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)
43 * Default ctor, dtor, copy ctor, assignment operator and helpers
46 pseries::pseries() : basic(TINFO_pseries)
48 debugmsg("pseries default ctor", LOGLEVEL_CONSTRUCT);
51 void pseries::copy(const pseries &other)
53 inherited::copy(other);
59 void pseries::destroy(bool call_parent)
62 inherited::destroy(call_parent);
70 /** Construct pseries from a vector of coefficients and powers.
71 * expair.rest holds the coefficient, expair.coeff holds the power.
72 * The powers must be integers (positive or negative) and in ascending order;
73 * the last coefficient can be Order(_ex1()) to represent a truncated,
74 * non-terminating series.
76 * @param rel_ expansion variable and point (must hold a relational)
77 * @param ops_ vector of {coefficient, power} pairs (coefficient must not be zero)
78 * @return newly constructed pseries */
79 pseries::pseries(const ex &rel_, const epvector &ops_) : basic(TINFO_pseries), seq(ops_)
81 debugmsg("pseries ctor from ex,epvector", LOGLEVEL_CONSTRUCT);
82 GINAC_ASSERT(is_ex_exactly_of_type(rel_, relational));
83 GINAC_ASSERT(is_ex_exactly_of_type(rel_.lhs(),symbol));
85 var = *static_cast<symbol *>(rel_.lhs().bp);
93 /** Construct object from archive_node. */
94 pseries::pseries(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
96 debugmsg("pseries ctor from archive_node", LOGLEVEL_CONSTRUCT);
97 for (unsigned int i=0; true; ++i) {
100 if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
101 seq.push_back(expair(rest, coeff));
105 n.find_ex("var", var, sym_lst);
106 n.find_ex("point", point, sym_lst);
109 /** Unarchive the object. */
110 ex pseries::unarchive(const archive_node &n, const lst &sym_lst)
112 return (new pseries(n, sym_lst))->setflag(status_flags::dynallocated);
115 /** Archive the object. */
116 void pseries::archive(archive_node &n) const
118 inherited::archive(n);
119 epvector::const_iterator i = seq.begin(), iend = seq.end();
121 n.add_ex("coeff", i->rest);
122 n.add_ex("power", i->coeff);
125 n.add_ex("var", var);
126 n.add_ex("point", point);
130 // functions overriding virtual functions from bases classes
133 void pseries::print(std::ostream &os, unsigned upper_precedence) const
135 debugmsg("pseries print", LOGLEVEL_PRINT);
136 if (precedence<=upper_precedence) os << "(";
137 // objects of type pseries must not have any zero entries, so the
138 // trivial (zero) pseries needs a special treatment here:
141 for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
142 // print a sign, if needed
145 if (!is_order_function(i->rest)) {
146 // print 'rest', i.e. the expansion coefficient
147 if (i->rest.info(info_flags::numeric) &&
148 i->rest.info(info_flags::positive)) {
151 os << "(" << i->rest << ')';
152 // print 'coeff', something like (x-1)^42
153 if (!i->coeff.is_zero()) {
155 if (!point.is_zero())
156 os << '(' << var-point << ')';
159 if (i->coeff.compare(_ex1())) {
161 if (i->coeff.info(info_flags::negative))
162 os << '(' << i->coeff << ')';
168 os << Order(power(var-point,i->coeff));
171 if (precedence<=upper_precedence) os << ")";
175 void pseries::printraw(std::ostream &os) const
177 debugmsg("pseries printraw", LOGLEVEL_PRINT);
178 os << "pseries(" << var << ";" << point << ";";
179 for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i)
180 os << "(" << (*i).rest << "," << (*i).coeff << "),";
185 void pseries::printtree(std::ostream & os, unsigned indent) const
187 debugmsg("pseries printtree",LOGLEVEL_PRINT);
188 os << std::string(indent,' ') << "pseries "
189 << ", hash=" << hashvalue
190 << " (0x" << std::hex << hashvalue << std::dec << ")"
191 << ", flags=" << flags << std::endl;
192 for (unsigned i=0; i<seq.size(); ++i) {
193 seq[i].rest.printtree(os,indent+delta_indent);
194 seq[i].coeff.printtree(os,indent+delta_indent);
196 os << std::string(indent+delta_indent,' ') << "-----" << std::endl;
198 var.printtree(os, indent+delta_indent);
199 point.printtree(os, indent+delta_indent);
202 int pseries::compare_same_type(const basic & other) const
204 GINAC_ASSERT(is_of_type(other, pseries));
205 const pseries &o = static_cast<const pseries &>(other);
207 int cmpval = var.compare(o.var);
210 cmpval = point.compare(o.point);
214 epvector::const_iterator it1 = seq.begin(), it2 = o.seq.begin(), it1end = seq.end(), it2end = o.seq.end();
215 while ((it1 != it1end) && (it2 != it2end)) {
216 cmpval = it1->compare(*it2);
222 return it2 == it2end ? 0 : -1;
227 /** Return the number of operands including a possible order term. */
228 unsigned pseries::nops(void) const
234 /** Return the ith term in the series when represented as a sum. */
235 ex pseries::op(int i) const
237 if (i < 0 || unsigned(i) >= seq.size())
238 throw (std::out_of_range("op() out of range"));
239 return seq[i].rest * power(var - point, seq[i].coeff);
243 ex &pseries::let_op(int i)
245 throw (std::logic_error("let_op not defined for pseries"));
249 /** Return degree of highest power of the series. This is usually the exponent
250 * of the Order term. If s is not the expansion variable of the series, the
251 * series is examined termwise. */
252 int pseries::degree(const symbol &s) const
254 if (var.is_equal(s)) {
255 // Return last exponent
257 return ex_to_numeric((*(seq.end() - 1)).coeff).to_int();
261 epvector::const_iterator it = seq.begin(), itend = seq.end();
264 int max_pow = INT_MIN;
265 while (it != itend) {
266 int pow = it->rest.degree(s);
275 /** Return degree of lowest power of the series. This is usually the exponent
276 * of the leading term. If s is not the expansion variable of the series, the
277 * series is examined termwise. If s is the expansion variable but the
278 * expansion point is not zero the series is not expanded to find the degree.
279 * I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
280 int pseries::ldegree(const symbol &s) const
282 if (var.is_equal(s)) {
283 // Return first exponent
285 return ex_to_numeric((*(seq.begin())).coeff).to_int();
289 epvector::const_iterator it = seq.begin(), itend = seq.end();
292 int min_pow = INT_MAX;
293 while (it != itend) {
294 int pow = it->rest.ldegree(s);
303 /** Return coefficient of degree n in power series if s is the expansion
304 * variable. If the expansion point is nonzero, by definition the n=1
305 * coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
306 * the expansion took place in the s in the first place).
307 * If s is not the expansion variable, an attempt is made to convert the
308 * series to a polynomial and return the corresponding coefficient from
310 ex pseries::coeff(const symbol &s, int n) const
312 if (var.is_equal(s)) {
316 // Binary search in sequence for given power
317 numeric looking_for = numeric(n);
318 int lo = 0, hi = seq.size() - 1;
320 int mid = (lo + hi) / 2;
321 GINAC_ASSERT(is_ex_exactly_of_type(seq[mid].coeff, numeric));
322 int cmp = ex_to_numeric(seq[mid].coeff).compare(looking_for);
328 return seq[mid].rest;
333 throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
338 return convert_to_poly().coeff(s, n);
342 ex pseries::collect(const symbol &s) const
348 /** Evaluate coefficients. */
349 ex pseries::eval(int level) const
354 if (level == -max_recursion_level)
355 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
357 // Construct a new series with evaluated coefficients
359 new_seq.reserve(seq.size());
360 epvector::const_iterator it = seq.begin(), itend = seq.end();
361 while (it != itend) {
362 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
365 return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
369 /** Evaluate coefficients numerically. */
370 ex pseries::evalf(int level) const
375 if (level == -max_recursion_level)
376 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
378 // Construct a new series with evaluated coefficients
380 new_seq.reserve(seq.size());
381 epvector::const_iterator it = seq.begin(), itend = seq.end();
382 while (it != itend) {
383 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
386 return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
390 ex pseries::subs(const lst & ls, const lst & lr) const
392 // If expansion variable is being substituted, convert the series to a
393 // polynomial and do the substitution there because the result might
394 // no longer be a power series
396 return convert_to_poly(true).subs(ls, lr);
398 // Otherwise construct a new series with substituted coefficients and
401 newseq.reserve(seq.size());
402 epvector::const_iterator it = seq.begin(), itend = seq.end();
403 while (it != itend) {
404 newseq.push_back(expair(it->rest.subs(ls, lr), it->coeff));
407 return (new pseries(relational(var,point.subs(ls, lr)), newseq))->setflag(status_flags::dynallocated);
411 /** Implementation of ex::expand() for a power series. It expands all the
412 * terms individually and returns the resulting series as a new pseries. */
413 ex pseries::expand(unsigned options) const
416 for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
417 ex restexp = i->rest.expand();
418 if (!restexp.is_zero())
419 newseq.push_back(expair(restexp, i->coeff));
421 return (new pseries(relational(var,point), newseq))
422 ->setflag(status_flags::dynallocated | status_flags::expanded);
426 /** Implementation of ex::diff() for a power series. It treats the series as a
429 ex pseries::derivative(const symbol & s) const
433 epvector::const_iterator it = seq.begin(), itend = seq.end();
435 // FIXME: coeff might depend on var
436 while (it != itend) {
437 if (is_order_function(it->rest)) {
438 new_seq.push_back(expair(it->rest, it->coeff - 1));
440 ex c = it->rest * it->coeff;
442 new_seq.push_back(expair(c, it->coeff - 1));
446 return pseries(relational(var,point), new_seq);
453 /** Convert a pseries object to an ordinary polynomial.
455 * @param no_order flag: discard higher order terms */
456 ex pseries::convert_to_poly(bool no_order) const
459 epvector::const_iterator it = seq.begin(), itend = seq.end();
461 while (it != itend) {
462 if (is_order_function(it->rest)) {
464 e += Order(power(var - point, it->coeff));
466 e += it->rest * power(var - point, it->coeff);
473 /** Returns true if there is no order term, i.e. the series terminates and
474 * false otherwise. */
475 bool pseries::is_terminating(void) const
477 return seq.size() == 0 || !is_order_function((seq.end()-1)->rest);
482 * Implementations of series expansion
485 /** Default implementation of ex::series(). This performs Taylor expansion.
487 ex basic::series(const relational & r, int order, unsigned options) const
492 ex coeff = deriv.subs(r);
493 const symbol *s = static_cast<symbol *>(r.lhs().bp);
495 if (!coeff.is_zero())
496 seq.push_back(expair(coeff, numeric(0)));
499 for (n=1; n<order; ++n) {
500 fac = fac.mul(numeric(n));
501 deriv = deriv.diff(*s).expand();
502 if (deriv.is_zero()) {
504 return pseries(r, seq);
506 coeff = deriv.subs(r);
507 if (!coeff.is_zero())
508 seq.push_back(expair(fac.inverse() * coeff, numeric(n)));
511 // Higher-order terms, if present
512 deriv = deriv.diff(*s);
513 if (!deriv.expand().is_zero())
514 seq.push_back(expair(Order(_ex1()), numeric(n)));
515 return pseries(r, seq);
519 /** Implementation of ex::series() for symbols.
521 ex symbol::series(const relational & r, int order, unsigned options) const
524 const ex point = r.rhs();
525 GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
526 const symbol *s = static_cast<symbol *>(r.lhs().bp);
528 if (this->is_equal(*s)) {
529 if (order > 0 && !point.is_zero())
530 seq.push_back(expair(point, _ex0()));
532 seq.push_back(expair(_ex1(), _ex1()));
534 seq.push_back(expair(Order(_ex1()), numeric(order)));
536 seq.push_back(expair(*this, _ex0()));
537 return pseries(r, seq);
541 /** Add one series object to another, producing a pseries object that
542 * represents the sum.
544 * @param other pseries object to add with
545 * @return the sum as a pseries */
546 ex pseries::add_series(const pseries &other) const
548 // Adding two series with different variables or expansion points
549 // results in an empty (constant) series
550 if (!is_compatible_to(other)) {
552 nul.push_back(expair(Order(_ex1()), _ex0()));
553 return pseries(relational(var,point), nul);
558 epvector::const_iterator a = seq.begin();
559 epvector::const_iterator b = other.seq.begin();
560 epvector::const_iterator a_end = seq.end();
561 epvector::const_iterator b_end = other.seq.end();
562 int pow_a = INT_MAX, pow_b = INT_MAX;
564 // If a is empty, fill up with elements from b and stop
567 new_seq.push_back(*b);
572 pow_a = ex_to_numeric((*a).coeff).to_int();
574 // If b is empty, fill up with elements from a and stop
577 new_seq.push_back(*a);
582 pow_b = ex_to_numeric((*b).coeff).to_int();
584 // a and b are non-empty, compare powers
586 // a has lesser power, get coefficient from a
587 new_seq.push_back(*a);
588 if (is_order_function((*a).rest))
591 } else if (pow_b < pow_a) {
592 // b has lesser power, get coefficient from b
593 new_seq.push_back(*b);
594 if (is_order_function((*b).rest))
598 // Add coefficient of a and b
599 if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
600 new_seq.push_back(expair(Order(_ex1()), (*a).coeff));
601 break; // Order term ends the sequence
603 ex sum = (*a).rest + (*b).rest;
604 if (!(sum.is_zero()))
605 new_seq.push_back(expair(sum, numeric(pow_a)));
611 return pseries(relational(var,point), new_seq);
615 /** Implementation of ex::series() for sums. This performs series addition when
616 * adding pseries objects.
618 ex add::series(const relational & r, int order, unsigned options) const
620 ex acc; // Series accumulator
622 // Get first term from overall_coeff
623 acc = overall_coeff.series(r, order, options);
625 // Add remaining terms
626 epvector::const_iterator it = seq.begin();
627 epvector::const_iterator itend = seq.end();
628 for (; it!=itend; ++it) {
630 if (is_ex_exactly_of_type(it->rest, pseries))
633 op = it->rest.series(r, order, options);
634 if (!it->coeff.is_equal(_ex1()))
635 op = ex_to_pseries(op).mul_const(ex_to_numeric(it->coeff));
638 acc = ex_to_pseries(acc).add_series(ex_to_pseries(op));
644 /** Multiply a pseries object with a numeric constant, producing a pseries
645 * object that represents the product.
647 * @param other constant to multiply with
648 * @return the product as a pseries */
649 ex pseries::mul_const(const numeric &other) const
652 new_seq.reserve(seq.size());
654 epvector::const_iterator it = seq.begin(), itend = seq.end();
655 while (it != itend) {
656 if (!is_order_function(it->rest))
657 new_seq.push_back(expair(it->rest * other, it->coeff));
659 new_seq.push_back(*it);
662 return pseries(relational(var,point), new_seq);
666 /** Multiply one pseries object to another, producing a pseries object that
667 * represents the product.
669 * @param other pseries object to multiply with
670 * @return the product as a pseries */
671 ex pseries::mul_series(const pseries &other) const
673 // Multiplying two series with different variables or expansion points
674 // results in an empty (constant) series
675 if (!is_compatible_to(other)) {
677 nul.push_back(expair(Order(_ex1()), _ex0()));
678 return pseries(relational(var,point), nul);
681 // Series multiplication
684 const symbol *s = static_cast<symbol *>(var.bp);
685 int a_max = degree(*s);
686 int b_max = other.degree(*s);
687 int a_min = ldegree(*s);
688 int b_min = other.ldegree(*s);
689 int cdeg_min = a_min + b_min;
690 int cdeg_max = a_max + b_max;
692 int higher_order_a = INT_MAX;
693 int higher_order_b = INT_MAX;
694 if (is_order_function(coeff(*s, a_max)))
695 higher_order_a = a_max + b_min;
696 if (is_order_function(other.coeff(*s, b_max)))
697 higher_order_b = b_max + a_min;
698 int higher_order_c = std::min(higher_order_a, higher_order_b);
699 if (cdeg_max >= higher_order_c)
700 cdeg_max = higher_order_c - 1;
702 for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
704 // c(i)=a(0)b(i)+...+a(i)b(0)
705 for (int i=a_min; cdeg-i>=b_min; ++i) {
706 ex a_coeff = coeff(*s, i);
707 ex b_coeff = other.coeff(*s, cdeg-i);
708 if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
709 co += a_coeff * b_coeff;
712 new_seq.push_back(expair(co, numeric(cdeg)));
714 if (higher_order_c < INT_MAX)
715 new_seq.push_back(expair(Order(_ex1()), numeric(higher_order_c)));
716 return pseries(relational(var,point), new_seq);
720 /** Implementation of ex::series() for product. This performs series
721 * multiplication when multiplying series.
723 ex mul::series(const relational & r, int order, unsigned options) const
725 ex acc; // Series accumulator
727 // Get first term from overall_coeff
728 acc = overall_coeff.series(r, order, options);
730 // Multiply with remaining terms
731 epvector::const_iterator it = seq.begin();
732 epvector::const_iterator itend = seq.end();
733 for (; it!=itend; ++it) {
735 if (op.info(info_flags::numeric)) {
736 // series * const (special case, faster)
737 ex f = power(op, it->coeff);
738 acc = ex_to_pseries(acc).mul_const(ex_to_numeric(f));
740 } else if (!is_ex_exactly_of_type(op, pseries))
741 op = op.series(r, order, options);
742 if (!it->coeff.is_equal(_ex1()))
743 op = ex_to_pseries(op).power_const(ex_to_numeric(it->coeff), order);
745 // Series multiplication
746 acc = ex_to_pseries(acc).mul_series(ex_to_pseries(op));
752 /** Compute the p-th power of a series.
754 * @param p power to compute
755 * @param deg truncation order of series calculation */
756 ex pseries::power_const(const numeric &p, int deg) const
759 // let A(x) be this series and for the time being let it start with a
760 // constant (later we'll generalize):
761 // A(x) = a_0 + a_1*x + a_2*x^2 + ...
762 // We want to compute
764 // C(x) = c_0 + c_1*x + c_2*x^2 + ...
765 // Taking the derivative on both sides and multiplying with A(x) one
766 // immediately arrives at
767 // C'(x)*A(x) = p*C(x)*A'(x)
768 // Multiplying this out and comparing coefficients we get the recurrence
770 // c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
771 // ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
772 // which can easily be solved given the starting value c_0 = (a_0)^p.
773 // For the more general case where the leading coefficient of A(x) is not
774 // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
775 // repeat the above derivation. The leading power of C2(x) = A2(x)^2 is
776 // then of course x^(p*m) but the recurrence formula still holds.
779 // as a spacial case, handle the empty (zero) series honoring the
780 // usual power laws such as implemented in power::eval()
781 if (p.real().is_zero())
782 throw (std::domain_error("pseries::power_const(): pow(0,I) is undefined"));
783 else if (p.real().is_negative())
784 throw (pole_error("pseries::power_const(): division by zero",1));
789 const symbol *s = static_cast<symbol *>(var.bp);
790 int ldeg = ldegree(*s);
792 // Compute coefficients of the powered series
795 co.push_back(power(coeff(*s, ldeg), p));
796 bool all_sums_zero = true;
797 for (int i=1; i<deg; ++i) {
799 for (int j=1; j<=i; ++j) {
800 ex c = coeff(*s, j + ldeg);
801 if (is_order_function(c)) {
802 co.push_back(Order(_ex1()));
805 sum += (p * j - (i - j)) * co[i - j] * c;
808 all_sums_zero = false;
809 co.push_back(sum / coeff(*s, ldeg) / numeric(i));
812 // Construct new series (of non-zero coefficients)
814 bool higher_order = false;
815 for (int i=0; i<deg; ++i) {
816 if (!co[i].is_zero())
817 new_seq.push_back(expair(co[i], numeric(i) + p * ldeg));
818 if (is_order_function(co[i])) {
823 if (!higher_order && !all_sums_zero)
824 new_seq.push_back(expair(Order(_ex1()), numeric(deg) + p * ldeg));
825 return pseries(relational(var,point), new_seq);
829 /** Return a new pseries object with the powers shifted by deg. */
830 pseries pseries::shift_exponents(int deg) const
832 epvector newseq(seq);
833 for (epvector::iterator i=newseq.begin(); i!=newseq.end(); ++i)
834 i->coeff = i->coeff + deg;
835 return pseries(relational(var, point), newseq);
839 /** Implementation of ex::series() for powers. This performs Laurent expansion
840 * of reciprocals of series at singularities.
842 ex power::series(const relational & r, int order, unsigned options) const
845 if (!is_ex_exactly_of_type(basis, pseries)) {
846 // Basis is not a series, may there be a singularity?
847 bool must_expand_basis = false;
850 } catch (pole_error) {
851 must_expand_basis = true;
854 // Is the expression of type something^(-int)?
855 if (!must_expand_basis && !exponent.info(info_flags::negint))
856 return basic::series(r, order, options);
858 // Is the expression of type 0^something?
859 if (!must_expand_basis && !basis.subs(r).is_zero())
860 return basic::series(r, order, options);
862 // Singularity encountered, expand basis into series
863 e = basis.series(r, order, options);
870 return ex_to_pseries(e).power_const(ex_to_numeric(exponent), order);
874 /** Re-expansion of a pseries object. */
875 ex pseries::series(const relational & r, int order, unsigned options) const
877 const ex p = r.rhs();
878 GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
879 const symbol *s = static_cast<symbol *>(r.lhs().bp);
881 if (var.is_equal(*s) && point.is_equal(p)) {
882 if (order > degree(*s))
886 epvector::const_iterator it = seq.begin(), itend = seq.end();
887 while (it != itend) {
888 int o = ex_to_numeric(it->coeff).to_int();
890 new_seq.push_back(expair(Order(_ex1()), o));
893 new_seq.push_back(*it);
896 return pseries(r, new_seq);
899 return convert_to_poly().series(r, order, options);
903 /** Compute the truncated series expansion of an expression.
904 * This function returns an expression containing an object of class pseries
905 * to represent the series. If the series does not terminate within the given
906 * truncation order, the last term of the series will be an order term.
908 * @param r expansion relation, lhs holds variable and rhs holds point
909 * @param order truncation order of series calculations
910 * @param options of class series_options
911 * @return an expression holding a pseries object */
912 ex ex::series(const ex & r, int order, unsigned options) const
918 if (is_ex_exactly_of_type(r,relational))
919 rel_ = ex_to_relational(r);
920 else if (is_ex_exactly_of_type(r,symbol))
921 rel_ = relational(r,_ex0());
923 throw (std::logic_error("ex::series(): expansion point has unknown type"));
926 e = bp->series(rel_, order, options);
927 } catch (std::exception &x) {
928 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));
934 // static member variables
939 unsigned pseries::precedence = 38; // for clarity just below add::precedence