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"
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_exactly_a<relational>(rel_));
78 GINAC_ASSERT(is_exactly_a<symbol>(rel_.lhs()));
88 pseries::pseries(const archive_node &n, const 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 unsigned num = seq.size();
130 for (unsigned 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 unsigned num = seq.size();
145 for (unsigned 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 unsigned pseries::nops(void) const
262 /** Return the ith term in the series when represented as a sum. */
263 ex pseries::op(int i) const
265 if (i < 0 || unsigned(i) >= seq.size())
266 throw (std::out_of_range("op() out of range"));
267 return seq[i].rest * power(var - point, seq[i].coeff);
270 ex &pseries::let_op(int i)
272 throw (std::logic_error("let_op not defined for pseries"));
275 /** Return degree of highest power of the series. This is usually the exponent
276 * of the Order term. If s is not the expansion variable of the series, the
277 * series is examined termwise. */
278 int pseries::degree(const ex &s) const
280 if (var.is_equal(s)) {
281 // Return last exponent
283 return ex_to<numeric>((seq.end()-1)->coeff).to_int();
287 epvector::const_iterator it = seq.begin(), itend = seq.end();
290 int max_pow = INT_MIN;
291 while (it != itend) {
292 int pow = it->rest.degree(s);
301 /** Return degree of lowest power of the series. This is usually the exponent
302 * of the leading term. If s is not the expansion variable of the series, the
303 * series is examined termwise. If s is the expansion variable but the
304 * expansion point is not zero the series is not expanded to find the degree.
305 * I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
306 int pseries::ldegree(const ex &s) const
308 if (var.is_equal(s)) {
309 // Return first exponent
311 return ex_to<numeric>((seq.begin())->coeff).to_int();
315 epvector::const_iterator it = seq.begin(), itend = seq.end();
318 int min_pow = INT_MAX;
319 while (it != itend) {
320 int pow = it->rest.ldegree(s);
329 /** Return coefficient of degree n in power series if s is the expansion
330 * variable. If the expansion point is nonzero, by definition the n=1
331 * coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
332 * the expansion took place in the s in the first place).
333 * If s is not the expansion variable, an attempt is made to convert the
334 * series to a polynomial and return the corresponding coefficient from
336 ex pseries::coeff(const ex &s, int n) const
338 if (var.is_equal(s)) {
342 // Binary search in sequence for given power
343 numeric looking_for = numeric(n);
344 int lo = 0, hi = seq.size() - 1;
346 int mid = (lo + hi) / 2;
347 GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
348 int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
354 return seq[mid].rest;
359 throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
364 return convert_to_poly().coeff(s, n);
368 ex pseries::collect(const ex &s, bool distributed) const
373 /** Perform coefficient-wise automatic term rewriting rules in this class. */
374 ex pseries::eval(int level) const
379 if (level == -max_recursion_level)
380 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
382 // Construct a new series with evaluated coefficients
384 new_seq.reserve(seq.size());
385 epvector::const_iterator it = seq.begin(), itend = seq.end();
386 while (it != itend) {
387 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
390 return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
393 /** Evaluate coefficients numerically. */
394 ex pseries::evalf(int level) const
399 if (level == -max_recursion_level)
400 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
402 // Construct a new series with evaluated coefficients
404 new_seq.reserve(seq.size());
405 epvector::const_iterator it = seq.begin(), itend = seq.end();
406 while (it != itend) {
407 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
410 return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
413 ex pseries::subs(const lst & ls, const lst & lr, bool no_pattern) const
415 // If expansion variable is being substituted, convert the series to a
416 // polynomial and do the substitution there because the result might
417 // no longer be a power series
419 return convert_to_poly(true).subs(ls, lr, no_pattern);
421 // Otherwise construct a new series with substituted coefficients and
424 newseq.reserve(seq.size());
425 epvector::const_iterator it = seq.begin(), itend = seq.end();
426 while (it != itend) {
427 newseq.push_back(expair(it->rest.subs(ls, lr, no_pattern), it->coeff));
430 return (new pseries(relational(var,point.subs(ls, lr, no_pattern)), newseq))->setflag(status_flags::dynallocated);
433 /** Implementation of ex::expand() for a power series. It expands all the
434 * terms individually and returns the resulting series as a new pseries. */
435 ex pseries::expand(unsigned options) const
438 epvector::const_iterator i = seq.begin(), end = seq.end();
440 ex restexp = i->rest.expand();
441 if (!restexp.is_zero())
442 newseq.push_back(expair(restexp, i->coeff));
445 return (new pseries(relational(var,point), newseq))
446 ->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
449 /** Implementation of ex::diff() for a power series. It treats the series as a
452 ex pseries::derivative(const symbol & s) const
456 epvector::const_iterator it = seq.begin(), itend = seq.end();
458 // FIXME: coeff might depend on var
459 while (it != itend) {
460 if (is_order_function(it->rest)) {
461 new_seq.push_back(expair(it->rest, it->coeff - 1));
463 ex c = it->rest * it->coeff;
465 new_seq.push_back(expair(c, it->coeff - 1));
469 return pseries(relational(var,point), new_seq);
475 ex pseries::convert_to_poly(bool no_order) const
478 epvector::const_iterator it = seq.begin(), itend = seq.end();
480 while (it != itend) {
481 if (is_order_function(it->rest)) {
483 e += Order(power(var - point, it->coeff));
485 e += it->rest * power(var - point, it->coeff);
491 bool pseries::is_terminating(void) const
493 return seq.empty() || !is_order_function((seq.end()-1)->rest);
498 * Implementations of series expansion
501 /** Default implementation of ex::series(). This performs Taylor expansion.
503 ex basic::series(const relational & r, int order, unsigned options) const
508 ex coeff = deriv.subs(r);
509 const symbol &s = ex_to<symbol>(r.lhs());
511 if (!coeff.is_zero())
512 seq.push_back(expair(coeff, _ex0));
515 for (n=1; n<order; ++n) {
517 // We need to test for zero in order to see if the series terminates.
518 // The problem is that there is no such thing as a perfect test for
519 // zero. Expanding the term occasionally helps a little...
520 deriv = deriv.diff(s).expand();
521 if (deriv.is_zero()) // Series terminates
522 return pseries(r, seq);
524 coeff = deriv.subs(r);
525 if (!coeff.is_zero())
526 seq.push_back(expair(fac.inverse() * coeff, n));
529 // Higher-order terms, if present
530 deriv = deriv.diff(s);
531 if (!deriv.expand().is_zero())
532 seq.push_back(expair(Order(_ex1), n));
533 return pseries(r, seq);
537 /** Implementation of ex::series() for symbols.
539 ex symbol::series(const relational & r, int order, unsigned options) const
542 const ex point = r.rhs();
543 GINAC_ASSERT(is_exactly_a<symbol>(r.lhs()));
545 if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
546 if (order > 0 && !point.is_zero())
547 seq.push_back(expair(point, _ex0));
549 seq.push_back(expair(_ex1, _ex1));
551 seq.push_back(expair(Order(_ex1), numeric(order)));
553 seq.push_back(expair(*this, _ex0));
554 return pseries(r, seq);
558 /** Add one series object to another, producing a pseries object that
559 * represents the sum.
561 * @param other pseries object to add with
562 * @return the sum as a pseries */
563 ex pseries::add_series(const pseries &other) const
565 // Adding two series with different variables or expansion points
566 // results in an empty (constant) series
567 if (!is_compatible_to(other)) {
569 nul.push_back(expair(Order(_ex1), _ex0));
570 return pseries(relational(var,point), nul);
575 epvector::const_iterator a = seq.begin();
576 epvector::const_iterator b = other.seq.begin();
577 epvector::const_iterator a_end = seq.end();
578 epvector::const_iterator b_end = other.seq.end();
579 int pow_a = INT_MAX, pow_b = INT_MAX;
581 // If a is empty, fill up with elements from b and stop
584 new_seq.push_back(*b);
589 pow_a = ex_to<numeric>((*a).coeff).to_int();
591 // If b is empty, fill up with elements from a and stop
594 new_seq.push_back(*a);
599 pow_b = ex_to<numeric>((*b).coeff).to_int();
601 // a and b are non-empty, compare powers
603 // a has lesser power, get coefficient from a
604 new_seq.push_back(*a);
605 if (is_order_function((*a).rest))
608 } else if (pow_b < pow_a) {
609 // b has lesser power, get coefficient from b
610 new_seq.push_back(*b);
611 if (is_order_function((*b).rest))
615 // Add coefficient of a and b
616 if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
617 new_seq.push_back(expair(Order(_ex1), (*a).coeff));
618 break; // Order term ends the sequence
620 ex sum = (*a).rest + (*b).rest;
621 if (!(sum.is_zero()))
622 new_seq.push_back(expair(sum, numeric(pow_a)));
628 return pseries(relational(var,point), new_seq);
632 /** Implementation of ex::series() for sums. This performs series addition when
633 * adding pseries objects.
635 ex add::series(const relational & r, int order, unsigned options) const
637 ex acc; // Series accumulator
639 // Get first term from overall_coeff
640 acc = overall_coeff.series(r, order, options);
642 // Add remaining terms
643 epvector::const_iterator it = seq.begin();
644 epvector::const_iterator itend = seq.end();
645 for (; it!=itend; ++it) {
647 if (is_exactly_a<pseries>(it->rest))
650 op = it->rest.series(r, order, options);
651 if (!it->coeff.is_equal(_ex1))
652 op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it->coeff));
655 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
661 /** Multiply a pseries object with a numeric constant, producing a pseries
662 * object that represents the product.
664 * @param other constant to multiply with
665 * @return the product as a pseries */
666 ex pseries::mul_const(const numeric &other) const
669 new_seq.reserve(seq.size());
671 epvector::const_iterator it = seq.begin(), itend = seq.end();
672 while (it != itend) {
673 if (!is_order_function(it->rest))
674 new_seq.push_back(expair(it->rest * other, it->coeff));
676 new_seq.push_back(*it);
679 return pseries(relational(var,point), new_seq);
683 /** Multiply one pseries object to another, producing a pseries object that
684 * represents the product.
686 * @param other pseries object to multiply with
687 * @return the product as a pseries */
688 ex pseries::mul_series(const pseries &other) const
690 // Multiplying two series with different variables or expansion points
691 // results in an empty (constant) series
692 if (!is_compatible_to(other)) {
694 nul.push_back(expair(Order(_ex1), _ex0));
695 return pseries(relational(var,point), nul);
698 // Series multiplication
700 int a_max = degree(var);
701 int b_max = other.degree(var);
702 int a_min = ldegree(var);
703 int b_min = other.ldegree(var);
704 int cdeg_min = a_min + b_min;
705 int cdeg_max = a_max + b_max;
707 int higher_order_a = INT_MAX;
708 int higher_order_b = INT_MAX;
709 if (is_order_function(coeff(var, a_max)))
710 higher_order_a = a_max + b_min;
711 if (is_order_function(other.coeff(var, b_max)))
712 higher_order_b = b_max + a_min;
713 int higher_order_c = std::min(higher_order_a, higher_order_b);
714 if (cdeg_max >= higher_order_c)
715 cdeg_max = higher_order_c - 1;
717 for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
719 // c(i)=a(0)b(i)+...+a(i)b(0)
720 for (int i=a_min; cdeg-i>=b_min; ++i) {
721 ex a_coeff = coeff(var, i);
722 ex b_coeff = other.coeff(var, cdeg-i);
723 if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
724 co += a_coeff * b_coeff;
727 new_seq.push_back(expair(co, numeric(cdeg)));
729 if (higher_order_c < INT_MAX)
730 new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c)));
731 return pseries(relational(var, point), new_seq);
735 /** Implementation of ex::series() for product. This performs series
736 * multiplication when multiplying series.
738 ex mul::series(const relational & r, int order, unsigned options) const
740 pseries acc; // Series accumulator
742 // Multiply with remaining terms
743 const epvector::const_iterator itbeg = seq.begin();
744 const epvector::const_iterator itend = seq.end();
745 for (epvector::const_iterator it=itbeg; it!=itend; ++it) {
746 ex op = recombine_pair_to_ex(*it).series(r, order, options);
748 // Series multiplication
750 acc = ex_to<pseries>(op);
752 acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
754 return acc.mul_const(ex_to<numeric>(overall_coeff));
758 /** Compute the p-th power of a series.
760 * @param p power to compute
761 * @param deg truncation order of series calculation */
762 ex pseries::power_const(const numeric &p, int deg) const
765 // (due to Leonhard Euler)
766 // let A(x) be this series and for the time being let it start with a
767 // constant (later we'll generalize):
768 // A(x) = a_0 + a_1*x + a_2*x^2 + ...
769 // We want to compute
771 // C(x) = c_0 + c_1*x + c_2*x^2 + ...
772 // Taking the derivative on both sides and multiplying with A(x) one
773 // immediately arrives at
774 // C'(x)*A(x) = p*C(x)*A'(x)
775 // Multiplying this out and comparing coefficients we get the recurrence
777 // c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
778 // ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
779 // which can easily be solved given the starting value c_0 = (a_0)^p.
780 // For the more general case where the leading coefficient of A(x) is not
781 // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
782 // repeat the above derivation. The leading power of C2(x) = A2(x)^2 is
783 // then of course x^(p*m) but the recurrence formula still holds.
786 // as a special case, handle the empty (zero) series honoring the
787 // usual power laws such as implemented in power::eval()
788 if (p.real().is_zero())
789 throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
790 else if (p.real().is_negative())
791 throw pole_error("pseries::power_const(): division by zero",1);
796 const int ldeg = ldegree(var);
797 if (!(p*ldeg).is_integer())
798 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
800 // O(x^n)^(-m) is undefined
801 if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
802 throw pole_error("pseries::power_const(): division by zero",1);
804 // Compute coefficients of the powered series
807 co.push_back(power(coeff(var, ldeg), p));
808 bool all_sums_zero = true;
809 for (int i=1; i<deg; ++i) {
811 for (int j=1; j<=i; ++j) {
812 ex c = coeff(var, j + ldeg);
813 if (is_order_function(c)) {
814 co.push_back(Order(_ex1));
817 sum += (p * j - (i - j)) * co[i - j] * c;
820 all_sums_zero = false;
821 co.push_back(sum / coeff(var, ldeg) / i);
824 // Construct new series (of non-zero coefficients)
826 bool higher_order = false;
827 for (int i=0; i<deg; ++i) {
828 if (!co[i].is_zero())
829 new_seq.push_back(expair(co[i], p * ldeg + i));
830 if (is_order_function(co[i])) {
835 if (!higher_order && !all_sums_zero)
836 new_seq.push_back(expair(Order(_ex1), p * ldeg + deg));
837 return pseries(relational(var,point), new_seq);
841 /** Return a new pseries object with the powers shifted by deg. */
842 pseries pseries::shift_exponents(int deg) const
844 epvector newseq = seq;
845 epvector::iterator i = newseq.begin(), end = newseq.end();
850 return pseries(relational(var, point), newseq);
854 /** Implementation of ex::series() for powers. This performs Laurent expansion
855 * of reciprocals of series at singularities.
857 ex power::series(const relational & r, int order, unsigned options) const
859 // If basis is already a series, just power it
860 if (is_exactly_a<pseries>(basis))
861 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
863 // Basis is not a series, may there be a singularity?
864 bool must_expand_basis = false;
867 } catch (pole_error) {
868 must_expand_basis = true;
871 // Is the expression of type something^(-int)?
872 if (!must_expand_basis && !exponent.info(info_flags::negint))
873 return basic::series(r, order, options);
875 // Is the expression of type 0^something?
876 if (!must_expand_basis && !basis.subs(r).is_zero())
877 return basic::series(r, order, options);
879 // Singularity encountered, is the basis equal to (var - point)?
880 if (basis.is_equal(r.lhs() - r.rhs())) {
882 if (ex_to<numeric>(exponent).to_int() < order)
883 new_seq.push_back(expair(_ex1, exponent));
885 new_seq.push_back(expair(Order(_ex1), exponent));
886 return pseries(r, new_seq);
889 // No, expand basis into series
890 ex e = basis.series(r, order, options);
891 return ex_to<pseries>(e).power_const(ex_to<numeric>(exponent), order);
895 /** Re-expansion of a pseries object. */
896 ex pseries::series(const relational & r, int order, unsigned options) const
898 const ex p = r.rhs();
899 GINAC_ASSERT(is_exactly_a<symbol>(r.lhs()));
900 const symbol &s = ex_to<symbol>(r.lhs());
902 if (var.is_equal(s) && point.is_equal(p)) {
903 if (order > degree(s))
907 epvector::const_iterator it = seq.begin(), itend = seq.end();
908 while (it != itend) {
909 int o = ex_to<numeric>(it->coeff).to_int();
911 new_seq.push_back(expair(Order(_ex1), o));
914 new_seq.push_back(*it);
917 return pseries(r, new_seq);
920 return convert_to_poly().series(r, order, options);
924 /** Compute the truncated series expansion of an expression.
925 * This function returns an expression containing an object of class pseries
926 * to represent the series. If the series does not terminate within the given
927 * truncation order, the last term of the series will be an order term.
929 * @param r expansion relation, lhs holds variable and rhs holds point
930 * @param order truncation order of series calculations
931 * @param options of class series_options
932 * @return an expression holding a pseries object */
933 ex ex::series(const ex & r, int order, unsigned options) const
939 if (is_exactly_a<relational>(r))
940 rel_ = ex_to<relational>(r);
941 else if (is_exactly_a<symbol>(r))
942 rel_ = relational(r,_ex0);
944 throw (std::logic_error("ex::series(): expansion point has unknown type"));
947 e = bp->series(rel_, order, options);
948 } catch (std::exception &x) {
949 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));