3 * Implementation of class for extended truncated power series and
4 * methods for series expansion. */
7 * GiNaC Copyright (C) 1999-2001 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
28 #include "inifcns.h" // for Order function
32 #include "relational.h"
41 GINAC_IMPLEMENT_REGISTERED_CLASS(pseries, basic)
45 * Default ctor, dtor, copy ctor, assignment operator and helpers
48 pseries::pseries() : basic(TINFO_pseries)
50 debugmsg("pseries default ctor", LOGLEVEL_CONSTRUCT);
53 void pseries::copy(const pseries &other)
55 inherited::copy(other);
61 DEFAULT_DESTROY(pseries)
68 /** Construct pseries from a vector of coefficients and powers.
69 * expair.rest holds the coefficient, expair.coeff holds the power.
70 * The powers must be integers (positive or negative) and in ascending order;
71 * the last coefficient can be Order(_ex1()) to represent a truncated,
72 * non-terminating series.
74 * @param rel_ expansion variable and point (must hold a relational)
75 * @param ops_ vector of {coefficient, power} pairs (coefficient must not be zero)
76 * @return newly constructed pseries */
77 pseries::pseries(const ex &rel_, const epvector &ops_) : basic(TINFO_pseries), seq(ops_)
79 debugmsg("pseries ctor from ex,epvector", LOGLEVEL_CONSTRUCT);
80 GINAC_ASSERT(is_exactly_a<relational>(rel_));
81 GINAC_ASSERT(is_exactly_a<symbol>(rel_.lhs()));
91 pseries::pseries(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
93 debugmsg("pseries ctor from archive_node", LOGLEVEL_CONSTRUCT);
94 for (unsigned int i=0; true; ++i) {
97 if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
98 seq.push_back(expair(rest, coeff));
102 n.find_ex("var", var, sym_lst);
103 n.find_ex("point", point, sym_lst);
106 void pseries::archive(archive_node &n) const
108 inherited::archive(n);
109 epvector::const_iterator i = seq.begin(), iend = seq.end();
111 n.add_ex("coeff", i->rest);
112 n.add_ex("power", i->coeff);
115 n.add_ex("var", var);
116 n.add_ex("point", point);
119 DEFAULT_UNARCHIVE(pseries)
122 // functions overriding virtual functions from base classes
125 void pseries::print(const print_context & c, unsigned level) const
127 debugmsg("pseries print", LOGLEVEL_PRINT);
129 if (is_a<print_tree>(c)) {
131 c.s << std::string(level, ' ') << class_name()
132 << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
134 unsigned delta_indent = static_cast<const print_tree &>(c).delta_indent;
135 unsigned num = seq.size();
136 for (unsigned i=0; i<num; ++i) {
137 seq[i].rest.print(c, level + delta_indent);
138 seq[i].coeff.print(c, level + delta_indent);
139 c.s << std::string(level + delta_indent, ' ') << "-----" << std::endl;
141 var.print(c, level + delta_indent);
142 point.print(c, level + delta_indent);
146 if (precedence() <= level)
149 std::string par_open = is_a<print_latex>(c) ? "{(" : "(";
150 std::string par_close = is_a<print_latex>(c) ? ")}" : ")";
152 // objects of type pseries must not have any zero entries, so the
153 // trivial (zero) pseries needs a special treatment here:
156 epvector::const_iterator i = seq.begin(), end = seq.end();
158 // print a sign, if needed
159 if (i != seq.begin())
161 if (!is_order_function(i->rest)) {
162 // print 'rest', i.e. the expansion coefficient
163 if (i->rest.info(info_flags::numeric) &&
164 i->rest.info(info_flags::positive)) {
171 // print 'coeff', something like (x-1)^42
172 if (!i->coeff.is_zero()) {
173 if (is_a<print_latex>(c))
177 if (!point.is_zero()) {
179 (var-point).print(c);
183 if (i->coeff.compare(_ex1())) {
185 if (i->coeff.info(info_flags::negative)) {
190 if (is_a<print_latex>(c)) {
200 Order(power(var-point,i->coeff)).print(c);
204 if (precedence() <= level)
209 int pseries::compare_same_type(const basic & other) const
211 GINAC_ASSERT(is_a<pseries>(other));
212 const pseries &o = static_cast<const pseries &>(other);
214 // first compare the lengths of the series...
215 if (seq.size()>o.seq.size())
217 if (seq.size()<o.seq.size())
220 // ...then the expansion point...
221 int cmpval = var.compare(o.var);
224 cmpval = point.compare(o.point);
228 // ...and if that failed the individual elements
229 epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
230 while (it!=seq.end() && o_it!=o.seq.end()) {
231 cmpval = it->compare(*o_it);
238 // so they are equal.
242 /** Return the number of operands including a possible order term. */
243 unsigned pseries::nops(void) const
248 /** Return the ith term in the series when represented as a sum. */
249 ex pseries::op(int i) const
251 if (i < 0 || unsigned(i) >= seq.size())
252 throw (std::out_of_range("op() out of range"));
253 return seq[i].rest * power(var - point, seq[i].coeff);
256 ex &pseries::let_op(int i)
258 throw (std::logic_error("let_op not defined for pseries"));
261 /** Return degree of highest power of the series. This is usually the exponent
262 * of the Order term. If s is not the expansion variable of the series, the
263 * series is examined termwise. */
264 int pseries::degree(const ex &s) const
266 if (var.is_equal(s)) {
267 // Return last exponent
269 return ex_to<numeric>((seq.end()-1)->coeff).to_int();
273 epvector::const_iterator it = seq.begin(), itend = seq.end();
276 int max_pow = INT_MIN;
277 while (it != itend) {
278 int pow = it->rest.degree(s);
287 /** Return degree of lowest power of the series. This is usually the exponent
288 * of the leading term. If s is not the expansion variable of the series, the
289 * series is examined termwise. If s is the expansion variable but the
290 * expansion point is not zero the series is not expanded to find the degree.
291 * I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
292 int pseries::ldegree(const ex &s) const
294 if (var.is_equal(s)) {
295 // Return first exponent
297 return ex_to<numeric>((seq.begin())->coeff).to_int();
301 epvector::const_iterator it = seq.begin(), itend = seq.end();
304 int min_pow = INT_MAX;
305 while (it != itend) {
306 int pow = it->rest.ldegree(s);
315 /** Return coefficient of degree n in power series if s is the expansion
316 * variable. If the expansion point is nonzero, by definition the n=1
317 * coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
318 * the expansion took place in the s in the first place).
319 * If s is not the expansion variable, an attempt is made to convert the
320 * series to a polynomial and return the corresponding coefficient from
322 ex pseries::coeff(const ex &s, int n) const
324 if (var.is_equal(s)) {
328 // Binary search in sequence for given power
329 numeric looking_for = numeric(n);
330 int lo = 0, hi = seq.size() - 1;
332 int mid = (lo + hi) / 2;
333 GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
334 int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
340 return seq[mid].rest;
345 throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
350 return convert_to_poly().coeff(s, n);
354 ex pseries::collect(const ex &s, bool distributed) const
359 /** Perform coefficient-wise automatic term rewriting rules in this class. */
360 ex pseries::eval(int level) const
365 if (level == -max_recursion_level)
366 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
368 // Construct a new series with evaluated coefficients
370 new_seq.reserve(seq.size());
371 epvector::const_iterator it = seq.begin(), itend = seq.end();
372 while (it != itend) {
373 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
376 return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
379 /** Evaluate coefficients numerically. */
380 ex pseries::evalf(int level) const
385 if (level == -max_recursion_level)
386 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
388 // Construct a new series with evaluated coefficients
390 new_seq.reserve(seq.size());
391 epvector::const_iterator it = seq.begin(), itend = seq.end();
392 while (it != itend) {
393 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
396 return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
399 ex pseries::subs(const lst & ls, const lst & lr, bool no_pattern) const
401 // If expansion variable is being substituted, convert the series to a
402 // polynomial and do the substitution there because the result might
403 // no longer be a power series
405 return convert_to_poly(true).subs(ls, lr, no_pattern);
407 // Otherwise construct a new series with substituted coefficients and
410 newseq.reserve(seq.size());
411 epvector::const_iterator it = seq.begin(), itend = seq.end();
412 while (it != itend) {
413 newseq.push_back(expair(it->rest.subs(ls, lr, no_pattern), it->coeff));
416 return (new pseries(relational(var,point.subs(ls, lr, no_pattern)), newseq))->setflag(status_flags::dynallocated);
419 /** Implementation of ex::expand() for a power series. It expands all the
420 * terms individually and returns the resulting series as a new pseries. */
421 ex pseries::expand(unsigned options) const
424 epvector::const_iterator i = seq.begin(), end = seq.end();
426 ex restexp = i->rest.expand();
427 if (!restexp.is_zero())
428 newseq.push_back(expair(restexp, i->coeff));
431 return (new pseries(relational(var,point), newseq))
432 ->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
435 /** Implementation of ex::diff() for a power series. It treats the series as a
438 ex pseries::derivative(const symbol & s) const
442 epvector::const_iterator it = seq.begin(), itend = seq.end();
444 // FIXME: coeff might depend on var
445 while (it != itend) {
446 if (is_order_function(it->rest)) {
447 new_seq.push_back(expair(it->rest, it->coeff - 1));
449 ex c = it->rest * it->coeff;
451 new_seq.push_back(expair(c, it->coeff - 1));
455 return pseries(relational(var,point), new_seq);
461 ex pseries::convert_to_poly(bool no_order) const
464 epvector::const_iterator it = seq.begin(), itend = seq.end();
466 while (it != itend) {
467 if (is_order_function(it->rest)) {
469 e += Order(power(var - point, it->coeff));
471 e += it->rest * power(var - point, it->coeff);
477 bool pseries::is_terminating(void) const
479 return seq.empty() || !is_order_function((seq.end()-1)->rest);
484 * Implementations of series expansion
487 /** Default implementation of ex::series(). This performs Taylor expansion.
489 ex basic::series(const relational & r, int order, unsigned options) const
494 ex coeff = deriv.subs(r);
495 const symbol &s = ex_to<symbol>(r.lhs());
497 if (!coeff.is_zero())
498 seq.push_back(expair(coeff, _ex0()));
501 for (n=1; n<order; ++n) {
503 // We need to test for zero in order to see if the series terminates.
504 // The problem is that there is no such thing as a perfect test for
505 // zero. Expanding the term occasionally helps a little...
506 deriv = deriv.diff(s).expand();
507 if (deriv.is_zero()) // Series terminates
508 return pseries(r, seq);
510 coeff = deriv.subs(r);
511 if (!coeff.is_zero())
512 seq.push_back(expair(fac.inverse() * coeff, n));
515 // Higher-order terms, if present
516 deriv = deriv.diff(s);
517 if (!deriv.expand().is_zero())
518 seq.push_back(expair(Order(_ex1()), n));
519 return pseries(r, seq);
523 /** Implementation of ex::series() for symbols.
525 ex symbol::series(const relational & r, int order, unsigned options) const
528 const ex point = r.rhs();
529 GINAC_ASSERT(is_exactly_a<symbol>(r.lhs()));
531 if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
532 if (order > 0 && !point.is_zero())
533 seq.push_back(expair(point, _ex0()));
535 seq.push_back(expair(_ex1(), _ex1()));
537 seq.push_back(expair(Order(_ex1()), numeric(order)));
539 seq.push_back(expair(*this, _ex0()));
540 return pseries(r, seq);
544 /** Add one series object to another, producing a pseries object that
545 * represents the sum.
547 * @param other pseries object to add with
548 * @return the sum as a pseries */
549 ex pseries::add_series(const pseries &other) const
551 // Adding two series with different variables or expansion points
552 // results in an empty (constant) series
553 if (!is_compatible_to(other)) {
555 nul.push_back(expair(Order(_ex1()), _ex0()));
556 return pseries(relational(var,point), nul);
561 epvector::const_iterator a = seq.begin();
562 epvector::const_iterator b = other.seq.begin();
563 epvector::const_iterator a_end = seq.end();
564 epvector::const_iterator b_end = other.seq.end();
565 int pow_a = INT_MAX, pow_b = INT_MAX;
567 // If a is empty, fill up with elements from b and stop
570 new_seq.push_back(*b);
575 pow_a = ex_to<numeric>((*a).coeff).to_int();
577 // If b is empty, fill up with elements from a and stop
580 new_seq.push_back(*a);
585 pow_b = ex_to<numeric>((*b).coeff).to_int();
587 // a and b are non-empty, compare powers
589 // a has lesser power, get coefficient from a
590 new_seq.push_back(*a);
591 if (is_order_function((*a).rest))
594 } else if (pow_b < pow_a) {
595 // b has lesser power, get coefficient from b
596 new_seq.push_back(*b);
597 if (is_order_function((*b).rest))
601 // Add coefficient of a and b
602 if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
603 new_seq.push_back(expair(Order(_ex1()), (*a).coeff));
604 break; // Order term ends the sequence
606 ex sum = (*a).rest + (*b).rest;
607 if (!(sum.is_zero()))
608 new_seq.push_back(expair(sum, numeric(pow_a)));
614 return pseries(relational(var,point), new_seq);
618 /** Implementation of ex::series() for sums. This performs series addition when
619 * adding pseries objects.
621 ex add::series(const relational & r, int order, unsigned options) const
623 ex acc; // Series accumulator
625 // Get first term from overall_coeff
626 acc = overall_coeff.series(r, order, options);
628 // Add remaining terms
629 epvector::const_iterator it = seq.begin();
630 epvector::const_iterator itend = seq.end();
631 for (; it!=itend; ++it) {
633 if (is_ex_exactly_of_type(it->rest, pseries))
636 op = it->rest.series(r, order, options);
637 if (!it->coeff.is_equal(_ex1()))
638 op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it->coeff));
641 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
647 /** Multiply a pseries object with a numeric constant, producing a pseries
648 * object that represents the product.
650 * @param other constant to multiply with
651 * @return the product as a pseries */
652 ex pseries::mul_const(const numeric &other) const
655 new_seq.reserve(seq.size());
657 epvector::const_iterator it = seq.begin(), itend = seq.end();
658 while (it != itend) {
659 if (!is_order_function(it->rest))
660 new_seq.push_back(expair(it->rest * other, it->coeff));
662 new_seq.push_back(*it);
665 return pseries(relational(var,point), new_seq);
669 /** Multiply one pseries object to another, producing a pseries object that
670 * represents the product.
672 * @param other pseries object to multiply with
673 * @return the product as a pseries */
674 ex pseries::mul_series(const pseries &other) const
676 // Multiplying two series with different variables or expansion points
677 // results in an empty (constant) series
678 if (!is_compatible_to(other)) {
680 nul.push_back(expair(Order(_ex1()), _ex0()));
681 return pseries(relational(var,point), nul);
684 // Series multiplication
686 int a_max = degree(var);
687 int b_max = other.degree(var);
688 int a_min = ldegree(var);
689 int b_min = other.ldegree(var);
690 int cdeg_min = a_min + b_min;
691 int cdeg_max = a_max + b_max;
693 int higher_order_a = INT_MAX;
694 int higher_order_b = INT_MAX;
695 if (is_order_function(coeff(var, a_max)))
696 higher_order_a = a_max + b_min;
697 if (is_order_function(other.coeff(var, b_max)))
698 higher_order_b = b_max + a_min;
699 int higher_order_c = std::min(higher_order_a, higher_order_b);
700 if (cdeg_max >= higher_order_c)
701 cdeg_max = higher_order_c - 1;
703 for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
705 // c(i)=a(0)b(i)+...+a(i)b(0)
706 for (int i=a_min; cdeg-i>=b_min; ++i) {
707 ex a_coeff = coeff(var, i);
708 ex b_coeff = other.coeff(var, cdeg-i);
709 if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
710 co += a_coeff * b_coeff;
713 new_seq.push_back(expair(co, numeric(cdeg)));
715 if (higher_order_c < INT_MAX)
716 new_seq.push_back(expair(Order(_ex1()), numeric(higher_order_c)));
717 return pseries(relational(var, point), new_seq);
721 /** Implementation of ex::series() for product. This performs series
722 * multiplication when multiplying series.
724 ex mul::series(const relational & r, int order, unsigned options) const
726 pseries acc; // Series accumulator
728 // Multiply with remaining terms
729 const epvector::const_iterator itbeg = seq.begin();
730 const epvector::const_iterator itend = seq.end();
731 for (epvector::const_iterator it=itbeg; it!=itend; ++it) {
732 ex op = recombine_pair_to_ex(*it).series(r, order, options);
734 // Series multiplication
736 acc = ex_to<pseries>(op);
738 acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
740 return acc.mul_const(ex_to<numeric>(overall_coeff));
744 /** Compute the p-th power of a series.
746 * @param p power to compute
747 * @param deg truncation order of series calculation */
748 ex pseries::power_const(const numeric &p, int deg) const
751 // (due to Leonhard Euler)
752 // let A(x) be this series and for the time being let it start with a
753 // constant (later we'll generalize):
754 // A(x) = a_0 + a_1*x + a_2*x^2 + ...
755 // We want to compute
757 // C(x) = c_0 + c_1*x + c_2*x^2 + ...
758 // Taking the derivative on both sides and multiplying with A(x) one
759 // immediately arrives at
760 // C'(x)*A(x) = p*C(x)*A'(x)
761 // Multiplying this out and comparing coefficients we get the recurrence
763 // c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
764 // ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
765 // which can easily be solved given the starting value c_0 = (a_0)^p.
766 // For the more general case where the leading coefficient of A(x) is not
767 // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
768 // repeat the above derivation. The leading power of C2(x) = A2(x)^2 is
769 // then of course x^(p*m) but the recurrence formula still holds.
772 // as a special case, handle the empty (zero) series honoring the
773 // usual power laws such as implemented in power::eval()
774 if (p.real().is_zero())
775 throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
776 else if (p.real().is_negative())
777 throw pole_error("pseries::power_const(): division by zero",1);
782 const int ldeg = ldegree(var);
783 if (!(p*ldeg).is_integer())
784 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
786 // O(x^n)^(-m) is undefined
787 if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
788 throw pole_error("pseries::power_const(): division by zero",1);
790 // Compute coefficients of the powered series
793 co.push_back(power(coeff(var, ldeg), p));
794 bool all_sums_zero = true;
795 for (int i=1; i<deg; ++i) {
797 for (int j=1; j<=i; ++j) {
798 ex c = coeff(var, j + ldeg);
799 if (is_order_function(c)) {
800 co.push_back(Order(_ex1()));
803 sum += (p * j - (i - j)) * co[i - j] * c;
806 all_sums_zero = false;
807 co.push_back(sum / coeff(var, ldeg) / i);
810 // Construct new series (of non-zero coefficients)
812 bool higher_order = false;
813 for (int i=0; i<deg; ++i) {
814 if (!co[i].is_zero())
815 new_seq.push_back(expair(co[i], p * ldeg + i));
816 if (is_order_function(co[i])) {
821 if (!higher_order && !all_sums_zero)
822 new_seq.push_back(expair(Order(_ex1()), p * ldeg + deg));
823 return pseries(relational(var,point), new_seq);
827 /** Return a new pseries object with the powers shifted by deg. */
828 pseries pseries::shift_exponents(int deg) const
830 epvector newseq = seq;
831 epvector::iterator i = newseq.begin(), end = newseq.end();
836 return pseries(relational(var, point), newseq);
840 /** Implementation of ex::series() for powers. This performs Laurent expansion
841 * of reciprocals of series at singularities.
843 ex power::series(const relational & r, int order, unsigned options) const
845 // If basis is already a series, just power it
846 if (is_ex_exactly_of_type(basis, pseries))
847 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
849 // Basis is not a series, may there be a singularity?
850 bool must_expand_basis = false;
853 } catch (pole_error) {
854 must_expand_basis = true;
857 // Is the expression of type something^(-int)?
858 if (!must_expand_basis && !exponent.info(info_flags::negint))
859 return basic::series(r, order, options);
861 // Is the expression of type 0^something?
862 if (!must_expand_basis && !basis.subs(r).is_zero())
863 return basic::series(r, order, options);
865 // Singularity encountered, is the basis equal to (var - point)?
866 if (basis.is_equal(r.lhs() - r.rhs())) {
868 if (ex_to<numeric>(exponent).to_int() < order)
869 new_seq.push_back(expair(_ex1(), exponent));
871 new_seq.push_back(expair(Order(_ex1()), exponent));
872 return pseries(r, new_seq);
875 // No, expand basis into series
876 ex e = basis.series(r, order, options);
877 return ex_to<pseries>(e).power_const(ex_to<numeric>(exponent), order);
881 /** Re-expansion of a pseries object. */
882 ex pseries::series(const relational & r, int order, unsigned options) const
884 const ex p = r.rhs();
885 GINAC_ASSERT(is_exactly_a<symbol>(r.lhs()));
886 const symbol &s = ex_to<symbol>(r.lhs());
888 if (var.is_equal(s) && point.is_equal(p)) {
889 if (order > degree(s))
893 epvector::const_iterator it = seq.begin(), itend = seq.end();
894 while (it != itend) {
895 int o = ex_to<numeric>(it->coeff).to_int();
897 new_seq.push_back(expair(Order(_ex1()), o));
900 new_seq.push_back(*it);
903 return pseries(r, new_seq);
906 return convert_to_poly().series(r, order, options);
910 /** Compute the truncated series expansion of an expression.
911 * This function returns an expression containing an object of class pseries
912 * to represent the series. If the series does not terminate within the given
913 * truncation order, the last term of the series will be an order term.
915 * @param r expansion relation, lhs holds variable and rhs holds point
916 * @param order truncation order of series calculations
917 * @param options of class series_options
918 * @return an expression holding a pseries object */
919 ex ex::series(const ex & r, int order, unsigned options) const
925 if (is_ex_exactly_of_type(r,relational))
926 rel_ = ex_to<relational>(r);
927 else if (is_ex_exactly_of_type(r,symbol))
928 rel_ = relational(r,_ex0());
930 throw (std::logic_error("ex::series(): expansion point has unknown type"));
933 e = bp->series(rel_, order, options);
934 } catch (std::exception &x) {
935 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));