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