7f77cc664bec67025f62df51070adcbda2db25c2
[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) 1999-2002 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 "symbol.h"
35 #include "print.h"
36 #include "archive.h"
37 #include "utils.h"
38
39 namespace GiNaC {
40
41 GINAC_IMPLEMENT_REGISTERED_CLASS(pseries, basic)
42
43
44 /*
45  *  Default ctor, dtor, copy ctor, assignment operator and helpers
46  */
47
48 pseries::pseries() : inherited(TINFO_pseries) { }
49
50 void pseries::copy(const pseries &other)
51 {
52         inherited::copy(other);
53         seq = other.seq;
54         var = other.var;
55         point = other.point;
56 }
57
58 DEFAULT_DESTROY(pseries)
59
60
61 /*
62  *  Other ctors
63  */
64
65 /** Construct pseries from a vector of coefficients and powers.
66  *  expair.rest holds the coefficient, expair.coeff holds the power.
67  *  The powers must be integers (positive or negative) and in ascending order;
68  *  the last coefficient can be Order(_ex1) to represent a truncated,
69  *  non-terminating series.
70  *
71  *  @param rel_  expansion variable and point (must hold a relational)
72  *  @param ops_  vector of {coefficient, power} pairs (coefficient must not be zero)
73  *  @return newly constructed pseries */
74 pseries::pseries(const ex &rel_, const epvector &ops_) : basic(TINFO_pseries), seq(ops_)
75 {
76         GINAC_ASSERT(is_exactly_a<relational>(rel_));
77         GINAC_ASSERT(is_exactly_a<symbol>(rel_.lhs()));
78         point = rel_.rhs();
79         var = rel_.lhs();
80 }
81
82
83 /*
84  *  Archiving
85  */
86
87 pseries::pseries(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
88 {
89         for (unsigned int i=0; true; ++i) {
90                 ex rest;
91                 ex coeff;
92                 if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
93                         seq.push_back(expair(rest, coeff));
94                 else
95                         break;
96         }
97         n.find_ex("var", var, sym_lst);
98         n.find_ex("point", point, sym_lst);
99 }
100
101 void pseries::archive(archive_node &n) const
102 {
103         inherited::archive(n);
104         epvector::const_iterator i = seq.begin(), iend = seq.end();
105         while (i != iend) {
106                 n.add_ex("coeff", i->rest);
107                 n.add_ex("power", i->coeff);
108                 ++i;
109         }
110         n.add_ex("var", var);
111         n.add_ex("point", point);
112 }
113
114 DEFAULT_UNARCHIVE(pseries)
115
116 //////////
117 // functions overriding virtual functions from base classes
118 //////////
119
120 void pseries::print(const print_context & c, unsigned level) const
121 {
122         if (is_a<print_tree>(c)) {
123
124                 c.s << std::string(level, ' ') << class_name()
125                     << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
126                     << std::endl;
127                 unsigned delta_indent = static_cast<const print_tree &>(c).delta_indent;
128                 unsigned num = seq.size();
129                 for (unsigned i=0; i<num; ++i) {
130                         seq[i].rest.print(c, level + delta_indent);
131                         seq[i].coeff.print(c, level + delta_indent);
132                         c.s << std::string(level + delta_indent, ' ') << "-----" << std::endl;
133                 }
134                 var.print(c, level + delta_indent);
135                 point.print(c, level + delta_indent);
136
137         } else if (is_a<print_python_repr>(c)) {
138                 c.s << class_name() << "(relational(";
139                 var.print(c);
140                 c.s << ',';
141                 point.print(c);
142                 c.s << "),[";
143                 unsigned num = seq.size();
144                 for (unsigned i=0; i<num; ++i) {
145                         if (i)
146                                 c.s << ',';
147                         c.s << '(';
148                         seq[i].rest.print(c);
149                         c.s << ',';
150                         seq[i].coeff.print(c);
151                         c.s << ')';
152                 }
153                 c.s << "])";
154         } else {
155
156                 if (precedence() <= level)
157                         c.s << "(";
158                 
159                 std::string par_open = is_a<print_latex>(c) ? "{(" : "(";
160                 std::string par_close = is_a<print_latex>(c) ? ")}" : ")";
161                 
162                 // objects of type pseries must not have any zero entries, so the
163                 // trivial (zero) pseries needs a special treatment here:
164                 if (seq.empty())
165                         c.s << '0';
166                 epvector::const_iterator i = seq.begin(), end = seq.end();
167                 while (i != end) {
168                         // print a sign, if needed
169                         if (i != seq.begin())
170                                 c.s << '+';
171                         if (!is_order_function(i->rest)) {
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 << par_open;
178                                         i->rest.print(c);
179                                         c.s << par_close;
180                                 }
181                                 // print 'coeff', something like (x-1)^42
182                                 if (!i->coeff.is_zero()) {
183                                         if (is_a<print_latex>(c))
184                                                 c.s << ' ';
185                                         else
186                                                 c.s << '*';
187                                         if (!point.is_zero()) {
188                                                 c.s << par_open;
189                                                 (var-point).print(c);
190                                                 c.s << par_close;
191                                         } else
192                                                 var.print(c);
193                                         if (i->coeff.compare(_ex1)) {
194                                                 if (is_a<print_python>(c))
195                                                         c.s << "**";
196                                                 else
197                                                         c.s << '^';
198                                                 if (i->coeff.info(info_flags::negative)) {
199                                                         c.s << par_open;
200                                                         i->coeff.print(c);
201                                                         c.s << par_close;
202                                                 } else {
203                                                         if (is_a<print_latex>(c)) {
204                                                                 c.s << '{';
205                                                                 i->coeff.print(c);
206                                                                 c.s << '}';
207                                                         } else
208                                                                 i->coeff.print(c);
209                                                 }
210                                         }
211                                 }
212                         } else
213                                 Order(power(var-point,i->coeff)).print(c);
214                         ++i;
215                 }
216
217                 if (precedence() <= level)
218                         c.s << ")";
219         }
220 }
221
222 int pseries::compare_same_type(const basic & other) const
223 {
224         GINAC_ASSERT(is_a<pseries>(other));
225         const pseries &o = static_cast<const pseries &>(other);
226         
227         // first compare the lengths of the series...
228         if (seq.size()>o.seq.size())
229                 return 1;
230         if (seq.size()<o.seq.size())
231                 return -1;
232         
233         // ...then the expansion point...
234         int cmpval = var.compare(o.var);
235         if (cmpval)
236                 return cmpval;
237         cmpval = point.compare(o.point);
238         if (cmpval)
239                 return cmpval;
240         
241         // ...and if that failed the individual elements
242         epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
243         while (it!=seq.end() && o_it!=o.seq.end()) {
244                 cmpval = it->compare(*o_it);
245                 if (cmpval)
246                         return cmpval;
247                 ++it;
248                 ++o_it;
249         }
250
251         // so they are equal.
252         return 0;
253 }
254
255 /** Return the number of operands including a possible order term. */
256 unsigned pseries::nops(void) const
257 {
258         return seq.size();
259 }
260
261 /** Return the ith term in the series when represented as a sum. */
262 ex pseries::op(int i) const
263 {
264         if (i < 0 || unsigned(i) >= seq.size())
265                 throw (std::out_of_range("op() out of range"));
266         return seq[i].rest * power(var - point, seq[i].coeff);
267 }
268
269 ex &pseries::let_op(int i)
270 {
271         throw (std::logic_error("let_op not defined for pseries"));
272 }
273
274 /** Return degree of highest power of the series.  This is usually the exponent
275  *  of the Order term.  If s is not the expansion variable of the series, the
276  *  series is examined termwise. */
277 int pseries::degree(const ex &s) const
278 {
279         if (var.is_equal(s)) {
280                 // Return last exponent
281                 if (seq.size())
282                         return ex_to<numeric>((seq.end()-1)->coeff).to_int();
283                 else
284                         return 0;
285         } else {
286                 epvector::const_iterator it = seq.begin(), itend = seq.end();
287                 if (it == itend)
288                         return 0;
289                 int max_pow = INT_MIN;
290                 while (it != itend) {
291                         int pow = it->rest.degree(s);
292                         if (pow > max_pow)
293                                 max_pow = pow;
294                         ++it;
295                 }
296                 return max_pow;
297         }
298 }
299
300 /** Return degree of lowest power of the series.  This is usually the exponent
301  *  of the leading term.  If s is not the expansion variable of the series, the
302  *  series is examined termwise.  If s is the expansion variable but the
303  *  expansion point is not zero the series is not expanded to find the degree.
304  *  I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
305 int pseries::ldegree(const ex &s) const
306 {
307         if (var.is_equal(s)) {
308                 // Return first exponent
309                 if (seq.size())
310                         return ex_to<numeric>((seq.begin())->coeff).to_int();
311                 else
312                         return 0;
313         } else {
314                 epvector::const_iterator it = seq.begin(), itend = seq.end();
315                 if (it == itend)
316                         return 0;
317                 int min_pow = INT_MAX;
318                 while (it != itend) {
319                         int pow = it->rest.ldegree(s);
320                         if (pow < min_pow)
321                                 min_pow = pow;
322                         ++it;
323                 }
324                 return min_pow;
325         }
326 }
327
328 /** Return coefficient of degree n in power series if s is the expansion
329  *  variable.  If the expansion point is nonzero, by definition the n=1
330  *  coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
331  *  the expansion took place in the s in the first place).
332  *  If s is not the expansion variable, an attempt is made to convert the
333  *  series to a polynomial and return the corresponding coefficient from
334  *  there. */
335 ex pseries::coeff(const ex &s, int n) const
336 {
337         if (var.is_equal(s)) {
338                 if (seq.empty())
339                         return _ex0;
340                 
341                 // Binary search in sequence for given power
342                 numeric looking_for = numeric(n);
343                 int lo = 0, hi = seq.size() - 1;
344                 while (lo <= hi) {
345                         int mid = (lo + hi) / 2;
346                         GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
347                         int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
348                         switch (cmp) {
349                                 case -1:
350                                         lo = mid + 1;
351                                         break;
352                                 case 0:
353                                         return seq[mid].rest;
354                                 case 1:
355                                         hi = mid - 1;
356                                         break;
357                                 default:
358                                         throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
359                         }
360                 }
361                 return _ex0;
362         } else
363                 return convert_to_poly().coeff(s, n);
364 }
365
366 /** Does nothing. */
367 ex pseries::collect(const ex &s, bool distributed) const
368 {
369         return *this;
370 }
371
372 /** Perform coefficient-wise automatic term rewriting rules in this class. */
373 ex pseries::eval(int level) const
374 {
375         if (level == 1)
376                 return this->hold();
377         
378         if (level == -max_recursion_level)
379                 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
380         
381         // Construct a new series with evaluated coefficients
382         epvector new_seq;
383         new_seq.reserve(seq.size());
384         epvector::const_iterator it = seq.begin(), itend = seq.end();
385         while (it != itend) {
386                 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
387                 ++it;
388         }
389         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
390 }
391
392 /** Evaluate coefficients numerically. */
393 ex pseries::evalf(int level) const
394 {
395         if (level == 1)
396                 return *this;
397         
398         if (level == -max_recursion_level)
399                 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
400         
401         // Construct a new series with evaluated coefficients
402         epvector new_seq;
403         new_seq.reserve(seq.size());
404         epvector::const_iterator it = seq.begin(), itend = seq.end();
405         while (it != itend) {
406                 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
407                 ++it;
408         }
409         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
410 }
411
412 ex pseries::subs(const lst & ls, const lst & lr, bool no_pattern) const
413 {
414         // If expansion variable is being substituted, convert the series to a
415         // polynomial and do the substitution there because the result might
416         // no longer be a power series
417         if (ls.has(var))
418                 return convert_to_poly(true).subs(ls, lr, no_pattern);
419         
420         // Otherwise construct a new series with substituted coefficients and
421         // expansion point
422         epvector newseq;
423         newseq.reserve(seq.size());
424         epvector::const_iterator it = seq.begin(), itend = seq.end();
425         while (it != itend) {
426                 newseq.push_back(expair(it->rest.subs(ls, lr, no_pattern), it->coeff));
427                 ++it;
428         }
429         return (new pseries(relational(var,point.subs(ls, lr, no_pattern)), newseq))->setflag(status_flags::dynallocated);
430 }
431
432 /** Implementation of ex::expand() for a power series.  It expands all the
433  *  terms individually and returns the resulting series as a new pseries. */
434 ex pseries::expand(unsigned options) const
435 {
436         epvector newseq;
437         epvector::const_iterator i = seq.begin(), end = seq.end();
438         while (i != end) {
439                 ex restexp = i->rest.expand();
440                 if (!restexp.is_zero())
441                         newseq.push_back(expair(restexp, i->coeff));
442                 ++i;
443         }
444         return (new pseries(relational(var,point), newseq))
445                 ->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
446 }
447
448 /** Implementation of ex::diff() for a power series.
449  *  @see ex::diff */
450 ex pseries::derivative(const symbol & s) const
451 {
452         epvector new_seq;
453         epvector::const_iterator it = seq.begin(), itend = seq.end();
454
455         if (s == var) {
456                 
457                 // FIXME: coeff might depend on var
458                 while (it != itend) {
459                         if (is_order_function(it->rest)) {
460                                 new_seq.push_back(expair(it->rest, it->coeff - 1));
461                         } else {
462                                 ex c = it->rest * it->coeff;
463                                 if (!c.is_zero())
464                                         new_seq.push_back(expair(c, it->coeff - 1));
465                         }
466                         ++it;
467                 }
468
469         } else {
470
471                 while (it != itend) {
472                         if (is_order_function(it->rest)) {
473                                 new_seq.push_back(*it);
474                         } else {
475                                 ex c = it->rest.diff(s);
476                                 if (!c.is_zero())
477                                         new_seq.push_back(expair(c, it->coeff));
478                         }
479                         ++it;
480                 }
481         }
482
483         return pseries(relational(var,point), new_seq);
484 }
485
486 ex pseries::convert_to_poly(bool no_order) const
487 {
488         ex e;
489         epvector::const_iterator it = seq.begin(), itend = seq.end();
490         
491         while (it != itend) {
492                 if (is_order_function(it->rest)) {
493                         if (!no_order)
494                                 e += Order(power(var - point, it->coeff));
495                 } else
496                         e += it->rest * power(var - point, it->coeff);
497                 ++it;
498         }
499         return e;
500 }
501
502 bool pseries::is_terminating(void) const
503 {
504         return seq.empty() || !is_order_function((seq.end()-1)->rest);
505 }
506
507
508 /*
509  *  Implementations of series expansion
510  */
511
512 /** Default implementation of ex::series(). This performs Taylor expansion.
513  *  @see ex::series */
514 ex basic::series(const relational & r, int order, unsigned options) const
515 {
516         epvector seq;
517         numeric fac = 1;
518         ex deriv = *this;
519         ex coeff = deriv.subs(r);
520         const symbol &s = ex_to<symbol>(r.lhs());
521         
522         if (!coeff.is_zero())
523                 seq.push_back(expair(coeff, _ex0));
524         
525         int n;
526         for (n=1; n<order; ++n) {
527                 fac = fac.mul(n);
528                 // We need to test for zero in order to see if the series terminates.
529                 // The problem is that there is no such thing as a perfect test for
530                 // zero.  Expanding the term occasionally helps a little...
531                 deriv = deriv.diff(s).expand();
532                 if (deriv.is_zero())  // Series terminates
533                         return pseries(r, seq);
534
535                 coeff = deriv.subs(r);
536                 if (!coeff.is_zero())
537                         seq.push_back(expair(fac.inverse() * coeff, n));
538         }
539         
540         // Higher-order terms, if present
541         deriv = deriv.diff(s);
542         if (!deriv.expand().is_zero())
543                 seq.push_back(expair(Order(_ex1), n));
544         return pseries(r, seq);
545 }
546
547
548 /** Implementation of ex::series() for symbols.
549  *  @see ex::series */
550 ex symbol::series(const relational & r, int order, unsigned options) const
551 {
552         epvector seq;
553         const ex point = r.rhs();
554         GINAC_ASSERT(is_exactly_a<symbol>(r.lhs()));
555
556         if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
557                 if (order > 0 && !point.is_zero())
558                         seq.push_back(expair(point, _ex0));
559                 if (order > 1)
560                         seq.push_back(expair(_ex1, _ex1));
561                 else
562                         seq.push_back(expair(Order(_ex1), numeric(order)));
563         } else
564                 seq.push_back(expair(*this, _ex0));
565         return pseries(r, seq);
566 }
567
568
569 /** Add one series object to another, producing a pseries object that
570  *  represents the sum.
571  *
572  *  @param other  pseries object to add with
573  *  @return the sum as a pseries */
574 ex pseries::add_series(const pseries &other) const
575 {
576         // Adding two series with different variables or expansion points
577         // results in an empty (constant) series 
578         if (!is_compatible_to(other)) {
579                 epvector nul;
580                 nul.push_back(expair(Order(_ex1), _ex0));
581                 return pseries(relational(var,point), nul);
582         }
583         
584         // Series addition
585         epvector new_seq;
586         epvector::const_iterator a = seq.begin();
587         epvector::const_iterator b = other.seq.begin();
588         epvector::const_iterator a_end = seq.end();
589         epvector::const_iterator b_end = other.seq.end();
590         int pow_a = INT_MAX, pow_b = INT_MAX;
591         for (;;) {
592                 // If a is empty, fill up with elements from b and stop
593                 if (a == a_end) {
594                         while (b != b_end) {
595                                 new_seq.push_back(*b);
596                                 ++b;
597                         }
598                         break;
599                 } else
600                         pow_a = ex_to<numeric>((*a).coeff).to_int();
601                 
602                 // If b is empty, fill up with elements from a and stop
603                 if (b == b_end) {
604                         while (a != a_end) {
605                                 new_seq.push_back(*a);
606                                 ++a;
607                         }
608                         break;
609                 } else
610                         pow_b = ex_to<numeric>((*b).coeff).to_int();
611                 
612                 // a and b are non-empty, compare powers
613                 if (pow_a < pow_b) {
614                         // a has lesser power, get coefficient from a
615                         new_seq.push_back(*a);
616                         if (is_order_function((*a).rest))
617                                 break;
618                         ++a;
619                 } else if (pow_b < pow_a) {
620                         // b has lesser power, get coefficient from b
621                         new_seq.push_back(*b);
622                         if (is_order_function((*b).rest))
623                                 break;
624                         ++b;
625                 } else {
626                         // Add coefficient of a and b
627                         if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
628                                 new_seq.push_back(expair(Order(_ex1), (*a).coeff));
629                                 break;  // Order term ends the sequence
630                         } else {
631                                 ex sum = (*a).rest + (*b).rest;
632                                 if (!(sum.is_zero()))
633                                         new_seq.push_back(expair(sum, numeric(pow_a)));
634                                 ++a;
635                                 ++b;
636                         }
637                 }
638         }
639         return pseries(relational(var,point), new_seq);
640 }
641
642
643 /** Implementation of ex::series() for sums. This performs series addition when
644  *  adding pseries objects.
645  *  @see ex::series */
646 ex add::series(const relational & r, int order, unsigned options) const
647 {
648         ex acc; // Series accumulator
649         
650         // Get first term from overall_coeff
651         acc = overall_coeff.series(r, order, options);
652         
653         // Add remaining terms
654         epvector::const_iterator it = seq.begin();
655         epvector::const_iterator itend = seq.end();
656         for (; it!=itend; ++it) {
657                 ex op;
658                 if (is_ex_exactly_of_type(it->rest, pseries))
659                         op = it->rest;
660                 else
661                         op = it->rest.series(r, order, options);
662                 if (!it->coeff.is_equal(_ex1))
663                         op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it->coeff));
664                 
665                 // Series addition
666                 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
667         }
668         return acc;
669 }
670
671
672 /** Multiply a pseries object with a numeric constant, producing a pseries
673  *  object that represents the product.
674  *
675  *  @param other  constant to multiply with
676  *  @return the product as a pseries */
677 ex pseries::mul_const(const numeric &other) const
678 {
679         epvector new_seq;
680         new_seq.reserve(seq.size());
681         
682         epvector::const_iterator it = seq.begin(), itend = seq.end();
683         while (it != itend) {
684                 if (!is_order_function(it->rest))
685                         new_seq.push_back(expair(it->rest * other, it->coeff));
686                 else
687                         new_seq.push_back(*it);
688                 ++it;
689         }
690         return pseries(relational(var,point), new_seq);
691 }
692
693
694 /** Multiply one pseries object to another, producing a pseries object that
695  *  represents the product.
696  *
697  *  @param other  pseries object to multiply with
698  *  @return the product as a pseries */
699 ex pseries::mul_series(const pseries &other) const
700 {
701         // Multiplying two series with different variables or expansion points
702         // results in an empty (constant) series 
703         if (!is_compatible_to(other)) {
704                 epvector nul;
705                 nul.push_back(expair(Order(_ex1), _ex0));
706                 return pseries(relational(var,point), nul);
707         }
708         
709         // Series multiplication
710         epvector new_seq;
711         int a_max = degree(var);
712         int b_max = other.degree(var);
713         int a_min = ldegree(var);
714         int b_min = other.ldegree(var);
715         int cdeg_min = a_min + b_min;
716         int cdeg_max = a_max + b_max;
717         
718         int higher_order_a = INT_MAX;
719         int higher_order_b = INT_MAX;
720         if (is_order_function(coeff(var, a_max)))
721                 higher_order_a = a_max + b_min;
722         if (is_order_function(other.coeff(var, b_max)))
723                 higher_order_b = b_max + a_min;
724         int higher_order_c = std::min(higher_order_a, higher_order_b);
725         if (cdeg_max >= higher_order_c)
726                 cdeg_max = higher_order_c - 1;
727         
728         for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
729                 ex co = _ex0;
730                 // c(i)=a(0)b(i)+...+a(i)b(0)
731                 for (int i=a_min; cdeg-i>=b_min; ++i) {
732                         ex a_coeff = coeff(var, i);
733                         ex b_coeff = other.coeff(var, cdeg-i);
734                         if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
735                                 co += a_coeff * b_coeff;
736                 }
737                 if (!co.is_zero())
738                         new_seq.push_back(expair(co, numeric(cdeg)));
739         }
740         if (higher_order_c < INT_MAX)
741                 new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c)));
742         return pseries(relational(var, point), new_seq);
743 }
744
745
746 /** Implementation of ex::series() for product. This performs series
747  *  multiplication when multiplying series.
748  *  @see ex::series */
749 ex mul::series(const relational & r, int order, unsigned options) const
750 {
751         pseries acc; // Series accumulator
752
753         // Multiply with remaining terms
754         const epvector::const_iterator itbeg = seq.begin();
755         const epvector::const_iterator itend = seq.end();
756         for (epvector::const_iterator it=itbeg; it!=itend; ++it) {
757                 ex op = recombine_pair_to_ex(*it).series(r, order, options);
758
759                 // Series multiplication
760                 if (it==itbeg)
761                         acc = ex_to<pseries>(op);
762                 else
763                         acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
764         }
765         return acc.mul_const(ex_to<numeric>(overall_coeff));
766 }
767
768
769 /** Compute the p-th power of a series.
770  *
771  *  @param p  power to compute
772  *  @param deg  truncation order of series calculation */
773 ex pseries::power_const(const numeric &p, int deg) const
774 {
775         // method:
776         // (due to Leonhard Euler)
777         // let A(x) be this series and for the time being let it start with a
778         // constant (later we'll generalize):
779         //     A(x) = a_0 + a_1*x + a_2*x^2 + ...
780         // We want to compute
781         //     C(x) = A(x)^p
782         //     C(x) = c_0 + c_1*x + c_2*x^2 + ...
783         // Taking the derivative on both sides and multiplying with A(x) one
784         // immediately arrives at
785         //     C'(x)*A(x) = p*C(x)*A'(x)
786         // Multiplying this out and comparing coefficients we get the recurrence
787         // formula
788         //     c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
789         //                    ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
790         // which can easily be solved given the starting value c_0 = (a_0)^p.
791         // For the more general case where the leading coefficient of A(x) is not
792         // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
793         // repeat the above derivation.  The leading power of C2(x) = A2(x)^2 is
794         // then of course x^(p*m) but the recurrence formula still holds.
795         
796         if (seq.empty()) {
797                 // as a special case, handle the empty (zero) series honoring the
798                 // usual power laws such as implemented in power::eval()
799                 if (p.real().is_zero())
800                         throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
801                 else if (p.real().is_negative())
802                         throw pole_error("pseries::power_const(): division by zero",1);
803                 else
804                         return *this;
805         }
806         
807         const int ldeg = ldegree(var);
808         if (!(p*ldeg).is_integer())
809                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
810
811         // O(x^n)^(-m) is undefined
812         if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
813                 throw pole_error("pseries::power_const(): division by zero",1);
814         
815         // Compute coefficients of the powered series
816         exvector co;
817         co.reserve(deg);
818         co.push_back(power(coeff(var, ldeg), p));
819         bool all_sums_zero = true;
820         for (int i=1; i<deg; ++i) {
821                 ex sum = _ex0;
822                 for (int j=1; j<=i; ++j) {
823                         ex c = coeff(var, j + ldeg);
824                         if (is_order_function(c)) {
825                                 co.push_back(Order(_ex1));
826                                 break;
827                         } else
828                                 sum += (p * j - (i - j)) * co[i - j] * c;
829                 }
830                 if (!sum.is_zero())
831                         all_sums_zero = false;
832                 co.push_back(sum / coeff(var, ldeg) / i);
833         }
834         
835         // Construct new series (of non-zero coefficients)
836         epvector new_seq;
837         bool higher_order = false;
838         for (int i=0; i<deg; ++i) {
839                 if (!co[i].is_zero())
840                         new_seq.push_back(expair(co[i], p * ldeg + i));
841                 if (is_order_function(co[i])) {
842                         higher_order = true;
843                         break;
844                 }
845         }
846         if (!higher_order && !all_sums_zero)
847                 new_seq.push_back(expair(Order(_ex1), p * ldeg + deg));
848         return pseries(relational(var,point), new_seq);
849 }
850
851
852 /** Return a new pseries object with the powers shifted by deg. */
853 pseries pseries::shift_exponents(int deg) const
854 {
855         epvector newseq = seq;
856         epvector::iterator i = newseq.begin(), end  = newseq.end();
857         while (i != end) {
858                 i->coeff += deg;
859                 ++i;
860         }
861         return pseries(relational(var, point), newseq);
862 }
863
864
865 /** Implementation of ex::series() for powers. This performs Laurent expansion
866  *  of reciprocals of series at singularities.
867  *  @see ex::series */
868 ex power::series(const relational & r, int order, unsigned options) const
869 {
870         // If basis is already a series, just power it
871         if (is_ex_exactly_of_type(basis, pseries))
872                 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
873
874         // Basis is not a series, may there be a singularity?
875         bool must_expand_basis = false;
876         try {
877                 basis.subs(r);
878         } catch (pole_error) {
879                 must_expand_basis = true;
880         }
881                 
882         // Is the expression of type something^(-int)?
883         if (!must_expand_basis && !exponent.info(info_flags::negint))
884                 return basic::series(r, order, options);
885                 
886         // Is the expression of type 0^something?
887         if (!must_expand_basis && !basis.subs(r).is_zero())
888                 return basic::series(r, order, options);
889
890         // Singularity encountered, is the basis equal to (var - point)?
891         if (basis.is_equal(r.lhs() - r.rhs())) {
892                 epvector new_seq;
893                 if (ex_to<numeric>(exponent).to_int() < order)
894                         new_seq.push_back(expair(_ex1, exponent));
895                 else
896                         new_seq.push_back(expair(Order(_ex1), exponent));
897                 return pseries(r, new_seq);
898         }
899
900         // No, expand basis into series
901         ex e = basis.series(r, order, options);
902         return ex_to<pseries>(e).power_const(ex_to<numeric>(exponent), order);
903 }
904
905
906 /** Re-expansion of a pseries object. */
907 ex pseries::series(const relational & r, int order, unsigned options) const
908 {
909         const ex p = r.rhs();
910         GINAC_ASSERT(is_exactly_a<symbol>(r.lhs()));
911         const symbol &s = ex_to<symbol>(r.lhs());
912         
913         if (var.is_equal(s) && point.is_equal(p)) {
914                 if (order > degree(s))
915                         return *this;
916                 else {
917                         epvector new_seq;
918                         epvector::const_iterator it = seq.begin(), itend = seq.end();
919                         while (it != itend) {
920                                 int o = ex_to<numeric>(it->coeff).to_int();
921                                 if (o >= order) {
922                                         new_seq.push_back(expair(Order(_ex1), o));
923                                         break;
924                                 }
925                                 new_seq.push_back(*it);
926                                 ++it;
927                         }
928                         return pseries(r, new_seq);
929                 }
930         } else
931                 return convert_to_poly().series(r, order, options);
932 }
933
934
935 /** Compute the truncated series expansion of an expression.
936  *  This function returns an expression containing an object of class pseries 
937  *  to represent the series. If the series does not terminate within the given
938  *  truncation order, the last term of the series will be an order term.
939  *
940  *  @param r  expansion relation, lhs holds variable and rhs holds point
941  *  @param order  truncation order of series calculations
942  *  @param options  of class series_options
943  *  @return an expression holding a pseries object */
944 ex ex::series(const ex & r, int order, unsigned options) const
945 {
946         GINAC_ASSERT(bp!=0);
947         ex e;
948         relational rel_;
949         
950         if (is_ex_exactly_of_type(r,relational))
951                 rel_ = ex_to<relational>(r);
952         else if (is_ex_exactly_of_type(r,symbol))
953                 rel_ = relational(r,_ex0);
954         else
955                 throw (std::logic_error("ex::series(): expansion point has unknown type"));
956         
957         try {
958                 e = bp->series(rel_, order, options);
959         } catch (std::exception &x) {
960                 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));
961         }
962         return e;
963 }
964
965 } // namespace GiNaC