X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fpseries.cpp;h=443969c1d766176d6a330cd60c481332f84bfd9a;hp=720fb9c732daef58311ac1b46d296246c2ce8288;hb=db81420a61959540b71507fe0d3737d06eee54dc;hpb=c378e979c7a9b9807941b27adbac690699abe70c diff --git a/ginac/pseries.cpp b/ginac/pseries.cpp index 720fb9c7..443969c1 100644 --- a/ginac/pseries.cpp +++ b/ginac/pseries.cpp @@ -4,7 +4,7 @@ * methods for series expansion. */ /* - * GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany + * GiNaC Copyright (C) 1999-2004 Johannes Gutenberg University Mainz, Germany * * This program is free software; you can redistribute it and/or modify * it under the terms of the GNU General Public License as published by @@ -21,6 +21,7 @@ * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */ +#include #include #include "pseries.h" @@ -30,35 +31,26 @@ #include "mul.h" #include "power.h" #include "relational.h" +#include "operators.h" #include "symbol.h" -#include "print.h" #include "archive.h" #include "utils.h" -#include "debugmsg.h" namespace GiNaC { -GINAC_IMPLEMENT_REGISTERED_CLASS(pseries, basic) +GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(pseries, basic, + print_func(&pseries::do_print). + print_func(&pseries::do_print_latex). + print_func(&pseries::do_print_tree). + print_func(&pseries::do_print_python). + print_func(&pseries::do_print_python_repr)) /* - * Default ctor, dtor, copy ctor, assignment operator and helpers + * Default constructor */ -pseries::pseries() : basic(TINFO_pseries) -{ - debugmsg("pseries default ctor", LOGLEVEL_CONSTRUCT); -} - -void pseries::copy(const pseries &other) -{ - inherited::copy(other); - seq = other.seq; - var = other.var; - point = other.point; -} - -DEFAULT_DESTROY(pseries) +pseries::pseries() : inherited(TINFO_pseries) { } /* @@ -68,7 +60,7 @@ DEFAULT_DESTROY(pseries) /** Construct pseries from a vector of coefficients and powers. * expair.rest holds the coefficient, expair.coeff holds the power. * The powers must be integers (positive or negative) and in ascending order; - * the last coefficient can be Order(_ex1()) to represent a truncated, + * the last coefficient can be Order(_ex1) to represent a truncated, * non-terminating series. * * @param rel_ expansion variable and point (must hold a relational) @@ -76,11 +68,10 @@ DEFAULT_DESTROY(pseries) * @return newly constructed pseries */ pseries::pseries(const ex &rel_, const epvector &ops_) : basic(TINFO_pseries), seq(ops_) { - debugmsg("pseries ctor from ex,epvector", LOGLEVEL_CONSTRUCT); - GINAC_ASSERT(is_ex_exactly_of_type(rel_, relational)); - GINAC_ASSERT(is_ex_exactly_of_type(rel_.lhs(),symbol)); + GINAC_ASSERT(is_a(rel_)); + GINAC_ASSERT(is_a(rel_.lhs())); point = rel_.rhs(); - var = *static_cast(rel_.lhs().bp); + var = rel_.lhs(); } @@ -88,9 +79,8 @@ pseries::pseries(const ex &rel_, const epvector &ops_) : basic(TINFO_pseries), s * Archiving */ -pseries::pseries(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst) +pseries::pseries(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst) { - debugmsg("pseries ctor from archive_node", LOGLEVEL_CONSTRUCT); for (unsigned int i=0; true; ++i) { ex rest; ex coeff; @@ -122,93 +112,118 @@ DEFAULT_UNARCHIVE(pseries) // functions overriding virtual functions from base classes ////////// -void pseries::print(const print_context & c, unsigned level) const +void pseries::print_series(const print_context & c, const char *openbrace, const char *closebrace, const char *mul_sym, const char *pow_sym, unsigned level) const { - debugmsg("pseries print", LOGLEVEL_PRINT); + if (precedence() <= level) + c.s << '('; + + // objects of type pseries must not have any zero entries, so the + // trivial (zero) pseries needs a special treatment here: + if (seq.empty()) + c.s << '0'; - if (is_a(c)) { + epvector::const_iterator i = seq.begin(), end = seq.end(); + while (i != end) { - c.s << std::string(level, ' ') << class_name() - << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec - << std::endl; - unsigned delta_indent = static_cast(c).delta_indent; - unsigned num = seq.size(); - for (unsigned i=0; irest)) { - if (precedence() <= level) - c.s << "("; - - std::string par_open = is_a(c) ? "{(" : "("; - std::string par_close = is_a(c) ? ")}" : ")"; - - // objects of type pseries must not have any zero entries, so the - // trivial (zero) pseries needs a special treatment here: - if (seq.empty()) - c.s << '0'; - epvector::const_iterator i = seq.begin(), end = seq.end(); - while (i != end) { - // print a sign, if needed - if (i != seq.begin()) - c.s << '+'; - if (!is_order_function(i->rest)) { - // print 'rest', i.e. the expansion coefficient - if (i->rest.info(info_flags::numeric) && - i->rest.info(info_flags::positive)) { - i->rest.print(c); - } else { - c.s << par_open; - i->rest.print(c); - c.s << par_close; - } - // print 'coeff', something like (x-1)^42 - if (!i->coeff.is_zero()) { - if (is_a(c)) - c.s << ' '; - else - c.s << '*'; - if (!point.is_zero()) { - c.s << par_open; - (var-point).print(c); - c.s << par_close; + // print 'rest', i.e. the expansion coefficient + if (i->rest.info(info_flags::numeric) && + i->rest.info(info_flags::positive)) { + i->rest.print(c); + } else { + c.s << openbrace << '('; + i->rest.print(c); + c.s << ')' << closebrace; + } + + // print 'coeff', something like (x-1)^42 + if (!i->coeff.is_zero()) { + c.s << mul_sym; + if (!point.is_zero()) { + c.s << openbrace << '('; + (var-point).print(c); + c.s << ')' << closebrace; + } else + var.print(c); + if (i->coeff.compare(_ex1)) { + c.s << pow_sym; + c.s << openbrace; + if (i->coeff.info(info_flags::negative)) { + c.s << '('; + i->coeff.print(c); + c.s << ')'; } else - var.print(c); - if (i->coeff.compare(_ex1())) { - c.s << '^'; - if (i->coeff.info(info_flags::negative)) { - c.s << par_open; - i->coeff.print(c); - c.s << par_close; - } else { - if (is_a(c)) { - c.s << '{'; - i->coeff.print(c); - c.s << '}'; - } else - i->coeff.print(c); - } - } + i->coeff.print(c); + c.s << closebrace; } - } else - Order(power(var-point,i->coeff)).print(c); - ++i; - } + } + } else + Order(power(var-point,i->coeff)).print(c); + ++i; + } + + if (precedence() <= level) + c.s << ')'; +} + +void pseries::do_print(const print_context & c, unsigned level) const +{ + print_series(c, "", "", "*", "^", level); +} + +void pseries::do_print_latex(const print_latex & c, unsigned level) const +{ + print_series(c, "{", "}", " ", "^", level); +} + +void pseries::do_print_python(const print_python & c, unsigned level) const +{ + print_series(c, "", "", "*", "**", level); +} - if (precedence() <= level) - c.s << ")"; +void pseries::do_print_tree(const print_tree & c, unsigned level) const +{ + c.s << std::string(level, ' ') << class_name() << " @" << this + << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec + << std::endl; + size_t num = seq.size(); + for (size_t i=0; i(other)); const pseries &o = static_cast(other); // first compare the lengths of the series... @@ -240,22 +255,18 @@ int pseries::compare_same_type(const basic & other) const } /** Return the number of operands including a possible order term. */ -unsigned pseries::nops(void) const +size_t pseries::nops() const { return seq.size(); } /** Return the ith term in the series when represented as a sum. */ -ex pseries::op(int i) const +ex pseries::op(size_t i) const { - if (i < 0 || unsigned(i) >= seq.size()) + if (i >= seq.size()) throw (std::out_of_range("op() out of range")); - return seq[i].rest * power(var - point, seq[i].coeff); -} -ex &pseries::let_op(int i) -{ - throw (std::logic_error("let_op not defined for pseries")); + return seq[i].rest * power(var - point, seq[i].coeff); } /** Return degree of highest power of the series. This is usually the exponent @@ -266,7 +277,7 @@ int pseries::degree(const ex &s) const if (var.is_equal(s)) { // Return last exponent if (seq.size()) - return ex_to((*(seq.end() - 1)).coeff).to_int(); + return ex_to((seq.end()-1)->coeff).to_int(); else return 0; } else { @@ -294,7 +305,7 @@ int pseries::ldegree(const ex &s) const if (var.is_equal(s)) { // Return first exponent if (seq.size()) - return ex_to((*(seq.begin())).coeff).to_int(); + return ex_to((seq.begin())->coeff).to_int(); else return 0; } else { @@ -323,14 +334,14 @@ ex pseries::coeff(const ex &s, int n) const { if (var.is_equal(s)) { if (seq.empty()) - return _ex0(); + return _ex0; // Binary search in sequence for given power numeric looking_for = numeric(n); int lo = 0, hi = seq.size() - 1; while (lo <= hi) { int mid = (lo + hi) / 2; - GINAC_ASSERT(is_ex_exactly_of_type(seq[mid].coeff, numeric)); + GINAC_ASSERT(is_exactly_a(seq[mid].coeff)); int cmp = ex_to(seq[mid].coeff).compare(looking_for); switch (cmp) { case -1: @@ -345,7 +356,7 @@ ex pseries::coeff(const ex &s, int n) const throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1")); } } - return _ex0(); + return _ex0; } else return convert_to_poly().coeff(s, n); } @@ -396,13 +407,30 @@ ex pseries::evalf(int level) const return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated); } -ex pseries::subs(const lst & ls, const lst & lr, bool no_pattern) const +ex pseries::conjugate() const +{ + epvector * newseq = conjugateepvector(seq); + ex newvar = var.conjugate(); + ex newpoint = point.conjugate(); + + if (!newseq && are_ex_trivially_equal(newvar, var) && are_ex_trivially_equal(point, newpoint)) { + return *this; + } + + ex result = (new pseries(newvar==newpoint, newseq ? *newseq : seq))->setflag(status_flags::dynallocated); + if (newseq) { + delete newseq; + } + return result; +} + +ex pseries::subs(const exmap & m, unsigned options) const { // If expansion variable is being substituted, convert the series to a // polynomial and do the substitution there because the result might // no longer be a power series - if (ls.has(var)) - return convert_to_poly(true).subs(ls, lr, no_pattern); + if (m.find(var) != m.end()) + return convert_to_poly(true).subs(m, options); // Otherwise construct a new series with substituted coefficients and // expansion point @@ -410,10 +438,10 @@ ex pseries::subs(const lst & ls, const lst & lr, bool no_pattern) const newseq.reserve(seq.size()); epvector::const_iterator it = seq.begin(), itend = seq.end(); while (it != itend) { - newseq.push_back(expair(it->rest.subs(ls, lr, no_pattern), it->coeff)); + newseq.push_back(expair(it->rest.subs(m, options), it->coeff)); ++it; } - return (new pseries(relational(var,point.subs(ls, lr, no_pattern)), newseq))->setflag(status_flags::dynallocated); + return (new pseries(relational(var,point.subs(m, options)), newseq))->setflag(status_flags::dynallocated); } /** Implementation of ex::expand() for a power series. It expands all the @@ -432,14 +460,14 @@ ex pseries::expand(unsigned options) const ->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0)); } -/** Implementation of ex::diff() for a power series. It treats the series as a - * polynomial. +/** Implementation of ex::diff() for a power series. * @see ex::diff */ ex pseries::derivative(const symbol & s) const { + epvector new_seq; + epvector::const_iterator it = seq.begin(), itend = seq.end(); + if (s == var) { - epvector new_seq; - epvector::const_iterator it = seq.begin(), itend = seq.end(); // FIXME: coeff might depend on var while (it != itend) { @@ -452,10 +480,22 @@ ex pseries::derivative(const symbol & s) const } ++it; } - return pseries(relational(var,point), new_seq); + } else { - return *this; + + while (it != itend) { + if (is_order_function(it->rest)) { + new_seq.push_back(*it); + } else { + ex c = it->rest.diff(s); + if (!c.is_zero()) + new_seq.push_back(expair(c, it->coeff)); + } + ++it; + } } + + return pseries(relational(var,point), new_seq); } ex pseries::convert_to_poly(bool no_order) const @@ -474,7 +514,7 @@ ex pseries::convert_to_poly(bool no_order) const return e; } -bool pseries::is_terminating(void) const +bool pseries::is_terminating() const { return seq.empty() || !is_order_function((seq.end()-1)->rest); } @@ -489,31 +529,42 @@ bool pseries::is_terminating(void) const ex basic::series(const relational & r, int order, unsigned options) const { epvector seq; - numeric fac(1); + const symbol &s = ex_to(r.lhs()); + + // default for order-values that make no sense for Taylor expansion + if ((order <= 0) && this->has(s)) { + seq.push_back(expair(Order(_ex1), order)); + return pseries(r, seq); + } + + // do Taylor expansion + numeric fac = 1; ex deriv = *this; - ex coeff = deriv.subs(r); - const symbol &s = static_cast(*r.lhs().bp); - - if (!coeff.is_zero()) - seq.push_back(expair(coeff, _ex0())); - + ex coeff = deriv.subs(r, subs_options::no_pattern); + + if (!coeff.is_zero()) { + seq.push_back(expair(coeff, _ex0)); + } + int n; for (n=1; nis_equal(*s.bp)) { + GINAC_ASSERT(is_a(r.lhs())); + + if (this->is_equal_same_type(ex_to(r.lhs()))) { if (order > 0 && !point.is_zero()) - seq.push_back(expair(point, _ex0())); + seq.push_back(expair(point, _ex0)); if (order > 1) - seq.push_back(expair(_ex1(), _ex1())); + seq.push_back(expair(_ex1, _ex1)); else - seq.push_back(expair(Order(_ex1()), numeric(order))); + seq.push_back(expair(Order(_ex1), numeric(order))); } else - seq.push_back(expair(*this, _ex0())); + seq.push_back(expair(*this, _ex0)); return pseries(r, seq); } @@ -551,7 +601,7 @@ ex pseries::add_series(const pseries &other) const // results in an empty (constant) series if (!is_compatible_to(other)) { epvector nul; - nul.push_back(expair(Order(_ex1()), _ex0())); + nul.push_back(expair(Order(_ex1), _ex0)); return pseries(relational(var,point), nul); } @@ -599,7 +649,7 @@ ex pseries::add_series(const pseries &other) const } else { // Add coefficient of a and b if (is_order_function((*a).rest) || is_order_function((*b).rest)) { - new_seq.push_back(expair(Order(_ex1()), (*a).coeff)); + new_seq.push_back(expair(Order(_ex1), (*a).coeff)); break; // Order term ends the sequence } else { ex sum = (*a).rest + (*b).rest; @@ -629,11 +679,11 @@ ex add::series(const relational & r, int order, unsigned options) const epvector::const_iterator itend = seq.end(); for (; it!=itend; ++it) { ex op; - if (is_ex_exactly_of_type(it->rest, pseries)) + if (is_exactly_a(it->rest)) op = it->rest; else op = it->rest.series(r, order, options); - if (!it->coeff.is_equal(_ex1())) + if (!it->coeff.is_equal(_ex1)) op = ex_to(op).mul_const(ex_to(it->coeff)); // Series addition @@ -676,13 +726,12 @@ ex pseries::mul_series(const pseries &other) const // results in an empty (constant) series if (!is_compatible_to(other)) { epvector nul; - nul.push_back(expair(Order(_ex1()), _ex0())); + nul.push_back(expair(Order(_ex1), _ex0)); return pseries(relational(var,point), nul); } // Series multiplication epvector new_seq; - int a_max = degree(var); int b_max = other.degree(var); int a_min = ldegree(var); @@ -701,7 +750,7 @@ ex pseries::mul_series(const pseries &other) const cdeg_max = higher_order_c - 1; for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) { - ex co = _ex0(); + ex co = _ex0; // c(i)=a(0)b(i)+...+a(i)b(0) for (int i=a_min; cdeg-i>=b_min; ++i) { ex a_coeff = coeff(var, i); @@ -713,7 +762,7 @@ ex pseries::mul_series(const pseries &other) const new_seq.push_back(expair(co, numeric(cdeg))); } if (higher_order_c < INT_MAX) - new_seq.push_back(expair(Order(_ex1()), numeric(higher_order_c))); + new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c))); return pseries(relational(var, point), new_seq); } @@ -723,30 +772,68 @@ ex pseries::mul_series(const pseries &other) const * @see ex::series */ ex mul::series(const relational & r, int order, unsigned options) const { - ex acc; // Series accumulator - - // Get first term from overall_coeff - acc = overall_coeff.series(r, order, options); - + pseries acc; // Series accumulator + + GINAC_ASSERT(is_a(r.lhs())); + const ex& sym = r.lhs(); + + // holds ldegrees of the series of individual factors + std::vector ldegrees; + + // find minimal degrees + const epvector::const_iterator itbeg = seq.begin(); + const epvector::const_iterator itend = seq.end(); + for (epvector::const_iterator it=itbeg; it!=itend; ++it) { + + ex expon = it->coeff; + int factor = 1; + ex buf; + if (expon.info(info_flags::integer)) { + buf = it->rest; + factor = ex_to(expon).to_int(); + } else { + buf = recombine_pair_to_ex(*it); + } + + int real_ldegree = 0; + try { + real_ldegree = buf.expand().ldegree(sym-r.rhs()); + } catch (std::runtime_error) {} + + if (real_ldegree == 0) { + int orderloop = 0; + do { + orderloop++; + real_ldegree = buf.series(r, orderloop, options).ldegree(sym); + } while (real_ldegree == orderloop); + } + + ldegrees.push_back(factor * real_ldegree); + } + + int degsum = std::accumulate(ldegrees.begin(), ldegrees.end(), 0); + + if (degsum >= order) { + epvector epv; + epv.push_back(expair(Order(_ex1), order)); + return (new pseries(r, epv))->setflag(status_flags::dynallocated); + } + // Multiply with remaining terms - epvector::const_iterator it = seq.begin(); - epvector::const_iterator itend = seq.end(); - for (; it!=itend; ++it) { - ex op = it->rest; - if (op.info(info_flags::numeric)) { - // series * const (special case, faster) - ex f = power(op, it->coeff); - acc = ex_to(acc).mul_const(ex_to(f)); - continue; - } else if (!is_ex_exactly_of_type(op, pseries)) - op = op.series(r, order, options); - if (!it->coeff.is_equal(_ex1())) - op = ex_to(op).power_const(ex_to(it->coeff), order); + std::vector::const_iterator itd = ldegrees.begin(); + for (epvector::const_iterator it=itbeg; it!=itend; ++it, ++itd) { + + // do series expansion with adjusted order + ex op = recombine_pair_to_ex(*it).series(r, order-degsum+(*itd), options); // Series multiplication - acc = ex_to(acc).mul_series(ex_to(op)); + if (it == itbeg) + acc = ex_to(op); + else + acc = ex_to(acc.mul_series(ex_to(op))); } - return acc; + + return acc.mul_const(ex_to(overall_coeff)); } @@ -778,36 +865,42 @@ ex pseries::power_const(const numeric &p, int deg) const // then of course x^(p*m) but the recurrence formula still holds. if (seq.empty()) { - // as a spacial case, handle the empty (zero) series honoring the + // as a special case, handle the empty (zero) series honoring the // usual power laws such as implemented in power::eval() if (p.real().is_zero()) - throw (std::domain_error("pseries::power_const(): pow(0,I) is undefined")); + throw std::domain_error("pseries::power_const(): pow(0,I) is undefined"); else if (p.real().is_negative()) - throw (pole_error("pseries::power_const(): division by zero",1)); + throw pole_error("pseries::power_const(): division by zero",1); else return *this; } - int ldeg = ldegree(var); + const int ldeg = ldegree(var); + if (!(p*ldeg).is_integer()) + throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series"); + + // adjust number of coefficients + deg = deg - (p*ldeg).to_int(); + + // O(x^n)^(-m) is undefined + if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative()) + throw pole_error("pseries::power_const(): division by zero",1); // Compute coefficients of the powered series exvector co; co.reserve(deg); co.push_back(power(coeff(var, ldeg), p)); - bool all_sums_zero = true; for (int i=1; i(basis)) + return ex_to(basis).power_const(ex_to(exponent), order); + + // Basis is not a series, may there be a singularity? + bool must_expand_basis = false; + try { + basis.subs(r, subs_options::no_pattern); + } catch (pole_error) { + must_expand_basis = true; + } + + // Is the expression of type something^(-int)? + if (!must_expand_basis && !exponent.info(info_flags::negint) && !is_a(basis)) + return basic::series(r, order, options); + + // Is the expression of type 0^something? + if (!must_expand_basis && !basis.subs(r, subs_options::no_pattern).is_zero() && !is_a(basis)) + return basic::series(r, order, options); + + // Singularity encountered, is the basis equal to (var - point)? + if (basis.is_equal(r.lhs() - r.rhs())) { + epvector new_seq; + if (ex_to(exponent).to_int() < order) + new_seq.push_back(expair(_ex1, exponent)); + else + new_seq.push_back(expair(Order(_ex1), exponent)); + return pseries(r, new_seq); + } + + // No, expand basis into series + + numeric numexp = ex_to(exponent); + const ex& sym = r.lhs(); + // find existing minimal degree + int real_ldegree = basis.expand().ldegree(sym-r.rhs()); + if (real_ldegree == 0) { + int orderloop = 0; + do { + orderloop++; + real_ldegree = basis.series(r, orderloop, options).ldegree(sym); + } while (real_ldegree == orderloop); } + + if (!(real_ldegree*numexp).is_integer()) + throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series"); + ex e = basis.series(r, (order + real_ldegree*(1-numexp)).to_int(), options); - // Power e - return ex_to(e).power_const(ex_to(exponent), order); + ex result; + try { + result = ex_to(e).power_const(numexp, order); + } catch (pole_error) { + epvector ser; + ser.push_back(expair(Order(_ex1), order)); + result = pseries(r, ser); + } + + return result; } @@ -879,8 +1004,8 @@ ex power::series(const relational & r, int order, unsigned options) const ex pseries::series(const relational & r, int order, unsigned options) const { const ex p = r.rhs(); - GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol)); - const symbol &s = static_cast(*r.lhs().bp); + GINAC_ASSERT(is_a(r.lhs())); + const symbol &s = ex_to(r.lhs()); if (var.is_equal(s) && point.is_equal(p)) { if (order > degree(s)) @@ -891,7 +1016,7 @@ ex pseries::series(const relational & r, int order, unsigned options) const while (it != itend) { int o = ex_to(it->coeff).to_int(); if (o >= order) { - new_seq.push_back(expair(Order(_ex1()), o)); + new_seq.push_back(expair(Order(_ex1), o)); break; } new_seq.push_back(*it); @@ -915,14 +1040,13 @@ ex pseries::series(const relational & r, int order, unsigned options) const * @return an expression holding a pseries object */ ex ex::series(const ex & r, int order, unsigned options) const { - GINAC_ASSERT(bp!=0); ex e; relational rel_; - if (is_ex_exactly_of_type(r,relational)) + if (is_a(r)) rel_ = ex_to(r); - else if (is_ex_exactly_of_type(r,symbol)) - rel_ = relational(r,_ex0()); + else if (is_a(r)) + rel_ = relational(r,_ex0); else throw (std::logic_error("ex::series(): expansion point has unknown type"));