3b2ac7010e051a5d2b49fb2dd90e65aab7c24b82
[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) 1999-2001 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 <stdexcept>
25
26 #include "pseries.h"
27 #include "add.h"
28 #include "inifcns.h"
29 #include "lst.h"
30 #include "mul.h"
31 #include "power.h"
32 #include "relational.h"
33 #include "symbol.h"
34 #include "archive.h"
35 #include "utils.h"
36 #include "debugmsg.h"
37
38 namespace GiNaC {
39
40 GINAC_IMPLEMENT_REGISTERED_CLASS(pseries, basic)
41
42 /*
43  *  Default ctor, dtor, copy ctor, assignment operator and helpers
44  */
45
46 pseries::pseries() : basic(TINFO_pseries)
47 {
48         debugmsg("pseries default ctor", LOGLEVEL_CONSTRUCT);
49 }
50
51 void pseries::copy(const pseries &other)
52 {
53         inherited::copy(other);
54         seq = other.seq;
55         var = other.var;
56         point = other.point;
57 }
58
59 void pseries::destroy(bool call_parent)
60 {
61         if (call_parent)
62                 inherited::destroy(call_parent);
63 }
64
65
66 /*
67  *  Other ctors
68  */
69
70 /** Construct pseries from a vector of coefficients and powers.
71  *  expair.rest holds the coefficient, expair.coeff holds the power.
72  *  The powers must be integers (positive or negative) and in ascending order;
73  *  the last coefficient can be Order(_ex1()) to represent a truncated,
74  *  non-terminating series.
75  *
76  *  @param rel_  expansion variable and point (must hold a relational)
77  *  @param ops_  vector of {coefficient, power} pairs (coefficient must not be zero)
78  *  @return newly constructed pseries */
79 pseries::pseries(const ex &rel_, const epvector &ops_) : basic(TINFO_pseries), seq(ops_)
80 {
81         debugmsg("pseries ctor from ex,epvector", LOGLEVEL_CONSTRUCT);
82         GINAC_ASSERT(is_ex_exactly_of_type(rel_, relational));
83         GINAC_ASSERT(is_ex_exactly_of_type(rel_.lhs(),symbol));
84         point = rel_.rhs();
85         var = *static_cast<symbol *>(rel_.lhs().bp);
86 }
87
88
89 /*
90  *  Archiving
91  */
92
93 /** Construct object from archive_node. */
94 pseries::pseries(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
95 {
96         debugmsg("pseries ctor from archive_node", LOGLEVEL_CONSTRUCT);
97         for (unsigned int i=0; true; ++i) {
98                 ex rest;
99                 ex coeff;
100                 if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
101                         seq.push_back(expair(rest, coeff));
102                 else
103                         break;
104         }
105         n.find_ex("var", var, sym_lst);
106         n.find_ex("point", point, sym_lst);
107 }
108
109 /** Unarchive the object. */
110 ex pseries::unarchive(const archive_node &n, const lst &sym_lst)
111 {
112         return (new pseries(n, sym_lst))->setflag(status_flags::dynallocated);
113 }
114
115 /** Archive the object. */
116 void pseries::archive(archive_node &n) const
117 {
118         inherited::archive(n);
119         epvector::const_iterator i = seq.begin(), iend = seq.end();
120         while (i != iend) {
121                 n.add_ex("coeff", i->rest);
122                 n.add_ex("power", i->coeff);
123                 ++i;
124         }
125         n.add_ex("var", var);
126         n.add_ex("point", point);
127 }
128
129 //////////
130 // functions overriding virtual functions from bases classes
131 //////////
132
133 void pseries::print(std::ostream &os, unsigned upper_precedence) const
134 {
135         debugmsg("pseries print", LOGLEVEL_PRINT);
136         if (precedence<=upper_precedence) os << "(";
137         // objects of type pseries must not have any zero entries, so the
138         // trivial (zero) pseries needs a special treatment here:
139         if (seq.size()==0)
140                 os << '0';
141         for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
142                 // print a sign, if needed
143                 if (i!=seq.begin())
144                         os << '+';
145                 if (!is_order_function(i->rest)) {
146                         // print 'rest', i.e. the expansion coefficient
147                         if (i->rest.info(info_flags::numeric) &&
148                                 i->rest.info(info_flags::positive)) {
149                                 os << i->rest;
150                         } else
151                                 os << "(" << i->rest << ')';
152                         // print 'coeff', something like (x-1)^42
153                         if (!i->coeff.is_zero()) {
154                                 os << '*';
155                                 if (!point.is_zero())
156                                         os << '(' << var-point << ')';
157                                 else
158                                         os << var;
159                                 if (i->coeff.compare(_ex1())) {
160                                         os << '^';
161                                         if (i->coeff.info(info_flags::negative))
162                                                 os << '(' << i->coeff << ')';
163                                         else
164                                                 os << i->coeff;
165                                 }
166                         }
167                 } else {
168                         os << Order(power(var-point,i->coeff));
169                 }
170         }
171         if (precedence<=upper_precedence) os << ")";
172 }
173
174
175 void pseries::printraw(std::ostream &os) const
176 {
177         debugmsg("pseries printraw", LOGLEVEL_PRINT);
178         os << class_name() << "(" << var << ";" << point << ";";
179         for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i)
180                 os << "(" << (*i).rest << "," << (*i).coeff << "),";
181         os << ")";
182 }
183
184
185 void pseries::printtree(std::ostream & os, unsigned indent) const
186 {
187         debugmsg("pseries printtree",LOGLEVEL_PRINT);
188         os << std::string(indent,' ') << class_name()
189            << ", hash=" << hashvalue
190            << " (0x" << std::hex << hashvalue << std::dec << ")"
191            << ", flags=" << flags << std::endl;
192         for (unsigned i=0; i<seq.size(); ++i) {
193                 seq[i].rest.printtree(os,indent+delta_indent);
194                 seq[i].coeff.printtree(os,indent+delta_indent);
195                 if (i!=seq.size()-1)
196                         os << std::string(indent+delta_indent,' ') << "-----" << std::endl;
197         }
198         var.printtree(os, indent+delta_indent);
199         point.printtree(os, indent+delta_indent);
200 }
201
202 int pseries::compare_same_type(const basic & other) const
203 {
204         GINAC_ASSERT(is_of_type(other, pseries));
205         const pseries &o = static_cast<const pseries &>(other);
206         
207         // first compare the lengths of the series...
208         if (seq.size()>o.seq.size())
209                 return 1;
210         if (seq.size()<o.seq.size())
211                 return -1;
212         
213         // ...then the expansion point...
214         int cmpval = var.compare(o.var);
215         if (cmpval)
216                 return cmpval;
217         cmpval = point.compare(o.point);
218         if (cmpval)
219                 return cmpval;
220         
221         // ...and if that failed the individual elements
222         epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
223         while (it!=seq.end() && o_it!=o.seq.end()) {
224                 cmpval = it->compare(*o_it);
225                 if (cmpval)
226                         return cmpval;
227                 ++it;
228                 ++o_it;
229         }
230
231         // so they are equal.
232         return 0;
233 }
234
235 /** Return the number of operands including a possible order term. */
236 unsigned pseries::nops(void) const
237 {
238         return seq.size();
239 }
240
241
242 /** Return the ith term in the series when represented as a sum. */
243 ex pseries::op(int i) const
244 {
245         if (i < 0 || unsigned(i) >= seq.size())
246                 throw (std::out_of_range("op() out of range"));
247         return seq[i].rest * power(var - point, seq[i].coeff);
248 }
249
250
251 ex &pseries::let_op(int i)
252 {
253         throw (std::logic_error("let_op not defined for pseries"));
254 }
255
256
257 /** Return degree of highest power of the series.  This is usually the exponent
258  *  of the Order term.  If s is not the expansion variable of the series, the
259  *  series is examined termwise. */
260 int pseries::degree(const symbol &s) const
261 {
262         if (var.is_equal(s)) {
263                 // Return last exponent
264                 if (seq.size())
265                         return ex_to_numeric((*(seq.end() - 1)).coeff).to_int();
266                 else
267                         return 0;
268         } else {
269                 epvector::const_iterator it = seq.begin(), itend = seq.end();
270                 if (it == itend)
271                         return 0;
272                 int max_pow = INT_MIN;
273                 while (it != itend) {
274                         int pow = it->rest.degree(s);
275                         if (pow > max_pow)
276                                 max_pow = pow;
277                         ++it;
278                 }
279                 return max_pow;
280         }
281 }
282
283 /** Return degree of lowest power of the series.  This is usually the exponent
284  *  of the leading term.  If s is not the expansion variable of the series, the
285  *  series is examined termwise.  If s is the expansion variable but the
286  *  expansion point is not zero the series is not expanded to find the degree.
287  *  I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
288 int pseries::ldegree(const symbol &s) const
289 {
290         if (var.is_equal(s)) {
291                 // Return first exponent
292                 if (seq.size())
293                         return ex_to_numeric((*(seq.begin())).coeff).to_int();
294                 else
295                         return 0;
296         } else {
297                 epvector::const_iterator it = seq.begin(), itend = seq.end();
298                 if (it == itend)
299                         return 0;
300                 int min_pow = INT_MAX;
301                 while (it != itend) {
302                         int pow = it->rest.ldegree(s);
303                         if (pow < min_pow)
304                                 min_pow = pow;
305                         ++it;
306                 }
307                 return min_pow;
308         }
309 }
310
311 /** Return coefficient of degree n in power series if s is the expansion
312  *  variable.  If the expansion point is nonzero, by definition the n=1
313  *  coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
314  *  the expansion took place in the s in the first place).
315  *  If s is not the expansion variable, an attempt is made to convert the
316  *  series to a polynomial and return the corresponding coefficient from
317  *  there. */
318 ex pseries::coeff(const symbol &s, int n) const
319 {
320         if (var.is_equal(s)) {
321                 if (seq.size() == 0)
322                         return _ex0();
323                 
324                 // Binary search in sequence for given power
325                 numeric looking_for = numeric(n);
326                 int lo = 0, hi = seq.size() - 1;
327                 while (lo <= hi) {
328                         int mid = (lo + hi) / 2;
329                         GINAC_ASSERT(is_ex_exactly_of_type(seq[mid].coeff, numeric));
330                         int cmp = ex_to_numeric(seq[mid].coeff).compare(looking_for);
331                         switch (cmp) {
332                                 case -1:
333                                         lo = mid + 1;
334                                         break;
335                                 case 0:
336                                         return seq[mid].rest;
337                                 case 1:
338                                         hi = mid - 1;
339                                         break;
340                                 default:
341                                         throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
342                         }
343                 }
344                 return _ex0();
345         } else
346                 return convert_to_poly().coeff(s, n);
347 }
348
349 /** Does nothing. */
350 ex pseries::collect(const symbol &s) const
351 {
352         return *this;
353 }
354
355
356 /** Evaluate coefficients. */
357 ex pseries::eval(int level) const
358 {
359         if (level == 1)
360                 return this->hold();
361         
362         if (level == -max_recursion_level)
363                 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
364         
365         // Construct a new series with evaluated coefficients
366         epvector new_seq;
367         new_seq.reserve(seq.size());
368         epvector::const_iterator it = seq.begin(), itend = seq.end();
369         while (it != itend) {
370                 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
371                 ++it;
372         }
373         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
374 }
375
376
377 /** Evaluate coefficients numerically. */
378 ex pseries::evalf(int level) const
379 {
380         if (level == 1)
381                 return *this;
382         
383         if (level == -max_recursion_level)
384                 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
385         
386         // Construct a new series with evaluated coefficients
387         epvector new_seq;
388         new_seq.reserve(seq.size());
389         epvector::const_iterator it = seq.begin(), itend = seq.end();
390         while (it != itend) {
391                 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
392                 ++it;
393         }
394         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
395 }
396
397
398 ex pseries::subs(const lst & ls, const lst & lr) const
399 {
400         // If expansion variable is being substituted, convert the series to a
401         // polynomial and do the substitution there because the result might
402         // no longer be a power series
403         if (ls.has(var))
404                 return convert_to_poly(true).subs(ls, lr);
405         
406         // Otherwise construct a new series with substituted coefficients and
407         // expansion point
408         epvector newseq;
409         newseq.reserve(seq.size());
410         epvector::const_iterator it = seq.begin(), itend = seq.end();
411         while (it != itend) {
412                 newseq.push_back(expair(it->rest.subs(ls, lr), it->coeff));
413                 ++it;
414         }
415         return (new pseries(relational(var,point.subs(ls, lr)), newseq))->setflag(status_flags::dynallocated);
416 }
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         for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
425                 ex restexp = i->rest.expand();
426                 if (!restexp.is_zero())
427                         newseq.push_back(expair(restexp, i->coeff));
428         }
429         return (new pseries(relational(var,point), newseq))
430                 ->setflag(status_flags::dynallocated | status_flags::expanded);
431 }
432
433
434 /** Implementation of ex::diff() for a power series.  It treats the series as a
435  *  polynomial.
436  *  @see ex::diff */
437 ex pseries::derivative(const symbol & s) const
438 {
439         if (s == var) {
440                 epvector new_seq;
441                 epvector::const_iterator it = seq.begin(), itend = seq.end();
442                 
443                 // FIXME: coeff might depend on var
444                 while (it != itend) {
445                         if (is_order_function(it->rest)) {
446                                 new_seq.push_back(expair(it->rest, it->coeff - 1));
447                         } else {
448                                 ex c = it->rest * it->coeff;
449                                 if (!c.is_zero())
450                                         new_seq.push_back(expair(c, it->coeff - 1));
451                         }
452                         ++it;
453                 }
454                 return pseries(relational(var,point), new_seq);
455         } else {
456                 return *this;
457         }
458 }
459
460
461 /** Convert a pseries object to an ordinary polynomial.
462  *
463  *  @param no_order flag: discard higher order terms */
464 ex pseries::convert_to_poly(bool no_order) const
465 {
466         ex e;
467         epvector::const_iterator it = seq.begin(), itend = seq.end();
468         
469         while (it != itend) {
470                 if (is_order_function(it->rest)) {
471                         if (!no_order)
472                                 e += Order(power(var - point, it->coeff));
473                 } else
474                         e += it->rest * power(var - point, it->coeff);
475                 ++it;
476         }
477         return e;
478 }
479
480
481 /** Returns true if there is no order term, i.e. the series terminates and
482  *  false otherwise. */
483 bool pseries::is_terminating(void) const
484 {
485         return seq.size() == 0 || !is_order_function((seq.end()-1)->rest);
486 }
487
488
489 /*
490  *  Implementations of series expansion
491  */
492
493 /** Default implementation of ex::series(). This performs Taylor expansion.
494  *  @see ex::series */
495 ex basic::series(const relational & r, int order, unsigned options) const
496 {
497         epvector seq;
498         numeric fac(1);
499         ex deriv = *this;
500         ex coeff = deriv.subs(r);
501         const symbol *s = static_cast<symbol *>(r.lhs().bp);
502         
503         if (!coeff.is_zero())
504                 seq.push_back(expair(coeff, numeric(0)));
505         
506         int n;
507         for (n=1; n<order; ++n) {
508                 fac = fac.mul(numeric(n));
509                 deriv = deriv.diff(*s).expand();
510                 if (deriv.is_zero()) {
511                         // Series terminates
512                         return pseries(r, seq);
513                 }
514                 coeff = deriv.subs(r);
515                 if (!coeff.is_zero())
516                         seq.push_back(expair(fac.inverse() * coeff, numeric(n)));
517         }
518         
519         // Higher-order terms, if present
520         deriv = deriv.diff(*s);
521         if (!deriv.expand().is_zero())
522                 seq.push_back(expair(Order(_ex1()), numeric(n)));
523         return pseries(r, seq);
524 }
525
526
527 /** Implementation of ex::series() for symbols.
528  *  @see ex::series */
529 ex symbol::series(const relational & r, int order, unsigned options) const
530 {
531         epvector seq;
532         const ex point = r.rhs();
533         GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
534         const symbol *s = static_cast<symbol *>(r.lhs().bp);
535         
536         if (this->is_equal(*s)) {
537                 if (order > 0 && !point.is_zero())
538                         seq.push_back(expair(point, _ex0()));
539                 if (order > 1)
540                         seq.push_back(expair(_ex1(), _ex1()));
541                 else
542                         seq.push_back(expair(Order(_ex1()), numeric(order)));
543         } else
544                 seq.push_back(expair(*this, _ex0()));
545         return pseries(r, seq);
546 }
547
548
549 /** Add one series object to another, producing a pseries object that
550  *  represents the sum.
551  *
552  *  @param other  pseries object to add with
553  *  @return the sum as a pseries */
554 ex pseries::add_series(const pseries &other) const
555 {
556         // Adding two series with different variables or expansion points
557         // results in an empty (constant) series 
558         if (!is_compatible_to(other)) {
559                 epvector nul;
560                 nul.push_back(expair(Order(_ex1()), _ex0()));
561                 return pseries(relational(var,point), nul);
562         }
563         
564         // Series addition
565         epvector new_seq;
566         epvector::const_iterator a = seq.begin();
567         epvector::const_iterator b = other.seq.begin();
568         epvector::const_iterator a_end = seq.end();
569         epvector::const_iterator b_end = other.seq.end();
570         int pow_a = INT_MAX, pow_b = INT_MAX;
571         for (;;) {
572                 // If a is empty, fill up with elements from b and stop
573                 if (a == a_end) {
574                         while (b != b_end) {
575                                 new_seq.push_back(*b);
576                                 ++b;
577                         }
578                         break;
579                 } else
580                         pow_a = ex_to_numeric((*a).coeff).to_int();
581                 
582                 // If b is empty, fill up with elements from a and stop
583                 if (b == b_end) {
584                         while (a != a_end) {
585                                 new_seq.push_back(*a);
586                                 ++a;
587                         }
588                         break;
589                 } else
590                         pow_b = ex_to_numeric((*b).coeff).to_int();
591                 
592                 // a and b are non-empty, compare powers
593                 if (pow_a < pow_b) {
594                         // a has lesser power, get coefficient from a
595                         new_seq.push_back(*a);
596                         if (is_order_function((*a).rest))
597                                 break;
598                         ++a;
599                 } else if (pow_b < pow_a) {
600                         // b has lesser power, get coefficient from b
601                         new_seq.push_back(*b);
602                         if (is_order_function((*b).rest))
603                                 break;
604                         ++b;
605                 } else {
606                         // Add coefficient of a and b
607                         if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
608                                 new_seq.push_back(expair(Order(_ex1()), (*a).coeff));
609                                 break;  // Order term ends the sequence
610                         } else {
611                                 ex sum = (*a).rest + (*b).rest;
612                                 if (!(sum.is_zero()))
613                                         new_seq.push_back(expair(sum, numeric(pow_a)));
614                                 ++a;
615                                 ++b;
616                         }
617                 }
618         }
619         return pseries(relational(var,point), new_seq);
620 }
621
622
623 /** Implementation of ex::series() for sums. This performs series addition when
624  *  adding pseries objects.
625  *  @see ex::series */
626 ex add::series(const relational & r, int order, unsigned options) const
627 {
628         ex acc; // Series accumulator
629         
630         // Get first term from overall_coeff
631         acc = overall_coeff.series(r, order, options);
632         
633         // Add remaining terms
634         epvector::const_iterator it = seq.begin();
635         epvector::const_iterator itend = seq.end();
636         for (; it!=itend; ++it) {
637                 ex op;
638                 if (is_ex_exactly_of_type(it->rest, pseries))
639                         op = it->rest;
640                 else
641                         op = it->rest.series(r, order, options);
642                 if (!it->coeff.is_equal(_ex1()))
643                         op = ex_to_pseries(op).mul_const(ex_to_numeric(it->coeff));
644                 
645                 // Series addition
646                 acc = ex_to_pseries(acc).add_series(ex_to_pseries(op));
647         }
648         return acc;
649 }
650
651
652 /** Multiply a pseries object with a numeric constant, producing a pseries
653  *  object that represents the product.
654  *
655  *  @param other  constant to multiply with
656  *  @return the product as a pseries */
657 ex pseries::mul_const(const numeric &other) const
658 {
659         epvector new_seq;
660         new_seq.reserve(seq.size());
661         
662         epvector::const_iterator it = seq.begin(), itend = seq.end();
663         while (it != itend) {
664                 if (!is_order_function(it->rest))
665                         new_seq.push_back(expair(it->rest * other, it->coeff));
666                 else
667                         new_seq.push_back(*it);
668                 ++it;
669         }
670         return pseries(relational(var,point), new_seq);
671 }
672
673
674 /** Multiply one pseries object to another, producing a pseries object that
675  *  represents the product.
676  *
677  *  @param other  pseries object to multiply with
678  *  @return the product as a pseries */
679 ex pseries::mul_series(const pseries &other) const
680 {
681         // Multiplying two series with different variables or expansion points
682         // results in an empty (constant) series 
683         if (!is_compatible_to(other)) {
684                 epvector nul;
685                 nul.push_back(expair(Order(_ex1()), _ex0()));
686                 return pseries(relational(var,point), nul);
687         }
688         
689         // Series multiplication
690         epvector new_seq;
691         
692         const symbol *s = static_cast<symbol *>(var.bp);
693         int a_max = degree(*s);
694         int b_max = other.degree(*s);
695         int a_min = ldegree(*s);
696         int b_min = other.ldegree(*s);
697         int cdeg_min = a_min + b_min;
698         int cdeg_max = a_max + b_max;
699         
700         int higher_order_a = INT_MAX;
701         int higher_order_b = INT_MAX;
702         if (is_order_function(coeff(*s, a_max)))
703                 higher_order_a = a_max + b_min;
704         if (is_order_function(other.coeff(*s, b_max)))
705                 higher_order_b = b_max + a_min;
706         int higher_order_c = std::min(higher_order_a, higher_order_b);
707         if (cdeg_max >= higher_order_c)
708                 cdeg_max = higher_order_c - 1;
709         
710         for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
711                 ex co = _ex0();
712                 // c(i)=a(0)b(i)+...+a(i)b(0)
713                 for (int i=a_min; cdeg-i>=b_min; ++i) {
714                         ex a_coeff = coeff(*s, i);
715                         ex b_coeff = other.coeff(*s, cdeg-i);
716                         if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
717                                 co += a_coeff * b_coeff;
718                 }
719                 if (!co.is_zero())
720                         new_seq.push_back(expair(co, numeric(cdeg)));
721         }
722         if (higher_order_c < INT_MAX)
723                 new_seq.push_back(expair(Order(_ex1()), numeric(higher_order_c)));
724         return pseries(relational(var,point), new_seq);
725 }
726
727
728 /** Implementation of ex::series() for product. This performs series
729  *  multiplication when multiplying series.
730  *  @see ex::series */
731 ex mul::series(const relational & r, int order, unsigned options) const
732 {
733         ex acc; // Series accumulator
734         
735         // Get first term from overall_coeff
736         acc = overall_coeff.series(r, order, options);
737         
738         // Multiply with remaining terms
739         epvector::const_iterator it = seq.begin();
740         epvector::const_iterator itend = seq.end();
741         for (; it!=itend; ++it) {
742                 ex op = it->rest;
743                 if (op.info(info_flags::numeric)) {
744                         // series * const (special case, faster)
745                         ex f = power(op, it->coeff);
746                         acc = ex_to_pseries(acc).mul_const(ex_to_numeric(f));
747                         continue;
748                 } else if (!is_ex_exactly_of_type(op, pseries))
749                         op = op.series(r, order, options);
750                 if (!it->coeff.is_equal(_ex1()))
751                         op = ex_to_pseries(op).power_const(ex_to_numeric(it->coeff), order);
752
753                 // Series multiplication
754                 acc = ex_to_pseries(acc).mul_series(ex_to_pseries(op));
755         }
756         return acc;
757 }
758
759
760 /** Compute the p-th power of a series.
761  *
762  *  @param p  power to compute
763  *  @param deg  truncation order of series calculation */
764 ex pseries::power_const(const numeric &p, int deg) const
765 {
766         // method:
767         // let A(x) be this series and for the time being let it start with a
768         // constant (later we'll generalize):
769         //     A(x) = a_0 + a_1*x + a_2*x^2 + ...
770         // We want to compute
771         //     C(x) = A(x)^p
772         //     C(x) = c_0 + c_1*x + c_2*x^2 + ...
773         // Taking the derivative on both sides and multiplying with A(x) one
774         // immediately arrives at
775         //     C'(x)*A(x) = p*C(x)*A'(x)
776         // Multiplying this out and comparing coefficients we get the recurrence
777         // formula
778         //     c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
779         //                    ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
780         // which can easily be solved given the starting value c_0 = (a_0)^p.
781         // For the more general case where the leading coefficient of A(x) is not
782         // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
783         // repeat the above derivation.  The leading power of C2(x) = A2(x)^2 is
784         // then of course x^(p*m) but the recurrence formula still holds.
785         
786         if (seq.size()==0) {
787                 // as a spacial case, handle the empty (zero) series honoring the
788                 // usual power laws such as implemented in power::eval()
789                 if (p.real().is_zero())
790                         throw (std::domain_error("pseries::power_const(): pow(0,I) is undefined"));
791                 else if (p.real().is_negative())
792                         throw (pole_error("pseries::power_const(): division by zero",1));
793                 else
794                         return *this;
795         }
796         
797         const symbol *s = static_cast<symbol *>(var.bp);
798         int ldeg = ldegree(*s);
799         
800         // Compute coefficients of the powered series
801         exvector co;
802         co.reserve(deg);
803         co.push_back(power(coeff(*s, ldeg), p));
804         bool all_sums_zero = true;
805         for (int i=1; i<deg; ++i) {
806                 ex sum = _ex0();
807                 for (int j=1; j<=i; ++j) {
808                         ex c = coeff(*s, j + ldeg);
809                         if (is_order_function(c)) {
810                                 co.push_back(Order(_ex1()));
811                                 break;
812                         } else
813                                 sum += (p * j - (i - j)) * co[i - j] * c;
814                 }
815                 if (!sum.is_zero())
816                         all_sums_zero = false;
817                 co.push_back(sum / coeff(*s, ldeg) / numeric(i));
818         }
819         
820         // Construct new series (of non-zero coefficients)
821         epvector new_seq;
822         bool higher_order = false;
823         for (int i=0; i<deg; ++i) {
824                 if (!co[i].is_zero())
825                         new_seq.push_back(expair(co[i], numeric(i) + p * ldeg));
826                 if (is_order_function(co[i])) {
827                         higher_order = true;
828                         break;
829                 }
830         }
831         if (!higher_order && !all_sums_zero)
832                 new_seq.push_back(expair(Order(_ex1()), numeric(deg) + p * ldeg));
833         return pseries(relational(var,point), new_seq);
834 }
835
836
837 /** Return a new pseries object with the powers shifted by deg. */
838 pseries pseries::shift_exponents(int deg) const
839 {
840         epvector newseq(seq);
841         for (epvector::iterator i=newseq.begin(); i!=newseq.end(); ++i)
842                 i->coeff = i->coeff + deg;
843         return pseries(relational(var, point), newseq);
844 }
845
846
847 /** Implementation of ex::series() for powers. This performs Laurent expansion
848  *  of reciprocals of series at singularities.
849  *  @see ex::series */
850 ex power::series(const relational & r, int order, unsigned options) const
851 {
852         ex e;
853         if (!is_ex_exactly_of_type(basis, pseries)) {
854                 // Basis is not a series, may there be a singularity?
855                 bool must_expand_basis = false;
856                 try {
857                         basis.subs(r);
858                 } catch (pole_error) {
859                         must_expand_basis = true;
860                 }
861                 
862                 // Is the expression of type something^(-int)?
863                 if (!must_expand_basis && !exponent.info(info_flags::negint))
864                         return basic::series(r, order, options);
865                 
866                 // Is the expression of type 0^something?
867                 if (!must_expand_basis && !basis.subs(r).is_zero())
868                         return basic::series(r, order, options);
869                 
870                 // Singularity encountered, expand basis into series
871                 e = basis.series(r, order, options);
872         } else {
873                 // Basis is a series
874                 e = basis;
875         }
876         
877         // Power e
878         return ex_to_pseries(e).power_const(ex_to_numeric(exponent), order);
879 }
880
881
882 /** Re-expansion of a pseries object. */
883 ex pseries::series(const relational & r, int order, unsigned options) const
884 {
885         const ex p = r.rhs();
886         GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
887         const symbol *s = static_cast<symbol *>(r.lhs().bp);
888         
889         if (var.is_equal(*s) && point.is_equal(p)) {
890                 if (order > degree(*s))
891                         return *this;
892                 else {
893                         epvector new_seq;
894                         epvector::const_iterator it = seq.begin(), itend = seq.end();
895                         while (it != itend) {
896                                 int o = ex_to_numeric(it->coeff).to_int();
897                                 if (o >= order) {
898                                         new_seq.push_back(expair(Order(_ex1()), o));
899                                         break;
900                                 }
901                                 new_seq.push_back(*it);
902                                 ++it;
903                         }
904                         return pseries(r, new_seq);
905                 }
906         } else
907                 return convert_to_poly().series(r, order, options);
908 }
909
910
911 /** Compute the truncated series expansion of an expression.
912  *  This function returns an expression containing an object of class pseries 
913  *  to represent the series. If the series does not terminate within the given
914  *  truncation order, the last term of the series will be an order term.
915  *
916  *  @param r  expansion relation, lhs holds variable and rhs holds point
917  *  @param order  truncation order of series calculations
918  *  @param options  of class series_options
919  *  @return an expression holding a pseries object */
920 ex ex::series(const ex & r, int order, unsigned options) const
921 {
922         GINAC_ASSERT(bp!=0);
923         ex e;
924         relational rel_;
925         
926         if (is_ex_exactly_of_type(r,relational))
927                 rel_ = ex_to_relational(r);
928         else if (is_ex_exactly_of_type(r,symbol))
929                 rel_ = relational(r,_ex0());
930         else
931                 throw (std::logic_error("ex::series(): expansion point has unknown type"));
932         
933         try {
934                 e = bp->series(rel_, order, options);
935         } catch (std::exception &x) {
936                 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));
937         }
938         return e;
939 }
940
941 //////////
942 // static member variables
943 //////////
944
945 // protected
946
947 unsigned pseries::precedence = 38;  // for clarity just below add::precedence
948
949 } // namespace GiNaC