3 * Implementation of class for extended truncated power series and
4 * methods for series expansion. */
7 * GiNaC Copyright (C) 1999-2003 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"
41 GINAC_IMPLEMENT_REGISTERED_CLASS(pseries, basic)
48 pseries::pseries() : inherited(TINFO_pseries) { }
55 /** Construct pseries from a vector of coefficients and powers.
56 * expair.rest holds the coefficient, expair.coeff holds the power.
57 * The powers must be integers (positive or negative) and in ascending order;
58 * the last coefficient can be Order(_ex1) to represent a truncated,
59 * non-terminating series.
61 * @param rel_ expansion variable and point (must hold a relational)
62 * @param ops_ vector of {coefficient, power} pairs (coefficient must not be zero)
63 * @return newly constructed pseries */
64 pseries::pseries(const ex &rel_, const epvector &ops_) : basic(TINFO_pseries), seq(ops_)
66 GINAC_ASSERT(is_a<relational>(rel_));
67 GINAC_ASSERT(is_a<symbol>(rel_.lhs()));
77 pseries::pseries(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
79 for (unsigned int i=0; true; ++i) {
82 if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
83 seq.push_back(expair(rest, coeff));
87 n.find_ex("var", var, sym_lst);
88 n.find_ex("point", point, sym_lst);
91 void pseries::archive(archive_node &n) const
93 inherited::archive(n);
94 epvector::const_iterator i = seq.begin(), iend = seq.end();
96 n.add_ex("coeff", i->rest);
97 n.add_ex("power", i->coeff);
100 n.add_ex("var", var);
101 n.add_ex("point", point);
104 DEFAULT_UNARCHIVE(pseries)
107 // functions overriding virtual functions from base classes
110 void pseries::print(const print_context & c, unsigned level) const
112 if (is_a<print_tree>(c)) {
114 c.s << std::string(level, ' ') << class_name()
115 << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
117 unsigned delta_indent = static_cast<const print_tree &>(c).delta_indent;
118 size_t num = seq.size();
119 for (size_t i=0; i<num; ++i) {
120 seq[i].rest.print(c, level + delta_indent);
121 seq[i].coeff.print(c, level + delta_indent);
122 c.s << std::string(level + delta_indent, ' ') << "-----" << std::endl;
124 var.print(c, level + delta_indent);
125 point.print(c, level + delta_indent);
127 } else if (is_a<print_python_repr>(c)) {
128 c.s << class_name() << "(relational(";
133 size_t num = seq.size();
134 for (size_t i=0; i<num; ++i) {
138 seq[i].rest.print(c);
140 seq[i].coeff.print(c);
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)) {
184 if (is_a<print_python>(c))
188 if (i->coeff.info(info_flags::negative)) {
193 if (is_a<print_latex>(c)) {
203 Order(power(var-point,i->coeff)).print(c);
207 if (precedence() <= level)
212 int pseries::compare_same_type(const basic & other) const
214 GINAC_ASSERT(is_a<pseries>(other));
215 const pseries &o = static_cast<const pseries &>(other);
217 // first compare the lengths of the series...
218 if (seq.size()>o.seq.size())
220 if (seq.size()<o.seq.size())
223 // ...then the expansion point...
224 int cmpval = var.compare(o.var);
227 cmpval = point.compare(o.point);
231 // ...and if that failed the individual elements
232 epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
233 while (it!=seq.end() && o_it!=o.seq.end()) {
234 cmpval = it->compare(*o_it);
241 // so they are equal.
245 /** Return the number of operands including a possible order term. */
246 size_t pseries::nops() const
251 /** Return the ith term in the series when represented as a sum. */
252 ex pseries::op(size_t i) const
255 throw (std::out_of_range("op() out of range"));
257 return seq[i].rest * power(var - point, seq[i].coeff);
260 /** Return degree of highest power of the series. This is usually the exponent
261 * of the Order term. If s is not the expansion variable of the series, the
262 * series is examined termwise. */
263 int pseries::degree(const ex &s) const
265 if (var.is_equal(s)) {
266 // Return last exponent
268 return ex_to<numeric>((seq.end()-1)->coeff).to_int();
272 epvector::const_iterator it = seq.begin(), itend = seq.end();
275 int max_pow = INT_MIN;
276 while (it != itend) {
277 int pow = it->rest.degree(s);
286 /** Return degree of lowest power of the series. This is usually the exponent
287 * of the leading term. If s is not the expansion variable of the series, the
288 * series is examined termwise. If s is the expansion variable but the
289 * expansion point is not zero the series is not expanded to find the degree.
290 * I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
291 int pseries::ldegree(const ex &s) const
293 if (var.is_equal(s)) {
294 // Return first exponent
296 return ex_to<numeric>((seq.begin())->coeff).to_int();
300 epvector::const_iterator it = seq.begin(), itend = seq.end();
303 int min_pow = INT_MAX;
304 while (it != itend) {
305 int pow = it->rest.ldegree(s);
314 /** Return coefficient of degree n in power series if s is the expansion
315 * variable. If the expansion point is nonzero, by definition the n=1
316 * coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
317 * the expansion took place in the s in the first place).
318 * If s is not the expansion variable, an attempt is made to convert the
319 * series to a polynomial and return the corresponding coefficient from
321 ex pseries::coeff(const ex &s, int n) const
323 if (var.is_equal(s)) {
327 // Binary search in sequence for given power
328 numeric looking_for = numeric(n);
329 int lo = 0, hi = seq.size() - 1;
331 int mid = (lo + hi) / 2;
332 GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
333 int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
339 return seq[mid].rest;
344 throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
349 return convert_to_poly().coeff(s, n);
353 ex pseries::collect(const ex &s, bool distributed) const
358 /** Perform coefficient-wise automatic term rewriting rules in this class. */
359 ex pseries::eval(int level) const
364 if (level == -max_recursion_level)
365 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
367 // Construct a new series with evaluated coefficients
369 new_seq.reserve(seq.size());
370 epvector::const_iterator it = seq.begin(), itend = seq.end();
371 while (it != itend) {
372 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
375 return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
378 /** Evaluate coefficients numerically. */
379 ex pseries::evalf(int level) const
384 if (level == -max_recursion_level)
385 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
387 // Construct a new series with evaluated coefficients
389 new_seq.reserve(seq.size());
390 epvector::const_iterator it = seq.begin(), itend = seq.end();
391 while (it != itend) {
392 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
395 return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
398 ex pseries::subs(const exmap & m, unsigned options) const
400 // If expansion variable is being substituted, convert the series to a
401 // polynomial and do the substitution there because the result might
402 // no longer be a power series
403 if (m.find(var) != m.end())
404 return convert_to_poly(true).subs(m, options);
406 // Otherwise construct a new series with substituted coefficients and
409 newseq.reserve(seq.size());
410 epvector::const_iterator it = seq.begin(), itend = seq.end();
411 while (it != itend) {
412 newseq.push_back(expair(it->rest.subs(m, options), it->coeff));
415 return (new pseries(relational(var,point.subs(m, options)), newseq))->setflag(status_flags::dynallocated);
418 /** Implementation of ex::expand() for a power series. It expands all the
419 * terms individually and returns the resulting series as a new pseries. */
420 ex pseries::expand(unsigned options) const
423 epvector::const_iterator i = seq.begin(), end = seq.end();
425 ex restexp = i->rest.expand();
426 if (!restexp.is_zero())
427 newseq.push_back(expair(restexp, i->coeff));
430 return (new pseries(relational(var,point), newseq))
431 ->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
434 /** Implementation of ex::diff() for a power series.
436 ex pseries::derivative(const symbol & s) const
439 epvector::const_iterator it = seq.begin(), itend = seq.end();
443 // FIXME: coeff might depend on var
444 while (it != itend) {
445 if (is_order_function(it->rest)) {
446 new_seq.push_back(expair(it->rest, it->coeff - 1));
448 ex c = it->rest * it->coeff;
450 new_seq.push_back(expair(c, it->coeff - 1));
457 while (it != itend) {
458 if (is_order_function(it->rest)) {
459 new_seq.push_back(*it);
461 ex c = it->rest.diff(s);
463 new_seq.push_back(expair(c, it->coeff));
469 return pseries(relational(var,point), new_seq);
472 ex pseries::convert_to_poly(bool no_order) const
475 epvector::const_iterator it = seq.begin(), itend = seq.end();
477 while (it != itend) {
478 if (is_order_function(it->rest)) {
480 e += Order(power(var - point, it->coeff));
482 e += it->rest * power(var - point, it->coeff);
488 bool pseries::is_terminating() const
490 return seq.empty() || !is_order_function((seq.end()-1)->rest);
495 * Implementations of series expansion
498 /** Default implementation of ex::series(). This performs Taylor expansion.
500 ex basic::series(const relational & r, int order, unsigned options) const
505 ex coeff = deriv.subs(r, subs_options::no_pattern);
506 const symbol &s = ex_to<symbol>(r.lhs());
508 if (!coeff.is_zero())
509 seq.push_back(expair(coeff, _ex0));
512 for (n=1; n<order; ++n) {
514 // We need to test for zero in order to see if the series terminates.
515 // The problem is that there is no such thing as a perfect test for
516 // zero. Expanding the term occasionally helps a little...
517 deriv = deriv.diff(s).expand();
518 if (deriv.is_zero()) // Series terminates
519 return pseries(r, seq);
521 coeff = deriv.subs(r, subs_options::no_pattern);
522 if (!coeff.is_zero())
523 seq.push_back(expair(fac.inverse() * coeff, n));
526 // Higher-order terms, if present
527 deriv = deriv.diff(s);
528 if (!deriv.expand().is_zero())
529 seq.push_back(expair(Order(_ex1), n));
530 return pseries(r, seq);
534 /** Implementation of ex::series() for symbols.
536 ex symbol::series(const relational & r, int order, unsigned options) const
539 const ex point = r.rhs();
540 GINAC_ASSERT(is_a<symbol>(r.lhs()));
542 if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
543 if (order > 0 && !point.is_zero())
544 seq.push_back(expair(point, _ex0));
546 seq.push_back(expair(_ex1, _ex1));
548 seq.push_back(expair(Order(_ex1), numeric(order)));
550 seq.push_back(expair(*this, _ex0));
551 return pseries(r, seq);
555 /** Add one series object to another, producing a pseries object that
556 * represents the sum.
558 * @param other pseries object to add with
559 * @return the sum as a pseries */
560 ex pseries::add_series(const pseries &other) const
562 // Adding two series with different variables or expansion points
563 // results in an empty (constant) series
564 if (!is_compatible_to(other)) {
566 nul.push_back(expair(Order(_ex1), _ex0));
567 return pseries(relational(var,point), nul);
572 epvector::const_iterator a = seq.begin();
573 epvector::const_iterator b = other.seq.begin();
574 epvector::const_iterator a_end = seq.end();
575 epvector::const_iterator b_end = other.seq.end();
576 int pow_a = INT_MAX, pow_b = INT_MAX;
578 // If a is empty, fill up with elements from b and stop
581 new_seq.push_back(*b);
586 pow_a = ex_to<numeric>((*a).coeff).to_int();
588 // If b is empty, fill up with elements from a and stop
591 new_seq.push_back(*a);
596 pow_b = ex_to<numeric>((*b).coeff).to_int();
598 // a and b are non-empty, compare powers
600 // a has lesser power, get coefficient from a
601 new_seq.push_back(*a);
602 if (is_order_function((*a).rest))
605 } else if (pow_b < pow_a) {
606 // b has lesser power, get coefficient from b
607 new_seq.push_back(*b);
608 if (is_order_function((*b).rest))
612 // Add coefficient of a and b
613 if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
614 new_seq.push_back(expair(Order(_ex1), (*a).coeff));
615 break; // Order term ends the sequence
617 ex sum = (*a).rest + (*b).rest;
618 if (!(sum.is_zero()))
619 new_seq.push_back(expair(sum, numeric(pow_a)));
625 return pseries(relational(var,point), new_seq);
629 /** Implementation of ex::series() for sums. This performs series addition when
630 * adding pseries objects.
632 ex add::series(const relational & r, int order, unsigned options) const
634 ex acc; // Series accumulator
636 // Get first term from overall_coeff
637 acc = overall_coeff.series(r, order, options);
639 // Add remaining terms
640 epvector::const_iterator it = seq.begin();
641 epvector::const_iterator itend = seq.end();
642 for (; it!=itend; ++it) {
644 if (is_exactly_a<pseries>(it->rest))
647 op = it->rest.series(r, order, options);
648 if (!it->coeff.is_equal(_ex1))
649 op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it->coeff));
652 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
658 /** Multiply a pseries object with a numeric constant, producing a pseries
659 * object that represents the product.
661 * @param other constant to multiply with
662 * @return the product as a pseries */
663 ex pseries::mul_const(const numeric &other) const
666 new_seq.reserve(seq.size());
668 epvector::const_iterator it = seq.begin(), itend = seq.end();
669 while (it != itend) {
670 if (!is_order_function(it->rest))
671 new_seq.push_back(expair(it->rest * other, it->coeff));
673 new_seq.push_back(*it);
676 return pseries(relational(var,point), new_seq);
680 /** Multiply one pseries object to another, producing a pseries object that
681 * represents the product.
683 * @param other pseries object to multiply with
684 * @return the product as a pseries */
685 ex pseries::mul_series(const pseries &other) const
687 // Multiplying two series with different variables or expansion points
688 // results in an empty (constant) series
689 if (!is_compatible_to(other)) {
691 nul.push_back(expair(Order(_ex1), _ex0));
692 return pseries(relational(var,point), nul);
695 // Series multiplication
697 int a_max = degree(var);
698 int b_max = other.degree(var);
699 int a_min = ldegree(var);
700 int b_min = other.ldegree(var);
701 int cdeg_min = a_min + b_min;
702 int cdeg_max = a_max + b_max;
704 int higher_order_a = INT_MAX;
705 int higher_order_b = INT_MAX;
706 if (is_order_function(coeff(var, a_max)))
707 higher_order_a = a_max + b_min;
708 if (is_order_function(other.coeff(var, b_max)))
709 higher_order_b = b_max + a_min;
710 int higher_order_c = std::min(higher_order_a, higher_order_b);
711 if (cdeg_max >= higher_order_c)
712 cdeg_max = higher_order_c - 1;
714 for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
716 // c(i)=a(0)b(i)+...+a(i)b(0)
717 for (int i=a_min; cdeg-i>=b_min; ++i) {
718 ex a_coeff = coeff(var, i);
719 ex b_coeff = other.coeff(var, cdeg-i);
720 if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
721 co += a_coeff * b_coeff;
724 new_seq.push_back(expair(co, numeric(cdeg)));
726 if (higher_order_c < INT_MAX)
727 new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c)));
728 return pseries(relational(var, point), new_seq);
732 /** Implementation of ex::series() for product. This performs series
733 * multiplication when multiplying series.
735 ex mul::series(const relational & r, int order, unsigned options) const
737 pseries acc; // Series accumulator
739 // Multiply with remaining terms
740 const epvector::const_iterator itbeg = seq.begin();
741 const epvector::const_iterator itend = seq.end();
742 for (epvector::const_iterator it=itbeg; it!=itend; ++it) {
743 ex op = recombine_pair_to_ex(*it).series(r, order, options);
745 // Series multiplication
747 acc = ex_to<pseries>(op);
749 acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
751 return acc.mul_const(ex_to<numeric>(overall_coeff));
755 /** Compute the p-th power of a series.
757 * @param p power to compute
758 * @param deg truncation order of series calculation */
759 ex pseries::power_const(const numeric &p, int deg) const
762 // (due to Leonhard Euler)
763 // let A(x) be this series and for the time being let it start with a
764 // constant (later we'll generalize):
765 // A(x) = a_0 + a_1*x + a_2*x^2 + ...
766 // We want to compute
768 // C(x) = c_0 + c_1*x + c_2*x^2 + ...
769 // Taking the derivative on both sides and multiplying with A(x) one
770 // immediately arrives at
771 // C'(x)*A(x) = p*C(x)*A'(x)
772 // Multiplying this out and comparing coefficients we get the recurrence
774 // c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
775 // ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
776 // which can easily be solved given the starting value c_0 = (a_0)^p.
777 // For the more general case where the leading coefficient of A(x) is not
778 // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
779 // repeat the above derivation. The leading power of C2(x) = A2(x)^2 is
780 // then of course x^(p*m) but the recurrence formula still holds.
783 // as a special case, handle the empty (zero) series honoring the
784 // usual power laws such as implemented in power::eval()
785 if (p.real().is_zero())
786 throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
787 else if (p.real().is_negative())
788 throw pole_error("pseries::power_const(): division by zero",1);
793 const int ldeg = ldegree(var);
794 if (!(p*ldeg).is_integer())
795 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
797 // O(x^n)^(-m) is undefined
798 if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
799 throw pole_error("pseries::power_const(): division by zero",1);
801 // Compute coefficients of the powered series
804 co.push_back(power(coeff(var, ldeg), p));
805 bool all_sums_zero = true;
806 for (int i=1; i<deg; ++i) {
808 for (int j=1; j<=i; ++j) {
809 ex c = coeff(var, j + ldeg);
810 if (is_order_function(c)) {
811 co.push_back(Order(_ex1));
814 sum += (p * j - (i - j)) * co[i - j] * c;
817 all_sums_zero = false;
818 co.push_back(sum / coeff(var, ldeg) / i);
821 // Construct new series (of non-zero coefficients)
823 bool higher_order = false;
824 for (int i=0; i<deg; ++i) {
825 if (!co[i].is_zero())
826 new_seq.push_back(expair(co[i], p * ldeg + i));
827 if (is_order_function(co[i])) {
832 if (!higher_order && !all_sums_zero)
833 new_seq.push_back(expair(Order(_ex1), p * ldeg + deg));
834 return pseries(relational(var,point), new_seq);
838 /** Return a new pseries object with the powers shifted by deg. */
839 pseries pseries::shift_exponents(int deg) const
841 epvector newseq = seq;
842 epvector::iterator i = newseq.begin(), end = newseq.end();
847 return pseries(relational(var, point), newseq);
851 /** Implementation of ex::series() for powers. This performs Laurent expansion
852 * of reciprocals of series at singularities.
854 ex power::series(const relational & r, int order, unsigned options) const
856 // If basis is already a series, just power it
857 if (is_exactly_a<pseries>(basis))
858 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
860 // Basis is not a series, may there be a singularity?
861 bool must_expand_basis = false;
863 basis.subs(r, subs_options::no_pattern);
864 } catch (pole_error) {
865 must_expand_basis = true;
868 // Is the expression of type something^(-int)?
869 if (!must_expand_basis && !exponent.info(info_flags::negint))
870 return basic::series(r, order, options);
872 // Is the expression of type 0^something?
873 if (!must_expand_basis && !basis.subs(r, subs_options::no_pattern).is_zero())
874 return basic::series(r, order, options);
876 // Singularity encountered, is the basis equal to (var - point)?
877 if (basis.is_equal(r.lhs() - r.rhs())) {
879 if (ex_to<numeric>(exponent).to_int() < order)
880 new_seq.push_back(expair(_ex1, exponent));
882 new_seq.push_back(expair(Order(_ex1), exponent));
883 return pseries(r, new_seq);
886 // No, expand basis into series
887 ex e = basis.series(r, order, options);
888 return ex_to<pseries>(e).power_const(ex_to<numeric>(exponent), order);
892 /** Re-expansion of a pseries object. */
893 ex pseries::series(const relational & r, int order, unsigned options) const
895 const ex p = r.rhs();
896 GINAC_ASSERT(is_a<symbol>(r.lhs()));
897 const symbol &s = ex_to<symbol>(r.lhs());
899 if (var.is_equal(s) && point.is_equal(p)) {
900 if (order > degree(s))
904 epvector::const_iterator it = seq.begin(), itend = seq.end();
905 while (it != itend) {
906 int o = ex_to<numeric>(it->coeff).to_int();
908 new_seq.push_back(expair(Order(_ex1), o));
911 new_seq.push_back(*it);
914 return pseries(r, new_seq);
917 return convert_to_poly().series(r, order, options);
921 /** Compute the truncated series expansion of an expression.
922 * This function returns an expression containing an object of class pseries
923 * to represent the series. If the series does not terminate within the given
924 * truncation order, the last term of the series will be an order term.
926 * @param r expansion relation, lhs holds variable and rhs holds point
927 * @param order truncation order of series calculations
928 * @param options of class series_options
929 * @return an expression holding a pseries object */
930 ex ex::series(const ex & r, int order, unsigned options) const
935 if (is_a<relational>(r))
936 rel_ = ex_to<relational>(r);
937 else if (is_a<symbol>(r))
938 rel_ = relational(r,_ex0);
940 throw (std::logic_error("ex::series(): expansion point has unknown type"));
943 e = bp->series(rel_, order, options);
944 } catch (std::exception &x) {
945 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));