currcoeff = currcoeff.series(r, orderforf);
1240                 ex term = ex_to<pseries>(bseries).power_const(ex_to<numeric>(currexpon+1),order);
1241                 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(1/(currexpon+1)));
1242                 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1243                 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1244         }
1245
1246         return result;
1247 }
1248
1249
1250 /** Compute the truncated series expansion of an expression.
1251  *  This function returns an expression containing an object of class pseries 
1252  *  to represent the series. If the series does not terminate within the given
1253  *  truncation order, the last term of the series will be an order term.
1254  *
1255  *  @param r  expansion relation, lhs holds variable and rhs holds point
1256  *  @param order  truncation order of series calculations
1257  *  @param options  of class series_options
1258  *  @return an expression holding a pseries object */
1259 ex ex::series(const ex & r, int order, unsigned options) const
1260 {
1261         ex e;
1262         relational rel_;
1263         
1264         if (is_a<relational>(r))
1265                 rel_ = ex_to<relational>(r);
1266         else if (is_a<symbol>(r))
1267                 rel_ = relational(r,_ex0);
1268         else
1269                 throw (std::logic_error("ex::series(): expansion point has unknown type"));
1270         
1271         e = bp->series(rel_, order, options);
1272         return e;
1273 }
1274
1275 GINAC_BIND_UNARCHIVER(pseries);
1276
1277 } // namespace GiNaC