3 * Implementation of class for extended truncated power series and
4 * methods for series expansion. */
7 * GiNaC Copyright (C) 1999-2000 Johannes Gutenberg University Mainz, Germany
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.
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.
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
32 #include "relational.h"
38 #ifndef NO_NAMESPACE_GINAC
40 #endif // ndef NO_NAMESPACE_GINAC
42 GINAC_IMPLEMENT_REGISTERED_CLASS(pseries, basic)
45 * Default constructor, destructor, copy constructor, assignment operator and helpers
48 pseries::pseries() : basic(TINFO_pseries)
50 debugmsg("pseries default constructor", LOGLEVEL_CONSTRUCT);
55 debugmsg("pseries destructor", LOGLEVEL_DESTRUCT);
59 pseries::pseries(const pseries &other)
61 debugmsg("pseries copy constructor", LOGLEVEL_CONSTRUCT);
65 const pseries &pseries::operator=(const pseries & other)
67 debugmsg("pseries operator=", LOGLEVEL_ASSIGNMENT);
75 void pseries::copy(const pseries &other)
77 inherited::copy(other);
83 void pseries::destroy(bool call_parent)
86 inherited::destroy(call_parent);
94 /** Construct pseries from a vector of coefficients and powers.
95 * expair.rest holds the coefficient, expair.coeff holds the power.
96 * The powers must be integers (positive or negative) and in ascending order;
97 * the last coefficient can be Order(_ex1()) to represent a truncated,
98 * non-terminating series.
100 * @param rel_ expansion variable and point (must hold a relational)
101 * @param ops_ vector of {coefficient, power} pairs (coefficient must not be zero)
102 * @return newly constructed pseries */
103 pseries::pseries(const ex &rel_, const epvector &ops_)
104 : basic(TINFO_pseries), seq(ops_)
106 debugmsg("pseries constructor from ex,epvector", LOGLEVEL_CONSTRUCT);
107 GINAC_ASSERT(is_ex_exactly_of_type(rel_, relational));
108 GINAC_ASSERT(is_ex_exactly_of_type(rel_.lhs(),symbol));
110 var = *static_cast<symbol *>(rel_.lhs().bp);
118 /** Construct object from archive_node. */
119 pseries::pseries(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
121 debugmsg("pseries constructor from archive_node", LOGLEVEL_CONSTRUCT);
122 for (unsigned int i=0; true; ++i) {
125 if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
126 seq.push_back(expair(rest, coeff));
130 n.find_ex("var", var, sym_lst);
131 n.find_ex("point", point, sym_lst);
134 /** Unarchive the object. */
135 ex pseries::unarchive(const archive_node &n, const lst &sym_lst)
137 return (new pseries(n, sym_lst))->setflag(status_flags::dynallocated);
140 /** Archive the object. */
141 void pseries::archive(archive_node &n) const
143 inherited::archive(n);
144 epvector::const_iterator i = seq.begin(), iend = seq.end();
146 n.add_ex("coeff", i->rest);
147 n.add_ex("power", i->coeff);
150 n.add_ex("var", var);
151 n.add_ex("point", point);
155 // functions overriding virtual functions from bases classes
158 basic *pseries::duplicate() const
160 debugmsg("pseries duplicate", LOGLEVEL_DUPLICATE);
161 return new pseries(*this);
164 void pseries::print(std::ostream &os, unsigned upper_precedence) const
166 debugmsg("pseries print", LOGLEVEL_PRINT);
167 for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
169 if (i->rest.is_zero())
171 // print a sign, if needed
174 if (!is_order_function(i->rest)) {
175 // print 'rest', i.e. the expansion coefficient
176 if (i->rest.info(info_flags::numeric) &&
177 i->rest.info(info_flags::positive)) {
180 os << "(" << i->rest << ')';
181 // print 'coeff', something like (x-1)^42
182 if (!i->coeff.is_zero()) {
184 if (!point.is_zero())
185 os << '(' << var-point << ')';
188 if (i->coeff.compare(_ex1())) {
190 if (i->coeff.info(info_flags::negative))
191 os << '(' << i->coeff << ')';
197 os << Order(power(var-point,i->coeff));
203 void pseries::printraw(std::ostream &os) const
205 debugmsg("pseries printraw", LOGLEVEL_PRINT);
206 os << "pseries(" << var << ";" << point << ";";
207 for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
208 os << "(" << (*i).rest << "," << (*i).coeff << "),";
214 void pseries::printtree(std::ostream & os, unsigned indent) const
216 debugmsg("pseries printtree",LOGLEVEL_PRINT);
217 os << std::string(indent,' ') << "pseries "
218 << ", hash=" << hashvalue
219 << " (0x" << std::hex << hashvalue << std::dec << ")"
220 << ", flags=" << flags << std::endl;
221 for (unsigned i=0; i<seq.size(); ++i) {
222 seq[i].rest.printtree(os,indent+delta_indent);
223 seq[i].coeff.printtree(os,indent+delta_indent);
225 os << std::string(indent+delta_indent,' ') << "-----" << std::endl;
227 var.printtree(os, indent+delta_indent);
228 point.printtree(os, indent+delta_indent);
231 /** Return the number of operands including a possible order term. */
232 unsigned pseries::nops(void) const
238 /** Return the ith term in the series when represented as a sum. */
239 ex pseries::op(int i) const
241 if (i < 0 || unsigned(i) >= seq.size())
242 throw (std::out_of_range("op() out of range"));
243 return seq[i].rest * power(var - point, seq[i].coeff);
247 ex &pseries::let_op(int i)
249 throw (std::logic_error("let_op not defined for pseries"));
253 /** Return degree of highest power of the series. This is usually the exponent
254 * of the Order term. If s is not the expansion variable of the series, the
255 * series is examined termwise. */
256 int pseries::degree(const symbol &s) const
258 if (var.is_equal(s)) {
259 // Return last exponent
261 return ex_to_numeric((*(seq.end() - 1)).coeff).to_int();
265 epvector::const_iterator it = seq.begin(), itend = seq.end();
268 int max_pow = INT_MIN;
269 while (it != itend) {
270 int pow = it->rest.degree(s);
279 /** Return degree of lowest power of the series. This is usually the exponent
280 * of the leading term. If s is not the expansion variable of the series, the
281 * series is examined termwise. If s is the expansion variable but the
282 * expansion point is not zero the series is not expanded to find the degree.
283 * I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
284 int pseries::ldegree(const symbol &s) const
286 if (var.is_equal(s)) {
287 // Return first exponent
289 return ex_to_numeric((*(seq.begin())).coeff).to_int();
293 epvector::const_iterator it = seq.begin(), itend = seq.end();
296 int min_pow = INT_MAX;
297 while (it != itend) {
298 int pow = it->rest.ldegree(s);
307 ex pseries::coeff(const symbol &s, int n) const
309 if (var.is_equal(s)) {
313 // Binary search in sequence for given power
314 numeric looking_for = numeric(n);
315 int lo = 0, hi = seq.size() - 1;
317 int mid = (lo + hi) / 2;
318 GINAC_ASSERT(is_ex_exactly_of_type(seq[mid].coeff, numeric));
319 int cmp = ex_to_numeric(seq[mid].coeff).compare(looking_for);
325 return seq[mid].rest;
330 throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
335 return convert_to_poly().coeff(s, n);
339 ex pseries::collect(const symbol &s) const
345 /** Evaluate coefficients. */
346 ex pseries::eval(int level) const
351 if (level == -max_recursion_level)
352 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
354 // Construct a new series with evaluated coefficients
356 new_seq.reserve(seq.size());
357 epvector::const_iterator it = seq.begin(), itend = seq.end();
358 while (it != itend) {
359 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
362 return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
366 /** Evaluate coefficients numerically. */
367 ex pseries::evalf(int level) const
372 if (level == -max_recursion_level)
373 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
375 // Construct a new series with evaluated coefficients
377 new_seq.reserve(seq.size());
378 epvector::const_iterator it = seq.begin(), itend = seq.end();
379 while (it != itend) {
380 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
383 return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
387 ex pseries::subs(const lst & ls, const lst & lr) const
389 // If expansion variable is being substituted, convert the series to a
390 // polynomial and do the substitution there because the result might
391 // no longer be a power series
393 return convert_to_poly(true).subs(ls, lr);
395 // Otherwise construct a new series with substituted coefficients and
398 newseq.reserve(seq.size());
399 epvector::const_iterator it = seq.begin(), itend = seq.end();
400 while (it != itend) {
401 newseq.push_back(expair(it->rest.subs(ls, lr), it->coeff));
404 return (new pseries(relational(var,point.subs(ls, lr)), newseq))->setflag(status_flags::dynallocated);
408 /** Implementation of ex::expand() for a power series. It expands all the
409 * terms individually and returns the resulting series as a new pseries.
411 ex pseries::expand(unsigned options) const
414 newseq.reserve(seq.size());
415 for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i)
416 newseq.push_back(expair(i->rest.expand(), i->coeff));
417 return (new pseries(relational(var,point), newseq))
418 ->setflag(status_flags::dynallocated |
419 status_flags::expanded);
423 /** Implementation of ex::diff() for a power series. It treats the series as a
426 ex pseries::derivative(const symbol & s) const
430 epvector::const_iterator it = seq.begin(), itend = seq.end();
432 // FIXME: coeff might depend on var
433 while (it != itend) {
434 if (is_order_function(it->rest)) {
435 new_seq.push_back(expair(it->rest, it->coeff - 1));
437 ex c = it->rest * it->coeff;
439 new_seq.push_back(expair(c, it->coeff - 1));
443 return pseries(relational(var,point), new_seq);
451 * Construct ordinary polynomial out of series
454 /** Convert a pseries object to an ordinary polynomial.
456 * @param no_order flag: discard higher order terms */
457 ex pseries::convert_to_poly(bool no_order) const
460 epvector::const_iterator it = seq.begin(), itend = seq.end();
462 while (it != itend) {
463 if (is_order_function(it->rest)) {
465 e += Order(power(var - point, it->coeff));
467 e += it->rest * power(var - point, it->coeff);
473 /** Returns true if there is no order term, i.e. the series terminates and
474 * false otherwise. */
475 bool pseries::is_terminating(void) const
477 return !is_order_function((seq.end()-1)->rest);
482 * Implementation of series expansion
485 /** Default implementation of ex::series(). This performs Taylor expansion.
487 ex basic::series(const relational & r, int order, unsigned options) const
492 ex coeff = deriv.subs(r);
493 const symbol *s = static_cast<symbol *>(r.lhs().bp);
495 if (!coeff.is_zero())
496 seq.push_back(expair(coeff, numeric(0)));
499 for (n=1; n<order; ++n) {
500 fac = fac.mul(numeric(n));
501 deriv = deriv.diff(*s).expand();
502 if (deriv.is_zero()) {
504 return pseries(r, seq);
506 coeff = deriv.subs(r);
507 if (!coeff.is_zero())
508 seq.push_back(expair(fac.inverse() * coeff, numeric(n)));
511 // Higher-order terms, if present
512 deriv = deriv.diff(*s);
513 if (!deriv.expand().is_zero())
514 seq.push_back(expair(Order(_ex1()), numeric(n)));
515 return pseries(r, seq);
519 /** Implementation of ex::series() for symbols.
521 ex symbol::series(const relational & r, int order, unsigned options) const
524 const ex point = r.rhs();
525 GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
526 const symbol *s = static_cast<symbol *>(r.lhs().bp);
528 if (this->is_equal(*s)) {
529 if (order > 0 && !point.is_zero())
530 seq.push_back(expair(point, _ex0()));
532 seq.push_back(expair(_ex1(), _ex1()));
534 seq.push_back(expair(Order(_ex1()), numeric(order)));
536 seq.push_back(expair(*this, _ex0()));
537 return pseries(r, seq);
541 /** Add one series object to another, producing a pseries object that
542 * represents the sum.
544 * @param other pseries object to add with
545 * @return the sum as a pseries */
546 ex pseries::add_series(const pseries &other) const
548 // Adding two series with different variables or expansion points
549 // results in an empty (constant) series
550 if (!is_compatible_to(other)) {
552 nul.push_back(expair(Order(_ex1()), _ex0()));
553 return pseries(relational(var,point), nul);
558 epvector::const_iterator a = seq.begin();
559 epvector::const_iterator b = other.seq.begin();
560 epvector::const_iterator a_end = seq.end();
561 epvector::const_iterator b_end = other.seq.end();
562 int pow_a = INT_MAX, pow_b = INT_MAX;
564 // If a is empty, fill up with elements from b and stop
567 new_seq.push_back(*b);
572 pow_a = ex_to_numeric((*a).coeff).to_int();
574 // If b is empty, fill up with elements from a and stop
577 new_seq.push_back(*a);
582 pow_b = ex_to_numeric((*b).coeff).to_int();
584 // a and b are non-empty, compare powers
586 // a has lesser power, get coefficient from a
587 new_seq.push_back(*a);
588 if (is_order_function((*a).rest))
591 } else if (pow_b < pow_a) {
592 // b has lesser power, get coefficient from b
593 new_seq.push_back(*b);
594 if (is_order_function((*b).rest))
598 // Add coefficient of a and b
599 if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
600 new_seq.push_back(expair(Order(_ex1()), (*a).coeff));
601 break; // Order term ends the sequence
603 ex sum = (*a).rest + (*b).rest;
604 if (!(sum.is_zero()))
605 new_seq.push_back(expair(sum, numeric(pow_a)));
611 return pseries(relational(var,point), new_seq);
615 /** Implementation of ex::series() for sums. This performs series addition when
616 * adding pseries objects.
618 ex add::series(const relational & r, int order, unsigned options) const
620 ex acc; // Series accumulator
622 // Get first term from overall_coeff
623 acc = overall_coeff.series(r, order, options);
625 // Add remaining terms
626 epvector::const_iterator it = seq.begin();
627 epvector::const_iterator itend = seq.end();
628 for (; it!=itend; ++it) {
630 if (is_ex_exactly_of_type(it->rest, pseries))
633 op = it->rest.series(r, order, options);
634 if (!it->coeff.is_equal(_ex1()))
635 op = ex_to_pseries(op).mul_const(ex_to_numeric(it->coeff));
638 acc = ex_to_pseries(acc).add_series(ex_to_pseries(op));
644 /** Multiply a pseries object with a numeric constant, producing a pseries
645 * object that represents the product.
647 * @param other constant to multiply with
648 * @return the product as a pseries */
649 ex pseries::mul_const(const numeric &other) const
652 new_seq.reserve(seq.size());
654 epvector::const_iterator it = seq.begin(), itend = seq.end();
655 while (it != itend) {
656 if (!is_order_function(it->rest))
657 new_seq.push_back(expair(it->rest * other, it->coeff));
659 new_seq.push_back(*it);
662 return pseries(relational(var,point), new_seq);
666 /** Multiply one pseries object to another, producing a pseries object that
667 * represents the product.
669 * @param other pseries object to multiply with
670 * @return the product as a pseries */
671 ex pseries::mul_series(const pseries &other) const
673 // Multiplying two series with different variables or expansion points
674 // results in an empty (constant) series
675 if (!is_compatible_to(other)) {
677 nul.push_back(expair(Order(_ex1()), _ex0()));
678 return pseries(relational(var,point), nul);
681 // Series multiplication
684 const symbol *s = static_cast<symbol *>(var.bp);
685 int a_max = degree(*s);
686 int b_max = other.degree(*s);
687 int a_min = ldegree(*s);
688 int b_min = other.ldegree(*s);
689 int cdeg_min = a_min + b_min;
690 int cdeg_max = a_max + b_max;
692 int higher_order_a = INT_MAX;
693 int higher_order_b = INT_MAX;
694 if (is_order_function(coeff(*s, a_max)))
695 higher_order_a = a_max + b_min;
696 if (is_order_function(other.coeff(*s, b_max)))
697 higher_order_b = b_max + a_min;
698 int higher_order_c = std::min(higher_order_a, higher_order_b);
699 if (cdeg_max >= higher_order_c)
700 cdeg_max = higher_order_c - 1;
702 for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
704 // c(i)=a(0)b(i)+...+a(i)b(0)
705 for (int i=a_min; cdeg-i>=b_min; ++i) {
706 ex a_coeff = coeff(*s, i);
707 ex b_coeff = other.coeff(*s, cdeg-i);
708 if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
709 co += a_coeff * b_coeff;
712 new_seq.push_back(expair(co, numeric(cdeg)));
714 if (higher_order_c < INT_MAX)
715 new_seq.push_back(expair(Order(_ex1()), numeric(higher_order_c)));
716 return pseries(relational(var,point), new_seq);
720 /** Implementation of ex::series() for product. This performs series
721 * multiplication when multiplying series.
723 ex mul::series(const relational & r, int order, unsigned options) const
725 ex acc; // Series accumulator
727 // Get first term from overall_coeff
728 acc = overall_coeff.series(r, order, options);
730 // Multiply with remaining terms
731 epvector::const_iterator it = seq.begin();
732 epvector::const_iterator itend = seq.end();
733 for (; it!=itend; ++it) {
735 if (op.info(info_flags::numeric)) {
736 // series * const (special case, faster)
737 ex f = power(op, it->coeff);
738 acc = ex_to_pseries(acc).mul_const(ex_to_numeric(f));
740 } else if (!is_ex_exactly_of_type(op, pseries))
741 op = op.series(r, order, options);
742 if (!it->coeff.is_equal(_ex1()))
743 op = ex_to_pseries(op).power_const(ex_to_numeric(it->coeff), order);
745 // Series multiplication
746 acc = ex_to_pseries(acc).mul_series(ex_to_pseries(op));
752 /** Compute the p-th power of a series.
754 * @param p power to compute
755 * @param deg truncation order of series calculation */
756 ex pseries::power_const(const numeric &p, int deg) const
759 const symbol *s = static_cast<symbol *>(var.bp);
760 int ldeg = ldegree(*s);
762 // Calculate coefficients of powered series
766 co.push_back(co0 = power(coeff(*s, ldeg), p));
767 bool all_sums_zero = true;
768 for (i=1; i<deg; ++i) {
770 for (int j=1; j<=i; ++j) {
771 ex c = coeff(*s, j + ldeg);
772 if (is_order_function(c)) {
773 co.push_back(Order(_ex1()));
776 sum += (p * j - (i - j)) * co[i - j] * c;
779 all_sums_zero = false;
780 co.push_back(co0 * sum / numeric(i));
783 // Construct new series (of non-zero coefficients)
785 bool higher_order = false;
786 for (i=0; i<deg; ++i) {
787 if (!co[i].is_zero())
788 new_seq.push_back(expair(co[i], numeric(i) + p * ldeg));
789 if (is_order_function(co[i])) {
794 if (!higher_order && !all_sums_zero)
795 new_seq.push_back(expair(Order(_ex1()), numeric(deg) + p * ldeg));
796 return pseries(relational(var,point), new_seq);
800 /** Return a new pseries object with the powers shifted by deg. */
801 pseries pseries::shift_exponents(int deg) const
803 epvector newseq(seq);
804 for (epvector::iterator i=newseq.begin(); i!=newseq.end(); ++i)
805 i->coeff = i->coeff + deg;
806 return pseries(relational(var, point), newseq);
810 /** Implementation of ex::series() for powers. This performs Laurent expansion
811 * of reciprocals of series at singularities.
813 ex power::series(const relational & r, int order, unsigned options) const
816 if (!is_ex_exactly_of_type(basis, pseries)) {
817 // Basis is not a series, may there be a singulary?
818 if (!exponent.info(info_flags::negint))
819 return basic::series(r, order, options);
821 // Expression is of type something^(-int), check for singularity
822 if (!basis.subs(r).is_zero())
823 return basic::series(r, order, options);
825 // Singularity encountered, expand basis into series
826 e = basis.series(r, order, options);
833 return ex_to_pseries(e).power_const(ex_to_numeric(exponent), order);
837 /** Re-expansion of a pseries object. */
838 ex pseries::series(const relational & r, int order, unsigned options) const
840 const ex p = r.rhs();
841 GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
842 const symbol *s = static_cast<symbol *>(r.lhs().bp);
844 if (var.is_equal(*s) && point.is_equal(p)) {
845 if (order > degree(*s))
849 epvector::const_iterator it = seq.begin(), itend = seq.end();
850 while (it != itend) {
851 int o = ex_to_numeric(it->coeff).to_int();
853 new_seq.push_back(expair(Order(_ex1()), o));
856 new_seq.push_back(*it);
859 return pseries(r, new_seq);
862 return convert_to_poly().series(r, order, options);
866 /** Compute the truncated series expansion of an expression.
867 * This function returns an expression containing an object of class pseries
868 * to represent the series. If the series does not terminate within the given
869 * truncation order, the last term of the series will be an order term.
871 * @param r expansion relation, lhs holds variable and rhs holds point
872 * @param order truncation order of series calculations
873 * @param options of class series_options
874 * @return an expression holding a pseries object */
875 ex ex::series(const ex & r, int order, unsigned options) const
881 if (is_ex_exactly_of_type(r,relational))
882 rel_ = ex_to_relational(r);
883 else if (is_ex_exactly_of_type(r,symbol))
884 rel_ = relational(r,_ex0());
886 throw (std::logic_error("ex::series(): expansion point has unknown type"));
889 e = bp->series(rel_, order, options);
890 } catch (std::exception &x) {
891 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));
898 const pseries some_pseries;
899 const type_info & typeid_pseries = typeid(some_pseries);
901 #ifndef NO_NAMESPACE_GINAC
903 #endif // ndef NO_NAMESPACE_GINAC