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
32 #include "relational.h"
40 GINAC_IMPLEMENT_REGISTERED_CLASS(pseries, basic)
44 * Default ctor, dtor, copy ctor, assignment operator and helpers
47 pseries::pseries() : basic(TINFO_pseries)
49 debugmsg("pseries default ctor", LOGLEVEL_CONSTRUCT);
52 void pseries::copy(const pseries &other)
54 inherited::copy(other);
60 DEFAULT_DESTROY(pseries)
67 /** Construct pseries from a vector of coefficients and powers.
68 * expair.rest holds the coefficient, expair.coeff holds the power.
69 * The powers must be integers (positive or negative) and in ascending order;
70 * the last coefficient can be Order(_ex1()) to represent a truncated,
71 * non-terminating series.
73 * @param rel_ expansion variable and point (must hold a relational)
74 * @param ops_ vector of {coefficient, power} pairs (coefficient must not be zero)
75 * @return newly constructed pseries */
76 pseries::pseries(const ex &rel_, const epvector &ops_) : basic(TINFO_pseries), seq(ops_)
78 debugmsg("pseries ctor from ex,epvector", LOGLEVEL_CONSTRUCT);
79 GINAC_ASSERT(is_ex_exactly_of_type(rel_, relational));
80 GINAC_ASSERT(is_ex_exactly_of_type(rel_.lhs(),symbol));
82 var = *static_cast<symbol *>(rel_.lhs().bp);
90 pseries::pseries(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
92 debugmsg("pseries ctor from archive_node", LOGLEVEL_CONSTRUCT);
93 for (unsigned int i=0; true; ++i) {
96 if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
97 seq.push_back(expair(rest, coeff));
101 n.find_ex("var", var, sym_lst);
102 n.find_ex("point", point, sym_lst);
105 void pseries::archive(archive_node &n) const
107 inherited::archive(n);
108 epvector::const_iterator i = seq.begin(), iend = seq.end();
110 n.add_ex("coeff", i->rest);
111 n.add_ex("power", i->coeff);
114 n.add_ex("var", var);
115 n.add_ex("point", point);
118 DEFAULT_UNARCHIVE(pseries)
121 // functions overriding virtual functions from bases classes
124 void pseries::print(std::ostream &os, unsigned upper_precedence) const
126 debugmsg("pseries print", LOGLEVEL_PRINT);
127 if (precedence<=upper_precedence) os << "(";
128 // objects of type pseries must not have any zero entries, so the
129 // trivial (zero) pseries needs a special treatment here:
132 for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
133 // print a sign, if needed
136 if (!is_order_function(i->rest)) {
137 // print 'rest', i.e. the expansion coefficient
138 if (i->rest.info(info_flags::numeric) &&
139 i->rest.info(info_flags::positive)) {
142 os << "(" << i->rest << ')';
143 // print 'coeff', something like (x-1)^42
144 if (!i->coeff.is_zero()) {
146 if (!point.is_zero())
147 os << '(' << var-point << ')';
150 if (i->coeff.compare(_ex1())) {
152 if (i->coeff.info(info_flags::negative))
153 os << '(' << i->coeff << ')';
159 os << Order(power(var-point,i->coeff));
162 if (precedence<=upper_precedence) os << ")";
166 void pseries::printraw(std::ostream &os) const
168 debugmsg("pseries printraw", LOGLEVEL_PRINT);
169 os << class_name() << "(" << var << ";" << point << ";";
170 for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i)
171 os << "(" << (*i).rest << "," << (*i).coeff << "),";
176 void pseries::printtree(std::ostream & os, unsigned indent) const
178 debugmsg("pseries printtree",LOGLEVEL_PRINT);
179 os << std::string(indent,' ') << class_name()
180 << ", hash=" << hashvalue
181 << " (0x" << std::hex << hashvalue << std::dec << ")"
182 << ", flags=" << flags << std::endl;
183 for (unsigned i=0; i<seq.size(); ++i) {
184 seq[i].rest.printtree(os,indent+delta_indent);
185 seq[i].coeff.printtree(os,indent+delta_indent);
187 os << std::string(indent+delta_indent,' ') << "-----" << std::endl;
189 var.printtree(os, indent+delta_indent);
190 point.printtree(os, indent+delta_indent);
193 int pseries::compare_same_type(const basic & other) const
195 GINAC_ASSERT(is_of_type(other, pseries));
196 const pseries &o = static_cast<const pseries &>(other);
198 // first compare the lengths of the series...
199 if (seq.size()>o.seq.size())
201 if (seq.size()<o.seq.size())
204 // ...then the expansion point...
205 int cmpval = var.compare(o.var);
208 cmpval = point.compare(o.point);
212 // ...and if that failed the individual elements
213 epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
214 while (it!=seq.end() && o_it!=o.seq.end()) {
215 cmpval = it->compare(*o_it);
222 // so they are equal.
226 /** Return the number of operands including a possible order term. */
227 unsigned pseries::nops(void) const
233 /** Return the ith term in the series when represented as a sum. */
234 ex pseries::op(int i) const
236 if (i < 0 || unsigned(i) >= seq.size())
237 throw (std::out_of_range("op() out of range"));
238 return seq[i].rest * power(var - point, seq[i].coeff);
242 ex &pseries::let_op(int i)
244 throw (std::logic_error("let_op not defined for pseries"));
248 /** Return degree of highest power of the series. This is usually the exponent
249 * of the Order term. If s is not the expansion variable of the series, the
250 * series is examined termwise. */
251 int pseries::degree(const ex &s) const
253 if (var.is_equal(s)) {
254 // Return last exponent
256 return ex_to_numeric((*(seq.end() - 1)).coeff).to_int();
260 epvector::const_iterator it = seq.begin(), itend = seq.end();
263 int max_pow = INT_MIN;
264 while (it != itend) {
265 int pow = it->rest.degree(s);
274 /** Return degree of lowest power of the series. This is usually the exponent
275 * of the leading term. If s is not the expansion variable of the series, the
276 * series is examined termwise. If s is the expansion variable but the
277 * expansion point is not zero the series is not expanded to find the degree.
278 * I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
279 int pseries::ldegree(const ex &s) const
281 if (var.is_equal(s)) {
282 // Return first exponent
284 return ex_to_numeric((*(seq.begin())).coeff).to_int();
288 epvector::const_iterator it = seq.begin(), itend = seq.end();
291 int min_pow = INT_MAX;
292 while (it != itend) {
293 int pow = it->rest.ldegree(s);
302 /** Return coefficient of degree n in power series if s is the expansion
303 * variable. If the expansion point is nonzero, by definition the n=1
304 * coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
305 * the expansion took place in the s in the first place).
306 * If s is not the expansion variable, an attempt is made to convert the
307 * series to a polynomial and return the corresponding coefficient from
309 ex pseries::coeff(const ex &s, int n) const
311 if (var.is_equal(s)) {
315 // Binary search in sequence for given power
316 numeric looking_for = numeric(n);
317 int lo = 0, hi = seq.size() - 1;
319 int mid = (lo + hi) / 2;
320 GINAC_ASSERT(is_ex_exactly_of_type(seq[mid].coeff, numeric));
321 int cmp = ex_to_numeric(seq[mid].coeff).compare(looking_for);
327 return seq[mid].rest;
332 throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
337 return convert_to_poly().coeff(s, n);
341 ex pseries::collect(const ex &s) const
347 /** Evaluate coefficients. */
348 ex pseries::eval(int level) const
353 if (level == -max_recursion_level)
354 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
356 // Construct a new series with evaluated coefficients
358 new_seq.reserve(seq.size());
359 epvector::const_iterator it = seq.begin(), itend = seq.end();
360 while (it != itend) {
361 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
364 return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
368 /** Evaluate coefficients numerically. */
369 ex pseries::evalf(int level) const
374 if (level == -max_recursion_level)
375 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
377 // Construct a new series with evaluated coefficients
379 new_seq.reserve(seq.size());
380 epvector::const_iterator it = seq.begin(), itend = seq.end();
381 while (it != itend) {
382 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
385 return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
389 ex pseries::subs(const lst & ls, const lst & lr) const
391 // If expansion variable is being substituted, convert the series to a
392 // polynomial and do the substitution there because the result might
393 // no longer be a power series
395 return convert_to_poly(true).subs(ls, lr);
397 // Otherwise construct a new series with substituted coefficients and
400 newseq.reserve(seq.size());
401 epvector::const_iterator it = seq.begin(), itend = seq.end();
402 while (it != itend) {
403 newseq.push_back(expair(it->rest.subs(ls, lr), it->coeff));
406 return (new pseries(relational(var,point.subs(ls, lr)), newseq))->setflag(status_flags::dynallocated);
410 /** Implementation of ex::expand() for a power series. It expands all the
411 * terms individually and returns the resulting series as a new pseries. */
412 ex pseries::expand(unsigned options) const
415 for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
416 ex restexp = i->rest.expand();
417 if (!restexp.is_zero())
418 newseq.push_back(expair(restexp, i->coeff));
420 return (new pseries(relational(var,point), newseq))
421 ->setflag(status_flags::dynallocated | status_flags::expanded);
425 /** Implementation of ex::diff() for a power series. It treats the series as a
428 ex pseries::derivative(const symbol & s) const
432 epvector::const_iterator it = seq.begin(), itend = seq.end();
434 // FIXME: coeff might depend on var
435 while (it != itend) {
436 if (is_order_function(it->rest)) {
437 new_seq.push_back(expair(it->rest, it->coeff - 1));
439 ex c = it->rest * it->coeff;
441 new_seq.push_back(expair(c, it->coeff - 1));
445 return pseries(relational(var,point), new_seq);
452 /** Convert a pseries object to an ordinary polynomial.
454 * @param no_order flag: discard higher order terms */
455 ex pseries::convert_to_poly(bool no_order) const
458 epvector::const_iterator it = seq.begin(), itend = seq.end();
460 while (it != itend) {
461 if (is_order_function(it->rest)) {
463 e += Order(power(var - point, it->coeff));
465 e += it->rest * power(var - point, it->coeff);
472 /** Returns true if there is no order term, i.e. the series terminates and
473 * false otherwise. */
474 bool pseries::is_terminating(void) const
476 return seq.size() == 0 || !is_order_function((seq.end()-1)->rest);
481 * Implementations of series expansion
484 /** Default implementation of ex::series(). This performs Taylor expansion.
486 ex basic::series(const relational & r, int order, unsigned options) const
491 ex coeff = deriv.subs(r);
492 const symbol &s = static_cast<symbol &>(*r.lhs().bp);
494 if (!coeff.is_zero())
495 seq.push_back(expair(coeff, numeric(0)));
498 for (n=1; n<order; ++n) {
499 fac = fac.mul(numeric(n));
500 deriv = deriv.diff(s).expand();
501 if (deriv.is_zero()) {
503 return pseries(r, seq);
505 coeff = deriv.subs(r);
506 if (!coeff.is_zero())
507 seq.push_back(expair(fac.inverse() * coeff, numeric(n)));
510 // Higher-order terms, if present
511 deriv = deriv.diff(s);
512 if (!deriv.expand().is_zero())
513 seq.push_back(expair(Order(_ex1()), numeric(n)));
514 return pseries(r, seq);
518 /** Implementation of ex::series() for symbols.
520 ex symbol::series(const relational & r, int order, unsigned options) const
523 const ex point = r.rhs();
524 GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
527 if (this->is_equal(*s.bp)) {
528 if (order > 0 && !point.is_zero())
529 seq.push_back(expair(point, _ex0()));
531 seq.push_back(expair(_ex1(), _ex1()));
533 seq.push_back(expair(Order(_ex1()), numeric(order)));
535 seq.push_back(expair(*this, _ex0()));
536 return pseries(r, seq);
540 /** Add one series object to another, producing a pseries object that
541 * represents the sum.
543 * @param other pseries object to add with
544 * @return the sum as a pseries */
545 ex pseries::add_series(const pseries &other) const
547 // Adding two series with different variables or expansion points
548 // results in an empty (constant) series
549 if (!is_compatible_to(other)) {
551 nul.push_back(expair(Order(_ex1()), _ex0()));
552 return pseries(relational(var,point), nul);
557 epvector::const_iterator a = seq.begin();
558 epvector::const_iterator b = other.seq.begin();
559 epvector::const_iterator a_end = seq.end();
560 epvector::const_iterator b_end = other.seq.end();
561 int pow_a = INT_MAX, pow_b = INT_MAX;
563 // If a is empty, fill up with elements from b and stop
566 new_seq.push_back(*b);
571 pow_a = ex_to_numeric((*a).coeff).to_int();
573 // If b is empty, fill up with elements from a and stop
576 new_seq.push_back(*a);
581 pow_b = ex_to_numeric((*b).coeff).to_int();
583 // a and b are non-empty, compare powers
585 // a has lesser power, get coefficient from a
586 new_seq.push_back(*a);
587 if (is_order_function((*a).rest))
590 } else if (pow_b < pow_a) {
591 // b has lesser power, get coefficient from b
592 new_seq.push_back(*b);
593 if (is_order_function((*b).rest))
597 // Add coefficient of a and b
598 if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
599 new_seq.push_back(expair(Order(_ex1()), (*a).coeff));
600 break; // Order term ends the sequence
602 ex sum = (*a).rest + (*b).rest;
603 if (!(sum.is_zero()))
604 new_seq.push_back(expair(sum, numeric(pow_a)));
610 return pseries(relational(var,point), new_seq);
614 /** Implementation of ex::series() for sums. This performs series addition when
615 * adding pseries objects.
617 ex add::series(const relational & r, int order, unsigned options) const
619 ex acc; // Series accumulator
621 // Get first term from overall_coeff
622 acc = overall_coeff.series(r, order, options);
624 // Add remaining terms
625 epvector::const_iterator it = seq.begin();
626 epvector::const_iterator itend = seq.end();
627 for (; it!=itend; ++it) {
629 if (is_ex_exactly_of_type(it->rest, pseries))
632 op = it->rest.series(r, order, options);
633 if (!it->coeff.is_equal(_ex1()))
634 op = ex_to_pseries(op).mul_const(ex_to_numeric(it->coeff));
637 acc = ex_to_pseries(acc).add_series(ex_to_pseries(op));
643 /** Multiply a pseries object with a numeric constant, producing a pseries
644 * object that represents the product.
646 * @param other constant to multiply with
647 * @return the product as a pseries */
648 ex pseries::mul_const(const numeric &other) const
651 new_seq.reserve(seq.size());
653 epvector::const_iterator it = seq.begin(), itend = seq.end();
654 while (it != itend) {
655 if (!is_order_function(it->rest))
656 new_seq.push_back(expair(it->rest * other, it->coeff));
658 new_seq.push_back(*it);
661 return pseries(relational(var,point), new_seq);
665 /** Multiply one pseries object to another, producing a pseries object that
666 * represents the product.
668 * @param other pseries object to multiply with
669 * @return the product as a pseries */
670 ex pseries::mul_series(const pseries &other) const
672 // Multiplying two series with different variables or expansion points
673 // results in an empty (constant) series
674 if (!is_compatible_to(other)) {
676 nul.push_back(expair(Order(_ex1()), _ex0()));
677 return pseries(relational(var,point), nul);
680 // Series multiplication
683 int a_max = degree(var);
684 int b_max = other.degree(var);
685 int a_min = ldegree(var);
686 int b_min = other.ldegree(var);
687 int cdeg_min = a_min + b_min;
688 int cdeg_max = a_max + b_max;
690 int higher_order_a = INT_MAX;
691 int higher_order_b = INT_MAX;
692 if (is_order_function(coeff(var, a_max)))
693 higher_order_a = a_max + b_min;
694 if (is_order_function(other.coeff(var, b_max)))
695 higher_order_b = b_max + a_min;
696 int higher_order_c = std::min(higher_order_a, higher_order_b);
697 if (cdeg_max >= higher_order_c)
698 cdeg_max = higher_order_c - 1;
700 for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
702 // c(i)=a(0)b(i)+...+a(i)b(0)
703 for (int i=a_min; cdeg-i>=b_min; ++i) {
704 ex a_coeff = coeff(var, i);
705 ex b_coeff = other.coeff(var, cdeg-i);
706 if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
707 co += a_coeff * b_coeff;
710 new_seq.push_back(expair(co, numeric(cdeg)));
712 if (higher_order_c < INT_MAX)
713 new_seq.push_back(expair(Order(_ex1()), numeric(higher_order_c)));
714 return pseries(relational(var, point), new_seq);
718 /** Implementation of ex::series() for product. This performs series
719 * multiplication when multiplying series.
721 ex mul::series(const relational & r, int order, unsigned options) const
723 ex acc; // Series accumulator
725 // Get first term from overall_coeff
726 acc = overall_coeff.series(r, order, options);
728 // Multiply with remaining terms
729 epvector::const_iterator it = seq.begin();
730 epvector::const_iterator itend = seq.end();
731 for (; it!=itend; ++it) {
733 if (op.info(info_flags::numeric)) {
734 // series * const (special case, faster)
735 ex f = power(op, it->coeff);
736 acc = ex_to_pseries(acc).mul_const(ex_to_numeric(f));
738 } else if (!is_ex_exactly_of_type(op, pseries))
739 op = op.series(r, order, options);
740 if (!it->coeff.is_equal(_ex1()))
741 op = ex_to_pseries(op).power_const(ex_to_numeric(it->coeff), order);
743 // Series multiplication
744 acc = ex_to_pseries(acc).mul_series(ex_to_pseries(op));
750 /** Compute the p-th power of a series.
752 * @param p power to compute
753 * @param deg truncation order of series calculation */
754 ex pseries::power_const(const numeric &p, int deg) const
757 // let A(x) be this series and for the time being let it start with a
758 // constant (later we'll generalize):
759 // A(x) = a_0 + a_1*x + a_2*x^2 + ...
760 // We want to compute
762 // C(x) = c_0 + c_1*x + c_2*x^2 + ...
763 // Taking the derivative on both sides and multiplying with A(x) one
764 // immediately arrives at
765 // C'(x)*A(x) = p*C(x)*A'(x)
766 // Multiplying this out and comparing coefficients we get the recurrence
768 // c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
769 // ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
770 // which can easily be solved given the starting value c_0 = (a_0)^p.
771 // For the more general case where the leading coefficient of A(x) is not
772 // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
773 // repeat the above derivation. The leading power of C2(x) = A2(x)^2 is
774 // then of course x^(p*m) but the recurrence formula still holds.
777 // as a spacial case, handle the empty (zero) series honoring the
778 // usual power laws such as implemented in power::eval()
779 if (p.real().is_zero())
780 throw (std::domain_error("pseries::power_const(): pow(0,I) is undefined"));
781 else if (p.real().is_negative())
782 throw (pole_error("pseries::power_const(): division by zero",1));
787 int ldeg = ldegree(var);
789 // Compute coefficients of the powered series
792 co.push_back(power(coeff(var, ldeg), p));
793 bool all_sums_zero = true;
794 for (int i=1; i<deg; ++i) {
796 for (int j=1; j<=i; ++j) {
797 ex c = coeff(var, j + ldeg);
798 if (is_order_function(c)) {
799 co.push_back(Order(_ex1()));
802 sum += (p * j - (i - j)) * co[i - j] * c;
805 all_sums_zero = false;
806 co.push_back(sum / coeff(var, ldeg) / numeric(i));
809 // Construct new series (of non-zero coefficients)
811 bool higher_order = false;
812 for (int i=0; i<deg; ++i) {
813 if (!co[i].is_zero())
814 new_seq.push_back(expair(co[i], numeric(i) + p * ldeg));
815 if (is_order_function(co[i])) {
820 if (!higher_order && !all_sums_zero)
821 new_seq.push_back(expair(Order(_ex1()), numeric(deg) + p * ldeg));
822 return pseries(relational(var,point), new_seq);
826 /** Return a new pseries object with the powers shifted by deg. */
827 pseries pseries::shift_exponents(int deg) const
829 epvector newseq(seq);
830 for (epvector::iterator i=newseq.begin(); i!=newseq.end(); ++i)
831 i->coeff = i->coeff + deg;
832 return pseries(relational(var, point), newseq);
836 /** Implementation of ex::series() for powers. This performs Laurent expansion
837 * of reciprocals of series at singularities.
839 ex power::series(const relational & r, int order, unsigned options) const
842 if (!is_ex_exactly_of_type(basis, pseries)) {
843 // Basis is not a series, may there be a singularity?
844 bool must_expand_basis = false;
847 } catch (pole_error) {
848 must_expand_basis = true;
851 // Is the expression of type something^(-int)?
852 if (!must_expand_basis && !exponent.info(info_flags::negint))
853 return basic::series(r, order, options);
855 // Is the expression of type 0^something?
856 if (!must_expand_basis && !basis.subs(r).is_zero())
857 return basic::series(r, order, options);
859 // Singularity encountered, expand basis into series
860 e = basis.series(r, order, options);
867 return ex_to_pseries(e).power_const(ex_to_numeric(exponent), order);
871 /** Re-expansion of a pseries object. */
872 ex pseries::series(const relational & r, int order, unsigned options) const
874 const ex p = r.rhs();
875 GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
876 const symbol &s = static_cast<symbol &>(*r.lhs().bp);
878 if (var.is_equal(s) && point.is_equal(p)) {
879 if (order > degree(s))
883 epvector::const_iterator it = seq.begin(), itend = seq.end();
884 while (it != itend) {
885 int o = ex_to_numeric(it->coeff).to_int();
887 new_seq.push_back(expair(Order(_ex1()), o));
890 new_seq.push_back(*it);
893 return pseries(r, new_seq);
896 return convert_to_poly().series(r, order, options);
900 /** Compute the truncated series expansion of an expression.
901 * This function returns an expression containing an object of class pseries
902 * to represent the series. If the series does not terminate within the given
903 * truncation order, the last term of the series will be an order term.
905 * @param r expansion relation, lhs holds variable and rhs holds point
906 * @param order truncation order of series calculations
907 * @param options of class series_options
908 * @return an expression holding a pseries object */
909 ex ex::series(const ex & r, int order, unsigned options) const
915 if (is_ex_exactly_of_type(r,relational))
916 rel_ = ex_to_relational(r);
917 else if (is_ex_exactly_of_type(r,symbol))
918 rel_ = relational(r,_ex0());
920 throw (std::logic_error("ex::series(): expansion point has unknown type"));
923 e = bp->series(rel_, order, options);
924 } catch (std::exception &x) {
925 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));
931 // static member variables
936 unsigned pseries::precedence = 38; // for clarity just below add::precedence