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[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) 1999-2004 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 <numeric>
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_OPT(pseries, basic,
42   print_func<print_context>(&pseries::do_print).
43   print_func<print_latex>(&pseries::do_print_latex).
44   print_func<print_tree>(&pseries::do_print_tree).
45   print_func<print_python>(&pseries::do_print_python).
46   print_func<print_python_repr>(&pseries::do_print_python_repr))
47
48
49 /*
50  *  Default constructor
51  */
52
53 pseries::pseries() : inherited(TINFO_pseries) { }
54
55
56 /*
57  *  Other ctors
58  */
59
60 /** Construct pseries from a vector of coefficients and powers.
61  *  expair.rest holds the coefficient, expair.coeff holds the power.
62  *  The powers must be integers (positive or negative) and in ascending order;
63  *  the last coefficient can be Order(_ex1) to represent a truncated,
64  *  non-terminating series.
65  *
66  *  @param rel_  expansion variable and point (must hold a relational)
67  *  @param ops_  vector of {coefficient, power} pairs (coefficient must not be zero)
68  *  @return newly constructed pseries */
69 pseries::pseries(const ex &rel_, const epvector &ops_) : basic(TINFO_pseries), seq(ops_)
70 {
71         GINAC_ASSERT(is_a<relational>(rel_));
72         GINAC_ASSERT(is_a<symbol>(rel_.lhs()));
73         point = rel_.rhs();
74         var = rel_.lhs();
75 }
76
77
78 /*
79  *  Archiving
80  */
81
82 pseries::pseries(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
83 {
84         for (unsigned int i=0; true; ++i) {
85                 ex rest;
86                 ex coeff;
87                 if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
88                         seq.push_back(expair(rest, coeff));
89                 else
90                         break;
91         }
92         n.find_ex("var", var, sym_lst);
93         n.find_ex("point", point, sym_lst);
94 }
95
96 void pseries::archive(archive_node &n) const
97 {
98         inherited::archive(n);
99         epvector::const_iterator i = seq.begin(), iend = seq.end();
100         while (i != iend) {
101                 n.add_ex("coeff", i->rest);
102                 n.add_ex("power", i->coeff);
103                 ++i;
104         }
105         n.add_ex("var", var);
106         n.add_ex("point", point);
107 }
108
109 DEFAULT_UNARCHIVE(pseries)
110
111 //////////
112 // functions overriding virtual functions from base classes
113 //////////
114
115 void pseries::print_series(const print_context & c, const char *openbrace, const char *closebrace, const char *mul_sym, const char *pow_sym, unsigned level) const
116 {
117         if (precedence() <= level)
118                 c.s << '(';
119                 
120         // objects of type pseries must not have any zero entries, so the
121         // trivial (zero) pseries needs a special treatment here:
122         if (seq.empty())
123                 c.s << '0';
124
125         epvector::const_iterator i = seq.begin(), end = seq.end();
126         while (i != end) {
127
128                 // print a sign, if needed
129                 if (i != seq.begin())
130                         c.s << '+';
131
132                 if (!is_order_function(i->rest)) {
133
134                         // print 'rest', i.e. the expansion coefficient
135                         if (i->rest.info(info_flags::numeric) &&
136                                 i->rest.info(info_flags::positive)) {
137                                 i->rest.print(c);
138                         } else {
139                                 c.s << openbrace << '(';
140                                 i->rest.print(c);
141                                 c.s << ')' << closebrace;
142                         }
143
144                         // print 'coeff', something like (x-1)^42
145                         if (!i->coeff.is_zero()) {
146                                 c.s << mul_sym;
147                                 if (!point.is_zero()) {
148                                         c.s << openbrace << '(';
149                                         (var-point).print(c);
150                                         c.s << ')' << closebrace;
151                                 } else
152                                         var.print(c);
153                                 if (i->coeff.compare(_ex1)) {
154                                         c.s << pow_sym;
155                                         c.s << openbrace;
156                                         if (i->coeff.info(info_flags::negative)) {
157                                                 c.s << '(';
158                                                 i->coeff.print(c);
159                                                 c.s << ')';
160                                         } else
161                                                 i->coeff.print(c);
162                                         c.s << closebrace;
163                                 }
164                         }
165                 } else
166                         Order(power(var-point,i->coeff)).print(c);
167                 ++i;
168         }
169
170         if (precedence() <= level)
171                 c.s << ')';
172 }
173
174 void pseries::do_print(const print_context & c, unsigned level) const
175 {
176         print_series(c, "", "", "*", "^", level);
177 }
178
179 void pseries::do_print_latex(const print_latex & c, unsigned level) const
180 {
181         print_series(c, "{", "}", " ", "^", level);
182 }
183
184 void pseries::do_print_python(const print_python & c, unsigned level) const
185 {
186         print_series(c, "", "", "*", "**", level);
187 }
188
189 void pseries::do_print_tree(const print_tree & c, unsigned level) const
190 {
191         c.s << std::string(level, ' ') << class_name() << " @" << this
192             << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
193             << std::endl;
194         size_t num = seq.size();
195         for (size_t i=0; i<num; ++i) {
196                 seq[i].rest.print(c, level + c.delta_indent);
197                 seq[i].coeff.print(c, level + c.delta_indent);
198                 c.s << std::string(level + c.delta_indent, ' ') << "-----" << std::endl;
199         }
200         var.print(c, level + c.delta_indent);
201         point.print(c, level + c.delta_indent);
202 }
203
204 void pseries::do_print_python_repr(const print_python_repr & c, unsigned level) const
205 {
206         c.s << class_name() << "(relational(";
207         var.print(c);
208         c.s << ',';
209         point.print(c);
210         c.s << "),[";
211         size_t num = seq.size();
212         for (size_t i=0; i<num; ++i) {
213                 if (i)
214                         c.s << ',';
215                 c.s << '(';
216                 seq[i].rest.print(c);
217                 c.s << ',';
218                 seq[i].coeff.print(c);
219                 c.s << ')';
220         }
221         c.s << "])";
222 }
223
224 int pseries::compare_same_type(const basic & other) const
225 {
226         GINAC_ASSERT(is_a<pseries>(other));
227         const pseries &o = static_cast<const pseries &>(other);
228         
229         // first compare the lengths of the series...
230         if (seq.size()>o.seq.size())
231                 return 1;
232         if (seq.size()<o.seq.size())
233                 return -1;
234         
235         // ...then the expansion point...
236         int cmpval = var.compare(o.var);
237         if (cmpval)
238                 return cmpval;
239         cmpval = point.compare(o.point);
240         if (cmpval)
241                 return cmpval;
242         
243         // ...and if that failed the individual elements
244         epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
245         while (it!=seq.end() && o_it!=o.seq.end()) {
246                 cmpval = it->compare(*o_it);
247                 if (cmpval)
248                         return cmpval;
249                 ++it;
250                 ++o_it;
251         }
252
253         // so they are equal.
254         return 0;
255 }
256
257 /** Return the number of operands including a possible order term. */
258 size_t pseries::nops() const
259 {
260         return seq.size();
261 }
262
263 /** Return the ith term in the series when represented as a sum. */
264 ex pseries::op(size_t i) const
265 {
266         if (i >= seq.size())
267                 throw (std::out_of_range("op() out of range"));
268
269         return seq[i].rest * power(var - point, seq[i].coeff);
270 }
271
272 /** Return degree of highest power of the series.  This is usually the exponent
273  *  of the Order term.  If s is not the expansion variable of the series, the
274  *  series is examined termwise. */
275 int pseries::degree(const ex &s) const
276 {
277         if (var.is_equal(s)) {
278                 // Return last exponent
279                 if (seq.size())
280                         return ex_to<numeric>((seq.end()-1)->coeff).to_int();
281                 else
282                         return 0;
283         } else {
284                 epvector::const_iterator it = seq.begin(), itend = seq.end();
285                 if (it == itend)
286                         return 0;
287                 int max_pow = INT_MIN;
288                 while (it != itend) {
289                         int pow = it->rest.degree(s);
290                         if (pow > max_pow)
291                                 max_pow = pow;
292                         ++it;
293                 }
294                 return max_pow;
295         }
296 }
297
298 /** Return degree of lowest power of the series.  This is usually the exponent
299  *  of the leading term.  If s is not the expansion variable of the series, the
300  *  series is examined termwise.  If s is the expansion variable but the
301  *  expansion point is not zero the series is not expanded to find the degree.
302  *  I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
303 int pseries::ldegree(const ex &s) const
304 {
305         if (var.is_equal(s)) {
306                 // Return first exponent
307                 if (seq.size())
308                         return ex_to<numeric>((seq.begin())->coeff).to_int();
309                 else
310                         return 0;
311         } else {
312                 epvector::const_iterator it = seq.begin(), itend = seq.end();
313                 if (it == itend)
314                         return 0;
315                 int min_pow = INT_MAX;
316                 while (it != itend) {
317                         int pow = it->rest.ldegree(s);
318                         if (pow < min_pow)
319                                 min_pow = pow;
320                         ++it;
321                 }
322                 return min_pow;
323         }
324 }
325
326 /** Return coefficient of degree n in power series if s is the expansion
327  *  variable.  If the expansion point is nonzero, by definition the n=1
328  *  coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
329  *  the expansion took place in the s in the first place).
330  *  If s is not the expansion variable, an attempt is made to convert the
331  *  series to a polynomial and return the corresponding coefficient from
332  *  there. */
333 ex pseries::coeff(const ex &s, int n) const
334 {
335         if (var.is_equal(s)) {
336                 if (seq.empty())
337                         return _ex0;
338                 
339                 // Binary search in sequence for given power
340                 numeric looking_for = numeric(n);
341                 int lo = 0, hi = seq.size() - 1;
342                 while (lo <= hi) {
343                         int mid = (lo + hi) / 2;
344                         GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
345                         int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
346                         switch (cmp) {
347                                 case -1:
348                                         lo = mid + 1;
349                                         break;
350                                 case 0:
351                                         return seq[mid].rest;
352                                 case 1:
353                                         hi = mid - 1;
354                                         break;
355                                 default:
356                                         throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
357                         }
358                 }
359                 return _ex0;
360         } else
361                 return convert_to_poly().coeff(s, n);
362 }
363
364 /** Does nothing. */
365 ex pseries::collect(const ex &s, bool distributed) const
366 {
367         return *this;
368 }
369
370 /** Perform coefficient-wise automatic term rewriting rules in this class. */
371 ex pseries::eval(int level) const
372 {
373         if (level == 1)
374                 return this->hold();
375         
376         if (level == -max_recursion_level)
377                 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
378         
379         // Construct a new series with evaluated coefficients
380         epvector new_seq;
381         new_seq.reserve(seq.size());
382         epvector::const_iterator it = seq.begin(), itend = seq.end();
383         while (it != itend) {
384                 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
385                 ++it;
386         }
387         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
388 }
389
390 /** Evaluate coefficients numerically. */
391 ex pseries::evalf(int level) const
392 {
393         if (level == 1)
394                 return *this;
395         
396         if (level == -max_recursion_level)
397                 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
398         
399         // Construct a new series with evaluated coefficients
400         epvector new_seq;
401         new_seq.reserve(seq.size());
402         epvector::const_iterator it = seq.begin(), itend = seq.end();
403         while (it != itend) {
404                 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
405                 ++it;
406         }
407         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
408 }
409
410 ex pseries::conjugate() const
411 {
412         epvector * newseq = conjugateepvector(seq);
413         ex newvar = var.conjugate();
414         ex newpoint = point.conjugate();
415
416         if (!newseq     && are_ex_trivially_equal(newvar, var) && are_ex_trivially_equal(point, newpoint)) {
417                 return *this;
418         }
419
420         ex result = (new pseries(newvar==newpoint, newseq ? *newseq : seq))->setflag(status_flags::dynallocated);
421         if (newseq) {
422                 delete newseq;
423         }
424         return result;
425 }
426
427 ex pseries::subs(const exmap & m, unsigned options) const
428 {
429         // If expansion variable is being substituted, convert the series to a
430         // polynomial and do the substitution there because the result might
431         // no longer be a power series
432         if (m.find(var) != m.end())
433                 return convert_to_poly(true).subs(m, options);
434         
435         // Otherwise construct a new series with substituted coefficients and
436         // expansion point
437         epvector newseq;
438         newseq.reserve(seq.size());
439         epvector::const_iterator it = seq.begin(), itend = seq.end();
440         while (it != itend) {
441                 newseq.push_back(expair(it->rest.subs(m, options), it->coeff));
442                 ++it;
443         }
444         return (new pseries(relational(var,point.subs(m, options)), newseq))->setflag(status_flags::dynallocated);
445 }
446
447 /** Implementation of ex::expand() for a power series.  It expands all the
448  *  terms individually and returns the resulting series as a new pseries. */
449 ex pseries::expand(unsigned options) const
450 {
451         epvector newseq;
452         epvector::const_iterator i = seq.begin(), end = seq.end();
453         while (i != end) {
454                 ex restexp = i->rest.expand();
455                 if (!restexp.is_zero())
456                         newseq.push_back(expair(restexp, i->coeff));
457                 ++i;
458         }
459         return (new pseries(relational(var,point), newseq))
460                 ->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
461 }
462
463 /** Implementation of ex::diff() for a power series.
464  *  @see ex::diff */
465 ex pseries::derivative(const symbol & s) const
466 {
467         epvector new_seq;
468         epvector::const_iterator it = seq.begin(), itend = seq.end();
469
470         if (s == var) {
471                 
472                 // FIXME: coeff might depend on var
473                 while (it != itend) {
474                         if (is_order_function(it->rest)) {
475                                 new_seq.push_back(expair(it->rest, it->coeff - 1));
476                         } else {
477                                 ex c = it->rest * it->coeff;
478                                 if (!c.is_zero())
479                                         new_seq.push_back(expair(c, it->coeff - 1));
480                         }
481                         ++it;
482                 }
483
484         } else {
485
486                 while (it != itend) {
487                         if (is_order_function(it->rest)) {
488                                 new_seq.push_back(*it);
489                         } else {
490                                 ex c = it->rest.diff(s);
491                                 if (!c.is_zero())
492                                         new_seq.push_back(expair(c, it->coeff));
493                         }
494                         ++it;
495                 }
496         }
497
498         return pseries(relational(var,point), new_seq);
499 }
500
501 ex pseries::convert_to_poly(bool no_order) const
502 {
503         ex e;
504         epvector::const_iterator it = seq.begin(), itend = seq.end();
505         
506         while (it != itend) {
507                 if (is_order_function(it->rest)) {
508                         if (!no_order)
509                                 e += Order(power(var - point, it->coeff));
510                 } else
511                         e += it->rest * power(var - point, it->coeff);
512                 ++it;
513         }
514         return e;
515 }
516
517 bool pseries::is_terminating() const
518 {
519         return seq.empty() || !is_order_function((seq.end()-1)->rest);
520 }
521
522
523 /*
524  *  Implementations of series expansion
525  */
526
527 /** Default implementation of ex::series(). This performs Taylor expansion.
528  *  @see ex::series */
529 ex basic::series(const relational & r, int order, unsigned options) const
530 {
531         epvector seq;
532         const symbol &s = ex_to<symbol>(r.lhs());
533
534         // default for order-values that make no sense for Taylor expansion
535         if ((order <= 0) && this->has(s)) {
536                 seq.push_back(expair(Order(_ex1), order));
537                 return pseries(r, seq);
538         }
539
540         // do Taylor expansion
541         numeric fac = 1;
542         ex deriv = *this;
543         ex coeff = deriv.subs(r, subs_options::no_pattern);
544
545         if (!coeff.is_zero()) {
546                 seq.push_back(expair(coeff, _ex0));
547         }
548
549         int n;
550         for (n=1; n<order; ++n) {
551                 fac = fac.mul(n);
552                 // We need to test for zero in order to see if the series terminates.
553                 // The problem is that there is no such thing as a perfect test for
554                 // zero.  Expanding the term occasionally helps a little...
555                 deriv = deriv.diff(s).expand();
556                 if (deriv.is_zero())  // Series terminates
557                         return pseries(r, seq);
558
559                 coeff = deriv.subs(r, subs_options::no_pattern);
560                 if (!coeff.is_zero())
561                         seq.push_back(expair(fac.inverse() * coeff, n));
562         }
563         
564         // Higher-order terms, if present
565         deriv = deriv.diff(s);
566         if (!deriv.expand().is_zero())
567                 seq.push_back(expair(Order(_ex1), n));
568         return pseries(r, seq);
569 }
570
571
572 /** Implementation of ex::series() for symbols.
573  *  @see ex::series */
574 ex symbol::series(const relational & r, int order, unsigned options) const
575 {
576         epvector seq;
577         const ex point = r.rhs();
578         GINAC_ASSERT(is_a<symbol>(r.lhs()));
579
580         if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
581                 if (order > 0 && !point.is_zero())
582                         seq.push_back(expair(point, _ex0));
583                 if (order > 1)
584                         seq.push_back(expair(_ex1, _ex1));
585                 else
586                         seq.push_back(expair(Order(_ex1), numeric(order)));
587         } else
588                 seq.push_back(expair(*this, _ex0));
589         return pseries(r, seq);
590 }
591
592
593 /** Add one series object to another, producing a pseries object that
594  *  represents the sum.
595  *
596  *  @param other  pseries object to add with
597  *  @return the sum as a pseries */
598 ex pseries::add_series(const pseries &other) const
599 {
600         // Adding two series with different variables or expansion points
601         // results in an empty (constant) series 
602         if (!is_compatible_to(other)) {
603                 epvector nul;
604                 nul.push_back(expair(Order(_ex1), _ex0));
605                 return pseries(relational(var,point), nul);
606         }
607         
608         // Series addition
609         epvector new_seq;
610         epvector::const_iterator a = seq.begin();
611         epvector::const_iterator b = other.seq.begin();
612         epvector::const_iterator a_end = seq.end();
613         epvector::const_iterator b_end = other.seq.end();
614         int pow_a = INT_MAX, pow_b = INT_MAX;
615         for (;;) {
616                 // If a is empty, fill up with elements from b and stop
617                 if (a == a_end) {
618                         while (b != b_end) {
619                                 new_seq.push_back(*b);
620                                 ++b;
621                         }
622                         break;
623                 } else
624                         pow_a = ex_to<numeric>((*a).coeff).to_int();
625                 
626                 // If b is empty, fill up with elements from a and stop
627                 if (b == b_end) {
628                         while (a != a_end) {
629                                 new_seq.push_back(*a);
630                                 ++a;
631                         }
632                         break;
633                 } else
634                         pow_b = ex_to<numeric>((*b).coeff).to_int();
635                 
636                 // a and b are non-empty, compare powers
637                 if (pow_a < pow_b) {
638                         // a has lesser power, get coefficient from a
639                         new_seq.push_back(*a);
640                         if (is_order_function((*a).rest))
641                                 break;
642                         ++a;
643                 } else if (pow_b < pow_a) {
644                         // b has lesser power, get coefficient from b
645                         new_seq.push_back(*b);
646                         if (is_order_function((*b).rest))
647                                 break;
648                         ++b;
649                 } else {
650                         // Add coefficient of a and b
651                         if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
652                                 new_seq.push_back(expair(Order(_ex1), (*a).coeff));
653                                 break;  // Order term ends the sequence
654                         } else {
655                                 ex sum = (*a).rest + (*b).rest;
656                                 if (!(sum.is_zero()))
657                                         new_seq.push_back(expair(sum, numeric(pow_a)));
658                                 ++a;
659                                 ++b;
660                         }
661                 }
662         }
663         return pseries(relational(var,point), new_seq);
664 }
665
666
667 /** Implementation of ex::series() for sums. This performs series addition when
668  *  adding pseries objects.
669  *  @see ex::series */
670 ex add::series(const relational & r, int order, unsigned options) const
671 {
672         ex acc; // Series accumulator
673         
674         // Get first term from overall_coeff
675         acc = overall_coeff.series(r, order, options);
676         
677         // Add remaining terms
678         epvector::const_iterator it = seq.begin();
679         epvector::const_iterator itend = seq.end();
680         for (; it!=itend; ++it) {
681                 ex op;
682                 if (is_exactly_a<pseries>(it->rest))
683                         op = it->rest;
684                 else
685                         op = it->rest.series(r, order, options);
686                 if (!it->coeff.is_equal(_ex1))
687                         op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it->coeff));
688                 
689                 // Series addition
690                 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
691         }
692         return acc;
693 }
694
695
696 /** Multiply a pseries object with a numeric constant, producing a pseries
697  *  object that represents the product.
698  *
699  *  @param other  constant to multiply with
700  *  @return the product as a pseries */
701 ex pseries::mul_const(const numeric &other) const
702 {
703         epvector new_seq;
704         new_seq.reserve(seq.size());
705         
706         epvector::const_iterator it = seq.begin(), itend = seq.end();
707         while (it != itend) {
708                 if (!is_order_function(it->rest))
709                         new_seq.push_back(expair(it->rest * other, it->coeff));
710                 else
711                         new_seq.push_back(*it);
712                 ++it;
713         }
714         return pseries(relational(var,point), new_seq);
715 }
716
717
718 /** Multiply one pseries object to another, producing a pseries object that
719  *  represents the product.
720  *
721  *  @param other  pseries object to multiply with
722  *  @return the product as a pseries */
723 ex pseries::mul_series(const pseries &other) const
724 {
725         // Multiplying two series with different variables or expansion points
726         // results in an empty (constant) series 
727         if (!is_compatible_to(other)) {
728                 epvector nul;
729                 nul.push_back(expair(Order(_ex1), _ex0));
730                 return pseries(relational(var,point), nul);
731         }
732         
733         // Series multiplication
734         epvector new_seq;
735         int a_max = degree(var);
736         int b_max = other.degree(var);
737         int a_min = ldegree(var);
738         int b_min = other.ldegree(var);
739         int cdeg_min = a_min + b_min;
740         int cdeg_max = a_max + b_max;
741         
742         int higher_order_a = INT_MAX;
743         int higher_order_b = INT_MAX;
744         if (is_order_function(coeff(var, a_max)))
745                 higher_order_a = a_max + b_min;
746         if (is_order_function(other.coeff(var, b_max)))
747                 higher_order_b = b_max + a_min;
748         int higher_order_c = std::min(higher_order_a, higher_order_b);
749         if (cdeg_max >= higher_order_c)
750                 cdeg_max = higher_order_c - 1;
751         
752         for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
753                 ex co = _ex0;
754                 // c(i)=a(0)b(i)+...+a(i)b(0)
755                 for (int i=a_min; cdeg-i>=b_min; ++i) {
756                         ex a_coeff = coeff(var, i);
757                         ex b_coeff = other.coeff(var, cdeg-i);
758                         if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
759                                 co += a_coeff * b_coeff;
760                 }
761                 if (!co.is_zero())
762                         new_seq.push_back(expair(co, numeric(cdeg)));
763         }
764         if (higher_order_c < INT_MAX)
765                 new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c)));
766         return pseries(relational(var, point), new_seq);
767 }
768
769
770 /** Implementation of ex::series() for product. This performs series
771  *  multiplication when multiplying series.
772  *  @see ex::series */
773 ex mul::series(const relational & r, int order, unsigned options) const
774 {
775         pseries acc; // Series accumulator
776
777         GINAC_ASSERT(is_a<symbol>(r.lhs()));
778         const ex& sym = r.lhs();
779                 
780         // holds ldegrees of the series of individual factors
781         std::vector<int> ldegrees;
782
783         // find minimal degrees
784         const epvector::const_iterator itbeg = seq.begin();
785         const epvector::const_iterator itend = seq.end();
786         for (epvector::const_iterator it=itbeg; it!=itend; ++it) {
787                 
788                 ex buf = recombine_pair_to_ex(*it);
789                 
790                 int real_ldegree = buf.expand().ldegree(sym-r.rhs());
791                 if (real_ldegree == 0) {
792                         int orderloop = 0;
793                         do {
794                                 orderloop++;
795                                 real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
796                         } while (real_ldegree == orderloop);
797                 }
798
799                 ldegrees.push_back(real_ldegree);
800         }
801
802         int degsum = std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
803                 
804         // Multiply with remaining terms
805         std::vector<int>::const_iterator itd = ldegrees.begin();
806         for (epvector::const_iterator it=itbeg; it!=itend; ++it, ++itd) {
807
808                 // do series expansion with adjusted order
809                 ex op = recombine_pair_to_ex(*it).series(r, order-degsum+(*itd), options);
810
811                 // Series multiplication
812                 if (it==itbeg)
813                         acc = ex_to<pseries>(op);
814                 else
815                         acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
816         }
817
818         return acc.mul_const(ex_to<numeric>(overall_coeff));
819 }
820
821
822 /** Compute the p-th power of a series.
823  *
824  *  @param p  power to compute
825  *  @param deg  truncation order of series calculation */
826 ex pseries::power_const(const numeric &p, int deg) const
827 {
828         // method:
829         // (due to Leonhard Euler)
830         // let A(x) be this series and for the time being let it start with a
831         // constant (later we'll generalize):
832         //     A(x) = a_0 + a_1*x + a_2*x^2 + ...
833         // We want to compute
834         //     C(x) = A(x)^p
835         //     C(x) = c_0 + c_1*x + c_2*x^2 + ...
836         // Taking the derivative on both sides and multiplying with A(x) one
837         // immediately arrives at
838         //     C'(x)*A(x) = p*C(x)*A'(x)
839         // Multiplying this out and comparing coefficients we get the recurrence
840         // formula
841         //     c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
842         //                    ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
843         // which can easily be solved given the starting value c_0 = (a_0)^p.
844         // For the more general case where the leading coefficient of A(x) is not
845         // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
846         // repeat the above derivation.  The leading power of C2(x) = A2(x)^2 is
847         // then of course x^(p*m) but the recurrence formula still holds.
848         
849         if (seq.empty()) {
850                 // as a special case, handle the empty (zero) series honoring the
851                 // usual power laws such as implemented in power::eval()
852                 if (p.real().is_zero())
853                         throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
854                 else if (p.real().is_negative())
855                         throw pole_error("pseries::power_const(): division by zero",1);
856                 else
857                         return *this;
858         }
859         
860         const int ldeg = ldegree(var);
861         if (!(p*ldeg).is_integer())
862                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
863
864         // adjust number of coefficients
865         deg = deg - p.to_int()*ldeg;
866         
867         // O(x^n)^(-m) is undefined
868         if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
869                 throw pole_error("pseries::power_const(): division by zero",1);
870         
871         // Compute coefficients of the powered series
872         exvector co;
873         co.reserve(deg);
874         co.push_back(power(coeff(var, ldeg), p));
875         for (int i=1; i<deg; ++i) {
876                 ex sum = _ex0;
877                 for (int j=1; j<=i; ++j) {
878                         ex c = coeff(var, j + ldeg);
879                         if (is_order_function(c)) {
880                                 co.push_back(Order(_ex1));
881                                 break;
882                         } else
883                                 sum += (p * j - (i - j)) * co[i - j] * c;
884                 }
885                 co.push_back(sum / coeff(var, ldeg) / i);
886         }
887         
888         // Construct new series (of non-zero coefficients)
889         epvector new_seq;
890         bool higher_order = false;
891         for (int i=0; i<deg; ++i) {
892                 if (!co[i].is_zero())
893                         new_seq.push_back(expair(co[i], p * ldeg + i));
894                 if (is_order_function(co[i])) {
895                         higher_order = true;
896                         break;
897                 }
898         }
899         if (!higher_order)
900                 new_seq.push_back(expair(Order(_ex1), p * ldeg + deg));
901
902         return pseries(relational(var,point), new_seq);
903 }
904
905
906 /** Return a new pseries object with the powers shifted by deg. */
907 pseries pseries::shift_exponents(int deg) const
908 {
909         epvector newseq = seq;
910         epvector::iterator i = newseq.begin(), end  = newseq.end();
911         while (i != end) {
912                 i->coeff += deg;
913                 ++i;
914         }
915         return pseries(relational(var, point), newseq);
916 }
917
918
919 /** Implementation of ex::series() for powers. This performs Laurent expansion
920  *  of reciprocals of series at singularities.
921  *  @see ex::series */
922 ex power::series(const relational & r, int order, unsigned options) const
923 {
924         // If basis is already a series, just power it
925         if (is_exactly_a<pseries>(basis))
926                 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
927
928         // Basis is not a series, may there be a singularity?
929         bool must_expand_basis = false;
930         try {
931                 basis.subs(r, subs_options::no_pattern);
932         } catch (pole_error) {
933                 must_expand_basis = true;
934         }
935
936         // Is the expression of type something^(-int)?
937         if (!must_expand_basis && !exponent.info(info_flags::negint))
938                 return basic::series(r, order, options);
939
940         // Is the expression of type 0^something?
941         if (!must_expand_basis && !basis.subs(r, subs_options::no_pattern).is_zero())
942                 return basic::series(r, order, options);
943
944         // Singularity encountered, is the basis equal to (var - point)?
945         if (basis.is_equal(r.lhs() - r.rhs())) {
946                 epvector new_seq;
947                 if (ex_to<numeric>(exponent).to_int() < order)
948                         new_seq.push_back(expair(_ex1, exponent));
949                 else
950                         new_seq.push_back(expair(Order(_ex1), exponent));
951                 return pseries(r, new_seq);
952         }
953
954         // No, expand basis into series
955
956         int intexp = ex_to<numeric>(exponent).to_int();
957         const ex& sym = r.lhs();
958         // find existing minimal degree
959         int real_ldegree = basis.expand().ldegree(sym-r.rhs());
960         if (real_ldegree == 0) {
961                 int orderloop = 0;
962                 do {
963                         orderloop++;
964                         real_ldegree = basis.series(r, orderloop, options).ldegree(sym);
965                 } while (real_ldegree == orderloop);
966         }
967
968         ex e = basis.series(r, order + real_ldegree*(1-intexp), options);
969         
970         ex result;
971         try {
972                 result = ex_to<pseries>(e).power_const(intexp, order);
973         }
974         catch (pole_error) {
975                 epvector ser;
976                 ser.push_back(expair(Order(_ex1), order));
977                 result = pseries(r, ser);
978         }
979
980         return result;
981 }
982
983
984 /** Re-expansion of a pseries object. */
985 ex pseries::series(const relational & r, int order, unsigned options) const
986 {
987         const ex p = r.rhs();
988         GINAC_ASSERT(is_a<symbol>(r.lhs()));
989         const symbol &s = ex_to<symbol>(r.lhs());
990         
991         if (var.is_equal(s) && point.is_equal(p)) {
992                 if (order > degree(s))
993                         return *this;
994                 else {
995                         epvector new_seq;
996                         epvector::const_iterator it = seq.begin(), itend = seq.end();
997                         while (it != itend) {
998                                 int o = ex_to<numeric>(it->coeff).to_int();
999                                 if (o >= order) {
1000                                         new_seq.push_back(expair(Order(_ex1), o));
1001                                         break;
1002                                 }
1003                                 new_seq.push_back(*it);
1004                                 ++it;
1005                         }
1006                         return pseries(r, new_seq);
1007                 }
1008         } else
1009                 return convert_to_poly().series(r, order, options);
1010 }
1011
1012
1013 /** Compute the truncated series expansion of an expression.
1014  *  This function returns an expression containing an object of class pseries 
1015  *  to represent the series. If the series does not terminate within the given
1016  *  truncation order, the last term of the series will be an order term.
1017  *
1018  *  @param r  expansion relation, lhs holds variable and rhs holds point
1019  *  @param order  truncation order of series calculations
1020  *  @param options  of class series_options
1021  *  @return an expression holding a pseries object */
1022 ex ex::series(const ex & r, int order, unsigned options) const
1023 {
1024         ex e;
1025         relational rel_;
1026         
1027         if (is_a<relational>(r))
1028                 rel_ = ex_to<relational>(r);
1029         else if (is_a<symbol>(r))
1030                 rel_ = relational(r,_ex0);
1031         else
1032                 throw (std::logic_error("ex::series(): expansion point has unknown type"));
1033         
1034         try {
1035                 e = bp->series(rel_, order, options);
1036         } catch (std::exception &x) {
1037                 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));
1038         }
1039         return e;
1040 }
1041
1042 } // namespace GiNaC