subs() and normal() use maps instead of lists, resulting in a huge performance
[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) 1999-2003 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., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
22  */
23
24 #include <iostream>
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 "print.h"
37 #include "archive.h"
38 #include "utils.h"
39
40 namespace GiNaC {
41
42 GINAC_IMPLEMENT_REGISTERED_CLASS(pseries, basic)
43
44
45 /*
46  *  Default constructor
47  */
48
49 pseries::pseries() : inherited(TINFO_pseries) { }
50
51
52 /*
53  *  Other ctors
54  */
55
56 /** Construct pseries from a vector of coefficients and powers.
57  *  expair.rest holds the coefficient, expair.coeff holds the power.
58  *  The powers must be integers (positive or negative) and in ascending order;
59  *  the last coefficient can be Order(_ex1) to represent a truncated,
60  *  non-terminating series.
61  *
62  *  @param rel_  expansion variable and point (must hold a relational)
63  *  @param ops_  vector of {coefficient, power} pairs (coefficient must not be zero)
64  *  @return newly constructed pseries */
65 pseries::pseries(const ex &rel_, const epvector &ops_) : basic(TINFO_pseries), seq(ops_)
66 {
67         GINAC_ASSERT(is_a<relational>(rel_));
68         GINAC_ASSERT(is_a<symbol>(rel_.lhs()));
69         point = rel_.rhs();
70         var = rel_.lhs();
71 }
72
73
74 /*
75  *  Archiving
76  */
77
78 pseries::pseries(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
79 {
80         for (unsigned int i=0; true; ++i) {
81                 ex rest;
82                 ex coeff;
83                 if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
84                         seq.push_back(expair(rest, coeff));
85                 else
86                         break;
87         }
88         n.find_ex("var", var, sym_lst);
89         n.find_ex("point", point, sym_lst);
90 }
91
92 void pseries::archive(archive_node &n) const
93 {
94         inherited::archive(n);
95         epvector::const_iterator i = seq.begin(), iend = seq.end();
96         while (i != iend) {
97                 n.add_ex("coeff", i->rest);
98                 n.add_ex("power", i->coeff);
99                 ++i;
100         }
101         n.add_ex("var", var);
102         n.add_ex("point", point);
103 }
104
105 DEFAULT_UNARCHIVE(pseries)
106
107 //////////
108 // functions overriding virtual functions from base classes
109 //////////
110
111 void pseries::print(const print_context & c, unsigned level) const
112 {
113         if (is_a<print_tree>(c)) {
114
115                 c.s << std::string(level, ' ') << class_name()
116                     << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
117                     << std::endl;
118                 unsigned delta_indent = static_cast<const print_tree &>(c).delta_indent;
119                 size_t num = seq.size();
120                 for (size_t i=0; i<num; ++i) {
121                         seq[i].rest.print(c, level + delta_indent);
122                         seq[i].coeff.print(c, level + delta_indent);
123                         c.s << std::string(level + delta_indent, ' ') << "-----" << std::endl;
124                 }
125                 var.print(c, level + delta_indent);
126                 point.print(c, level + delta_indent);
127
128         } else if (is_a<print_python_repr>(c)) {
129                 c.s << class_name() << "(relational(";
130                 var.print(c);
131                 c.s << ',';
132                 point.print(c);
133                 c.s << "),[";
134                 size_t num = seq.size();
135                 for (size_t i=0; i<num; ++i) {
136                         if (i)
137                                 c.s << ',';
138                         c.s << '(';
139                         seq[i].rest.print(c);
140                         c.s << ',';
141                         seq[i].coeff.print(c);
142                         c.s << ')';
143                 }
144                 c.s << "])";
145         } else {
146
147                 if (precedence() <= level)
148                         c.s << "(";
149                 
150                 std::string par_open = is_a<print_latex>(c) ? "{(" : "(";
151                 std::string par_close = is_a<print_latex>(c) ? ")}" : ")";
152                 
153                 // objects of type pseries must not have any zero entries, so the
154                 // trivial (zero) pseries needs a special treatment here:
155                 if (seq.empty())
156                         c.s << '0';
157                 epvector::const_iterator i = seq.begin(), end = seq.end();
158                 while (i != end) {
159                         // print a sign, if needed
160                         if (i != seq.begin())
161                                 c.s << '+';
162                         if (!is_order_function(i->rest)) {
163                                 // print 'rest', i.e. the expansion coefficient
164                                 if (i->rest.info(info_flags::numeric) &&
165                                         i->rest.info(info_flags::positive)) {
166                                         i->rest.print(c);
167                                 } else {
168                                         c.s << par_open;
169                                         i->rest.print(c);
170                                         c.s << par_close;
171                                 }
172                                 // print 'coeff', something like (x-1)^42
173                                 if (!i->coeff.is_zero()) {
174                                         if (is_a<print_latex>(c))
175                                                 c.s << ' ';
176                                         else
177                                                 c.s << '*';
178                                         if (!point.is_zero()) {
179                                                 c.s << par_open;
180                                                 (var-point).print(c);
181                                                 c.s << par_close;
182                                         } else
183                                                 var.print(c);
184                                         if (i->coeff.compare(_ex1)) {
185                                                 if (is_a<print_python>(c))
186                                                         c.s << "**";
187                                                 else
188                                                         c.s << '^';
189                                                 if (i->coeff.info(info_flags::negative)) {
190                                                         c.s << par_open;
191                                                         i->coeff.print(c);
192                                                         c.s << par_close;
193                                                 } else {
194                                                         if (is_a<print_latex>(c)) {
195                                                                 c.s << '{';
196                                                                 i->coeff.print(c);
197                                                                 c.s << '}';
198                                                         } else
199                                                                 i->coeff.print(c);
200                                                 }
201                                         }
202                                 }
203                         } else
204                                 Order(power(var-point,i->coeff)).print(c);
205                         ++i;
206                 }
207
208                 if (precedence() <= level)
209                         c.s << ")";
210         }
211 }
212
213 int pseries::compare_same_type(const basic & other) const
214 {
215         GINAC_ASSERT(is_a<pseries>(other));
216         const pseries &o = static_cast<const pseries &>(other);
217         
218         // first compare the lengths of the series...
219         if (seq.size()>o.seq.size())
220                 return 1;
221         if (seq.size()<o.seq.size())
222                 return -1;
223         
224         // ...then the expansion point...
225         int cmpval = var.compare(o.var);
226         if (cmpval)
227                 return cmpval;
228         cmpval = point.compare(o.point);
229         if (cmpval)
230                 return cmpval;
231         
232         // ...and if that failed the individual elements
233         epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
234         while (it!=seq.end() && o_it!=o.seq.end()) {
235                 cmpval = it->compare(*o_it);
236                 if (cmpval)
237                         return cmpval;
238                 ++it;
239                 ++o_it;
240         }
241
242         // so they are equal.
243         return 0;
244 }
245
246 /** Return the number of operands including a possible order term. */
247 size_t pseries::nops() const
248 {
249         return seq.size();
250 }
251
252 /** Return the ith term in the series when represented as a sum. */
253 ex pseries::op(size_t i) const
254 {
255         if (i >= seq.size())
256                 throw (std::out_of_range("op() out of range"));
257
258         return seq[i].rest * power(var - point, seq[i].coeff);
259 }
260
261 /** Return degree of highest power of the series.  This is usually the exponent
262  *  of the Order term.  If s is not the expansion variable of the series, the
263  *  series is examined termwise. */
264 int pseries::degree(const ex &s) const
265 {
266         if (var.is_equal(s)) {
267                 // Return last exponent
268                 if (seq.size())
269                         return ex_to<numeric>((seq.end()-1)->coeff).to_int();
270                 else
271                         return 0;
272         } else {
273                 epvector::const_iterator it = seq.begin(), itend = seq.end();
274                 if (it == itend)
275                         return 0;
276                 int max_pow = INT_MIN;
277                 while (it != itend) {
278                         int pow = it->rest.degree(s);
279                         if (pow > max_pow)
280                                 max_pow = pow;
281                         ++it;
282                 }
283                 return max_pow;
284         }
285 }
286
287 /** Return degree of lowest power of the series.  This is usually the exponent
288  *  of the leading term.  If s is not the expansion variable of the series, the
289  *  series is examined termwise.  If s is the expansion variable but the
290  *  expansion point is not zero the series is not expanded to find the degree.
291  *  I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
292 int pseries::ldegree(const ex &s) const
293 {
294         if (var.is_equal(s)) {
295                 // Return first exponent
296                 if (seq.size())
297                         return ex_to<numeric>((seq.begin())->coeff).to_int();
298                 else
299                         return 0;
300         } else {
301                 epvector::const_iterator it = seq.begin(), itend = seq.end();
302                 if (it == itend)
303                         return 0;
304                 int min_pow = INT_MAX;
305                 while (it != itend) {
306                         int pow = it->rest.ldegree(s);
307                         if (pow < min_pow)
308                                 min_pow = pow;
309                         ++it;
310                 }
311                 return min_pow;
312         }
313 }
314
315 /** Return coefficient of degree n in power series if s is the expansion
316  *  variable.  If the expansion point is nonzero, by definition the n=1
317  *  coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
318  *  the expansion took place in the s in the first place).
319  *  If s is not the expansion variable, an attempt is made to convert the
320  *  series to a polynomial and return the corresponding coefficient from
321  *  there. */
322 ex pseries::coeff(const ex &s, int n) const
323 {
324         if (var.is_equal(s)) {
325                 if (seq.empty())
326                         return _ex0;
327                 
328                 // Binary search in sequence for given power
329                 numeric looking_for = numeric(n);
330                 int lo = 0, hi = seq.size() - 1;
331                 while (lo <= hi) {
332                         int mid = (lo + hi) / 2;
333                         GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
334                         int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
335                         switch (cmp) {
336                                 case -1:
337                                         lo = mid + 1;
338                                         break;
339                                 case 0:
340                                         return seq[mid].rest;
341                                 case 1:
342                                         hi = mid - 1;
343                                         break;
344                                 default:
345                                         throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
346                         }
347                 }
348                 return _ex0;
349         } else
350                 return convert_to_poly().coeff(s, n);
351 }
352
353 /** Does nothing. */
354 ex pseries::collect(const ex &s, bool distributed) const
355 {
356         return *this;
357 }
358
359 /** Perform coefficient-wise automatic term rewriting rules in this class. */
360 ex pseries::eval(int level) const
361 {
362         if (level == 1)
363                 return this->hold();
364         
365         if (level == -max_recursion_level)
366                 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
367         
368         // Construct a new series with evaluated coefficients
369         epvector new_seq;
370         new_seq.reserve(seq.size());
371         epvector::const_iterator it = seq.begin(), itend = seq.end();
372         while (it != itend) {
373                 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
374                 ++it;
375         }
376         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
377 }
378
379 /** Evaluate coefficients numerically. */
380 ex pseries::evalf(int level) const
381 {
382         if (level == 1)
383                 return *this;
384         
385         if (level == -max_recursion_level)
386                 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
387         
388         // Construct a new series with evaluated coefficients
389         epvector new_seq;
390         new_seq.reserve(seq.size());
391         epvector::const_iterator it = seq.begin(), itend = seq.end();
392         while (it != itend) {
393                 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
394                 ++it;
395         }
396         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
397 }
398
399 ex pseries::subs(const exmap & m, unsigned options) const
400 {
401         // If expansion variable is being substituted, convert the series to a
402         // polynomial and do the substitution there because the result might
403         // no longer be a power series
404         if (m.find(var) != m.end())
405                 return convert_to_poly(true).subs(m, options);
406         
407         // Otherwise construct a new series with substituted coefficients and
408         // expansion point
409         epvector newseq;
410         newseq.reserve(seq.size());
411         epvector::const_iterator it = seq.begin(), itend = seq.end();
412         while (it != itend) {
413                 newseq.push_back(expair(it->rest.subs(m, options), it->coeff));
414                 ++it;
415         }
416         return (new pseries(relational(var,point.subs(m, options)), newseq))->setflag(status_flags::dynallocated);
417 }
418
419 /** Implementation of ex::expand() for a power series.  It expands all the
420  *  terms individually and returns the resulting series as a new pseries. */
421 ex pseries::expand(unsigned options) const
422 {
423         epvector newseq;
424         epvector::const_iterator i = seq.begin(), end = seq.end();
425         while (i != end) {
426                 ex restexp = i->rest.expand();
427                 if (!restexp.is_zero())
428                         newseq.push_back(expair(restexp, i->coeff));
429                 ++i;
430         }
431         return (new pseries(relational(var,point), newseq))
432                 ->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
433 }
434
435 /** Implementation of ex::diff() for a power series.
436  *  @see ex::diff */
437 ex pseries::derivative(const symbol & s) const
438 {
439         epvector new_seq;
440         epvector::const_iterator it = seq.begin(), itend = seq.end();
441
442         if (s == var) {
443                 
444                 // FIXME: coeff might depend on var
445                 while (it != itend) {
446                         if (is_order_function(it->rest)) {
447                                 new_seq.push_back(expair(it->rest, it->coeff - 1));
448                         } else {
449                                 ex c = it->rest * it->coeff;
450                                 if (!c.is_zero())
451                                         new_seq.push_back(expair(c, it->coeff - 1));
452                         }
453                         ++it;
454                 }
455
456         } else {
457
458                 while (it != itend) {
459                         if (is_order_function(it->rest)) {
460                                 new_seq.push_back(*it);
461                         } else {
462                                 ex c = it->rest.diff(s);
463                                 if (!c.is_zero())
464                                         new_seq.push_back(expair(c, it->coeff));
465                         }
466                         ++it;
467                 }
468         }
469
470         return pseries(relational(var,point), new_seq);
471 }
472
473 ex pseries::convert_to_poly(bool no_order) const
474 {
475         ex e;
476         epvector::const_iterator it = seq.begin(), itend = seq.end();
477         
478         while (it != itend) {
479                 if (is_order_function(it->rest)) {
480                         if (!no_order)
481                                 e += Order(power(var - point, it->coeff));
482                 } else
483                         e += it->rest * power(var - point, it->coeff);
484                 ++it;
485         }
486         return e;
487 }
488
489 bool pseries::is_terminating() const
490 {
491         return seq.empty() || !is_order_function((seq.end()-1)->rest);
492 }
493
494
495 /*
496  *  Implementations of series expansion
497  */
498
499 /** Default implementation of ex::series(). This performs Taylor expansion.
500  *  @see ex::series */
501 ex basic::series(const relational & r, int order, unsigned options) const
502 {
503         epvector seq;
504         numeric fac = 1;
505         ex deriv = *this;
506         ex coeff = deriv.subs(r);
507         const symbol &s = ex_to<symbol>(r.lhs());
508         
509         if (!coeff.is_zero())
510                 seq.push_back(expair(coeff, _ex0));
511         
512         int n;
513         for (n=1; n<order; ++n) {
514                 fac = fac.mul(n);
515                 // We need to test for zero in order to see if the series terminates.
516                 // The problem is that there is no such thing as a perfect test for
517                 // zero.  Expanding the term occasionally helps a little...
518                 deriv = deriv.diff(s).expand();
519                 if (deriv.is_zero())  // Series terminates
520                         return pseries(r, seq);
521
522                 coeff = deriv.subs(r);
523                 if (!coeff.is_zero())
524                         seq.push_back(expair(fac.inverse() * coeff, n));
525         }
526         
527         // Higher-order terms, if present
528         deriv = deriv.diff(s);
529         if (!deriv.expand().is_zero())
530                 seq.push_back(expair(Order(_ex1), n));
531         return pseries(r, seq);
532 }
533
534
535 /** Implementation of ex::series() for symbols.
536  *  @see ex::series */
537 ex symbol::series(const relational & r, int order, unsigned options) const
538 {
539         epvector seq;
540         const ex point = r.rhs();
541         GINAC_ASSERT(is_a<symbol>(r.lhs()));
542
543         if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
544                 if (order > 0 && !point.is_zero())
545                         seq.push_back(expair(point, _ex0));
546                 if (order > 1)
547                         seq.push_back(expair(_ex1, _ex1));
548                 else
549                         seq.push_back(expair(Order(_ex1), numeric(order)));
550         } else
551                 seq.push_back(expair(*this, _ex0));
552         return pseries(r, seq);
553 }
554
555
556 /** Add one series object to another, producing a pseries object that
557  *  represents the sum.
558  *
559  *  @param other  pseries object to add with
560  *  @return the sum as a pseries */
561 ex pseries::add_series(const pseries &other) const
562 {
563         // Adding two series with different variables or expansion points
564         // results in an empty (constant) series 
565         if (!is_compatible_to(other)) {
566                 epvector nul;
567                 nul.push_back(expair(Order(_ex1), _ex0));
568                 return pseries(relational(var,point), nul);
569         }
570         
571         // Series addition
572         epvector new_seq;
573         epvector::const_iterator a = seq.begin();
574         epvector::const_iterator b = other.seq.begin();
575         epvector::const_iterator a_end = seq.end();
576         epvector::const_iterator b_end = other.seq.end();
577         int pow_a = INT_MAX, pow_b = INT_MAX;
578         for (;;) {
579                 // If a is empty, fill up with elements from b and stop
580                 if (a == a_end) {
581                         while (b != b_end) {
582                                 new_seq.push_back(*b);
583                                 ++b;
584                         }
585                         break;
586                 } else
587                         pow_a = ex_to<numeric>((*a).coeff).to_int();
588                 
589                 // If b is empty, fill up with elements from a and stop
590                 if (b == b_end) {
591                         while (a != a_end) {
592                                 new_seq.push_back(*a);
593                                 ++a;
594                         }
595                         break;
596                 } else
597                         pow_b = ex_to<numeric>((*b).coeff).to_int();
598                 
599                 // a and b are non-empty, compare powers
600                 if (pow_a < pow_b) {
601                         // a has lesser power, get coefficient from a
602                         new_seq.push_back(*a);
603                         if (is_order_function((*a).rest))
604                                 break;
605                         ++a;
606                 } else if (pow_b < pow_a) {
607                         // b has lesser power, get coefficient from b
608                         new_seq.push_back(*b);
609                         if (is_order_function((*b).rest))
610                                 break;
611                         ++b;
612                 } else {
613                         // Add coefficient of a and b
614                         if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
615                                 new_seq.push_back(expair(Order(_ex1), (*a).coeff));
616                                 break;  // Order term ends the sequence
617                         } else {
618                                 ex sum = (*a).rest + (*b).rest;
619                                 if (!(sum.is_zero()))
620                                         new_seq.push_back(expair(sum, numeric(pow_a)));
621                                 ++a;
622                                 ++b;
623                         }
624                 }
625         }
626         return pseries(relational(var,point), new_seq);
627 }
628
629
630 /** Implementation of ex::series() for sums. This performs series addition when
631  *  adding pseries objects.
632  *  @see ex::series */
633 ex add::series(const relational & r, int order, unsigned options) const
634 {
635         ex acc; // Series accumulator
636         
637         // Get first term from overall_coeff
638         acc = overall_coeff.series(r, order, options);
639         
640         // Add remaining terms
641         epvector::const_iterator it = seq.begin();
642         epvector::const_iterator itend = seq.end();
643         for (; it!=itend; ++it) {
644                 ex op;
645                 if (is_exactly_a<pseries>(it->rest))
646                         op = it->rest;
647                 else
648                         op = it->rest.series(r, order, options);
649                 if (!it->coeff.is_equal(_ex1))
650                         op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it->coeff));
651                 
652                 // Series addition
653                 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
654         }
655         return acc;
656 }
657
658
659 /** Multiply a pseries object with a numeric constant, producing a pseries
660  *  object that represents the product.
661  *
662  *  @param other  constant to multiply with
663  *  @return the product as a pseries */
664 ex pseries::mul_const(const numeric &other) const
665 {
666         epvector new_seq;
667         new_seq.reserve(seq.size());
668         
669         epvector::const_iterator it = seq.begin(), itend = seq.end();
670         while (it != itend) {
671                 if (!is_order_function(it->rest))
672                         new_seq.push_back(expair(it->rest * other, it->coeff));
673                 else
674                         new_seq.push_back(*it);
675                 ++it;
676         }
677         return pseries(relational(var,point), new_seq);
678 }
679
680
681 /** Multiply one pseries object to another, producing a pseries object that
682  *  represents the product.
683  *
684  *  @param other  pseries object to multiply with
685  *  @return the product as a pseries */
686 ex pseries::mul_series(const pseries &other) const
687 {
688         // Multiplying two series with different variables or expansion points
689         // results in an empty (constant) series 
690         if (!is_compatible_to(other)) {
691                 epvector nul;
692                 nul.push_back(expair(Order(_ex1), _ex0));
693                 return pseries(relational(var,point), nul);
694         }
695         
696         // Series multiplication
697         epvector new_seq;
698         int a_max = degree(var);
699         int b_max = other.degree(var);
700         int a_min = ldegree(var);
701         int b_min = other.ldegree(var);
702         int cdeg_min = a_min + b_min;
703         int cdeg_max = a_max + b_max;
704         
705         int higher_order_a = INT_MAX;
706         int higher_order_b = INT_MAX;
707         if (is_order_function(coeff(var, a_max)))
708                 higher_order_a = a_max + b_min;
709         if (is_order_function(other.coeff(var, b_max)))
710                 higher_order_b = b_max + a_min;
711         int higher_order_c = std::min(higher_order_a, higher_order_b);
712         if (cdeg_max >= higher_order_c)
713                 cdeg_max = higher_order_c - 1;
714         
715         for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
716                 ex co = _ex0;
717                 // c(i)=a(0)b(i)+...+a(i)b(0)
718                 for (int i=a_min; cdeg-i>=b_min; ++i) {
719                         ex a_coeff = coeff(var, i);
720                         ex b_coeff = other.coeff(var, cdeg-i);
721                         if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
722                                 co += a_coeff * b_coeff;
723                 }
724                 if (!co.is_zero())
725                         new_seq.push_back(expair(co, numeric(cdeg)));
726         }
727         if (higher_order_c < INT_MAX)
728                 new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c)));
729         return pseries(relational(var, point), new_seq);
730 }
731
732
733 /** Implementation of ex::series() for product. This performs series
734  *  multiplication when multiplying series.
735  *  @see ex::series */
736 ex mul::series(const relational & r, int order, unsigned options) const
737 {
738         pseries acc; // Series accumulator
739
740         // Multiply with remaining terms
741         const epvector::const_iterator itbeg = seq.begin();
742         const epvector::const_iterator itend = seq.end();
743         for (epvector::const_iterator it=itbeg; it!=itend; ++it) {
744                 ex op = recombine_pair_to_ex(*it).series(r, order, options);
745
746                 // Series multiplication
747                 if (it==itbeg)
748                         acc = ex_to<pseries>(op);
749                 else
750                         acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
751         }
752         return acc.mul_const(ex_to<numeric>(overall_coeff));
753 }
754
755
756 /** Compute the p-th power of a series.
757  *
758  *  @param p  power to compute
759  *  @param deg  truncation order of series calculation */
760 ex pseries::power_const(const numeric &p, int deg) const
761 {
762         // method:
763         // (due to Leonhard Euler)
764         // let A(x) be this series and for the time being let it start with a
765         // constant (later we'll generalize):
766         //     A(x) = a_0 + a_1*x + a_2*x^2 + ...
767         // We want to compute
768         //     C(x) = A(x)^p
769         //     C(x) = c_0 + c_1*x + c_2*x^2 + ...
770         // Taking the derivative on both sides and multiplying with A(x) one
771         // immediately arrives at
772         //     C'(x)*A(x) = p*C(x)*A'(x)
773         // Multiplying this out and comparing coefficients we get the recurrence
774         // formula
775         //     c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
776         //                    ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
777         // which can easily be solved given the starting value c_0 = (a_0)^p.
778         // For the more general case where the leading coefficient of A(x) is not
779         // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
780         // repeat the above derivation.  The leading power of C2(x) = A2(x)^2 is
781         // then of course x^(p*m) but the recurrence formula still holds.
782         
783         if (seq.empty()) {
784                 // as a special case, handle the empty (zero) series honoring the
785                 // usual power laws such as implemented in power::eval()
786                 if (p.real().is_zero())
787                         throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
788                 else if (p.real().is_negative())
789                         throw pole_error("pseries::power_const(): division by zero",1);
790                 else
791                         return *this;
792         }
793         
794         const int ldeg = ldegree(var);
795         if (!(p*ldeg).is_integer())
796                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
797
798         // O(x^n)^(-m) is undefined
799         if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
800                 throw pole_error("pseries::power_const(): division by zero",1);
801         
802         // Compute coefficients of the powered series
803         exvector co;
804         co.reserve(deg);
805         co.push_back(power(coeff(var, ldeg), p));
806         bool all_sums_zero = true;
807         for (int i=1; i<deg; ++i) {
808                 ex sum = _ex0;
809                 for (int j=1; j<=i; ++j) {
810                         ex c = coeff(var, j + ldeg);
811                         if (is_order_function(c)) {
812                                 co.push_back(Order(_ex1));
813                                 break;
814                         } else
815                                 sum += (p * j - (i - j)) * co[i - j] * c;
816                 }
817                 if (!sum.is_zero())
818                         all_sums_zero = false;
819                 co.push_back(sum / coeff(var, ldeg) / i);
820         }
821         
822         // Construct new series (of non-zero coefficients)
823         epvector new_seq;
824         bool higher_order = false;
825         for (int i=0; i<deg; ++i) {
826                 if (!co[i].is_zero())
827                         new_seq.push_back(expair(co[i], p * ldeg + i));
828                 if (is_order_function(co[i])) {
829                         higher_order = true;
830                         break;
831                 }
832         }
833         if (!higher_order && !all_sums_zero)
834                 new_seq.push_back(expair(Order(_ex1), p * ldeg + deg));
835         return pseries(relational(var,point), new_seq);
836 }
837
838
839 /** Return a new pseries object with the powers shifted by deg. */
840 pseries pseries::shift_exponents(int deg) const
841 {
842         epvector newseq = seq;
843         epvector::iterator i = newseq.begin(), end  = newseq.end();
844         while (i != end) {
845                 i->coeff += deg;
846                 ++i;
847         }
848         return pseries(relational(var, point), newseq);
849 }
850
851
852 /** Implementation of ex::series() for powers. This performs Laurent expansion
853  *  of reciprocals of series at singularities.
854  *  @see ex::series */
855 ex power::series(const relational & r, int order, unsigned options) const
856 {
857         // If basis is already a series, just power it
858         if (is_exactly_a<pseries>(basis))
859                 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
860
861         // Basis is not a series, may there be a singularity?
862         bool must_expand_basis = false;
863         try {
864                 basis.subs(r);
865         } catch (pole_error) {
866                 must_expand_basis = true;
867         }
868                 
869         // Is the expression of type something^(-int)?
870         if (!must_expand_basis && !exponent.info(info_flags::negint))
871                 return basic::series(r, order, options);
872                 
873         // Is the expression of type 0^something?
874         if (!must_expand_basis && !basis.subs(r).is_zero())
875                 return basic::series(r, order, options);
876
877         // Singularity encountered, is the basis equal to (var - point)?
878         if (basis.is_equal(r.lhs() - r.rhs())) {
879                 epvector new_seq;
880                 if (ex_to<numeric>(exponent).to_int() < order)
881                         new_seq.push_back(expair(_ex1, exponent));
882                 else
883                         new_seq.push_back(expair(Order(_ex1), exponent));
884                 return pseries(r, new_seq);
885         }
886
887         // No, expand basis into series
888         ex e = basis.series(r, order, options);
889         return ex_to<pseries>(e).power_const(ex_to<numeric>(exponent), order);
890 }
891
892
893 /** Re-expansion of a pseries object. */
894 ex pseries::series(const relational & r, int order, unsigned options) const
895 {
896         const ex p = r.rhs();
897         GINAC_ASSERT(is_a<symbol>(r.lhs()));
898         const symbol &s = ex_to<symbol>(r.lhs());
899         
900         if (var.is_equal(s) && point.is_equal(p)) {
901                 if (order > degree(s))
902                         return *this;
903                 else {
904                         epvector new_seq;
905                         epvector::const_iterator it = seq.begin(), itend = seq.end();
906                         while (it != itend) {
907                                 int o = ex_to<numeric>(it->coeff).to_int();
908                                 if (o >= order) {
909                                         new_seq.push_back(expair(Order(_ex1), o));
910                                         break;
911                                 }
912                                 new_seq.push_back(*it);
913                                 ++it;
914                         }
915                         return pseries(r, new_seq);
916                 }
917         } else
918                 return convert_to_poly().series(r, order, options);
919 }
920
921
922 /** Compute the truncated series expansion of an expression.
923  *  This function returns an expression containing an object of class pseries 
924  *  to represent the series. If the series does not terminate within the given
925  *  truncation order, the last term of the series will be an order term.
926  *
927  *  @param r  expansion relation, lhs holds variable and rhs holds point
928  *  @param order  truncation order of series calculations
929  *  @param options  of class series_options
930  *  @return an expression holding a pseries object */
931 ex ex::series(const ex & r, int order, unsigned options) const
932 {
933         ex e;
934         relational rel_;
935         
936         if (is_a<relational>(r))
937                 rel_ = ex_to<relational>(r);
938         else if (is_a<symbol>(r))
939                 rel_ = relational(r,_ex0);
940         else
941                 throw (std::logic_error("ex::series(): expansion point has unknown type"));
942         
943         try {
944                 e = bp->series(rel_, order, options);
945         } catch (std::exception &x) {
946                 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));
947         }
948         return e;
949 }
950
951 } // namespace GiNaC