3 * Implementation of class for extended truncated power series and
4 * methods for series expansion. */
7 * GiNaC Copyright (C) 1999-2002 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"
41 GINAC_IMPLEMENT_REGISTERED_CLASS(pseries, basic)
45 * Default ctor, dtor, copy ctor, assignment operator and helpers
48 pseries::pseries() : inherited(TINFO_pseries) { }
50 void pseries::copy(const pseries &other)
52 inherited::copy(other);
58 DEFAULT_DESTROY(pseries)
65 /** Construct pseries from a vector of coefficients and powers.
66 * expair.rest holds the coefficient, expair.coeff holds the power.
67 * The powers must be integers (positive or negative) and in ascending order;
68 * the last coefficient can be Order(_ex1) to represent a truncated,
69 * non-terminating series.
71 * @param rel_ expansion variable and point (must hold a relational)
72 * @param ops_ vector of {coefficient, power} pairs (coefficient must not be zero)
73 * @return newly constructed pseries */
74 pseries::pseries(const ex &rel_, const epvector &ops_) : basic(TINFO_pseries), seq(ops_)
76 GINAC_ASSERT(is_exactly_a<relational>(rel_));
77 GINAC_ASSERT(is_exactly_a<symbol>(rel_.lhs()));
87 pseries::pseries(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
89 for (unsigned int i=0; true; ++i) {
92 if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
93 seq.push_back(expair(rest, coeff));
97 n.find_ex("var", var, sym_lst);
98 n.find_ex("point", point, sym_lst);
101 void pseries::archive(archive_node &n) const
103 inherited::archive(n);
104 epvector::const_iterator i = seq.begin(), iend = seq.end();
106 n.add_ex("coeff", i->rest);
107 n.add_ex("power", i->coeff);
110 n.add_ex("var", var);
111 n.add_ex("point", point);
114 DEFAULT_UNARCHIVE(pseries)
117 // functions overriding virtual functions from base classes
120 void pseries::print(const print_context & c, unsigned level) const
122 if (is_a<print_tree>(c)) {
124 c.s << std::string(level, ' ') << class_name()
125 << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
127 unsigned delta_indent = static_cast<const print_tree &>(c).delta_indent;
128 unsigned num = seq.size();
129 for (unsigned i=0; i<num; ++i) {
130 seq[i].rest.print(c, level + delta_indent);
131 seq[i].coeff.print(c, level + delta_indent);
132 c.s << std::string(level + delta_indent, ' ') << "-----" << std::endl;
134 var.print(c, level + delta_indent);
135 point.print(c, level + delta_indent);
137 } else if (is_a<print_python_repr>(c)) {
138 c.s << class_name() << "(relational(";
143 unsigned num = seq.size();
144 for (unsigned i=0; i<num; ++i) {
148 seq[i].rest.print(c);
150 seq[i].coeff.print(c);
156 if (precedence() <= level)
159 std::string par_open = is_a<print_latex>(c) ? "{(" : "(";
160 std::string par_close = is_a<print_latex>(c) ? ")}" : ")";
162 // objects of type pseries must not have any zero entries, so the
163 // trivial (zero) pseries needs a special treatment here:
166 epvector::const_iterator i = seq.begin(), end = seq.end();
168 // print a sign, if needed
169 if (i != seq.begin())
171 if (!is_order_function(i->rest)) {
172 // print 'rest', i.e. the expansion coefficient
173 if (i->rest.info(info_flags::numeric) &&
174 i->rest.info(info_flags::positive)) {
181 // print 'coeff', something like (x-1)^42
182 if (!i->coeff.is_zero()) {
183 if (is_a<print_latex>(c))
187 if (!point.is_zero()) {
189 (var-point).print(c);
193 if (i->coeff.compare(_ex1)) {
194 if (is_a<print_python>(c))
198 if (i->coeff.info(info_flags::negative)) {
203 if (is_a<print_latex>(c)) {
213 Order(power(var-point,i->coeff)).print(c);
217 if (precedence() <= level)
222 int pseries::compare_same_type(const basic & other) const
224 GINAC_ASSERT(is_a<pseries>(other));
225 const pseries &o = static_cast<const pseries &>(other);
227 // first compare the lengths of the series...
228 if (seq.size()>o.seq.size())
230 if (seq.size()<o.seq.size())
233 // ...then the expansion point...
234 int cmpval = var.compare(o.var);
237 cmpval = point.compare(o.point);
241 // ...and if that failed the individual elements
242 epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
243 while (it!=seq.end() && o_it!=o.seq.end()) {
244 cmpval = it->compare(*o_it);
251 // so they are equal.
255 /** Return the number of operands including a possible order term. */
256 unsigned pseries::nops(void) const
261 /** Return the ith term in the series when represented as a sum. */
262 ex pseries::op(int i) const
264 if (i < 0 || unsigned(i) >= seq.size())
265 throw (std::out_of_range("op() out of range"));
266 return seq[i].rest * power(var - point, seq[i].coeff);
269 ex &pseries::let_op(int i)
271 throw (std::logic_error("let_op not defined for pseries"));
274 /** Return degree of highest power of the series. This is usually the exponent
275 * of the Order term. If s is not the expansion variable of the series, the
276 * series is examined termwise. */
277 int pseries::degree(const ex &s) const
279 if (var.is_equal(s)) {
280 // Return last exponent
282 return ex_to<numeric>((seq.end()-1)->coeff).to_int();
286 epvector::const_iterator it = seq.begin(), itend = seq.end();
289 int max_pow = INT_MIN;
290 while (it != itend) {
291 int pow = it->rest.degree(s);
300 /** Return degree of lowest power of the series. This is usually the exponent
301 * of the leading term. If s is not the expansion variable of the series, the
302 * series is examined termwise. If s is the expansion variable but the
303 * expansion point is not zero the series is not expanded to find the degree.
304 * I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
305 int pseries::ldegree(const ex &s) const
307 if (var.is_equal(s)) {
308 // Return first exponent
310 return ex_to<numeric>((seq.begin())->coeff).to_int();
314 epvector::const_iterator it = seq.begin(), itend = seq.end();
317 int min_pow = INT_MAX;
318 while (it != itend) {
319 int pow = it->rest.ldegree(s);
328 /** Return coefficient of degree n in power series if s is the expansion
329 * variable. If the expansion point is nonzero, by definition the n=1
330 * coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
331 * the expansion took place in the s in the first place).
332 * If s is not the expansion variable, an attempt is made to convert the
333 * series to a polynomial and return the corresponding coefficient from
335 ex pseries::coeff(const ex &s, int n) const
337 if (var.is_equal(s)) {
341 // Binary search in sequence for given power
342 numeric looking_for = numeric(n);
343 int lo = 0, hi = seq.size() - 1;
345 int mid = (lo + hi) / 2;
346 GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
347 int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
353 return seq[mid].rest;
358 throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
363 return convert_to_poly().coeff(s, n);
367 ex pseries::collect(const ex &s, bool distributed) const
372 /** Perform coefficient-wise automatic term rewriting rules in this class. */
373 ex pseries::eval(int level) const
378 if (level == -max_recursion_level)
379 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
381 // Construct a new series with evaluated coefficients
383 new_seq.reserve(seq.size());
384 epvector::const_iterator it = seq.begin(), itend = seq.end();
385 while (it != itend) {
386 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
389 return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
392 /** Evaluate coefficients numerically. */
393 ex pseries::evalf(int level) const
398 if (level == -max_recursion_level)
399 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
401 // Construct a new series with evaluated coefficients
403 new_seq.reserve(seq.size());
404 epvector::const_iterator it = seq.begin(), itend = seq.end();
405 while (it != itend) {
406 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
409 return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
412 ex pseries::subs(const lst & ls, const lst & lr, bool no_pattern) const
414 // If expansion variable is being substituted, convert the series to a
415 // polynomial and do the substitution there because the result might
416 // no longer be a power series
418 return convert_to_poly(true).subs(ls, lr, no_pattern);
420 // Otherwise construct a new series with substituted coefficients and
423 newseq.reserve(seq.size());
424 epvector::const_iterator it = seq.begin(), itend = seq.end();
425 while (it != itend) {
426 newseq.push_back(expair(it->rest.subs(ls, lr, no_pattern), it->coeff));
429 return (new pseries(relational(var,point.subs(ls, lr, no_pattern)), newseq))->setflag(status_flags::dynallocated);
432 /** Implementation of ex::expand() for a power series. It expands all the
433 * terms individually and returns the resulting series as a new pseries. */
434 ex pseries::expand(unsigned options) const
437 epvector::const_iterator i = seq.begin(), end = seq.end();
439 ex restexp = i->rest.expand();
440 if (!restexp.is_zero())
441 newseq.push_back(expair(restexp, i->coeff));
444 return (new pseries(relational(var,point), newseq))
445 ->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
448 /** Implementation of ex::diff() for a power series. It treats the series as a
451 ex pseries::derivative(const symbol & s) const
455 epvector::const_iterator it = seq.begin(), itend = seq.end();
457 // FIXME: coeff might depend on var
458 while (it != itend) {
459 if (is_order_function(it->rest)) {
460 new_seq.push_back(expair(it->rest, it->coeff - 1));
462 ex c = it->rest * it->coeff;
464 new_seq.push_back(expair(c, it->coeff - 1));
468 return pseries(relational(var,point), new_seq);
474 ex pseries::convert_to_poly(bool no_order) const
477 epvector::const_iterator it = seq.begin(), itend = seq.end();
479 while (it != itend) {
480 if (is_order_function(it->rest)) {
482 e += Order(power(var - point, it->coeff));
484 e += it->rest * power(var - point, it->coeff);
490 bool pseries::is_terminating(void) const
492 return seq.empty() || !is_order_function((seq.end()-1)->rest);
497 * Implementations of series expansion
500 /** Default implementation of ex::series(). This performs Taylor expansion.
502 ex basic::series(const relational & r, int order, unsigned options) const
507 ex coeff = deriv.subs(r);
508 const symbol &s = ex_to<symbol>(r.lhs());
510 if (!coeff.is_zero())
511 seq.push_back(expair(coeff, _ex0));
514 for (n=1; n<order; ++n) {
516 // We need to test for zero in order to see if the series terminates.
517 // The problem is that there is no such thing as a perfect test for
518 // zero. Expanding the term occasionally helps a little...
519 deriv = deriv.diff(s).expand();
520 if (deriv.is_zero()) // Series terminates
521 return pseries(r, seq);
523 coeff = deriv.subs(r);
524 if (!coeff.is_zero())
525 seq.push_back(expair(fac.inverse() * coeff, n));
528 // Higher-order terms, if present
529 deriv = deriv.diff(s);
530 if (!deriv.expand().is_zero())
531 seq.push_back(expair(Order(_ex1), n));
532 return pseries(r, seq);
536 /** Implementation of ex::series() for symbols.
538 ex symbol::series(const relational & r, int order, unsigned options) const
541 const ex point = r.rhs();
542 GINAC_ASSERT(is_exactly_a<symbol>(r.lhs()));
544 if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
545 if (order > 0 && !point.is_zero())
546 seq.push_back(expair(point, _ex0));
548 seq.push_back(expair(_ex1, _ex1));
550 seq.push_back(expair(Order(_ex1), numeric(order)));
552 seq.push_back(expair(*this, _ex0));
553 return pseries(r, seq);
557 /** Add one series object to another, producing a pseries object that
558 * represents the sum.
560 * @param other pseries object to add with
561 * @return the sum as a pseries */
562 ex pseries::add_series(const pseries &other) const
564 // Adding two series with different variables or expansion points
565 // results in an empty (constant) series
566 if (!is_compatible_to(other)) {
568 nul.push_back(expair(Order(_ex1), _ex0));
569 return pseries(relational(var,point), nul);
574 epvector::const_iterator a = seq.begin();
575 epvector::const_iterator b = other.seq.begin();
576 epvector::const_iterator a_end = seq.end();
577 epvector::const_iterator b_end = other.seq.end();
578 int pow_a = INT_MAX, pow_b = INT_MAX;
580 // If a is empty, fill up with elements from b and stop
583 new_seq.push_back(*b);
588 pow_a = ex_to<numeric>((*a).coeff).to_int();
590 // If b is empty, fill up with elements from a and stop
593 new_seq.push_back(*a);
598 pow_b = ex_to<numeric>((*b).coeff).to_int();
600 // a and b are non-empty, compare powers
602 // a has lesser power, get coefficient from a
603 new_seq.push_back(*a);
604 if (is_order_function((*a).rest))
607 } else if (pow_b < pow_a) {
608 // b has lesser power, get coefficient from b
609 new_seq.push_back(*b);
610 if (is_order_function((*b).rest))
614 // Add coefficient of a and b
615 if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
616 new_seq.push_back(expair(Order(_ex1), (*a).coeff));
617 break; // Order term ends the sequence
619 ex sum = (*a).rest + (*b).rest;
620 if (!(sum.is_zero()))
621 new_seq.push_back(expair(sum, numeric(pow_a)));
627 return pseries(relational(var,point), new_seq);
631 /** Implementation of ex::series() for sums. This performs series addition when
632 * adding pseries objects.
634 ex add::series(const relational & r, int order, unsigned options) const
636 ex acc; // Series accumulator
638 // Get first term from overall_coeff
639 acc = overall_coeff.series(r, order, options);
641 // Add remaining terms
642 epvector::const_iterator it = seq.begin();
643 epvector::const_iterator itend = seq.end();
644 for (; it!=itend; ++it) {
646 if (is_exactly_a<pseries>(it->rest))
649 op = it->rest.series(r, order, options);
650 if (!it->coeff.is_equal(_ex1))
651 op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it->coeff));
654 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
660 /** Multiply a pseries object with a numeric constant, producing a pseries
661 * object that represents the product.
663 * @param other constant to multiply with
664 * @return the product as a pseries */
665 ex pseries::mul_const(const numeric &other) const
668 new_seq.reserve(seq.size());
670 epvector::const_iterator it = seq.begin(), itend = seq.end();
671 while (it != itend) {
672 if (!is_order_function(it->rest))
673 new_seq.push_back(expair(it->rest * other, it->coeff));
675 new_seq.push_back(*it);
678 return pseries(relational(var,point), new_seq);
682 /** Multiply one pseries object to another, producing a pseries object that
683 * represents the product.
685 * @param other pseries object to multiply with
686 * @return the product as a pseries */
687 ex pseries::mul_series(const pseries &other) const
689 // Multiplying two series with different variables or expansion points
690 // results in an empty (constant) series
691 if (!is_compatible_to(other)) {
693 nul.push_back(expair(Order(_ex1), _ex0));
694 return pseries(relational(var,point), nul);
697 // Series multiplication
699 int a_max = degree(var);
700 int b_max = other.degree(var);
701 int a_min = ldegree(var);
702 int b_min = other.ldegree(var);
703 int cdeg_min = a_min + b_min;
704 int cdeg_max = a_max + b_max;
706 int higher_order_a = INT_MAX;
707 int higher_order_b = INT_MAX;
708 if (is_order_function(coeff(var, a_max)))
709 higher_order_a = a_max + b_min;
710 if (is_order_function(other.coeff(var, b_max)))
711 higher_order_b = b_max + a_min;
712 int higher_order_c = std::min(higher_order_a, higher_order_b);
713 if (cdeg_max >= higher_order_c)
714 cdeg_max = higher_order_c - 1;
716 for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
718 // c(i)=a(0)b(i)+...+a(i)b(0)
719 for (int i=a_min; cdeg-i>=b_min; ++i) {
720 ex a_coeff = coeff(var, i);
721 ex b_coeff = other.coeff(var, cdeg-i);
722 if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
723 co += a_coeff * b_coeff;
726 new_seq.push_back(expair(co, numeric(cdeg)));
728 if (higher_order_c < INT_MAX)
729 new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c)));
730 return pseries(relational(var, point), new_seq);
734 /** Implementation of ex::series() for product. This performs series
735 * multiplication when multiplying series.
737 ex mul::series(const relational & r, int order, unsigned options) const
739 pseries acc; // Series accumulator
741 // Multiply with remaining terms
742 const epvector::const_iterator itbeg = seq.begin();
743 const epvector::const_iterator itend = seq.end();
744 for (epvector::const_iterator it=itbeg; it!=itend; ++it) {
745 ex op = recombine_pair_to_ex(*it).series(r, order, options);
747 // Series multiplication
749 acc = ex_to<pseries>(op);
751 acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
753 return acc.mul_const(ex_to<numeric>(overall_coeff));
757 /** Compute the p-th power of a series.
759 * @param p power to compute
760 * @param deg truncation order of series calculation */
761 ex pseries::power_const(const numeric &p, int deg) const
764 // (due to Leonhard Euler)
765 // let A(x) be this series and for the time being let it start with a
766 // constant (later we'll generalize):
767 // A(x) = a_0 + a_1*x + a_2*x^2 + ...
768 // We want to compute
770 // C(x) = c_0 + c_1*x + c_2*x^2 + ...
771 // Taking the derivative on both sides and multiplying with A(x) one
772 // immediately arrives at
773 // C'(x)*A(x) = p*C(x)*A'(x)
774 // Multiplying this out and comparing coefficients we get the recurrence
776 // c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
777 // ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
778 // which can easily be solved given the starting value c_0 = (a_0)^p.
779 // For the more general case where the leading coefficient of A(x) is not
780 // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
781 // repeat the above derivation. The leading power of C2(x) = A2(x)^2 is
782 // then of course x^(p*m) but the recurrence formula still holds.
785 // as a special case, handle the empty (zero) series honoring the
786 // usual power laws such as implemented in power::eval()
787 if (p.real().is_zero())
788 throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
789 else if (p.real().is_negative())
790 throw pole_error("pseries::power_const(): division by zero",1);
795 const int ldeg = ldegree(var);
796 if (!(p*ldeg).is_integer())
797 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
799 // O(x^n)^(-m) is undefined
800 if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
801 throw pole_error("pseries::power_const(): division by zero",1);
803 // Compute coefficients of the powered series
806 co.push_back(power(coeff(var, ldeg), p));
807 bool all_sums_zero = true;
808 for (int i=1; i<deg; ++i) {
810 for (int j=1; j<=i; ++j) {
811 ex c = coeff(var, j + ldeg);
812 if (is_order_function(c)) {
813 co.push_back(Order(_ex1));
816 sum += (p * j - (i - j)) * co[i - j] * c;
819 all_sums_zero = false;
820 co.push_back(sum / coeff(var, ldeg) / i);
823 // Construct new series (of non-zero coefficients)
825 bool higher_order = false;
826 for (int i=0; i<deg; ++i) {
827 if (!co[i].is_zero())
828 new_seq.push_back(expair(co[i], p * ldeg + i));
829 if (is_order_function(co[i])) {
834 if (!higher_order && !all_sums_zero)
835 new_seq.push_back(expair(Order(_ex1), p * ldeg + deg));
836 return pseries(relational(var,point), new_seq);
840 /** Return a new pseries object with the powers shifted by deg. */
841 pseries pseries::shift_exponents(int deg) const
843 epvector newseq = seq;
844 epvector::iterator i = newseq.begin(), end = newseq.end();
849 return pseries(relational(var, point), newseq);
853 /** Implementation of ex::series() for powers. This performs Laurent expansion
854 * of reciprocals of series at singularities.
856 ex power::series(const relational & r, int order, unsigned options) const
858 // If basis is already a series, just power it
859 if (is_exactly_a<pseries>(basis))
860 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
862 // Basis is not a series, may there be a singularity?
863 bool must_expand_basis = false;
866 } catch (pole_error) {
867 must_expand_basis = true;
870 // Is the expression of type something^(-int)?
871 if (!must_expand_basis && !exponent.info(info_flags::negint))
872 return basic::series(r, order, options);
874 // Is the expression of type 0^something?
875 if (!must_expand_basis && !basis.subs(r).is_zero())
876 return basic::series(r, order, options);
878 // Singularity encountered, is the basis equal to (var - point)?
879 if (basis.is_equal(r.lhs() - r.rhs())) {
881 if (ex_to<numeric>(exponent).to_int() < order)
882 new_seq.push_back(expair(_ex1, exponent));
884 new_seq.push_back(expair(Order(_ex1), exponent));
885 return pseries(r, new_seq);
888 // No, expand basis into series
889 ex e = basis.series(r, order, options);
890 return ex_to<pseries>(e).power_const(ex_to<numeric>(exponent), order);
894 /** Re-expansion of a pseries object. */
895 ex pseries::series(const relational & r, int order, unsigned options) const
897 const ex p = r.rhs();
898 GINAC_ASSERT(is_exactly_a<symbol>(r.lhs()));
899 const symbol &s = ex_to<symbol>(r.lhs());
901 if (var.is_equal(s) && point.is_equal(p)) {
902 if (order > degree(s))
906 epvector::const_iterator it = seq.begin(), itend = seq.end();
907 while (it != itend) {
908 int o = ex_to<numeric>(it->coeff).to_int();
910 new_seq.push_back(expair(Order(_ex1), o));
913 new_seq.push_back(*it);
916 return pseries(r, new_seq);
919 return convert_to_poly().series(r, order, options);
923 /** Compute the truncated series expansion of an expression.
924 * This function returns an expression containing an object of class pseries
925 * to represent the series. If the series does not terminate within the given
926 * truncation order, the last term of the series will be an order term.
928 * @param r expansion relation, lhs holds variable and rhs holds point
929 * @param order truncation order of series calculations
930 * @param options of class series_options
931 * @return an expression holding a pseries object */
932 ex ex::series(const ex & r, int order, unsigned options) const
938 if (is_exactly_a<relational>(r))
939 rel_ = ex_to<relational>(r);
940 else if (is_exactly_a<symbol>(r))
941 rel_ = relational(r,_ex0);
943 throw (std::logic_error("ex::series(): expansion point has unknown type"));
946 e = bp->series(rel_, order, options);
947 } catch (std::exception &x) {
948 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));