0fae94bc7360f348879ef3caff8be1e8f0c7eb10
[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" // for Order function
29 #include "lst.h"
30 #include "mul.h"
31 #include "power.h"
32 #include "relational.h"
33 #include "symbol.h"
34 #include "print.h"
35 #include "archive.h"
36 #include "utils.h"
37 #include "debugmsg.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() : basic(TINFO_pseries)
49 {
50         debugmsg("pseries default ctor", LOGLEVEL_CONSTRUCT);
51 }
52
53 void pseries::copy(const pseries &other)
54 {
55         inherited::copy(other);
56         seq = other.seq;
57         var = other.var;
58         point = other.point;
59 }
60
61 DEFAULT_DESTROY(pseries)
62
63
64 /*
65  *  Other ctors
66  */
67
68 /** Construct pseries from a vector of coefficients and powers.
69  *  expair.rest holds the coefficient, expair.coeff holds the power.
70  *  The powers must be integers (positive or negative) and in ascending order;
71  *  the last coefficient can be Order(_ex1()) to represent a truncated,
72  *  non-terminating series.
73  *
74  *  @param rel_  expansion variable and point (must hold a relational)
75  *  @param ops_  vector of {coefficient, power} pairs (coefficient must not be zero)
76  *  @return newly constructed pseries */
77 pseries::pseries(const ex &rel_, const epvector &ops_) : basic(TINFO_pseries), seq(ops_)
78 {
79         debugmsg("pseries ctor from ex,epvector", LOGLEVEL_CONSTRUCT);
80         GINAC_ASSERT(is_exactly_a<relational>(rel_));
81         GINAC_ASSERT(is_exactly_a<symbol>(rel_.lhs()));
82         point = rel_.rhs();
83         var = rel_.lhs();
84 }
85
86
87 /*
88  *  Archiving
89  */
90
91 pseries::pseries(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
92 {
93         debugmsg("pseries ctor from archive_node", LOGLEVEL_CONSTRUCT);
94         for (unsigned int i=0; true; ++i) {
95                 ex rest;
96                 ex coeff;
97                 if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
98                         seq.push_back(expair(rest, coeff));
99                 else
100                         break;
101         }
102         n.find_ex("var", var, sym_lst);
103         n.find_ex("point", point, sym_lst);
104 }
105
106 void pseries::archive(archive_node &n) const
107 {
108         inherited::archive(n);
109         epvector::const_iterator i = seq.begin(), iend = seq.end();
110         while (i != iend) {
111                 n.add_ex("coeff", i->rest);
112                 n.add_ex("power", i->coeff);
113                 ++i;
114         }
115         n.add_ex("var", var);
116         n.add_ex("point", point);
117 }
118
119 DEFAULT_UNARCHIVE(pseries)
120
121 //////////
122 // functions overriding virtual functions from base classes
123 //////////
124
125 void pseries::print(const print_context & c, unsigned level) const
126 {
127         debugmsg("pseries print", LOGLEVEL_PRINT);
128
129         if (is_a<print_tree>(c)) {
130
131                 c.s << std::string(level, ' ') << class_name()
132                     << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
133                     << std::endl;
134                 unsigned delta_indent = static_cast<const print_tree &>(c).delta_indent;
135                 unsigned num = seq.size();
136                 for (unsigned i=0; i<num; ++i) {
137                         seq[i].rest.print(c, level + delta_indent);
138                         seq[i].coeff.print(c, level + delta_indent);
139                         c.s << std::string(level + delta_indent, ' ') << "-----" << std::endl;
140                 }
141                 var.print(c, level + delta_indent);
142                 point.print(c, level + delta_indent);
143
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                                                 c.s << '^';
185                                                 if (i->coeff.info(info_flags::negative)) {
186                                                         c.s << par_open;
187                                                         i->coeff.print(c);
188                                                         c.s << par_close;
189                                                 } else {
190                                                         if (is_a<print_latex>(c)) {
191                                                                 c.s << '{';
192                                                                 i->coeff.print(c);
193                                                                 c.s << '}';
194                                                         } else
195                                                                 i->coeff.print(c);
196                                                 }
197                                         }
198                                 }
199                         } else
200                                 Order(power(var-point,i->coeff)).print(c);
201                         ++i;
202                 }
203
204                 if (precedence() <= level)
205                         c.s << ")";
206         }
207 }
208
209 int pseries::compare_same_type(const basic & other) const
210 {
211         GINAC_ASSERT(is_of_type(other, pseries));
212         const pseries &o = static_cast<const pseries &>(other);
213         
214         // first compare the lengths of the series...
215         if (seq.size()>o.seq.size())
216                 return 1;
217         if (seq.size()<o.seq.size())
218                 return -1;
219         
220         // ...then the expansion point...
221         int cmpval = var.compare(o.var);
222         if (cmpval)
223                 return cmpval;
224         cmpval = point.compare(o.point);
225         if (cmpval)
226                 return cmpval;
227         
228         // ...and if that failed the individual elements
229         epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
230         while (it!=seq.end() && o_it!=o.seq.end()) {
231                 cmpval = it->compare(*o_it);
232                 if (cmpval)
233                         return cmpval;
234                 ++it;
235                 ++o_it;
236         }
237
238         // so they are equal.
239         return 0;
240 }
241
242 /** Return the number of operands including a possible order term. */
243 unsigned pseries::nops(void) const
244 {
245         return seq.size();
246 }
247
248 /** Return the ith term in the series when represented as a sum. */
249 ex pseries::op(int i) const
250 {
251         if (i < 0 || unsigned(i) >= seq.size())
252                 throw (std::out_of_range("op() out of range"));
253         return seq[i].rest * power(var - point, seq[i].coeff);
254 }
255
256 ex &pseries::let_op(int i)
257 {
258         throw (std::logic_error("let_op not defined for pseries"));
259 }
260
261 /** Return degree of highest power of the series.  This is usually the exponent
262  *  of the Order term.  If s is not the expansion variable of the series, the
263  *  series is examined termwise. */
264 int pseries::degree(const ex &s) const
265 {
266         if (var.is_equal(s)) {
267                 // Return last exponent
268                 if (seq.size())
269                         return ex_to<numeric>((seq.end()-1)->coeff).to_int();
270                 else
271                         return 0;
272         } else {
273                 epvector::const_iterator it = seq.begin(), itend = seq.end();
274                 if (it == itend)
275                         return 0;
276                 int max_pow = INT_MIN;
277                 while (it != itend) {
278                         int pow = it->rest.degree(s);
279                         if (pow > max_pow)
280                                 max_pow = pow;
281                         ++it;
282                 }
283                 return max_pow;
284         }
285 }
286
287 /** Return degree of lowest power of the series.  This is usually the exponent
288  *  of the leading term.  If s is not the expansion variable of the series, the
289  *  series is examined termwise.  If s is the expansion variable but the
290  *  expansion point is not zero the series is not expanded to find the degree.
291  *  I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
292 int pseries::ldegree(const ex &s) const
293 {
294         if (var.is_equal(s)) {
295                 // Return first exponent
296                 if (seq.size())
297                         return ex_to<numeric>((seq.begin())->coeff).to_int();
298                 else
299                         return 0;
300         } else {
301                 epvector::const_iterator it = seq.begin(), itend = seq.end();
302                 if (it == itend)
303                         return 0;
304                 int min_pow = INT_MAX;
305                 while (it != itend) {
306                         int pow = it->rest.ldegree(s);
307                         if (pow < min_pow)
308                                 min_pow = pow;
309                         ++it;
310                 }
311                 return min_pow;
312         }
313 }
314
315 /** Return coefficient of degree n in power series if s is the expansion
316  *  variable.  If the expansion point is nonzero, by definition the n=1
317  *  coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
318  *  the expansion took place in the s in the first place).
319  *  If s is not the expansion variable, an attempt is made to convert the
320  *  series to a polynomial and return the corresponding coefficient from
321  *  there. */
322 ex pseries::coeff(const ex &s, int n) const
323 {
324         if (var.is_equal(s)) {
325                 if (seq.empty())
326                         return _ex0();
327                 
328                 // Binary search in sequence for given power
329                 numeric looking_for = numeric(n);
330                 int lo = 0, hi = seq.size() - 1;
331                 while (lo <= hi) {
332                         int mid = (lo + hi) / 2;
333                         GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
334                         int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
335                         switch (cmp) {
336                                 case -1:
337                                         lo = mid + 1;
338                                         break;
339                                 case 0:
340                                         return seq[mid].rest;
341                                 case 1:
342                                         hi = mid - 1;
343                                         break;
344                                 default:
345                                         throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
346                         }
347                 }
348                 return _ex0();
349         } else
350                 return convert_to_poly().coeff(s, n);
351 }
352
353 /** Does nothing. */
354 ex pseries::collect(const ex &s, bool distributed) const
355 {
356         return *this;
357 }
358
359 /** Perform coefficient-wise automatic term rewriting rules in this class. */
360 ex pseries::eval(int level) const
361 {
362         if (level == 1)
363                 return this->hold();
364         
365         if (level == -max_recursion_level)
366                 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
367         
368         // Construct a new series with evaluated coefficients
369         epvector new_seq;
370         new_seq.reserve(seq.size());
371         epvector::const_iterator it = seq.begin(), itend = seq.end();
372         while (it != itend) {
373                 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
374                 ++it;
375         }
376         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
377 }
378
379 /** Evaluate coefficients numerically. */
380 ex pseries::evalf(int level) const
381 {
382         if (level == 1)
383                 return *this;
384         
385         if (level == -max_recursion_level)
386                 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
387         
388         // Construct a new series with evaluated coefficients
389         epvector new_seq;
390         new_seq.reserve(seq.size());
391         epvector::const_iterator it = seq.begin(), itend = seq.end();
392         while (it != itend) {
393                 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
394                 ++it;
395         }
396         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
397 }
398
399 ex pseries::subs(const lst & ls, const lst & lr, bool no_pattern) const
400 {
401         // If expansion variable is being substituted, convert the series to a
402         // polynomial and do the substitution there because the result might
403         // no longer be a power series
404         if (ls.has(var))
405                 return convert_to_poly(true).subs(ls, lr, no_pattern);
406         
407         // Otherwise construct a new series with substituted coefficients and
408         // expansion point
409         epvector newseq;
410         newseq.reserve(seq.size());
411         epvector::const_iterator it = seq.begin(), itend = seq.end();
412         while (it != itend) {
413                 newseq.push_back(expair(it->rest.subs(ls, lr, no_pattern), it->coeff));
414                 ++it;
415         }
416         return (new pseries(relational(var,point.subs(ls, lr, no_pattern)), newseq))->setflag(status_flags::dynallocated);
417 }
418
419 /** Implementation of ex::expand() for a power series.  It expands all the
420  *  terms individually and returns the resulting series as a new pseries. */
421 ex pseries::expand(unsigned options) const
422 {
423         epvector newseq;
424         epvector::const_iterator i = seq.begin(), end = seq.end();
425         while (i != end) {
426                 ex restexp = i->rest.expand();
427                 if (!restexp.is_zero())
428                         newseq.push_back(expair(restexp, i->coeff));
429                 ++i;
430         }
431         return (new pseries(relational(var,point), newseq))
432                 ->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
433 }
434
435 /** Implementation of ex::diff() for a power series.  It treats the series as a
436  *  polynomial.
437  *  @see ex::diff */
438 ex pseries::derivative(const symbol & s) const
439 {
440         if (s == var) {
441                 epvector new_seq;
442                 epvector::const_iterator it = seq.begin(), itend = seq.end();
443                 
444                 // FIXME: coeff might depend on var
445                 while (it != itend) {
446                         if (is_order_function(it->rest)) {
447                                 new_seq.push_back(expair(it->rest, it->coeff - 1));
448                         } else {
449                                 ex c = it->rest * it->coeff;
450                                 if (!c.is_zero())
451                                         new_seq.push_back(expair(c, it->coeff - 1));
452                         }
453                         ++it;
454                 }
455                 return pseries(relational(var,point), new_seq);
456         } else {
457                 return *this;
458         }
459 }
460
461 ex pseries::convert_to_poly(bool no_order) const
462 {
463         ex e;
464         epvector::const_iterator it = seq.begin(), itend = seq.end();
465         
466         while (it != itend) {
467                 if (is_order_function(it->rest)) {
468                         if (!no_order)
469                                 e += Order(power(var - point, it->coeff));
470                 } else
471                         e += it->rest * power(var - point, it->coeff);
472                 ++it;
473         }
474         return e;
475 }
476
477 bool pseries::is_terminating(void) const
478 {
479         return seq.empty() || !is_order_function((seq.end()-1)->rest);
480 }
481
482
483 /*
484  *  Implementations of series expansion
485  */
486
487 /** Default implementation of ex::series(). This performs Taylor expansion.
488  *  @see ex::series */
489 ex basic::series(const relational & r, int order, unsigned options) const
490 {
491         epvector seq;
492         numeric fac(1);
493         ex deriv = *this;
494         ex coeff = deriv.subs(r);
495         const symbol &s = ex_to<symbol>(r.lhs());
496         
497         if (!coeff.is_zero())
498                 seq.push_back(expair(coeff, _ex0()));
499         
500         int n;
501         for (n=1; n<order; ++n) {
502                 fac = fac.mul(numeric(n));
503                 deriv = deriv.diff(s).expand();
504                 if (deriv.is_zero()) {
505                         // Series terminates
506                         return pseries(r, seq);
507                 }
508                 coeff = deriv.subs(r);
509                 if (!coeff.is_zero())
510                         seq.push_back(expair(fac.inverse() * coeff, numeric(n)));
511         }
512         
513         // Higher-order terms, if present
514         deriv = deriv.diff(s);
515         if (!deriv.expand().is_zero())
516                 seq.push_back(expair(Order(_ex1()), numeric(n)));
517         return pseries(r, seq);
518 }
519
520
521 /** Implementation of ex::series() for symbols.
522  *  @see ex::series */
523 ex symbol::series(const relational & r, int order, unsigned options) const
524 {
525         epvector seq;
526         const ex point = r.rhs();
527         GINAC_ASSERT(is_exactly_a<symbol>(r.lhs()));
528
529         if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
530                 if (order > 0 && !point.is_zero())
531                         seq.push_back(expair(point, _ex0()));
532                 if (order > 1)
533                         seq.push_back(expair(_ex1(), _ex1()));
534                 else
535                         seq.push_back(expair(Order(_ex1()), numeric(order)));
536         } else
537                 seq.push_back(expair(*this, _ex0()));
538         return pseries(r, seq);
539 }
540
541
542 /** Add one series object to another, producing a pseries object that
543  *  represents the sum.
544  *
545  *  @param other  pseries object to add with
546  *  @return the sum as a pseries */
547 ex pseries::add_series(const pseries &other) const
548 {
549         // Adding two series with different variables or expansion points
550         // results in an empty (constant) series 
551         if (!is_compatible_to(other)) {
552                 epvector nul;
553                 nul.push_back(expair(Order(_ex1()), _ex0()));
554                 return pseries(relational(var,point), nul);
555         }
556         
557         // Series addition
558         epvector new_seq;
559         epvector::const_iterator a = seq.begin();
560         epvector::const_iterator b = other.seq.begin();
561         epvector::const_iterator a_end = seq.end();
562         epvector::const_iterator b_end = other.seq.end();
563         int pow_a = INT_MAX, pow_b = INT_MAX;
564         for (;;) {
565                 // If a is empty, fill up with elements from b and stop
566                 if (a == a_end) {
567                         while (b != b_end) {
568                                 new_seq.push_back(*b);
569                                 ++b;
570                         }
571                         break;
572                 } else
573                         pow_a = ex_to<numeric>((*a).coeff).to_int();
574                 
575                 // If b is empty, fill up with elements from a and stop
576                 if (b == b_end) {
577                         while (a != a_end) {
578                                 new_seq.push_back(*a);
579                                 ++a;
580                         }
581                         break;
582                 } else
583                         pow_b = ex_to<numeric>((*b).coeff).to_int();
584                 
585                 // a and b are non-empty, compare powers
586                 if (pow_a < pow_b) {
587                         // a has lesser power, get coefficient from a
588                         new_seq.push_back(*a);
589                         if (is_order_function((*a).rest))
590                                 break;
591                         ++a;
592                 } else if (pow_b < pow_a) {
593                         // b has lesser power, get coefficient from b
594                         new_seq.push_back(*b);
595                         if (is_order_function((*b).rest))
596                                 break;
597                         ++b;
598                 } else {
599                         // Add coefficient of a and b
600                         if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
601                                 new_seq.push_back(expair(Order(_ex1()), (*a).coeff));
602                                 break;  // Order term ends the sequence
603                         } else {
604                                 ex sum = (*a).rest + (*b).rest;
605                                 if (!(sum.is_zero()))
606                                         new_seq.push_back(expair(sum, numeric(pow_a)));
607                                 ++a;
608                                 ++b;
609                         }
610                 }
611         }
612         return pseries(relational(var,point), new_seq);
613 }
614
615
616 /** Implementation of ex::series() for sums. This performs series addition when
617  *  adding pseries objects.
618  *  @see ex::series */
619 ex add::series(const relational & r, int order, unsigned options) const
620 {
621         ex acc; // Series accumulator
622         
623         // Get first term from overall_coeff
624         acc = overall_coeff.series(r, order, options);
625         
626         // Add remaining terms
627         epvector::const_iterator it = seq.begin();
628         epvector::const_iterator itend = seq.end();
629         for (; it!=itend; ++it) {
630                 ex op;
631                 if (is_ex_exactly_of_type(it->rest, pseries))
632                         op = it->rest;
633                 else
634                         op = it->rest.series(r, order, options);
635                 if (!it->coeff.is_equal(_ex1()))
636                         op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it->coeff));
637                 
638                 // Series addition
639                 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
640         }
641         return acc;
642 }
643
644
645 /** Multiply a pseries object with a numeric constant, producing a pseries
646  *  object that represents the product.
647  *
648  *  @param other  constant to multiply with
649  *  @return the product as a pseries */
650 ex pseries::mul_const(const numeric &other) const
651 {
652         epvector new_seq;
653         new_seq.reserve(seq.size());
654         
655         epvector::const_iterator it = seq.begin(), itend = seq.end();
656         while (it != itend) {
657                 if (!is_order_function(it->rest))
658                         new_seq.push_back(expair(it->rest * other, it->coeff));
659                 else
660                         new_seq.push_back(*it);
661                 ++it;
662         }
663         return pseries(relational(var,point), new_seq);
664 }
665
666
667 /** Multiply one pseries object to another, producing a pseries object that
668  *  represents the product.
669  *
670  *  @param other  pseries object to multiply with
671  *  @return the product as a pseries */
672 ex pseries::mul_series(const pseries &other) const
673 {
674         // Multiplying two series with different variables or expansion points
675         // results in an empty (constant) series 
676         if (!is_compatible_to(other)) {
677                 epvector nul;
678                 nul.push_back(expair(Order(_ex1()), _ex0()));
679                 return pseries(relational(var,point), nul);
680         }
681         
682         // Series multiplication
683         epvector new_seq;
684         int a_max = degree(var);
685         int b_max = other.degree(var);
686         int a_min = ldegree(var);
687         int b_min = other.ldegree(var);
688         int cdeg_min = a_min + b_min;
689         int cdeg_max = a_max + b_max;
690         
691         int higher_order_a = INT_MAX;
692         int higher_order_b = INT_MAX;
693         if (is_order_function(coeff(var, a_max)))
694                 higher_order_a = a_max + b_min;
695         if (is_order_function(other.coeff(var, b_max)))
696                 higher_order_b = b_max + a_min;
697         int higher_order_c = std::min(higher_order_a, higher_order_b);
698         if (cdeg_max >= higher_order_c)
699                 cdeg_max = higher_order_c - 1;
700         
701         for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
702                 ex co = _ex0();
703                 // c(i)=a(0)b(i)+...+a(i)b(0)
704                 for (int i=a_min; cdeg-i>=b_min; ++i) {
705                         ex a_coeff = coeff(var, i);
706                         ex b_coeff = other.coeff(var, cdeg-i);
707                         if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
708                                 co += a_coeff * b_coeff;
709                 }
710                 if (!co.is_zero())
711                         new_seq.push_back(expair(co, numeric(cdeg)));
712         }
713         if (higher_order_c < INT_MAX)
714                 new_seq.push_back(expair(Order(_ex1()), numeric(higher_order_c)));
715         return pseries(relational(var, point), new_seq);
716 }
717
718
719 /** Implementation of ex::series() for product. This performs series
720  *  multiplication when multiplying series.
721  *  @see ex::series */
722 ex mul::series(const relational & r, int order, unsigned options) const
723 {
724         pseries acc; // Series accumulator
725
726         // Multiply with remaining terms
727         const epvector::const_iterator itbeg = seq.begin();
728         const epvector::const_iterator itend = seq.end();
729         for (epvector::const_iterator it=itbeg; it!=itend; ++it) {
730                 ex op = recombine_pair_to_ex(*it).series(r, order, options);
731
732                 // Series multiplication
733                 if (it==itbeg)
734                         acc = ex_to<pseries>(op);
735                 else
736                         acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
737         }
738         return acc.mul_const(ex_to<numeric>(overall_coeff));
739 }
740
741
742 /** Compute the p-th power of a series.
743  *
744  *  @param p  power to compute
745  *  @param deg  truncation order of series calculation */
746 ex pseries::power_const(const numeric &p, int deg) const
747 {
748         // method:
749         // (due to Leonhard Euler)
750         // let A(x) be this series and for the time being let it start with a
751         // constant (later we'll generalize):
752         //     A(x) = a_0 + a_1*x + a_2*x^2 + ...
753         // We want to compute
754         //     C(x) = A(x)^p
755         //     C(x) = c_0 + c_1*x + c_2*x^2 + ...
756         // Taking the derivative on both sides and multiplying with A(x) one
757         // immediately arrives at
758         //     C'(x)*A(x) = p*C(x)*A'(x)
759         // Multiplying this out and comparing coefficients we get the recurrence
760         // formula
761         //     c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
762         //                    ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
763         // which can easily be solved given the starting value c_0 = (a_0)^p.
764         // For the more general case where the leading coefficient of A(x) is not
765         // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
766         // repeat the above derivation.  The leading power of C2(x) = A2(x)^2 is
767         // then of course x^(p*m) but the recurrence formula still holds.
768         
769         if (seq.empty()) {
770                 // as a special case, handle the empty (zero) series honoring the
771                 // usual power laws such as implemented in power::eval()
772                 if (p.real().is_zero())
773                         throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
774                 else if (p.real().is_negative())
775                         throw pole_error("pseries::power_const(): division by zero",1);
776                 else
777                         return *this;
778         }
779         
780         const int ldeg = ldegree(var);
781         if (!(p*ldeg).is_integer())
782                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
783
784         // O(x^n)^(-m) is undefined
785         if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
786                 throw pole_error("pseries::power_const(): division by zero",1);
787         
788         // Compute coefficients of the powered series
789         exvector co;
790         co.reserve(deg);
791         co.push_back(power(coeff(var, ldeg), p));
792         bool all_sums_zero = true;
793         for (int i=1; i<deg; ++i) {
794                 ex sum = _ex0();
795                 for (int j=1; j<=i; ++j) {
796                         ex c = coeff(var, j + ldeg);
797                         if (is_order_function(c)) {
798                                 co.push_back(Order(_ex1()));
799                                 break;
800                         } else
801                                 sum += (p * j - (i - j)) * co[i - j] * c;
802                 }
803                 if (!sum.is_zero())
804                         all_sums_zero = false;
805                 co.push_back(sum / coeff(var, ldeg) / i);
806         }
807         
808         // Construct new series (of non-zero coefficients)
809         epvector new_seq;
810         bool higher_order = false;
811         for (int i=0; i<deg; ++i) {
812                 if (!co[i].is_zero())
813                         new_seq.push_back(expair(co[i], p * ldeg + i));
814                 if (is_order_function(co[i])) {
815                         higher_order = true;
816                         break;
817                 }
818         }
819         if (!higher_order && !all_sums_zero)
820                 new_seq.push_back(expair(Order(_ex1()), p * ldeg + deg));
821         return pseries(relational(var,point), new_seq);
822 }
823
824
825 /** Return a new pseries object with the powers shifted by deg. */
826 pseries pseries::shift_exponents(int deg) const
827 {
828         epvector newseq = seq;
829         epvector::iterator i = newseq.begin(), end  = newseq.end();
830         while (i != end) {
831                 i->coeff += deg;
832                 ++i;
833         }
834         return pseries(relational(var, point), newseq);
835 }
836
837
838 /** Implementation of ex::series() for powers. This performs Laurent expansion
839  *  of reciprocals of series at singularities.
840  *  @see ex::series */
841 ex power::series(const relational & r, int order, unsigned options) const
842 {
843         // If basis is already a series, just power it
844         if (is_ex_exactly_of_type(basis, pseries))
845                 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
846
847         // Basis is not a series, may there be a singularity?
848         bool must_expand_basis = false;
849         try {
850                 basis.subs(r);
851         } catch (pole_error) {
852                 must_expand_basis = true;
853         }
854                 
855         // Is the expression of type something^(-int)?
856         if (!must_expand_basis && !exponent.info(info_flags::negint))
857                 return basic::series(r, order, options);
858                 
859         // Is the expression of type 0^something?
860         if (!must_expand_basis && !basis.subs(r).is_zero())
861                 return basic::series(r, order, options);
862
863         // Singularity encountered, is the basis equal to (var - point)?
864         if (basis.is_equal(r.lhs() - r.rhs())) {
865                 epvector new_seq;
866                 if (ex_to<numeric>(exponent).to_int() < order)
867                         new_seq.push_back(expair(_ex1(), exponent));
868                 else
869                         new_seq.push_back(expair(Order(_ex1()), exponent));
870                 return pseries(r, new_seq);
871         }
872
873         // No, expand basis into series
874         ex e = basis.series(r, order, options);
875         return ex_to<pseries>(e).power_const(ex_to<numeric>(exponent), order);
876 }
877
878
879 /** Re-expansion of a pseries object. */
880 ex pseries::series(const relational & r, int order, unsigned options) const
881 {
882         const ex p = r.rhs();
883         GINAC_ASSERT(is_exactly_a<symbol>(r.lhs()));
884         const symbol &s = ex_to<symbol>(r.lhs());
885         
886         if (var.is_equal(s) && point.is_equal(p)) {
887                 if (order > degree(s))
888                         return *this;
889                 else {
890                         epvector new_seq;
891                         epvector::const_iterator it = seq.begin(), itend = seq.end();
892                         while (it != itend) {
893                                 int o = ex_to<numeric>(it->coeff).to_int();
894                                 if (o >= order) {
895                                         new_seq.push_back(expair(Order(_ex1()), o));
896                                         break;
897                                 }
898                                 new_seq.push_back(*it);
899                                 ++it;
900                         }
901                         return pseries(r, new_seq);
902                 }
903         } else
904                 return convert_to_poly().series(r, order, options);
905 }
906
907
908 /** Compute the truncated series expansion of an expression.
909  *  This function returns an expression containing an object of class pseries 
910  *  to represent the series. If the series does not terminate within the given
911  *  truncation order, the last term of the series will be an order term.
912  *
913  *  @param r  expansion relation, lhs holds variable and rhs holds point
914  *  @param order  truncation order of series calculations
915  *  @param options  of class series_options
916  *  @return an expression holding a pseries object */
917 ex ex::series(const ex & r, int order, unsigned options) const
918 {
919         GINAC_ASSERT(bp!=0);
920         ex e;
921         relational rel_;
922         
923         if (is_ex_exactly_of_type(r,relational))
924                 rel_ = ex_to<relational>(r);
925         else if (is_ex_exactly_of_type(r,symbol))
926                 rel_ = relational(r,_ex0());
927         else
928                 throw (std::logic_error("ex::series(): expansion point has unknown type"));
929         
930         try {
931                 e = bp->series(rel_, order, options);
932         } catch (std::exception &x) {
933                 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));
934         }
935         return e;
936 }
937
938 } // namespace GiNaC