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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) 1999-2005 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 "integral.h"
37 #include "archive.h"
38 #include "utils.h"
39
40 namespace GiNaC {
41
42 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(pseries, basic,
43   print_func<print_context>(&pseries::do_print).
44   print_func<print_latex>(&pseries::do_print_latex).
45   print_func<print_tree>(&pseries::do_print_tree).
46   print_func<print_python>(&pseries::do_print_python).
47   print_func<print_python_repr>(&pseries::do_print_python_repr))
48
49
50 /*
51  *  Default constructor
52  */
53
54 pseries::pseries() : inherited(TINFO_pseries) { }
55
56
57 /*
58  *  Other ctors
59  */
60
61 /** Construct pseries from a vector of coefficients and powers.
62  *  expair.rest holds the coefficient, expair.coeff holds the power.
63  *  The powers must be integers (positive or negative) and in ascending order;
64  *  the last coefficient can be Order(_ex1) to represent a truncated,
65  *  non-terminating series.
66  *
67  *  @param rel_  expansion variable and point (must hold a relational)
68  *  @param ops_  vector of {coefficient, power} pairs (coefficient must not be zero)
69  *  @return newly constructed pseries */
70 pseries::pseries(const ex &rel_, const epvector &ops_) : basic(TINFO_pseries), seq(ops_)
71 {
72         GINAC_ASSERT(is_a<relational>(rel_));
73         GINAC_ASSERT(is_a<symbol>(rel_.lhs()));
74         point = rel_.rhs();
75         var = rel_.lhs();
76 }
77
78
79 /*
80  *  Archiving
81  */
82
83 pseries::pseries(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
84 {
85         for (unsigned int i=0; true; ++i) {
86                 ex rest;
87                 ex coeff;
88                 if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
89                         seq.push_back(expair(rest, coeff));
90                 else
91                         break;
92         }
93         n.find_ex("var", var, sym_lst);
94         n.find_ex("point", point, sym_lst);
95 }
96
97 void pseries::archive(archive_node &n) const
98 {
99         inherited::archive(n);
100         epvector::const_iterator i = seq.begin(), iend = seq.end();
101         while (i != iend) {
102                 n.add_ex("coeff", i->rest);
103                 n.add_ex("power", i->coeff);
104                 ++i;
105         }
106         n.add_ex("var", var);
107         n.add_ex("point", point);
108 }
109
110 DEFAULT_UNARCHIVE(pseries)
111
112 //////////
113 // functions overriding virtual functions from base classes
114 //////////
115
116 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
117 {
118         if (precedence() <= level)
119                 c.s << '(';
120                 
121         // objects of type pseries must not have any zero entries, so the
122         // trivial (zero) pseries needs a special treatment here:
123         if (seq.empty())
124                 c.s << '0';
125
126         epvector::const_iterator i = seq.begin(), end = seq.end();
127         while (i != end) {
128
129                 // print a sign, if needed
130                 if (i != seq.begin())
131                         c.s << '+';
132
133                 if (!is_order_function(i->rest)) {
134
135                         // print 'rest', i.e. the expansion coefficient
136                         if (i->rest.info(info_flags::numeric) &&
137                                 i->rest.info(info_flags::positive)) {
138                                 i->rest.print(c);
139                         } else {
140                                 c.s << openbrace << '(';
141                                 i->rest.print(c);
142                                 c.s << ')' << closebrace;
143                         }
144
145                         // print 'coeff', something like (x-1)^42
146                         if (!i->coeff.is_zero()) {
147                                 c.s << mul_sym;
148                                 if (!point.is_zero()) {
149                                         c.s << openbrace << '(';
150                                         (var-point).print(c);
151                                         c.s << ')' << closebrace;
152                                 } else
153                                         var.print(c);
154                                 if (i->coeff.compare(_ex1)) {
155                                         c.s << pow_sym;
156                                         c.s << openbrace;
157                                         if (i->coeff.info(info_flags::negative)) {
158                                                 c.s << '(';
159                                                 i->coeff.print(c);
160                                                 c.s << ')';
161                                         } else
162                                                 i->coeff.print(c);
163                                         c.s << closebrace;
164                                 }
165                         }
166                 } else
167                         Order(power(var-point,i->coeff)).print(c);
168                 ++i;
169         }
170
171         if (precedence() <= level)
172                 c.s << ')';
173 }
174
175 void pseries::do_print(const print_context & c, unsigned level) const
176 {
177         print_series(c, "", "", "*", "^", level);
178 }
179
180 void pseries::do_print_latex(const print_latex & c, unsigned level) const
181 {
182         print_series(c, "{", "}", " ", "^", level);
183 }
184
185 void pseries::do_print_python(const print_python & c, unsigned level) const
186 {
187         print_series(c, "", "", "*", "**", level);
188 }
189
190 void pseries::do_print_tree(const print_tree & c, unsigned level) const
191 {
192         c.s << std::string(level, ' ') << class_name() << " @" << this
193             << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
194             << std::endl;
195         size_t num = seq.size();
196         for (size_t i=0; i<num; ++i) {
197                 seq[i].rest.print(c, level + c.delta_indent);
198                 seq[i].coeff.print(c, level + c.delta_indent);
199                 c.s << std::string(level + c.delta_indent, ' ') << "-----" << std::endl;
200         }
201         var.print(c, level + c.delta_indent);
202         point.print(c, level + c.delta_indent);
203 }
204
205 void pseries::do_print_python_repr(const print_python_repr & c, unsigned level) const
206 {
207         c.s << class_name() << "(relational(";
208         var.print(c);
209         c.s << ',';
210         point.print(c);
211         c.s << "),[";
212         size_t num = seq.size();
213         for (size_t i=0; i<num; ++i) {
214                 if (i)
215                         c.s << ',';
216                 c.s << '(';
217                 seq[i].rest.print(c);
218                 c.s << ',';
219                 seq[i].coeff.print(c);
220                 c.s << ')';
221         }
222         c.s << "])";
223 }
224
225 int pseries::compare_same_type(const basic & other) const
226 {
227         GINAC_ASSERT(is_a<pseries>(other));
228         const pseries &o = static_cast<const pseries &>(other);
229         
230         // first compare the lengths of the series...
231         if (seq.size()>o.seq.size())
232                 return 1;
233         if (seq.size()<o.seq.size())
234                 return -1;
235         
236         // ...then the expansion point...
237         int cmpval = var.compare(o.var);
238         if (cmpval)
239                 return cmpval;
240         cmpval = point.compare(o.point);
241         if (cmpval)
242                 return cmpval;
243         
244         // ...and if that failed the individual elements
245         epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
246         while (it!=seq.end() && o_it!=o.seq.end()) {
247                 cmpval = it->compare(*o_it);
248                 if (cmpval)
249                         return cmpval;
250                 ++it;
251                 ++o_it;
252         }
253
254         // so they are equal.
255         return 0;
256 }
257
258 /** Return the number of operands including a possible order term. */
259 size_t pseries::nops() const
260 {
261         return seq.size();
262 }
263
264 /** Return the ith term in the series when represented as a sum. */
265 ex pseries::op(size_t i) const
266 {
267         if (i >= seq.size())
268                 throw (std::out_of_range("op() out of range"));
269
270         if (is_order_function(seq[i].rest))
271                 return Order(power(var-point, seq[i].coeff));
272         return seq[i].rest * power(var - point, seq[i].coeff);
273 }
274
275 /** Return degree of highest power of the series.  This is usually the exponent
276  *  of the Order term.  If s is not the expansion variable of the series, the
277  *  series is examined termwise. */
278 int pseries::degree(const ex &s) const
279 {
280         if (var.is_equal(s)) {
281                 // Return last exponent
282                 if (seq.size())
283                         return ex_to<numeric>((seq.end()-1)->coeff).to_int();
284                 else
285                         return 0;
286         } else {
287                 epvector::const_iterator it = seq.begin(), itend = seq.end();
288                 if (it == itend)
289                         return 0;
290                 int max_pow = INT_MIN;
291                 while (it != itend) {
292                         int pow = it->rest.degree(s);
293                         if (pow > max_pow)
294                                 max_pow = pow;
295                         ++it;
296                 }
297                 return max_pow;
298         }
299 }
300
301 /** Return degree of lowest power of the series.  This is usually the exponent
302  *  of the leading term.  If s is not the expansion variable of the series, the
303  *  series is examined termwise.  If s is the expansion variable but the
304  *  expansion point is not zero the series is not expanded to find the degree.
305  *  I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
306 int pseries::ldegree(const ex &s) const
307 {
308         if (var.is_equal(s)) {
309                 // Return first exponent
310                 if (seq.size())
311                         return ex_to<numeric>((seq.begin())->coeff).to_int();
312                 else
313                         return 0;
314         } else {
315                 epvector::const_iterator it = seq.begin(), itend = seq.end();
316                 if (it == itend)
317                         return 0;
318                 int min_pow = INT_MAX;
319                 while (it != itend) {
320                         int pow = it->rest.ldegree(s);
321                         if (pow < min_pow)
322                                 min_pow = pow;
323                         ++it;
324                 }
325                 return min_pow;
326         }
327 }
328
329 /** Return coefficient of degree n in power series if s is the expansion
330  *  variable.  If the expansion point is nonzero, by definition the n=1
331  *  coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
332  *  the expansion took place in the s in the first place).
333  *  If s is not the expansion variable, an attempt is made to convert the
334  *  series to a polynomial and return the corresponding coefficient from
335  *  there. */
336 ex pseries::coeff(const ex &s, int n) const
337 {
338         if (var.is_equal(s)) {
339                 if (seq.empty())
340                         return _ex0;
341                 
342                 // Binary search in sequence for given power
343                 numeric looking_for = numeric(n);
344                 int lo = 0, hi = seq.size() - 1;
345                 while (lo <= hi) {
346                         int mid = (lo + hi) / 2;
347                         GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
348                         int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
349                         switch (cmp) {
350                                 case -1:
351                                         lo = mid + 1;
352                                         break;
353                                 case 0:
354                                         return seq[mid].rest;
355                                 case 1:
356                                         hi = mid - 1;
357                                         break;
358                                 default:
359                                         throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
360                         }
361                 }
362                 return _ex0;
363         } else
364                 return convert_to_poly().coeff(s, n);
365 }
366
367 /** Does nothing. */
368 ex pseries::collect(const ex &s, bool distributed) const
369 {
370         return *this;
371 }
372
373 /** Perform coefficient-wise automatic term rewriting rules in this class. */
374 ex pseries::eval(int level) const
375 {
376         if (level == 1)
377                 return this->hold();
378         
379         if (level == -max_recursion_level)
380                 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
381         
382         // Construct a new series with evaluated coefficients
383         epvector new_seq;
384         new_seq.reserve(seq.size());
385         epvector::const_iterator it = seq.begin(), itend = seq.end();
386         while (it != itend) {
387                 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
388                 ++it;
389         }
390         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
391 }
392
393 /** Evaluate coefficients numerically. */
394 ex pseries::evalf(int level) const
395 {
396         if (level == 1)
397                 return *this;
398         
399         if (level == -max_recursion_level)
400                 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
401         
402         // Construct a new series with evaluated coefficients
403         epvector new_seq;
404         new_seq.reserve(seq.size());
405         epvector::const_iterator it = seq.begin(), itend = seq.end();
406         while (it != itend) {
407                 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
408                 ++it;
409         }
410         return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
411 }
412
413 ex pseries::conjugate() const
414 {
415         epvector * newseq = conjugateepvector(seq);
416         ex newvar = var.conjugate();
417         ex newpoint = point.conjugate();
418
419         if (!newseq     && are_ex_trivially_equal(newvar, var) && are_ex_trivially_equal(point, newpoint)) {
420                 return *this;
421         }
422
423         ex result = (new pseries(newvar==newpoint, newseq ? *newseq : seq))->setflag(status_flags::dynallocated);
424         if (newseq) {
425                 delete newseq;
426         }
427         return result;
428 }
429
430 ex pseries::eval_integ() const
431 {
432         epvector *newseq = NULL;
433         for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
434                 if (newseq) {
435                         newseq->push_back(expair(i->rest.eval_integ(), i->coeff));
436                         continue;
437                 }
438                 ex newterm = i->rest.eval_integ();
439                 if (!are_ex_trivially_equal(newterm, i->rest)) {
440                         newseq = new epvector;
441                         newseq->reserve(seq.size());
442                         for (epvector::const_iterator j=seq.begin(); j!=i; ++j)
443                                 newseq->push_back(*j);
444                         newseq->push_back(expair(newterm, i->coeff));
445                 }
446         }
447
448         ex newpoint = point.eval_integ();
449         if (newseq || !are_ex_trivially_equal(newpoint, point))
450                 return (new pseries(var==newpoint, *newseq))
451                        ->setflag(status_flags::dynallocated);
452         return *this;
453 }
454
455 ex pseries::subs(const exmap & m, unsigned options) const
456 {
457         // If expansion variable is being substituted, convert the series to a
458         // polynomial and do the substitution there because the result might
459         // no longer be a power series
460         if (m.find(var) != m.end())
461                 return convert_to_poly(true).subs(m, options);
462         
463         // Otherwise construct a new series with substituted coefficients and
464         // expansion point
465         epvector newseq;
466         newseq.reserve(seq.size());
467         epvector::const_iterator it = seq.begin(), itend = seq.end();
468         while (it != itend) {
469                 newseq.push_back(expair(it->rest.subs(m, options), it->coeff));
470                 ++it;
471         }
472         return (new pseries(relational(var,point.subs(m, options)), newseq))->setflag(status_flags::dynallocated);
473 }
474
475 /** Implementation of ex::expand() for a power series.  It expands all the
476  *  terms individually and returns the resulting series as a new pseries. */
477 ex pseries::expand(unsigned options) const
478 {
479         epvector newseq;
480         epvector::const_iterator i = seq.begin(), end = seq.end();
481         while (i != end) {
482                 ex restexp = i->rest.expand();
483                 if (!restexp.is_zero())
484                         newseq.push_back(expair(restexp, i->coeff));
485                 ++i;
486         }
487         return (new pseries(relational(var,point), newseq))
488                 ->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
489 }
490
491 /** Implementation of ex::diff() for a power series.
492  *  @see ex::diff */
493 ex pseries::derivative(const symbol & s) const
494 {
495         epvector new_seq;
496         epvector::const_iterator it = seq.begin(), itend = seq.end();
497
498         if (s == var) {
499                 
500                 // FIXME: coeff might depend on var
501                 while (it != itend) {
502                         if (is_order_function(it->rest)) {
503                                 new_seq.push_back(expair(it->rest, it->coeff - 1));
504                         } else {
505                                 ex c = it->rest * it->coeff;
506                                 if (!c.is_zero())
507                                         new_seq.push_back(expair(c, it->coeff - 1));
508                         }
509                         ++it;
510                 }
511
512         } else {
513
514                 while (it != itend) {
515                         if (is_order_function(it->rest)) {
516                                 new_seq.push_back(*it);
517                         } else {
518                                 ex c = it->rest.diff(s);
519                                 if (!c.is_zero())
520                                         new_seq.push_back(expair(c, it->coeff));
521                         }
522                         ++it;
523                 }
524         }
525
526         return pseries(relational(var,point), new_seq);
527 }
528
529 ex pseries::convert_to_poly(bool no_order) const
530 {
531         ex e;
532         epvector::const_iterator it = seq.begin(), itend = seq.end();
533         
534         while (it != itend) {
535                 if (is_order_function(it->rest)) {
536                         if (!no_order)
537                                 e += Order(power(var - point, it->coeff));
538                 } else
539                         e += it->rest * power(var - point, it->coeff);
540                 ++it;
541         }
542         return e;
543 }
544
545 bool pseries::is_terminating() const
546 {
547         return seq.empty() || !is_order_function((seq.end()-1)->rest);
548 }
549
550 ex pseries::coeffop(size_t i) const
551 {
552         if (i >=nops())
553                 throw (std::out_of_range("coeffop() out of range"));
554         return seq[i].rest;
555 }
556
557 ex pseries::exponop(size_t i) const
558 {
559         if (i >= nops())
560                 throw (std::out_of_range("exponop() out of range"));
561         return seq[i].coeff;
562 }
563
564
565 /*
566  *  Implementations of series expansion
567  */
568
569 /** Default implementation of ex::series(). This performs Taylor expansion.
570  *  @see ex::series */
571 ex basic::series(const relational & r, int order, unsigned options) const
572 {
573         epvector seq;
574         const symbol &s = ex_to<symbol>(r.lhs());
575
576         // default for order-values that make no sense for Taylor expansion
577         if ((order <= 0) && this->has(s)) {
578                 seq.push_back(expair(Order(_ex1), order));
579                 return pseries(r, seq);
580         }
581
582         // do Taylor expansion
583         numeric fac = 1;
584         ex deriv = *this;
585         ex coeff = deriv.subs(r, subs_options::no_pattern);
586
587         if (!coeff.is_zero()) {
588                 seq.push_back(expair(coeff, _ex0));
589         }
590
591         int n;
592         for (n=1; n<order; ++n) {
593                 fac = fac.mul(n);
594                 // We need to test for zero in order to see if the series terminates.
595                 // The problem is that there is no such thing as a perfect test for
596                 // zero.  Expanding the term occasionally helps a little...
597                 deriv = deriv.diff(s).expand();
598                 if (deriv.is_zero())  // Series terminates
599                         return pseries(r, seq);
600
601                 coeff = deriv.subs(r, subs_options::no_pattern);
602                 if (!coeff.is_zero())
603                         seq.push_back(expair(fac.inverse() * coeff, n));
604         }
605         
606         // Higher-order terms, if present
607         deriv = deriv.diff(s);
608         if (!deriv.expand().is_zero())
609                 seq.push_back(expair(Order(_ex1), n));
610         return pseries(r, seq);
611 }
612
613
614 /** Implementation of ex::series() for symbols.
615  *  @see ex::series */
616 ex symbol::series(const relational & r, int order, unsigned options) const
617 {
618         epvector seq;
619         const ex point = r.rhs();
620         GINAC_ASSERT(is_a<symbol>(r.lhs()));
621
622         if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
623                 if (order > 0 && !point.is_zero())
624                         seq.push_back(expair(point, _ex0));
625                 if (order > 1)
626                         seq.push_back(expair(_ex1, _ex1));
627                 else
628                         seq.push_back(expair(Order(_ex1), numeric(order)));
629         } else
630                 seq.push_back(expair(*this, _ex0));
631         return pseries(r, seq);
632 }
633
634
635 /** Add one series object to another, producing a pseries object that
636  *  represents the sum.
637  *
638  *  @param other  pseries object to add with
639  *  @return the sum as a pseries */
640 ex pseries::add_series(const pseries &other) const
641 {
642         // Adding two series with different variables or expansion points
643         // results in an empty (constant) series 
644         if (!is_compatible_to(other)) {
645                 epvector nul;
646                 nul.push_back(expair(Order(_ex1), _ex0));
647                 return pseries(relational(var,point), nul);
648         }
649         
650         // Series addition
651         epvector new_seq;
652         epvector::const_iterator a = seq.begin();
653         epvector::const_iterator b = other.seq.begin();
654         epvector::const_iterator a_end = seq.end();
655         epvector::const_iterator b_end = other.seq.end();
656         int pow_a = INT_MAX, pow_b = INT_MAX;
657         for (;;) {
658                 // If a is empty, fill up with elements from b and stop
659                 if (a == a_end) {
660                         while (b != b_end) {
661                                 new_seq.push_back(*b);
662                                 ++b;
663                         }
664                         break;
665                 } else
666                         pow_a = ex_to<numeric>((*a).coeff).to_int();
667                 
668                 // If b is empty, fill up with elements from a and stop
669                 if (b == b_end) {
670                         while (a != a_end) {
671                                 new_seq.push_back(*a);
672                                 ++a;
673                         }
674                         break;
675                 } else
676                         pow_b = ex_to<numeric>((*b).coeff).to_int();
677                 
678                 // a and b are non-empty, compare powers
679                 if (pow_a < pow_b) {
680                         // a has lesser power, get coefficient from a
681                         new_seq.push_back(*a);
682                         if (is_order_function((*a).rest))
683                                 break;
684                         ++a;
685                 } else if (pow_b < pow_a) {
686                         // b has lesser power, get coefficient from b
687                         new_seq.push_back(*b);
688                         if (is_order_function((*b).rest))
689                                 break;
690                         ++b;
691                 } else {
692                         // Add coefficient of a and b
693                         if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
694                                 new_seq.push_back(expair(Order(_ex1), (*a).coeff));
695                                 break;  // Order term ends the sequence
696                         } else {
697                                 ex sum = (*a).rest + (*b).rest;
698                                 if (!(sum.is_zero()))
699                                         new_seq.push_back(expair(sum, numeric(pow_a)));
700                                 ++a;
701                                 ++b;
702                         }
703                 }
704         }
705         return pseries(relational(var,point), new_seq);
706 }
707
708
709 /** Implementation of ex::series() for sums. This performs series addition when
710  *  adding pseries objects.
711  *  @see ex::series */
712 ex add::series(const relational & r, int order, unsigned options) const
713 {
714         ex acc; // Series accumulator
715         
716         // Get first term from overall_coeff
717         acc = overall_coeff.series(r, order, options);
718         
719         // Add remaining terms
720         epvector::const_iterator it = seq.begin();
721         epvector::const_iterator itend = seq.end();
722         for (; it!=itend; ++it) {
723                 ex op;
724                 if (is_exactly_a<pseries>(it->rest))
725                         op = it->rest;
726                 else
727                         op = it->rest.series(r, order, options);
728                 if (!it->coeff.is_equal(_ex1))
729                         op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it->coeff));
730                 
731                 // Series addition
732                 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
733         }
734         return acc;
735 }
736
737
738 /** Multiply a pseries object with a numeric constant, producing a pseries
739  *  object that represents the product.
740  *
741  *  @param other  constant to multiply with
742  *  @return the product as a pseries */
743 ex pseries::mul_const(const numeric &other) const
744 {
745         epvector new_seq;
746         new_seq.reserve(seq.size());
747         
748         epvector::const_iterator it = seq.begin(), itend = seq.end();
749         while (it != itend) {
750                 if (!is_order_function(it->rest))
751                         new_seq.push_back(expair(it->rest * other, it->coeff));
752                 else
753                         new_seq.push_back(*it);
754                 ++it;
755         }
756         return pseries(relational(var,point), new_seq);
757 }
758
759
760 /** Multiply one pseries object to another, producing a pseries object that
761  *  represents the product.
762  *
763  *  @param other  pseries object to multiply with
764  *  @return the product as a pseries */
765 ex pseries::mul_series(const pseries &other) const
766 {
767         // Multiplying two series with different variables or expansion points
768         // results in an empty (constant) series 
769         if (!is_compatible_to(other)) {
770                 epvector nul;
771                 nul.push_back(expair(Order(_ex1), _ex0));
772                 return pseries(relational(var,point), nul);
773         }
774
775         if (seq.empty() || other.seq.empty()) {
776                 return (new pseries(var==point, epvector()))
777                        ->setflag(status_flags::dynallocated);
778         }
779         
780         // Series multiplication
781         epvector new_seq;
782         int a_max = degree(var);
783         int b_max = other.degree(var);
784         int a_min = ldegree(var);
785         int b_min = other.ldegree(var);
786         int cdeg_min = a_min + b_min;
787         int cdeg_max = a_max + b_max;
788         
789         int higher_order_a = INT_MAX;
790         int higher_order_b = INT_MAX;
791         if (is_order_function(coeff(var, a_max)))
792                 higher_order_a = a_max + b_min;
793         if (is_order_function(other.coeff(var, b_max)))
794                 higher_order_b = b_max + a_min;
795         int higher_order_c = std::min(higher_order_a, higher_order_b);
796         if (cdeg_max >= higher_order_c)
797                 cdeg_max = higher_order_c - 1;
798         
799         for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
800                 ex co = _ex0;
801                 // c(i)=a(0)b(i)+...+a(i)b(0)
802                 for (int i=a_min; cdeg-i>=b_min; ++i) {
803                         ex a_coeff = coeff(var, i);
804                         ex b_coeff = other.coeff(var, cdeg-i);
805                         if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
806                                 co += a_coeff * b_coeff;
807                 }
808                 if (!co.is_zero())
809                         new_seq.push_back(expair(co, numeric(cdeg)));
810         }
811         if (higher_order_c < INT_MAX)
812                 new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c)));
813         return pseries(relational(var, point), new_seq);
814 }
815
816
817 /** Implementation of ex::series() for product. This performs series
818  *  multiplication when multiplying series.
819  *  @see ex::series */
820 ex mul::series(const relational & r, int order, unsigned options) const
821 {
822         pseries acc; // Series accumulator
823
824         GINAC_ASSERT(is_a<symbol>(r.lhs()));
825         const ex& sym = r.lhs();
826                 
827         // holds ldegrees of the series of individual factors
828         std::vector<int> ldegrees;
829
830         // find minimal degrees
831         const epvector::const_iterator itbeg = seq.begin();
832         const epvector::const_iterator itend = seq.end();
833         for (epvector::const_iterator it=itbeg; it!=itend; ++it) {
834
835                 ex expon = it->coeff;
836                 int factor = 1;
837                 ex buf;
838                 if (expon.info(info_flags::integer)) {
839                         buf = it->rest;
840                         factor = ex_to<numeric>(expon).to_int();
841                 } else {
842                         buf = recombine_pair_to_ex(*it);
843                 }
844
845                 int real_ldegree = 0;
846                 try {
847                         real_ldegree = buf.expand().ldegree(sym-r.rhs());
848                 } catch (std::runtime_error) {}
849
850                 if (real_ldegree == 0) {
851                         int orderloop = 0;
852                         do {
853                                 orderloop++;
854                                 real_ldegree = buf.series(r, orderloop, options).ldegree(sym);
855                         } while (real_ldegree == orderloop);
856                 }
857
858                 ldegrees.push_back(factor * real_ldegree);
859         }
860
861         int degsum = std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
862
863         if (degsum >= order) {
864                 epvector epv;
865                 epv.push_back(expair(Order(_ex1), order));
866                 return (new pseries(r, epv))->setflag(status_flags::dynallocated);
867         }
868
869         // Multiply with remaining terms
870         std::vector<int>::const_iterator itd = ldegrees.begin();
871         for (epvector::const_iterator it=itbeg; it!=itend; ++it, ++itd) {
872
873                 // do series expansion with adjusted order
874                 ex op = recombine_pair_to_ex(*it).series(r, order-degsum+(*itd), options);
875
876                 // Series multiplication
877                 if (it == itbeg)
878                         acc = ex_to<pseries>(op);
879                 else
880                         acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
881         }
882
883         return acc.mul_const(ex_to<numeric>(overall_coeff));
884 }
885
886
887 /** Compute the p-th power of a series.
888  *
889  *  @param p  power to compute
890  *  @param deg  truncation order of series calculation */
891 ex pseries::power_const(const numeric &p, int deg) const
892 {
893         // method:
894         // (due to Leonhard Euler)
895         // let A(x) be this series and for the time being let it start with a
896         // constant (later we'll generalize):
897         //     A(x) = a_0 + a_1*x + a_2*x^2 + ...
898         // We want to compute
899         //     C(x) = A(x)^p
900         //     C(x) = c_0 + c_1*x + c_2*x^2 + ...
901         // Taking the derivative on both sides and multiplying with A(x) one
902         // immediately arrives at
903         //     C'(x)*A(x) = p*C(x)*A'(x)
904         // Multiplying this out and comparing coefficients we get the recurrence
905         // formula
906         //     c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
907         //                    ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
908         // which can easily be solved given the starting value c_0 = (a_0)^p.
909         // For the more general case where the leading coefficient of A(x) is not
910         // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
911         // repeat the above derivation.  The leading power of C2(x) = A2(x)^2 is
912         // then of course x^(p*m) but the recurrence formula still holds.
913         
914         if (seq.empty()) {
915                 // as a special case, handle the empty (zero) series honoring the
916                 // usual power laws such as implemented in power::eval()
917                 if (p.real().is_zero())
918                         throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
919                 else if (p.real().is_negative())
920                         throw pole_error("pseries::power_const(): division by zero",1);
921                 else
922                         return *this;
923         }
924         
925         const int ldeg = ldegree(var);
926         if (!(p*ldeg).is_integer())
927                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
928
929         // adjust number of coefficients
930         int numcoeff = deg - (p*ldeg).to_int();
931         if (numcoeff <= 0) {
932                 epvector epv;
933                 epv.reserve(1);
934                 epv.push_back(expair(Order(_ex1), deg));
935                 return (new pseries(relational(var,point), epv))
936                        ->setflag(status_flags::dynallocated);
937         }
938         
939         // O(x^n)^(-m) is undefined
940         if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
941                 throw pole_error("pseries::power_const(): division by zero",1);
942         
943         // Compute coefficients of the powered series
944         exvector co;
945         co.reserve(numcoeff);
946         co.push_back(power(coeff(var, ldeg), p));
947         for (int i=1; i<numcoeff; ++i) {
948                 ex sum = _ex0;
949                 for (int j=1; j<=i; ++j) {
950                         ex c = coeff(var, j + ldeg);
951                         if (is_order_function(c)) {
952                                 co.push_back(Order(_ex1));
953                                 break;
954                         } else
955                                 sum += (p * j - (i - j)) * co[i - j] * c;
956                 }
957                 co.push_back(sum / coeff(var, ldeg) / i);
958         }
959         
960         // Construct new series (of non-zero coefficients)
961         epvector new_seq;
962         bool higher_order = false;
963         for (int i=0; i<numcoeff; ++i) {
964                 if (!co[i].is_zero())
965                         new_seq.push_back(expair(co[i], p * ldeg + i));
966                 if (is_order_function(co[i])) {
967                         higher_order = true;
968                         break;
969                 }
970         }
971         if (!higher_order)
972                 new_seq.push_back(expair(Order(_ex1), p * ldeg + numcoeff));
973
974         return pseries(relational(var,point), new_seq);
975 }
976
977
978 /** Return a new pseries object with the powers shifted by deg. */
979 pseries pseries::shift_exponents(int deg) const
980 {
981         epvector newseq = seq;
982         epvector::iterator i = newseq.begin(), end  = newseq.end();
983         while (i != end) {
984                 i->coeff += deg;
985                 ++i;
986         }
987         return pseries(relational(var, point), newseq);
988 }
989
990
991 /** Implementation of ex::series() for powers. This performs Laurent expansion
992  *  of reciprocals of series at singularities.
993  *  @see ex::series */
994 ex power::series(const relational & r, int order, unsigned options) const
995 {
996         // If basis is already a series, just power it
997         if (is_exactly_a<pseries>(basis))
998                 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
999
1000         // Basis is not a series, may there be a singularity?
1001         bool must_expand_basis = false;
1002         try {
1003                 basis.subs(r, subs_options::no_pattern);
1004         } catch (pole_error) {
1005                 must_expand_basis = true;
1006         }
1007
1008         // Is the expression of type something^(-int)?
1009         if (!must_expand_basis && !exponent.info(info_flags::negint)
1010          && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1011                 return basic::series(r, order, options);
1012
1013         // Is the expression of type 0^something?
1014         if (!must_expand_basis && !basis.subs(r, subs_options::no_pattern).is_zero()
1015          && (!is_a<add>(basis) || !is_a<numeric>(exponent)))
1016                 return basic::series(r, order, options);
1017
1018         // Singularity encountered, is the basis equal to (var - point)?
1019         if (basis.is_equal(r.lhs() - r.rhs())) {
1020                 epvector new_seq;
1021                 if (ex_to<numeric>(exponent).to_int() < order)
1022                         new_seq.push_back(expair(_ex1, exponent));
1023                 else
1024                         new_seq.push_back(expair(Order(_ex1), exponent));
1025                 return pseries(r, new_seq);
1026         }
1027
1028         // No, expand basis into series
1029
1030         numeric numexp;
1031         if (is_a<numeric>(exponent)) {
1032                 numexp = ex_to<numeric>(exponent);
1033         } else {
1034                 numexp = 0;
1035         }
1036         const ex& sym = r.lhs();
1037         // find existing minimal degree
1038         int real_ldegree = basis.expand().ldegree(sym-r.rhs());
1039         if (real_ldegree == 0) {
1040                 int orderloop = 0;
1041                 do {
1042                         orderloop++;
1043                         real_ldegree = basis.series(r, orderloop, options).ldegree(sym);
1044                 } while (real_ldegree == orderloop);
1045         }
1046
1047         if (!(real_ldegree*numexp).is_integer())
1048                 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
1049         ex e = basis.series(r, (order + real_ldegree*(1-numexp)).to_int(), options);
1050         
1051         ex result;
1052         try {
1053                 result = ex_to<pseries>(e).power_const(numexp, order);
1054         } catch (pole_error) {
1055                 epvector ser;
1056                 ser.push_back(expair(Order(_ex1), order));
1057                 result = pseries(r, ser);
1058         }
1059
1060         return result;
1061 }
1062
1063
1064 /** Re-expansion of a pseries object. */
1065 ex pseries::series(const relational & r, int order, unsigned options) const
1066 {
1067         const ex p = r.rhs();
1068         GINAC_ASSERT(is_a<symbol>(r.lhs()));
1069         const symbol &s = ex_to<symbol>(r.lhs());
1070         
1071         if (var.is_equal(s) && point.is_equal(p)) {
1072                 if (order > degree(s))
1073                         return *this;
1074                 else {
1075                         epvector new_seq;
1076                         epvector::const_iterator it = seq.begin(), itend = seq.end();
1077                         while (it != itend) {
1078                                 int o = ex_to<numeric>(it->coeff).to_int();
1079                                 if (o >= order) {
1080                                         new_seq.push_back(expair(Order(_ex1), o));
1081                                         break;
1082                                 }
1083                                 new_seq.push_back(*it);
1084                                 ++it;
1085                         }
1086                         return pseries(r, new_seq);
1087                 }
1088         } else
1089                 return convert_to_poly().series(r, order, options);
1090 }
1091
1092 ex integral::series(const relational & r, int order, unsigned options) const
1093 {
1094         if (x.subs(r) != x)
1095                 throw std::logic_error("Cannot series expand wrt dummy variable");
1096         
1097         // Expanding integrant with r substituted taken in boundaries.
1098         ex fseries = f.series(r, order, options);
1099         epvector fexpansion;
1100         fexpansion.reserve(fseries.nops());
1101         for (size_t i=0; i<fseries.nops(); ++i) {
1102                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1103                 currcoeff = (currcoeff == Order(_ex1))
1104                         ? currcoeff
1105                         : integral(x, a.subs(r), b.subs(r), currcoeff);
1106                 if (currcoeff != 0)
1107                         fexpansion.push_back(
1108                                 expair(currcoeff, ex_to<pseries>(fseries).exponop(i)));
1109         }
1110
1111         // Expanding lower boundary
1112         ex result = (new pseries(r, fexpansion))->setflag(status_flags::dynallocated);
1113         ex aseries = (a-a.subs(r)).series(r, order, options);
1114         fseries = f.series(x == (a.subs(r)), order, options);
1115         for (size_t i=0; i<fseries.nops(); ++i) {
1116                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1117                 if (is_order_function(currcoeff))
1118                         break;
1119                 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1120                 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1121                 currcoeff = currcoeff.series(r, orderforf);
1122                 ex term = ex_to<pseries>(aseries).power_const(ex_to<numeric>(currexpon+1),order);
1123                 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(-1/(currexpon+1)));
1124                 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1125                 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1126         }
1127
1128         // Expanding upper boundary
1129         ex bseries = (b-b.subs(r)).series(r, order, options);
1130         fseries = f.series(x == (b.subs(r)), order, options);
1131         for (size_t i=0; i<fseries.nops(); ++i) {
1132                 ex currcoeff = ex_to<pseries>(fseries).coeffop(i);
1133                 if (is_order_function(currcoeff))
1134                         break;
1135                 ex currexpon = ex_to<pseries>(fseries).exponop(i);
1136                 int orderforf = order-ex_to<numeric>(currexpon).to_int()-1;
1137                 currcoeff = currcoeff.series(r, orderforf);
1138                 ex term = ex_to<pseries>(bseries).power_const(ex_to<numeric>(currexpon+1),order);
1139                 term = ex_to<pseries>(term).mul_const(ex_to<numeric>(1/(currexpon+1)));
1140                 term = ex_to<pseries>(term).mul_series(ex_to<pseries>(currcoeff));
1141                 result = ex_to<pseries>(result).add_series(ex_to<pseries>(term));
1142         }
1143
1144         return result;
1145 }
1146
1147
1148 /** Compute the truncated series expansion of an expression.
1149  *  This function returns an expression containing an object of class pseries 
1150  *  to represent the series. If the series does not terminate within the given
1151  *  truncation order, the last term of the series will be an order term.
1152  *
1153  *  @param r  expansion relation, lhs holds variable and rhs holds point
1154  *  @param order  truncation order of series calculations
1155  *  @param options  of class series_options
1156  *  @return an expression holding a pseries object */
1157 ex ex::series(const ex & r, int order, unsigned options) const
1158 {
1159         ex e;
1160         relational rel_;
1161         
1162         if (is_a<relational>(r))
1163                 rel_ = ex_to<relational>(r);
1164         else if (is_a<symbol>(r))
1165                 rel_ = relational(r,_ex0);
1166         else
1167                 throw (std::logic_error("ex::series(): expansion point has unknown type"));
1168         
1169         try {
1170                 e = bp->series(rel_, order, options);
1171         } catch (std::exception &x) {
1172                 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));
1173         }
1174         return e;
1175 }
1176
1177 } // namespace GiNaC