3 * Implementation of class for extended truncated power series and
4 * methods for series expansion. */
7 * GiNaC Copyright (C) 1999-2004 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
29 #include "inifcns.h" // for Order function
33 #include "relational.h"
34 #include "operators.h"
42 GINAC_IMPLEMENT_REGISTERED_CLASS(pseries, basic)
46 * Default ctor, dtor, copy ctor, assignment operator and helpers
49 pseries::pseries() : inherited(TINFO_pseries) { }
51 void pseries::copy(const pseries &other)
53 inherited::copy(other);
59 DEFAULT_DESTROY(pseries)
66 /** Construct pseries from a vector of coefficients and powers.
67 * expair.rest holds the coefficient, expair.coeff holds the power.
68 * The powers must be integers (positive or negative) and in ascending order;
69 * the last coefficient can be Order(_ex1) to represent a truncated,
70 * non-terminating series.
72 * @param rel_ expansion variable and point (must hold a relational)
73 * @param ops_ vector of {coefficient, power} pairs (coefficient must not be zero)
74 * @return newly constructed pseries */
75 pseries::pseries(const ex &rel_, const epvector &ops_) : basic(TINFO_pseries), seq(ops_)
77 GINAC_ASSERT(is_a<relational>(rel_));
78 GINAC_ASSERT(is_a<symbol>(rel_.lhs()));
88 pseries::pseries(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
90 for (unsigned int i=0; true; ++i) {
93 if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
94 seq.push_back(expair(rest, coeff));
98 n.find_ex("var", var, sym_lst);
99 n.find_ex("point", point, sym_lst);
102 void pseries::archive(archive_node &n) const
104 inherited::archive(n);
105 epvector::const_iterator i = seq.begin(), iend = seq.end();
107 n.add_ex("coeff", i->rest);
108 n.add_ex("power", i->coeff);
111 n.add_ex("var", var);
112 n.add_ex("point", point);
115 DEFAULT_UNARCHIVE(pseries)
118 // functions overriding virtual functions from base classes
121 void pseries::print(const print_context & c, unsigned level) const
123 if (is_a<print_tree>(c)) {
125 c.s << std::string(level, ' ') << class_name()
126 << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
128 unsigned delta_indent = static_cast<const print_tree &>(c).delta_indent;
129 size_t num = seq.size();
130 for (size_t i=0; i<num; ++i) {
131 seq[i].rest.print(c, level + delta_indent);
132 seq[i].coeff.print(c, level + delta_indent);
133 c.s << std::string(level + delta_indent, ' ') << "-----" << std::endl;
135 var.print(c, level + delta_indent);
136 point.print(c, level + delta_indent);
138 } else if (is_a<print_python_repr>(c)) {
139 c.s << class_name() << "(relational(";
144 size_t num = seq.size();
145 for (size_t i=0; i<num; ++i) {
149 seq[i].rest.print(c);
151 seq[i].coeff.print(c);
157 if (precedence() <= level)
160 std::string par_open = is_a<print_latex>(c) ? "{(" : "(";
161 std::string par_close = is_a<print_latex>(c) ? ")}" : ")";
163 // objects of type pseries must not have any zero entries, so the
164 // trivial (zero) pseries needs a special treatment here:
167 epvector::const_iterator i = seq.begin(), end = seq.end();
169 // print a sign, if needed
170 if (i != seq.begin())
172 if (!is_order_function(i->rest)) {
173 // print 'rest', i.e. the expansion coefficient
174 if (i->rest.info(info_flags::numeric) &&
175 i->rest.info(info_flags::positive)) {
182 // print 'coeff', something like (x-1)^42
183 if (!i->coeff.is_zero()) {
184 if (is_a<print_latex>(c))
188 if (!point.is_zero()) {
190 (var-point).print(c);
194 if (i->coeff.compare(_ex1)) {
195 if (is_a<print_python>(c))
199 if (i->coeff.info(info_flags::negative)) {
204 if (is_a<print_latex>(c)) {
214 Order(power(var-point,i->coeff)).print(c);
218 if (precedence() <= level)
223 int pseries::compare_same_type(const basic & other) const
225 GINAC_ASSERT(is_a<pseries>(other));
226 const pseries &o = static_cast<const pseries &>(other);
228 // first compare the lengths of the series...
229 if (seq.size()>o.seq.size())
231 if (seq.size()<o.seq.size())
234 // ...then the expansion point...
235 int cmpval = var.compare(o.var);
238 cmpval = point.compare(o.point);
242 // ...and if that failed the individual elements
243 epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
244 while (it!=seq.end() && o_it!=o.seq.end()) {
245 cmpval = it->compare(*o_it);
252 // so they are equal.
256 /** Return the number of operands including a possible order term. */
257 size_t pseries::nops(void) const
262 /** Return the ith term in the series when represented as a sum. */
263 ex pseries::op(size_t i) const
266 throw (std::out_of_range("op() out of range"));
268 return seq[i].rest * power(var - point, seq[i].coeff);
271 /** Return degree of highest power of the series. This is usually the exponent
272 * of the Order term. If s is not the expansion variable of the series, the
273 * series is examined termwise. */
274 int pseries::degree(const ex &s) const
276 if (var.is_equal(s)) {
277 // Return last exponent
279 return ex_to<numeric>((seq.end()-1)->coeff).to_int();
283 epvector::const_iterator it = seq.begin(), itend = seq.end();
286 int max_pow = INT_MIN;
287 while (it != itend) {
288 int pow = it->rest.degree(s);
297 /** Return degree of lowest power of the series. This is usually the exponent
298 * of the leading term. If s is not the expansion variable of the series, the
299 * series is examined termwise. If s is the expansion variable but the
300 * expansion point is not zero the series is not expanded to find the degree.
301 * I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
302 int pseries::ldegree(const ex &s) const
304 if (var.is_equal(s)) {
305 // Return first exponent
307 return ex_to<numeric>((seq.begin())->coeff).to_int();
311 epvector::const_iterator it = seq.begin(), itend = seq.end();
314 int min_pow = INT_MAX;
315 while (it != itend) {
316 int pow = it->rest.ldegree(s);
325 /** Return coefficient of degree n in power series if s is the expansion
326 * variable. If the expansion point is nonzero, by definition the n=1
327 * coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
328 * the expansion took place in the s in the first place).
329 * If s is not the expansion variable, an attempt is made to convert the
330 * series to a polynomial and return the corresponding coefficient from
332 ex pseries::coeff(const ex &s, int n) const
334 if (var.is_equal(s)) {
338 // Binary search in sequence for given power
339 numeric looking_for = numeric(n);
340 int lo = 0, hi = seq.size() - 1;
342 int mid = (lo + hi) / 2;
343 GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
344 int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
350 return seq[mid].rest;
355 throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
360 return convert_to_poly().coeff(s, n);
364 ex pseries::collect(const ex &s, bool distributed) const
369 /** Perform coefficient-wise automatic term rewriting rules in this class. */
370 ex pseries::eval(int level) const
375 if (level == -max_recursion_level)
376 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
378 // Construct a new series with evaluated coefficients
380 new_seq.reserve(seq.size());
381 epvector::const_iterator it = seq.begin(), itend = seq.end();
382 while (it != itend) {
383 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
386 return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
389 /** Evaluate coefficients numerically. */
390 ex pseries::evalf(int level) const
395 if (level == -max_recursion_level)
396 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
398 // Construct a new series with evaluated coefficients
400 new_seq.reserve(seq.size());
401 epvector::const_iterator it = seq.begin(), itend = seq.end();
402 while (it != itend) {
403 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
406 return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
409 ex pseries::subs(const lst & ls, const lst & lr, unsigned options) const
411 // If expansion variable is being substituted, convert the series to a
412 // polynomial and do the substitution there because the result might
413 // no longer be a power series
415 return convert_to_poly(true).subs(ls, lr, options);
417 // Otherwise construct a new series with substituted coefficients and
420 newseq.reserve(seq.size());
421 epvector::const_iterator it = seq.begin(), itend = seq.end();
422 while (it != itend) {
423 newseq.push_back(expair(it->rest.subs(ls, lr, options), it->coeff));
426 return (new pseries(relational(var,point.subs(ls, lr, options)), newseq))->setflag(status_flags::dynallocated);
429 /** Implementation of ex::expand() for a power series. It expands all the
430 * terms individually and returns the resulting series as a new pseries. */
431 ex pseries::expand(unsigned options) const
434 epvector::const_iterator i = seq.begin(), end = seq.end();
436 ex restexp = i->rest.expand();
437 if (!restexp.is_zero())
438 newseq.push_back(expair(restexp, i->coeff));
441 return (new pseries(relational(var,point), newseq))
442 ->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
445 /** Implementation of ex::diff() for a power series.
447 ex pseries::derivative(const symbol & s) const
450 epvector::const_iterator it = seq.begin(), itend = seq.end();
454 // FIXME: coeff might depend on var
455 while (it != itend) {
456 if (is_order_function(it->rest)) {
457 new_seq.push_back(expair(it->rest, it->coeff - 1));
459 ex c = it->rest * it->coeff;
461 new_seq.push_back(expair(c, it->coeff - 1));
468 while (it != itend) {
469 if (is_order_function(it->rest)) {
470 new_seq.push_back(*it);
472 ex c = it->rest.diff(s);
474 new_seq.push_back(expair(c, it->coeff));
480 return pseries(relational(var,point), new_seq);
483 ex pseries::convert_to_poly(bool no_order) const
486 epvector::const_iterator it = seq.begin(), itend = seq.end();
488 while (it != itend) {
489 if (is_order_function(it->rest)) {
491 e += Order(power(var - point, it->coeff));
493 e += it->rest * power(var - point, it->coeff);
499 bool pseries::is_terminating(void) const
501 return seq.empty() || !is_order_function((seq.end()-1)->rest);
506 * Implementations of series expansion
509 /** Default implementation of ex::series(). This performs Taylor expansion.
511 ex basic::series(const relational & r, int order, unsigned options) const
516 ex coeff = deriv.subs(r, subs_options::no_pattern);
517 const symbol &s = ex_to<symbol>(r.lhs());
519 if (!coeff.is_zero())
520 seq.push_back(expair(coeff, _ex0));
523 for (n=1; n<=order; ++n) {
525 // We need to test for zero in order to see if the series terminates.
526 // The problem is that there is no such thing as a perfect test for
527 // zero. Expanding the term occasionally helps a little...
528 deriv = deriv.diff(s).expand();
529 if (deriv.is_zero()) // Series terminates
530 return pseries(r, seq);
532 coeff = deriv.subs(r, subs_options::no_pattern);
533 if (!coeff.is_zero())
534 seq.push_back(expair(fac.inverse() * coeff, n));
537 // Higher-order terms, if present
540 ldeg = std::abs(deriv.ldegree(s));
542 catch (std::runtime_error) {
547 if (!deriv.subs(r, subs_options::no_pattern).is_zero()) {
548 seq.push_back(expair(Order(_ex1), n));
550 seq.push_back(expair(Order(_ex1), n+ldeg-1));
553 // something more complicated -> loop until next coefficient is found
555 deriv = deriv.diff(s).expand();
556 if (deriv.is_zero()) {
559 if (!deriv.subs(r, subs_options::no_pattern).is_zero()) {
560 seq.push_back(expair(Order(_ex1), n));
566 return pseries(r, seq);
570 /** Implementation of ex::series() for symbols.
572 ex symbol::series(const relational & r, int order, unsigned options) const
575 const ex point = r.rhs();
576 GINAC_ASSERT(is_a<symbol>(r.lhs()));
578 if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
579 if (order > 0 && !point.is_zero())
580 seq.push_back(expair(point, _ex0));
582 seq.push_back(expair(_ex1, _ex1));
584 seq.push_back(expair(Order(_ex1), numeric(order)));
586 seq.push_back(expair(*this, _ex0));
587 return pseries(r, seq);
591 /** Add one series object to another, producing a pseries object that
592 * represents the sum.
594 * @param other pseries object to add with
595 * @return the sum as a pseries */
596 ex pseries::add_series(const pseries &other) const
598 // Adding two series with different variables or expansion points
599 // results in an empty (constant) series
600 if (!is_compatible_to(other)) {
602 nul.push_back(expair(Order(_ex1), _ex0));
603 return pseries(relational(var,point), nul);
608 epvector::const_iterator a = seq.begin();
609 epvector::const_iterator b = other.seq.begin();
610 epvector::const_iterator a_end = seq.end();
611 epvector::const_iterator b_end = other.seq.end();
612 int pow_a = INT_MAX, pow_b = INT_MAX;
614 // If a is empty, fill up with elements from b and stop
617 new_seq.push_back(*b);
622 pow_a = ex_to<numeric>((*a).coeff).to_int();
624 // If b is empty, fill up with elements from a and stop
627 new_seq.push_back(*a);
632 pow_b = ex_to<numeric>((*b).coeff).to_int();
634 // a and b are non-empty, compare powers
636 // a has lesser power, get coefficient from a
637 new_seq.push_back(*a);
638 if (is_order_function((*a).rest))
641 } else if (pow_b < pow_a) {
642 // b has lesser power, get coefficient from b
643 new_seq.push_back(*b);
644 if (is_order_function((*b).rest))
648 // Add coefficient of a and b
649 if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
650 new_seq.push_back(expair(Order(_ex1), (*a).coeff));
651 break; // Order term ends the sequence
653 ex sum = (*a).rest + (*b).rest;
654 if (!(sum.is_zero()))
655 new_seq.push_back(expair(sum, numeric(pow_a)));
661 return pseries(relational(var,point), new_seq);
665 /** Implementation of ex::series() for sums. This performs series addition when
666 * adding pseries objects.
668 ex add::series(const relational & r, int order, unsigned options) const
670 ex acc; // Series accumulator
672 // Get first term from overall_coeff
673 acc = overall_coeff.series(r, order, options);
675 // Add remaining terms
676 epvector::const_iterator it = seq.begin();
677 epvector::const_iterator itend = seq.end();
678 for (; it!=itend; ++it) {
680 if (is_exactly_a<pseries>(it->rest))
683 op = it->rest.series(r, order, options);
684 if (!it->coeff.is_equal(_ex1))
685 op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it->coeff));
688 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
694 /** Multiply a pseries object with a numeric constant, producing a pseries
695 * object that represents the product.
697 * @param other constant to multiply with
698 * @return the product as a pseries */
699 ex pseries::mul_const(const numeric &other) const
702 new_seq.reserve(seq.size());
704 epvector::const_iterator it = seq.begin(), itend = seq.end();
705 while (it != itend) {
706 if (!is_order_function(it->rest))
707 new_seq.push_back(expair(it->rest * other, it->coeff));
709 new_seq.push_back(*it);
712 return pseries(relational(var,point), new_seq);
716 /** Multiply one pseries object to another, producing a pseries object that
717 * represents the product.
719 * @param other pseries object to multiply with
720 * @return the product as a pseries */
721 ex pseries::mul_series(const pseries &other) const
723 // Multiplying two series with different variables or expansion points
724 // results in an empty (constant) series
725 if (!is_compatible_to(other)) {
727 nul.push_back(expair(Order(_ex1), _ex0));
728 return pseries(relational(var,point), nul);
731 // Series multiplication
733 int a_max = degree(var);
734 int b_max = other.degree(var);
735 int a_min = ldegree(var);
736 int b_min = other.ldegree(var);
737 int cdeg_min = a_min + b_min;
738 int cdeg_max = a_max + b_max;
740 int higher_order_a = INT_MAX;
741 int higher_order_b = INT_MAX;
742 if (is_order_function(coeff(var, a_max)))
743 higher_order_a = a_max + b_min;
744 if (is_order_function(other.coeff(var, b_max)))
745 higher_order_b = b_max + a_min;
746 int higher_order_c = std::min(higher_order_a, higher_order_b);
747 if (cdeg_max >= higher_order_c)
748 cdeg_max = higher_order_c - 1;
750 for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
752 // c(i)=a(0)b(i)+...+a(i)b(0)
753 for (int i=a_min; cdeg-i>=b_min; ++i) {
754 ex a_coeff = coeff(var, i);
755 ex b_coeff = other.coeff(var, cdeg-i);
756 if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
757 co += a_coeff * b_coeff;
760 new_seq.push_back(expair(co, numeric(cdeg)));
762 if (higher_order_c < INT_MAX)
763 new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c)));
764 return pseries(relational(var, point), new_seq);
768 /** Implementation of ex::series() for product. This performs series
769 * multiplication when multiplying series.
771 ex mul::series(const relational & r, int order, unsigned options) const
773 pseries acc; // Series accumulator
775 GINAC_ASSERT(is_a<symbol>(r.lhs()));
776 const ex& sym = r.lhs();
778 // holds ldegrees of the series of individual factors
779 std::vector<int> ldegrees;
780 // flag if series have to be re-calculated
781 bool negldegree = false;
783 // Multiply with remaining terms
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 ex op = recombine_pair_to_ex(*it).series(r, order, options);
789 int ldeg = op.ldegree(sym);
790 ldegrees.push_back(ldeg);
795 // Series multiplication
797 acc = ex_to<pseries>(op);
799 acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
802 if (seq.size() > 1) {
804 // re-calculation of series
808 const int degsum = std::accumulate(ldegrees.begin(), ldegrees.end(), 0);
810 // Multiply with remaining terms
811 const epvector::const_reverse_iterator itbeg = seq.rbegin();
812 const epvector::const_reverse_iterator itend = seq.rend();
813 std::vector<int>::const_reverse_iterator itd = ldegrees.rbegin();
814 for (epvector::const_reverse_iterator it=itbeg; it!=itend; ++it, ++itd) {
816 ex op = recombine_pair_to_ex(*it).series(r, order-degsum+(*itd), options);
818 // Series multiplication
820 newacc = ex_to<pseries>(op);
822 newacc = ex_to<pseries>(newacc.mul_series(ex_to<pseries>(op)));
824 return newacc.mul_const(ex_to<numeric>(overall_coeff));
826 return acc.mul_const(ex_to<numeric>(overall_coeff));
831 /** Compute the p-th power of a series.
833 * @param p power to compute
834 * @param deg truncation order of series calculation */
835 ex pseries::power_const(const numeric &p, int deg) const
838 // (due to Leonhard Euler)
839 // let A(x) be this series and for the time being let it start with a
840 // constant (later we'll generalize):
841 // A(x) = a_0 + a_1*x + a_2*x^2 + ...
842 // We want to compute
844 // C(x) = c_0 + c_1*x + c_2*x^2 + ...
845 // Taking the derivative on both sides and multiplying with A(x) one
846 // immediately arrives at
847 // C'(x)*A(x) = p*C(x)*A'(x)
848 // Multiplying this out and comparing coefficients we get the recurrence
850 // c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
851 // ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
852 // which can easily be solved given the starting value c_0 = (a_0)^p.
853 // For the more general case where the leading coefficient of A(x) is not
854 // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
855 // repeat the above derivation. The leading power of C2(x) = A2(x)^2 is
856 // then of course x^(p*m) but the recurrence formula still holds.
859 // as a special case, handle the empty (zero) series honoring the
860 // usual power laws such as implemented in power::eval()
861 if (p.real().is_zero())
862 throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
863 else if (p.real().is_negative())
864 throw pole_error("pseries::power_const(): division by zero",1);
869 const int ldeg = ldegree(var);
870 if (!(p*ldeg).is_integer())
871 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
873 // adjust number of coefficients
874 deg = deg - p.to_int()*ldeg;
876 // O(x^n)^(-m) is undefined
877 if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
878 throw pole_error("pseries::power_const(): division by zero",1);
880 // Compute coefficients of the powered series
883 co.push_back(power(coeff(var, ldeg), p));
884 bool all_sums_zero = true;
885 for (int i=1; i<=deg; ++i) {
887 for (int j=1; j<=i; ++j) {
888 ex c = coeff(var, j + ldeg);
889 if (is_order_function(c)) {
890 co.push_back(Order(_ex1));
893 sum += (p * j - (i - j)) * co[i - j] * c;
896 all_sums_zero = false;
897 co.push_back(sum / coeff(var, ldeg) / i);
900 // Construct new series (of non-zero coefficients)
902 bool higher_order = false;
903 for (int i=0; i<=deg; ++i) {
904 if (!co[i].is_zero())
905 new_seq.push_back(expair(co[i], p * ldeg + i));
906 if (is_order_function(co[i])) {
911 if (!higher_order && !all_sums_zero)
912 new_seq.push_back(expair(Order(_ex1), p * ldeg + deg + 1));
914 return pseries(relational(var,point), new_seq);
918 /** Return a new pseries object with the powers shifted by deg. */
919 pseries pseries::shift_exponents(int deg) const
921 epvector newseq = seq;
922 epvector::iterator i = newseq.begin(), end = newseq.end();
927 return pseries(relational(var, point), newseq);
931 /** Implementation of ex::series() for powers. This performs Laurent expansion
932 * of reciprocals of series at singularities.
934 ex power::series(const relational & r, int order, unsigned options) const
936 // If basis is already a series, just power it
937 if (is_exactly_a<pseries>(basis))
938 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
940 // Basis is not a series, may there be a singularity?
941 bool must_expand_basis = false;
943 basis.subs(r, subs_options::no_pattern);
944 } catch (pole_error) {
945 must_expand_basis = true;
948 // Is the expression of type something^(-int)?
949 if (!must_expand_basis && !exponent.info(info_flags::negint))
950 return basic::series(r, order, options);
952 // Is the expression of type 0^something?
953 if (!must_expand_basis && !basis.subs(r, subs_options::no_pattern).is_zero())
954 return basic::series(r, order, options);
956 // Singularity encountered, is the basis equal to (var - point)?
957 if (basis.is_equal(r.lhs() - r.rhs())) {
959 if (ex_to<numeric>(exponent).to_int() < order)
960 new_seq.push_back(expair(_ex1, exponent));
962 new_seq.push_back(expair(Order(_ex1), exponent));
963 return pseries(r, new_seq);
966 // No, expand basis into series
967 int intexp = ex_to<numeric>(exponent).to_int();
968 ex e = basis.series(r, order, options);
969 int ldeg = ex_to<pseries>(e).ldegree(r.lhs());
970 if (intexp * ldeg < 0) {
971 e = basis.series(r, order + ldeg*(1-intexp), options);
973 return ex_to<pseries>(e).power_const(intexp, order);
977 /** Re-expansion of a pseries object. */
978 ex pseries::series(const relational & r, int order, unsigned options) const
980 const ex p = r.rhs();
981 GINAC_ASSERT(is_a<symbol>(r.lhs()));
982 const symbol &s = ex_to<symbol>(r.lhs());
984 if (var.is_equal(s) && point.is_equal(p)) {
985 if (order > degree(s))
989 epvector::const_iterator it = seq.begin(), itend = seq.end();
990 while (it != itend) {
991 int o = ex_to<numeric>(it->coeff).to_int();
993 new_seq.push_back(expair(Order(_ex1), o));
996 new_seq.push_back(*it);
999 return pseries(r, new_seq);
1002 return convert_to_poly().series(r, order, options);
1006 /** Compute the truncated series expansion of an expression.
1007 * This function returns an expression containing an object of class pseries
1008 * to represent the series. If the series does not terminate within the given
1009 * truncation order, the last term of the series will be an order term.
1011 * @param r expansion relation, lhs holds variable and rhs holds point
1012 * @param order truncation order of series calculations
1013 * @param options of class series_options
1014 * @return an expression holding a pseries object */
1015 ex ex::series(const ex & r, int order, unsigned options) const
1017 GINAC_ASSERT(bp!=0);
1021 if (is_a<relational>(r))
1022 rel_ = ex_to<relational>(r);
1023 else if (is_a<symbol>(r))
1024 rel_ = relational(r,_ex0);
1026 throw (std::logic_error("ex::series(): expansion point has unknown type"));
1029 e = bp->series(rel_, order, options);
1030 } catch (std::exception &x) {
1031 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));
1036 } // namespace GiNaC