Added methods for taking real and imaginary parts.
[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) 1999-2006 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 <numeric>
25 #include <stdexcept>
26
27 #include "pseries.h"
28 #include "add.h"
29 #include "inifcns.h" // for Order function
30 #include "lst.h"
31 #include "mul.h"
32 #include "power.h"
33 #include "relational.h"
34 #include "operators.h"
35 #include "symbol.h"
36 #include "integral.h"
37 #include "archive.h"
38 #include "utils.h"
39
40 namespace GiNaC {
41
42 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(pseries, basic,
43   print_func<print_context>(&pseries::do_print).
44   print_func<print_latex>(&pseries::do_print_latex).
45   print_func<print_tree>(&pseries::do_print_tree).
46   print_func<print_python>(&pseries::do_print_python).
47   print_func<print_python_repr>(&pseries::do_print_python_repr))
48
49
50 /*
51  *  Default constructor
52  */
53
54 pseries::pseries() : inherited(&pseries::tinfo_static) { }
55
56
57 /*
58  *  Other ctors
59  */
60
61 /** Construct pseries from a vector of coefficients and powers.
62  *  expair.rest holds the coefficient, expair.coeff holds the power.
63  *  The powers must be integers (positive or negative) and in ascending order;
64  *  the last coefficient can be Order(_ex1) to represent a truncated,
65  *  non-terminating series.
66  *
67  *  @param rel_  expansion variable and point (must hold a relational)
68  *  @param ops_  vector of {coefficient, power} pairs (coefficient must not be zero)
69  *  @return newly constructed pseries */
70 pseries::pseries(const ex &rel_, const epvector &ops_) : basic(&pseries::tinfo_static), seq(ops_)
71 {
72         GINAC_ASSERT(is_a<relational>(rel_));
73         GINAC_ASSERT(is_a<symbol>(rel_.lhs()));
74         point = rel_.rhs();
75         var = rel_.lhs();
76 }
77
78
79 /*
80  *  Archiving
81  */
82
83 pseries::pseries(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
84 {
85         for (unsigned int i=0; true; ++i) {
86                 ex rest;
87                 ex coeff;
88                 if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
89                         seq.push_back(expair(rest, coeff));
90                 else
91                         break;
92         }
93         n.find_ex("var", var, sym_lst);
94         n.find_ex("point", point, sym_lst);
95 }
96
97 void pseries::archive(archive_node &n) const
98 {
99         inherited::archive(n);
100         epvector::const_iterator i = seq.begin(), iend = seq.end();
101         while (i != iend) {
102                 n.add_ex("coeff", i->rest);
103                 n.add_ex("power", i->coeff);
104                 ++i;
105         }
106         n.add_ex("var", var);
107         n.add_ex("point", point);
108 }
109
110 DEFAULT_UNARCHIVE(pseries)
111
112 //////////
113 // functions overriding virtual functions from base classes
114 //////////
115
116 void pseries::print_series(const print_context & c, const char *openbrace, const char *closebrace, const char *mul_sym, const char *pow_sym, unsigned level) const
117 {
118         if (precedence() <= level)
119                 c.s << '(';
120                 
121         // objects of type pseries must not have any zero entries, so the
122         // trivial (zero) pseries needs a special treatment here:
123         if (seq.empty())
124                 c.s << '0';
125
126         epvector::const_iterator i = seq.begin(), end = seq.end();
127         while (i != end) {
128
129                 // print a sign, if needed
130                 if (i != seq.begin())
131                         c.s << '+';
132
133                 if (!is_order_function(i->rest)) {
134
135                         // print 'rest', i.e. the expansion coefficient
136                         if (i->rest.info(info_flags::numeric) &&
137                                 i->rest.info(info_flags::positive)) {
138                                 i->rest.print(c);
139                         } else {
140                                 c.s << openbrace << '(';
141                                 i->rest.print(c);
142                                 c.s << ')' << closebrace;
143                         }
144
145                         // print 'coeff', something like (x-1)^42
146                         if (!i->coeff.is_zero()) {
147                                 c.s << mul_sym;
148                                 if (!point.is_zero()) {
149                                         c.s << openbrace << '(';
150                                         (var-point).print(c);
151                                         c.s << ')' << closebrace;
152                                 } else
153                                         var.print(c);
154                                 if (i->coeff.compare(_ex1)) {
155                                         c.s << pow_sym;
156                                         c.s << openbrace;
157                                         if (i->coeff.info(info_flags::negative)) {
158                                                 c.s << '(';
159                                                 i->coeff.print(c);
160                                                 c.s << ')';
161                                         } else
162                                                 i->coeff.print(c);
163                                         c.s << closebrace;
164                                 }
165                         }
166                 } else
167                         Order(power(var-point,i->coeff)).print(c);
168                 ++i;
169         }
170
171         if (precedence() <= level)
172                 c.s << ')';
173 }
174
175 void pseries::do_print(const print_context & c, unsigned level) const
176 {
177         print_series(c, "", "", "*", "^", level);
178 }
179
180 void pseries::do_print_latex(const print_latex & c, unsigned level) const
181 {
182         print_series(c, "{", "}", " ", "^", level);
183 }
184
185 void pseries::do_print_python(const print_python & c, unsigned level) const
186 {
187         print_series(c, "", "", "*", "**", level);
188 }
189
190 void pseries::do_print_tree(const print_tree & c, unsigned level) const
191 {
192         c.s << std::string(level, ' ') << class_name() << " @" << this
193             << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
194             << std::endl;
195         size_t num = seq.size();
196         for (size_t i=0; i<num; ++i) {
197                 seq[i].rest.print(c, level + c.delta_indent);
198                 seq[i].coeff.print(c, level + c.delta_indent);
199                 c.s << std::string(level + c.delta_indent, ' ') << "-----" << std::endl;
200         }
201         var.print(c, level + c.delta_indent);
202         point.print(c, level + c.delta_indent);
203 }
204
205 void pseries::do_print_python_repr(const print_python_repr & c, unsigned level) const
206 {
207         c.s << class_name() << "(relational(";
208         var.print(c);
209         c.s << ',';
210         point.print(c);
211         c.s << "),[";
212         size_t num = seq.size();
213         for (size_t i=0; i<num; ++i) {
214                 if (i)
215                         c.s << ',';
216                 c.s << '(';
217                 seq[i].rest.print(c);
218                 c.s << ',';
219                 seq[i].coeff.print(c);
220                 c.s << ')';
221         }
222         c.s << "])";
223 }
224
225 int pseries::compare_same_type(const basic & other) const
226 {
227         GINAC_ASSERT(is_a<pseries>(other));
228         const pseries &o = static_cast<const pseries &>(other);
229         
230         // first compare the lengths of the series...
231         if (seq.size()>o.seq.size())
232                 return 1;
233         if (seq.size()<o.seq.size())
234                 return -1;
235         
236         // ...then the expansion point...
237         int cmpval = var.compare(o.var);
238         if (cmpval)
239                 return cmpval;
240         cmpval = point.compare(o.point);
241         if (cmpval)
242                 return cmpval;
243         
244         // ...and if that failed the individual elements
245         epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
246         while (it!=seq.end() && o_it!=o.seq.end()) {
247                 cmpval = it->compare(*o_it);
248                 if (cmpval)
249                         return cmpval;
250                 ++it;
251                 ++o_it;
252         }
253
254         // so they are equal.
255         return 0;
256 }
257
258 /** Return the number of operands including a possible order term. */
259 size_t pseries::nops() const
260 {
261         return seq.size();
262 }
263
264 /** Return the ith term in the series when represented as a sum. */
265 ex pseries::op(size_t i) const
266 {
267         if (i >= seq.size())
268                 throw (std::out_of_range("op() out of range"));
269
270         if (is_order_function(seq[i].rest))
271                 return Order(power(var-point, seq[i].coeff));
272         return seq[i].rest * power(var - point, seq[i].coeff);
273 }
274
275 /** Return degree of highest power of the series.  This is usually the exponent
276  *  of the Order term.  If s is not the expansion variable of the series, the
277  *  series is examined termwise. */
278 int pseries::degree(const ex &s) const
279 {
280         if (var.is_equal(s)) {
281                 // Return last exponent
282                 if (seq.size())
283                         return ex_to<numeric>((seq.end()-1)->coeff).to_int();
284                 else
285                         return 0;
286         } else {
287                 epvector::const_iterator it = seq.begin(), itend = seq.end();
288                 if (it == itend)
289                         return 0;
290                 int max_pow = INT_MIN;
291                 while (it != itend) {
292                         int pow = it->rest.degree(s);
293                         if (pow > max_pow)
294                                 max_pow = pow;
295                         ++it;
296                 }
297                 return max_pow;
298         }
299 }
300
301 /** Return degree of lowest power of the series.  This is usually the exponent
302  *  of the leading term.  If s is not the expansion variable of the series, the
303  *  series is examined termwise.  If s is the expansion variable but the
304  *  expansion point is not zero the series is not expanded to find the degree.
305  *  I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
306 int pseries::ldegree(const ex &s) const
307 {
308         if (var.is_equal(s)) {
309                 // Return first exponent
310                 if (seq.size())
311                         return ex_to<numeric>((seq.begin())->coeff).to_int();
312                 else
313                         return 0;
314         } else {
315                 epvector::const_iterator it = seq.begin(), itend = seq.end();
316                 if (it == itend)
317                         return 0;
318                 int min_pow = INT_MAX;
319                 while (it != itend) {
320                         int pow = it->rest.ldegree(s);
321                         if (pow < min_pow)
322                                 min_pow = pow;
323                         ++it;
324                 }
325                 return min_pow;
326         }
327 }
328
329 /** Return coefficient of degree n in power series if s is the expansion
330  *  variable.  If the expansion point is nonzero, by definition the n=1
331  *  coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
332  *  the expansion took place in the s in the first place).
333  *  If s is not the expansion variable, an attempt is made to convert the
334  *  series to a polynomial and return the corresponding coefficient from
335  *  there. */
336 ex pseries::coeff(const ex &s, int n) const
337 {
338         if (var.is_equal(s)) {
339                 if (seq.empty())
340                         return _ex0;
341                 
342                 // Binary search in sequence for given power
343                 numeric looking_for = numeric(n);
344                 int lo = 0, hi = seq.size() - 1;
345                 while (lo <= hi) {
346                         int mid = (lo + hi) / 2;
347                         GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
348                         int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
349                         switch (cmp) {
350                                 case -1:
351                                         lo = mid + 1;
352                                         break;
353                                 case 0:
354                                         return seq[mid].rest;
355                                 case 1:
356                                         hi = mid - 1;
357                                         break;
358                                 default:
359                                         throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
360                         }
361                 }
362                 return _ex0;
363         } else
364                 return convert_to_poly().coeff(s, n);
365 }
366
367 /** Does nothing. */
368 ex pseries::collect(const ex &s, bool distributed) const
369 {
370         return *this;
371 }
372
373 /** Perform coefficient-wise automatic term rewriting rules in this class. */
374 ex pseries::eval(int level) const
375 {
376         if (level == 1)
377                 return this->hold();
378         
379         if (level == -max_recursion_level)
380                 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
381         
382         // Construct a new series with evaluated coefficients
383         epvector new_seq;
384         new_seq.reserve(seq.size());
385         epvector::const_iterator it = seq.begin(), itend = seq.end();
386         while (it != itend) {
387                 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
388                 ++it;
389         }
390         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
391 }
392
393 /** Evaluate coefficients numerically. */
394 ex pseries::evalf(int level) const
395 {
396         if (level == 1)
397                 return *this;
398         
399         if (level == -max_recursion_level)
400                 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
401         
402         // Construct a new series with evaluated coefficients
403         epvector new_seq;
404         new_seq.reserve(seq.size());
405         epvector::const_iterator it = seq.begin(), itend = seq.end();
406         while (it != itend) {
407                 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
408                 ++it;
409         }
410         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
411 }
412
413 ex pseries::conjugate() const
414 {
415         if(!var.info(info_flags::real))
416                 return conjugate_function(*this).hold();
417
418         epvector * newseq = conjugateepvector(seq);
419         ex newpoint = point.conjugate();
420
421         if (!newseq     && are_ex_trivially_equal(point, newpoint)) {
422                 return *this;
423         }
424
425         ex result = (new pseries(var==newpoint, newseq ? *newseq : seq))->setflag(status_flags::dynallocated);
426         if (newseq) {
427                 delete newseq;
428         }
429         return result;
430 }
431
432 ex pseries::real_part() const
433 {
434         if(!var.info(info_flags::real))
435                 return real_part_function(*this).hold();
436         ex newpoint = point.real_part();
437         if(newpoint != point)
438                 return real_part_function(*this).hold();
439
440         epvector v;
441         v.reserve(seq.size());
442         for(epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i)
443                 v.push_back(expair((i->rest).real_part(), i->coeff));
444         return (new pseries(var==point, v))->setflag(status_flags::dynallocated);
445 }
446
447 ex pseries::imag_part() const
448 {
449         if(!var.info(info_flags::real))
450                 return imag_part_function(*this).hold();
451         ex newpoint = point.real_part();
452         if(newpoint != point)
453                 return imag_part_function(*this).hold();
454
455         epvector v;
456         v.reserve(seq.size());
457         for(epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i)
458                 v.push_back(expair((i->rest).imag_part(), i->coeff));
459         return (new pseries(var==point, v))->setflag(status_flags::dynallocated);
460 }
461
462 ex pseries::eval_integ() const
463 {
464         epvector *newseq = NULL;
465         for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
466                 if (newseq) {
467                         newseq->push_back(expair(i->rest.eval_integ(), i->coeff));
468                         continue;
469                 }
470                 ex newterm = i->rest.eval_integ();
471                 if (!are_ex_trivially_equal(newterm, i->rest)) {
472                         newseq = new epvector;
473                         newseq->reserve(seq.size());
474                         for (epvector::const_iterator j=seq.begin(); j!=i; ++j)
475                                 newseq->push_back(*j);
476                         newseq->push_back(expair(newterm, i->coeff));
477                 }
478         }
479
480         ex newpoint = point.eval_integ();
481         if (newseq || !are_ex_trivially_equal(newpoint, point))
482                 return (new pseries(var==newpoint, *newseq))
483                        ->setflag(status_flags::dynallocated);
484         return *this;
485 }
486
487 ex pseries::subs(const exmap & m, unsigned options) const
488 {
489         // If expansion variable is being substituted, convert the series to a
490         // polynomial and do the substitution there because the result might
491         // no longer be a power series
492         if (m.find(var) != m.end())
493                 return convert_to_poly(true).subs(m, options);
494         
495         // Otherwise construct a new series with substituted coefficients and
496         // expansion point
497         epvector newseq;
498         newseq.reserve(seq.size());
499         epvector::const_iterator it = seq.begin(), itend = seq.end();
500         while (it != itend) {
501                 newseq.push_back(expair(it->rest.subs(m, options), it->coeff));
502                 ++it;
503         }
504         return (new pseries(relational(var,point.subs(m, options)), newseq))->setflag(status_flags::dynallocated);
505 }
506
507 /** Implementation of ex::expand() for a power series.  It expands all the
508  *  terms individually and returns the resulting series as a new pseries. */
509 ex pseries::expand(unsigned options) const
510 {
511         epvector newseq;
512         epvector::const_iterator i = seq.begin(), end = seq.end();
513         while (i != end) {
514                 ex restexp = i->rest.expand();
515                 if (!restexp.is_zero())
516                         newseq.push_back(expair(restexp, i->coeff));
517                 ++i;
518         }
519         return (new pseries(relational(var,point), newseq))
520                 ->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
521 }
522
523 /** Implementation of ex::diff() for a power series.
524  *  @see ex::diff */
525 ex pseries::derivative(const symbol & s) const
526 {
527         epvector new_seq;
528         epvector::const_iterator it = seq.begin(), itend = seq.end();
529
530         if (s == var) {
531                 
532                 // FIXME: coeff might depend on var
533                 while (it != itend) {
534                         if (is_order_function(it->rest)) {
535                                 new_seq.push_back(expair(it->rest, it->coeff - 1));
536                         } else {
537                                 ex c = it->rest * it->coeff;
538                                 if (!c.is_zero())
539                                         new_seq.push_back(expair(c, it->coeff - 1));
540                         }
541                         ++it;
542                 }
543
544         } else {
545
546                 while (it != itend) {
547                         if (is_order_function(it->rest)) {
548                                 new_seq.push_back(*it);
549                         } else {
550                                 ex c = it->rest.diff(s);
551                                 if (!c.is_zero())
552                                         new_seq.push_back(expair(c, it->coeff));
553                         }
554                         ++it;
555                 }
556         }
557
558         return pseries(relational(var,point), new_seq);
559 }
560
561 ex pseries::convert_to_poly(bool no_order) const
562 {
563         ex e;
564         epvector::const_iterator it = seq.begin(), itend = seq.end();
565         
566         while (it != itend) {
567                 if (is_order_function(it->rest)) {
568                         if (!no_order)
569                                 e += Order(power(var - point, it->coeff));
570                 } else
571                         e += it->rest * power(var - point, it->coeff);
572                 ++it;
573         }
574         return e;
575 }
576
577 bool pseries::is_terminating() const
578 {
579         return seq.empty() || !is_order_function((seq.end()-1)->rest);
580 }
581
582 ex pseries::coeffop(size_t i) const
583 {
584         if (i >=nops())
585                 throw (std::out_of_range("coeffop() out of range"));
586         return seq[i].rest;
587 }
588
589 ex pseries::exponop(size_t i) const
590 {
591         if (i >= nops())
592                 throw (std::out_of_range("exponop() out of range"));
593         return seq[i].coeff;
594 }
595
596
597 /*
598  *  Implementations of series expansion
599  */
600
601 /** Default implementation of ex::series(). This performs Taylor expansion.
602  *  @see ex::series */
603 ex basic::series(const relational & r, int order, unsigned options) const
604 {
605         epvector seq;
606         const symbol &s = ex_to<symbol>(r.lhs());
607
608         // default for order-values that make no sense for Taylor expansion
609         if ((order <= 0) && this->has(s)) {
610                 seq.push_back(expair(Order(_ex1), order));
611                 return pseries(r, seq);
612         }
613
614         // do Taylor expansion
615         numeric fac = 1;
616         ex deriv = *this;
617         ex coeff = deriv.subs(r, subs_options::no_pattern);
618
619         if (!coeff.is_zero()) {
620                 seq.push_back(expair(coeff, _ex0));
621         }
622
623         int n;
624         for (n=1; n<order; ++n) {
625                 fac = fac.mul(n);
626                 // We need to test for zero in order to see if the series terminates.
627                 // The problem is that there is no such thing as a perfect test for
628                 // zero.  Expanding the term occasionally helps a little...
629                 deriv = deriv.diff(s).expand();
630                 if (deriv.is_zero())  // Series terminates
631                         return pseries(r, seq);
632
633                 coeff = deriv.subs(r, subs_options::no_pattern);
634                 if (!coeff.is_zero())
635                         seq.push_back(expair(fac.inverse() * coeff, n));
636         }
637         
638         // Higher-order terms, if present
639         deriv = deriv.diff(s);
640         if (!deriv.expand().is_zero())
641                 seq.push_back(expair(Order(_ex1), n));
642         return pseries(r, seq);
643 }
644
645
646 /** Implementation of ex::series() for symbols.
647  *  @see ex::series */
648 ex symbol::series(const relational & r, int order, unsigned options) const
649 {
650         epvector seq;
651         const ex point = r.rhs();
652         GINAC_ASSERT(is_a<symbol>(r.lhs()));
653
654         if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
655                 if (order > 0 && !point.is_zero())
656                         seq.push_back(expair(point, _ex0));
657                 if (order > 1)
658                         seq.push_back(expair(_ex1, _ex1));
659                 else
660                         seq.push_back(expair(Order(_ex1), numeric(order)));
661         } else
662                 seq.push_back(expair(*this, _ex0));
663         return pseries(r, seq);
664 }
665
666
667 /** Add one series object to another, producing a pseries object that
668  *  represents the sum.
669  *
670  *  @param other  pseries object to add with
671  *  @return the sum as a pseries */
672 ex pseries::add_series(const pseries &other) const
673 {
674         // Adding two series with different variables or expansion points
675         // results in an empty (constant) series 
676         if (!is_compatible_to(other)) {
677                 epvector nul;
678                 nul.push_back(expair(Order(_ex1), _ex0));
679                 return pseries(relational(var,point), nul);
680         }
681         
682         // Series addition
683         epvector new_seq;
684         epvector::const_iterator a = seq.begin();
685         epvector::const_iterator b = other.seq.begin();
686         epvector::const_iterator a_end = seq.end();
687         epvector::const_iterator b_end = other.seq.end();
688         int pow_a = INT_MAX, pow_b = INT_MAX;
689         for (;;) {
690                 // If a is empty, fill up with elements from b and stop
691                 if (a == a_end) {
692                         while (b != b_end) {
693                                 new_seq.push_back(*b);
694                                 ++b;
695                         }
696                         break;
697                 } else
698                         pow_a = ex_to<numeric>((*a).coeff).to_int();
699                 
700                 // If b is empty, fill up with elements from a and stop
701                 if (b == b_end) {
702                         while (a != a_end) {
703                                 new_seq.push_back(*a);
704                                 ++a;
705                         }
706                         break;
707                 } else
708                         pow_b = ex_to<numeric>((*b).coeff).to_int();
709                 
710                 // a and b are non-empty, compare powers
711                 if (pow_a < pow_b) {
712                         // a has lesser power, get coefficient from a
713                         new_seq.push_back(*a);
714                         if (is_order_function((*a).rest))
715                                 break;
716                         ++a;
717                 } else if (pow_b < pow_a) {
718                         // b has lesser power, get coefficient from b
719                         new_seq.push_back(*b);
720                         if (is_order_function((*b).rest))
721                                 break;
722                         ++b;
723                 } else {
724                         // Add coefficient of a and b
725                         if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
726                                 new_seq.push_back(expair(Order(_ex1), (*a).coeff));
727                                 break;  // Order term ends the sequence
728                         } else {
729                                 ex sum = (*a).rest + (*b).rest;
730                                 if (!(sum.is_zero()))
731                                         new_seq.push_back(expair(sum, numeric(pow_a)));
732                                 ++a;
733                                 ++b;
734                         }
735                 }
736         }
737         return pseries(relational(var,point), new_seq);
738 }
739
740
741 /** Implementation of ex::series() for sums. This performs series addition when
742  *  adding pseries objects.
743  *  @see ex::series */
744 ex add::series(const relational & r, int order, unsigned options) const
745 {
746         ex acc; // Series accumulator
747         
748         // Get first term from overall_coeff
749         acc = overall_coeff.series(r, order, options);
750         
751         // Add remaining terms
752         epvector::const_iterator it = seq.begin();
753         epvector::const_iterator itend = seq.end();
754         for (; it!=itend; ++it) {
755                 ex op;
756                 if (is_exactly_a<pseries>(it->rest))
757                         op = it->rest;
758                 else
759                         op = it->rest.series(r, order, options);
760                 if (!it->coeff.is_equal(_ex1))
761                         op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it->coeff));
762                 
763                 // Series addition
764                 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
765         }
766         return acc;
767 }
768
769
770 /** Multiply a pseries object with a numeric constant, producing a pseries
771  *  object that represents the product.
772  *
773  *  @param other  constant to multiply with
774  *  @return the product as a pseries */
775 ex pseries::mul_const(const numeric &other) const
776 {
777         epvector new_seq;
778         new_seq.reserve(seq.size());
779         
780         epvector::const_iterator it = seq.begin(), itend = seq.end();
781         while (it != itend) {
782                 if (!is_order_function(it->rest))
783                         new_seq.push_back(expair(it->rest * other, it->coeff));
784                 else
785                         new_seq.push_back(*it);
786                 ++it;
787         }
788         return pseries(relational(var,point), new_seq);
789 }
790
791
792 /** Multiply one pseries object to another, producing a pseries object that
793  *  represents the product.
794  *
795  *  @param other  pseries object to multiply with
796  *  @return the product as a pseries */
797 ex pseries::mul_series(const pseries &other) const
798 {
799         // Multiplying two series with different variables or expansion points
800         // results in an empty (constant) series 
801         if (!is_compatible_to(other)) {
802                 epvector nul;
803                 nul.push_back(expair(Order(_ex1), _ex0));
804                 return pseries(relational(var,point), nul);
805         }
806
807         if (seq.empty() || other.seq.empty()) {
808                 return (new pseries(var==point, epvector()))
809                        ->setflag(status_flags::dynallocated);
810         }
811         
812         // Series multiplication
813         epvector new_seq;
814         int a_max = degree(var);
815         int b_max = other.degree(var);
816         int a_min = ldegree(var);
817         int b_min = other.ldegree(var);
818         int cdeg_min = a_min + b_min;
819         int cdeg_max = a_max + b_max;
820         
821         int higher_order_a = INT_MAX;
822         int higher_order_b = INT_MAX;
823         if (is_order_function(coeff(var, a_max)))
824                 higher_order_a = a_max + b_min;
825         if (is_order_function(other.coeff(var, b_max)))
826                 higher_order_b = b_max + a_min;
827         int higher_order_c = std::min(higher_order_a, higher_order_b);
828         if (cdeg_max >= higher_order_c)
829                 cdeg_max = higher_order_c - 1;
830         
831         for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
832                 ex co = _ex0;
833                 // c(i)=a(0)b(i)+...+a(i)b(0)
834                 for (int i=a_min; cdeg-i>=b_min; ++i) {
835                         ex a_coeff = coeff(var, i);
836                         ex b_coeff = other.coeff(var, cdeg-i);
837                         if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
838                                 co += a_coeff * b_coeff;
839                 }
840                 if (!co.is_zero())
841                         new_seq.push_back(expair(co, numeric(cdeg)));
842         }
843         if (higher_order_c < INT_MAX)
844                 new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c)));
845         return pseries(relational(var, point), new_seq);
846 }
847
848
849 /** Implementation of ex::series() for product. This performs series
850  *  multiplication when multiplying series.
851  *  @see ex::series */
852 ex mul::series(const relational & r, int order, unsigned options) const
853 {
854         pseries acc; // Series accumulator
855
856         GINAC_ASSERT(is_a<symbol>(r.lhs()));
857         const ex& sym = r.lhs();
858                 
859         // holds ldegrees of the series of individual factors
860         std::vector<int> ldegrees;
861
862         // find minimal degrees
863         const epvector::const_iterator itbeg = seq.begin();
864         const epvector::const_iterator itend = seq.end();
865         for (epvector::const_iterator it=itbeg; it!=itend; ++it) {
866
867                 ex expon = it->coeff;
868                 int factor = 1;
869                 ex buf;
870                 if (expon.info(info_flags::integer)) {
871                         buf = it->rest;
872                         factor = ex_to<numeric>(expon).to_int();
873                 } else {
874                         buf = recombine_pair_to_ex(*it);
875                 }
876
877                 int real_ldegree = 0;
878                 try {
879                         real_ldegree = buf.expand().ldegree(sym-r.rhs());
880                 } catch (std::runtime_error) {}
881
882                 if (real_ldegree == 0) {
883                         int orderloop = 0;
884                         do {
885                                 orderloop++;
886                                 real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
887                         } while (real_ldegree == orderloop);
888                 }
889
890                 ldegrees.push_back(factor * real_ldegree);
891         }
892
893         int degsum = std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
894
895         if (degsum >= order) {
896                 epvector epv;
897                 epv.push_back(expair(Order(_ex1), order));
898                 return (new pseries(r, epv))->setflag(status_flags::dynallocated);
899         }
900
901         // Multiply with remaining terms
902         std::vector<int>::const_iterator itd = ldegrees.begin();
903         for (epvector::const_iterator it=itbeg; it!=itend; ++it, ++itd) {
904
905                 // do series expansion with adjusted order
906                 ex op = recombine_pair_to_ex(*it).series(r, order-degsum+(*itd), options);
907
908                 // Series multiplication
909                 if (it == itbeg)
910                         acc = ex_to<pseries>(op);
911                 else
912                         acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
913         }
914
915         return acc.mul_const(ex_to<numeric>(overall_coeff));
916 }
917
918
919 /** Compute the p-th power of a series.
920  *
921  *  @param p  power to compute
922  *  @param deg  truncation order of series calculation */
923 ex pseries::power_const(const numeric &p, int deg) const
924 {
925         // method:
926         // (due to Leonhard Euler)
927         // let A(x) be this series and for the time being let it start with a
928         // constant (later we'll generalize):
929         //     A(x) = a_0 + a_1*x + a_2*x^2 + ...
930         // We want to compute
931         //     C(x) = A(x)^p
932         //     C(x) = c_0 + c_1*x + c_2*x^2 + ...
933         // Taking the derivative on both sides and multiplying with A(x) one
934         // immediately arrives at
935         //     C'(x)*A(x) = p*C(x)*A'(x)
936         // Multiplying this out and comparing coefficients we get the recurrence
937         // formula
938         //     c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
939         //                    ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
940         // which can easily be solved given the starting value c_0 = (a_0)^p.
941         // For the more general case where the leading coefficient of A(x) is not
942         // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
943         // repeat the above derivation.  The leading power of C2(x) = A2(x)^2 is
944         // then of course x^(p*m) but the recurrence formula still holds.
945         
946         if (seq.empty()) {
947                 // as a special case, handle the empty (zero) series honoring the
948                 // usual power laws such as implemented in power::eval()
949                 if (p.real().is_zero())
950                         throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
951                 else if (p.real().is_negative())
952                         throw pole_error("pseries::power_const(): division by zero",1);
953                 else
954                         return *this;
955         }
956         
957         const int ldeg = ldegree(var);
958         if (!(p*ldeg).is_integer())
959                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
960
961         // adjust number of coefficients
962         int numcoeff = deg - (p*ldeg).to_int();
963         if (numcoeff <= 0) {
964                 epvector epv;
965                 epv.reserve(1);
966                 epv.push_back(expair(Order(_ex1), deg));
967                 return (new pseries(relational(var,point), epv))
968                        ->setflag(status_flags::dynallocated);
969         }
970         
971         // O(x^n)^(-m) is undefined
972         if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
973                 throw pole_error("pseries::power_const(): division by zero",1);
974         
975         // Compute coefficients of the powered series
976         exvector co;
977         co.reserve(numcoeff);
978         co.push_back(power(coeff(var, ldeg), p));
979         for (int i=1; i<numcoeff; ++i) {
980                 ex sum = _ex0;
981                 for (int j=1; j<=i; ++j) {
982                         ex c = coeff(var, j + ldeg);
983                         if (is_order_function(c)) {
984                                 co.push_back(Order(_ex1));
985                                 break;
986                         } else
987                                 sum += (p * j - (i - j)) * co[i - j] * c;
988                 }
989                 co.push_back(sum / coeff(var, ldeg) / i);
990         }
991         
992         // Construct new series (of non-zero coefficients)
993         epvector new_seq;
994         bool higher_order = false;
995         for (int i=0; i<numcoeff; ++i) {
996                 if (!co[i].is_zero())
997                         new_seq.push_back(expair(co[i], p * ldeg + i));
998                 if (is_order_function(co[i])) {
999                         higher_order = true;
1000                         break;
1001                 }
1002         }
1003         if (!higher_order)
1004                 new_seq.push_back(expair(Order(_ex1), p * ldeg + numcoeff));
1005
1006         return pseries(relational(var,point), new_seq);
1007 }
1008
1009
1010 /** Return a new pseries object with the powers shifted by deg. */
1011 pseries pseries::shift_exponents(int deg) const
1012 {
1013         epvector newseq = seq;
1014         epvector::iterator i = newseq.begin(), end  = newseq.end();
1015         while (i != end) {
1016                 i->coeff += deg;
1017                 ++i;
1018         }
1019         return pseries(relational(var, point), newseq);
1020 }
1021
1022
1023 /** Implementation of ex::series() for powers. This performs Laurent expansion
1024  *  of reciprocals of series at singularities.
1025  *  @see ex::series */
1026 ex power::series(const relational & r, int order, unsigned options) const
1027 {
1028         // If basis is already a series, just power it
1029         if (is_exactly_a<pseries>(basis))
1030                 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
1031
1032         // Basis is not a series, may there be a singularity?
1033         bool must_expand_basis = false;
1034         try {
1035                 basis.subs(r, subs_options::no_pattern);
1036         } catch (pole_error) {
1037                 must_expand_basis = true;
1038         }
1039
1040         // Is the expression of type something^(-int)?
1041         if (!must_expand_basis && !exponent.info(info_flags::negint)
1042          && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1043                 return basic::series(r, order, options);
1044
1045         // Is the expression of type 0^something?
1046         if (!must_expand_basis && !basis.subs(r, subs_options::no_pattern).is_zero()
1047          && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1048                 return basic::series(r, order, options);
1049
1050         // Singularity encountered, is the basis equal to (var - point)?
1051         if (basis.is_equal(r.lhs() - r.rhs())) {
1052                 epvector new_seq;
1053                 if (ex_to<numeric>(exponent).to_int() < order)
1054                         new_seq.push_back(expair(_ex1, exponent));
1055                 else
1056                         new_seq.push_back(expair(Order(_ex1), exponent));
1057                 return pseries(r, new_seq);
1058         }
1059
1060         // No, expand basis into series
1061
1062         numeric numexp;
1063         if (is_a<numeric>(exponent)) {
1064                 numexp = ex_to<numeric>(exponent);
1065         } else {
1066                 numexp = 0;
1067         }
1068         const ex& sym = r.lhs();
1069         // find existing minimal degree
1070         int real_ldegree = basis.expand().ldegree(sym-r.rhs());
1071         if (real_ldegree == 0) {
1072                 int orderloop = 0;
1073                 do {
1074                         orderloop++;
1075                         real_ldegree = basis.series(r, orderloop, options).ldegree(sym);
1076                 } while (real_ldegree == orderloop);
1077         }
1078
1079         if (!(real_ldegree*numexp).is_integer())
1080                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
1081         ex e = basis.series(r, (order + real_ldegree*(1-numexp)).to_int(), options);
1082         
1083         ex result;
1084         try {
1085                 result = ex_to<pseries>(e).power_const(numexp, order);
1086         } catch (pole_error) {
1087                 epvector ser;
1088                 ser.push_back(expair(Order(_ex1), order));
1089                 result = pseries(r, ser);
1090         }
1091
1092         return result;
1093 }
1094
1095
1096 /** Re-expansion of a pseries object. */
1097 ex pseries::series(const relational & r, int order, unsigned options) const
1098 {
1099         const ex p = r.rhs();
1100         GINAC_ASSERT(is_a<symbol>(r.lhs()));
1101         const symbol &s = ex_to<symbol>(r.lhs());
1102         
1103         if (var.is_equal(s) && point.is_equal(p)) {
1104                 if (order > degree(s))
1105                         return *this;
1106                 else {
1107                         epvector new_seq;
1108                         epvector::const_iterator it = seq.begin(), itend = seq.end();
1109                         while (it != itend) {
1110                                 int o = ex_to<numeric>(it->coeff).to_int();
1111                                 if (o >= order) {
1112                                         new_seq.push_back(expair(Order(_ex1), o));
1113                                         break;
1114                                 }
1115                                 new_seq.push_back(*it);
1116                                 ++it;
1117                         }
1118                         return pseries(r, new_seq);
1119                 }
1120         } else
1121                 return convert_to_poly().series(r, order, options);
1122 }
1123
1124 ex integral::series(const relational & r, int order, unsigned options) const
1125 {
1126         if (x.subs(r) != x)
1127                 throw std::logic_error("Cannot series expand wrt dummy variable");
1128         
1129         // Expanding integrant with r substituted taken in boundaries.
1130         ex fseries = f.series(r, order, options);
1131         epvector fexpansion;
1132         fexpansion.reserve(fseries.nops());
1133         for (size_t i=0; i<fseries.nops(); ++i) {
1134                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1135                 currcoeff = (currcoeff == Order(_ex1))
1136                         ? currcoeff
1137                         : integral(x, a.subs(r), b.subs(r), currcoeff);
1138                 if (currcoeff != 0)
1139                         fexpansion.push_back(
1140                                 expair(currcoeff, ex_to<pseries>(fseries).exponop(i)));
1141         }
1142
1143         // Expanding lower boundary
1144         ex result = (new pseries(r, fexpansion))->setflag(status_flags::dynallocated);
1145         ex aseries = (a-a.subs(r)).series(r, order, options);
1146         fseries = f.series(x == (a.subs(r)), order, options);
1147         for (size_t i=0; i<fseries.nops(); ++i) {
1148                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1149                 if (is_order_function(currcoeff))
1150                         break;
1151                 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1152                 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1153                 currcoeff = currcoeff.series(r, orderforf);
1154                 ex term = ex_to<pseries>(aseries).power_const(ex_to<numeric>(currexpon+1),order);
1155                 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(-1/(currexpon+1)));
1156                 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1157                 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1158         }
1159
1160         // Expanding upper boundary
1161         ex bseries = (b-b.subs(r)).series(r, order, options);
1162         fseries = f.series(x == (b.subs(r)), order, options);
1163         for (size_t i=0; i<fseries.nops(); ++i) {
1164                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1165                 if (is_order_function(currcoeff))
1166                         break;
1167                 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1168                 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1169                 currcoeff = currcoeff.series(r, orderforf);
1170                 ex term = ex_to<pseries>(bseries).power_const(ex_to<numeric>(currexpon+1),order);
1171                 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(1/(currexpon+1)));
1172                 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1173                 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1174         }
1175
1176         return result;
1177 }
1178
1179
1180 /** Compute the truncated series expansion of an expression.
1181  *  This function returns an expression containing an object of class pseries 
1182  *  to represent the series. If the series does not terminate within the given
1183  *  truncation order, the last term of the series will be an order term.
1184  *
1185  *  @param r  expansion relation, lhs holds variable and rhs holds point
1186  *  @param order  truncation order of series calculations
1187  *  @param options  of class series_options
1188  *  @return an expression holding a pseries object */
1189 ex ex::series(const ex & r, int order, unsigned options) const
1190 {
1191         ex e;
1192         relational rel_;
1193         
1194         if (is_a<relational>(r))
1195                 rel_ = ex_to<relational>(r);
1196         else if (is_a<symbol>(r))
1197                 rel_ = relational(r,_ex0);
1198         else
1199                 throw (std::logic_error("ex::series(): expansion point has unknown type"));
1200         
1201         try {
1202                 e = bp->series(rel_, order, options);
1203         } catch (std::exception &x) {
1204                 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));
1205         }
1206         return e;
1207 }
1208
1209 } // namespace GiNaC