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 << class_name() << "(" << 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,' ') << class_name()
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 // first compare the lengths of the series...
208 if (seq.size()>o.seq.size())
210 if (seq.size()<o.seq.size())
213 // ...then the expansion point...
214 int cmpval = var.compare(o.var);
217 cmpval = point.compare(o.point);
221 // ...and if that failed the individual elements
222 epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
223 while (it!=seq.end() && o_it!=o.seq.end()) {
224 cmpval = it->compare(*o_it);
231 // so they are equal.
235 /** Return the number of operands including a possible order term. */
236 unsigned pseries::nops(void) const
242 /** Return the ith term in the series when represented as a sum. */
243 ex pseries::op(int i) const
245 if (i < 0 || unsigned(i) >= seq.size())
246 throw (std::out_of_range("op() out of range"));
247 return seq[i].rest * power(var - point, seq[i].coeff);
251 ex &pseries::let_op(int i)
253 throw (std::logic_error("let_op not defined for pseries"));
257 /** Return degree of highest power of the series. This is usually the exponent
258 * of the Order term. If s is not the expansion variable of the series, the
259 * series is examined termwise. */
260 int pseries::degree(const symbol &s) const
262 if (var.is_equal(s)) {
263 // Return last exponent
265 return ex_to_numeric((*(seq.end() - 1)).coeff).to_int();
269 epvector::const_iterator it = seq.begin(), itend = seq.end();
272 int max_pow = INT_MIN;
273 while (it != itend) {
274 int pow = it->rest.degree(s);
283 /** Return degree of lowest power of the series. This is usually the exponent
284 * of the leading term. If s is not the expansion variable of the series, the
285 * series is examined termwise. If s is the expansion variable but the
286 * expansion point is not zero the series is not expanded to find the degree.
287 * I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
288 int pseries::ldegree(const symbol &s) const
290 if (var.is_equal(s)) {
291 // Return first exponent
293 return ex_to_numeric((*(seq.begin())).coeff).to_int();
297 epvector::const_iterator it = seq.begin(), itend = seq.end();
300 int min_pow = INT_MAX;
301 while (it != itend) {
302 int pow = it->rest.ldegree(s);
311 /** Return coefficient of degree n in power series if s is the expansion
312 * variable. If the expansion point is nonzero, by definition the n=1
313 * coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
314 * the expansion took place in the s in the first place).
315 * If s is not the expansion variable, an attempt is made to convert the
316 * series to a polynomial and return the corresponding coefficient from
318 ex pseries::coeff(const symbol &s, int n) const
320 if (var.is_equal(s)) {
324 // Binary search in sequence for given power
325 numeric looking_for = numeric(n);
326 int lo = 0, hi = seq.size() - 1;
328 int mid = (lo + hi) / 2;
329 GINAC_ASSERT(is_ex_exactly_of_type(seq[mid].coeff, numeric));
330 int cmp = ex_to_numeric(seq[mid].coeff).compare(looking_for);
336 return seq[mid].rest;
341 throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
346 return convert_to_poly().coeff(s, n);
350 ex pseries::collect(const symbol &s) const
356 /** Evaluate coefficients. */
357 ex pseries::eval(int level) const
362 if (level == -max_recursion_level)
363 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
365 // Construct a new series with evaluated coefficients
367 new_seq.reserve(seq.size());
368 epvector::const_iterator it = seq.begin(), itend = seq.end();
369 while (it != itend) {
370 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
373 return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
377 /** Evaluate coefficients numerically. */
378 ex pseries::evalf(int level) const
383 if (level == -max_recursion_level)
384 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
386 // Construct a new series with evaluated coefficients
388 new_seq.reserve(seq.size());
389 epvector::const_iterator it = seq.begin(), itend = seq.end();
390 while (it != itend) {
391 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
394 return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
398 ex pseries::subs(const lst & ls, const lst & lr) 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
404 return convert_to_poly(true).subs(ls, lr);
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(ls, lr), it->coeff));
415 return (new pseries(relational(var,point.subs(ls, lr)), newseq))->setflag(status_flags::dynallocated);
419 /** Implementation of ex::expand() for a power series. It expands all the
420 * terms individually and returns the resulting series as a new pseries. */
421 ex pseries::expand(unsigned options) const
424 for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
425 ex restexp = i->rest.expand();
426 if (!restexp.is_zero())
427 newseq.push_back(expair(restexp, i->coeff));
429 return (new pseries(relational(var,point), newseq))
430 ->setflag(status_flags::dynallocated | status_flags::expanded);
434 /** Implementation of ex::diff() for a power series. It treats the series as a
437 ex pseries::derivative(const symbol & s) const
441 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));
454 return pseries(relational(var,point), new_seq);
461 /** Convert a pseries object to an ordinary polynomial.
463 * @param no_order flag: discard higher order terms */
464 ex pseries::convert_to_poly(bool no_order) const
467 epvector::const_iterator it = seq.begin(), itend = seq.end();
469 while (it != itend) {
470 if (is_order_function(it->rest)) {
472 e += Order(power(var - point, it->coeff));
474 e += it->rest * power(var - point, it->coeff);
481 /** Returns true if there is no order term, i.e. the series terminates and
482 * false otherwise. */
483 bool pseries::is_terminating(void) const
485 return seq.size() == 0 || !is_order_function((seq.end()-1)->rest);
490 * Implementations of series expansion
493 /** Default implementation of ex::series(). This performs Taylor expansion.
495 ex basic::series(const relational & r, int order, unsigned options) const
500 ex coeff = deriv.subs(r);
501 const symbol *s = static_cast<symbol *>(r.lhs().bp);
503 if (!coeff.is_zero())
504 seq.push_back(expair(coeff, numeric(0)));
507 for (n=1; n<order; ++n) {
508 fac = fac.mul(numeric(n));
509 deriv = deriv.diff(*s).expand();
510 if (deriv.is_zero()) {
512 return pseries(r, seq);
514 coeff = deriv.subs(r);
515 if (!coeff.is_zero())
516 seq.push_back(expair(fac.inverse() * coeff, numeric(n)));
519 // Higher-order terms, if present
520 deriv = deriv.diff(*s);
521 if (!deriv.expand().is_zero())
522 seq.push_back(expair(Order(_ex1()), numeric(n)));
523 return pseries(r, seq);
527 /** Implementation of ex::series() for symbols.
529 ex symbol::series(const relational & r, int order, unsigned options) const
532 const ex point = r.rhs();
533 GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
534 const symbol *s = static_cast<symbol *>(r.lhs().bp);
536 if (this->is_equal(*s)) {
537 if (order > 0 && !point.is_zero())
538 seq.push_back(expair(point, _ex0()));
540 seq.push_back(expair(_ex1(), _ex1()));
542 seq.push_back(expair(Order(_ex1()), numeric(order)));
544 seq.push_back(expair(*this, _ex0()));
545 return pseries(r, seq);
549 /** Add one series object to another, producing a pseries object that
550 * represents the sum.
552 * @param other pseries object to add with
553 * @return the sum as a pseries */
554 ex pseries::add_series(const pseries &other) const
556 // Adding two series with different variables or expansion points
557 // results in an empty (constant) series
558 if (!is_compatible_to(other)) {
560 nul.push_back(expair(Order(_ex1()), _ex0()));
561 return pseries(relational(var,point), nul);
566 epvector::const_iterator a = seq.begin();
567 epvector::const_iterator b = other.seq.begin();
568 epvector::const_iterator a_end = seq.end();
569 epvector::const_iterator b_end = other.seq.end();
570 int pow_a = INT_MAX, pow_b = INT_MAX;
572 // If a is empty, fill up with elements from b and stop
575 new_seq.push_back(*b);
580 pow_a = ex_to_numeric((*a).coeff).to_int();
582 // If b is empty, fill up with elements from a and stop
585 new_seq.push_back(*a);
590 pow_b = ex_to_numeric((*b).coeff).to_int();
592 // a and b are non-empty, compare powers
594 // a has lesser power, get coefficient from a
595 new_seq.push_back(*a);
596 if (is_order_function((*a).rest))
599 } else if (pow_b < pow_a) {
600 // b has lesser power, get coefficient from b
601 new_seq.push_back(*b);
602 if (is_order_function((*b).rest))
606 // Add coefficient of a and b
607 if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
608 new_seq.push_back(expair(Order(_ex1()), (*a).coeff));
609 break; // Order term ends the sequence
611 ex sum = (*a).rest + (*b).rest;
612 if (!(sum.is_zero()))
613 new_seq.push_back(expair(sum, numeric(pow_a)));
619 return pseries(relational(var,point), new_seq);
623 /** Implementation of ex::series() for sums. This performs series addition when
624 * adding pseries objects.
626 ex add::series(const relational & r, int order, unsigned options) const
628 ex acc; // Series accumulator
630 // Get first term from overall_coeff
631 acc = overall_coeff.series(r, order, options);
633 // Add remaining terms
634 epvector::const_iterator it = seq.begin();
635 epvector::const_iterator itend = seq.end();
636 for (; it!=itend; ++it) {
638 if (is_ex_exactly_of_type(it->rest, pseries))
641 op = it->rest.series(r, order, options);
642 if (!it->coeff.is_equal(_ex1()))
643 op = ex_to_pseries(op).mul_const(ex_to_numeric(it->coeff));
646 acc = ex_to_pseries(acc).add_series(ex_to_pseries(op));
652 /** Multiply a pseries object with a numeric constant, producing a pseries
653 * object that represents the product.
655 * @param other constant to multiply with
656 * @return the product as a pseries */
657 ex pseries::mul_const(const numeric &other) const
660 new_seq.reserve(seq.size());
662 epvector::const_iterator it = seq.begin(), itend = seq.end();
663 while (it != itend) {
664 if (!is_order_function(it->rest))
665 new_seq.push_back(expair(it->rest * other, it->coeff));
667 new_seq.push_back(*it);
670 return pseries(relational(var,point), new_seq);
674 /** Multiply one pseries object to another, producing a pseries object that
675 * represents the product.
677 * @param other pseries object to multiply with
678 * @return the product as a pseries */
679 ex pseries::mul_series(const pseries &other) const
681 // Multiplying two series with different variables or expansion points
682 // results in an empty (constant) series
683 if (!is_compatible_to(other)) {
685 nul.push_back(expair(Order(_ex1()), _ex0()));
686 return pseries(relational(var,point), nul);
689 // Series multiplication
692 const symbol *s = static_cast<symbol *>(var.bp);
693 int a_max = degree(*s);
694 int b_max = other.degree(*s);
695 int a_min = ldegree(*s);
696 int b_min = other.ldegree(*s);
697 int cdeg_min = a_min + b_min;
698 int cdeg_max = a_max + b_max;
700 int higher_order_a = INT_MAX;
701 int higher_order_b = INT_MAX;
702 if (is_order_function(coeff(*s, a_max)))
703 higher_order_a = a_max + b_min;
704 if (is_order_function(other.coeff(*s, b_max)))
705 higher_order_b = b_max + a_min;
706 int higher_order_c = std::min(higher_order_a, higher_order_b);
707 if (cdeg_max >= higher_order_c)
708 cdeg_max = higher_order_c - 1;
710 for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
712 // c(i)=a(0)b(i)+...+a(i)b(0)
713 for (int i=a_min; cdeg-i>=b_min; ++i) {
714 ex a_coeff = coeff(*s, i);
715 ex b_coeff = other.coeff(*s, cdeg-i);
716 if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
717 co += a_coeff * b_coeff;
720 new_seq.push_back(expair(co, numeric(cdeg)));
722 if (higher_order_c < INT_MAX)
723 new_seq.push_back(expair(Order(_ex1()), numeric(higher_order_c)));
724 return pseries(relational(var,point), new_seq);
728 /** Implementation of ex::series() for product. This performs series
729 * multiplication when multiplying series.
731 ex mul::series(const relational & r, int order, unsigned options) const
733 ex acc; // Series accumulator
735 // Get first term from overall_coeff
736 acc = overall_coeff.series(r, order, options);
738 // Multiply with remaining terms
739 epvector::const_iterator it = seq.begin();
740 epvector::const_iterator itend = seq.end();
741 for (; it!=itend; ++it) {
743 if (op.info(info_flags::numeric)) {
744 // series * const (special case, faster)
745 ex f = power(op, it->coeff);
746 acc = ex_to_pseries(acc).mul_const(ex_to_numeric(f));
748 } else if (!is_ex_exactly_of_type(op, pseries))
749 op = op.series(r, order, options);
750 if (!it->coeff.is_equal(_ex1()))
751 op = ex_to_pseries(op).power_const(ex_to_numeric(it->coeff), order);
753 // Series multiplication
754 acc = ex_to_pseries(acc).mul_series(ex_to_pseries(op));
760 /** Compute the p-th power of a series.
762 * @param p power to compute
763 * @param deg truncation order of series calculation */
764 ex pseries::power_const(const numeric &p, int deg) const
767 // let A(x) be this series and for the time being let it start with a
768 // constant (later we'll generalize):
769 // A(x) = a_0 + a_1*x + a_2*x^2 + ...
770 // We want to compute
772 // C(x) = c_0 + c_1*x + c_2*x^2 + ...
773 // Taking the derivative on both sides and multiplying with A(x) one
774 // immediately arrives at
775 // C'(x)*A(x) = p*C(x)*A'(x)
776 // Multiplying this out and comparing coefficients we get the recurrence
778 // c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
779 // ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
780 // which can easily be solved given the starting value c_0 = (a_0)^p.
781 // For the more general case where the leading coefficient of A(x) is not
782 // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
783 // repeat the above derivation. The leading power of C2(x) = A2(x)^2 is
784 // then of course x^(p*m) but the recurrence formula still holds.
787 // as a spacial case, handle the empty (zero) series honoring the
788 // usual power laws such as implemented in power::eval()
789 if (p.real().is_zero())
790 throw (std::domain_error("pseries::power_const(): pow(0,I) is undefined"));
791 else if (p.real().is_negative())
792 throw (pole_error("pseries::power_const(): division by zero",1));
797 const symbol *s = static_cast<symbol *>(var.bp);
798 int ldeg = ldegree(*s);
800 // Compute coefficients of the powered series
803 co.push_back(power(coeff(*s, ldeg), p));
804 bool all_sums_zero = true;
805 for (int i=1; i<deg; ++i) {
807 for (int j=1; j<=i; ++j) {
808 ex c = coeff(*s, j + ldeg);
809 if (is_order_function(c)) {
810 co.push_back(Order(_ex1()));
813 sum += (p * j - (i - j)) * co[i - j] * c;
816 all_sums_zero = false;
817 co.push_back(sum / coeff(*s, ldeg) / numeric(i));
820 // Construct new series (of non-zero coefficients)
822 bool higher_order = false;
823 for (int i=0; i<deg; ++i) {
824 if (!co[i].is_zero())
825 new_seq.push_back(expair(co[i], numeric(i) + p * ldeg));
826 if (is_order_function(co[i])) {
831 if (!higher_order && !all_sums_zero)
832 new_seq.push_back(expair(Order(_ex1()), numeric(deg) + p * ldeg));
833 return pseries(relational(var,point), new_seq);
837 /** Return a new pseries object with the powers shifted by deg. */
838 pseries pseries::shift_exponents(int deg) const
840 epvector newseq(seq);
841 for (epvector::iterator i=newseq.begin(); i!=newseq.end(); ++i)
842 i->coeff = i->coeff + deg;
843 return pseries(relational(var, point), newseq);
847 /** Implementation of ex::series() for powers. This performs Laurent expansion
848 * of reciprocals of series at singularities.
850 ex power::series(const relational & r, int order, unsigned options) const
853 if (!is_ex_exactly_of_type(basis, pseries)) {
854 // Basis is not a series, may there be a singularity?
855 bool must_expand_basis = false;
858 } catch (pole_error) {
859 must_expand_basis = true;
862 // Is the expression of type something^(-int)?
863 if (!must_expand_basis && !exponent.info(info_flags::negint))
864 return basic::series(r, order, options);
866 // Is the expression of type 0^something?
867 if (!must_expand_basis && !basis.subs(r).is_zero())
868 return basic::series(r, order, options);
870 // Singularity encountered, expand basis into series
871 e = basis.series(r, order, options);
878 return ex_to_pseries(e).power_const(ex_to_numeric(exponent), order);
882 /** Re-expansion of a pseries object. */
883 ex pseries::series(const relational & r, int order, unsigned options) const
885 const ex p = r.rhs();
886 GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
887 const symbol *s = static_cast<symbol *>(r.lhs().bp);
889 if (var.is_equal(*s) && point.is_equal(p)) {
890 if (order > degree(*s))
894 epvector::const_iterator it = seq.begin(), itend = seq.end();
895 while (it != itend) {
896 int o = ex_to_numeric(it->coeff).to_int();
898 new_seq.push_back(expair(Order(_ex1()), o));
901 new_seq.push_back(*it);
904 return pseries(r, new_seq);
907 return convert_to_poly().series(r, order, options);
911 /** Compute the truncated series expansion of an expression.
912 * This function returns an expression containing an object of class pseries
913 * to represent the series. If the series does not terminate within the given
914 * truncation order, the last term of the series will be an order term.
916 * @param r expansion relation, lhs holds variable and rhs holds point
917 * @param order truncation order of series calculations
918 * @param options of class series_options
919 * @return an expression holding a pseries object */
920 ex ex::series(const ex & r, int order, unsigned options) const
926 if (is_ex_exactly_of_type(r,relational))
927 rel_ = ex_to_relational(r);
928 else if (is_ex_exactly_of_type(r,symbol))
929 rel_ = relational(r,_ex0());
931 throw (std::logic_error("ex::series(): expansion point has unknown type"));
934 e = bp->series(rel_, order, options);
935 } catch (std::exception &x) {
936 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));
942 // static member variables
947 unsigned pseries::precedence = 38; // for clarity just below add::precedence