X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fpseries.cpp;h=993eacf119f7583903747f2a92577f59fa508841;hp=de0e45cc785f054ba1dbbbf718884936f8c584b8;hb=1acadaea2a80886aac7afa758d9b250880b6f186;hpb=383d5eb3b0f0506810d9105a268f939125bfc347 diff --git a/ginac/pseries.cpp b/ginac/pseries.cpp index de0e45cc..993eacf1 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 @@ -21,80 +21,46 @@ * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */ +#include #include #include "pseries.h" #include "add.h" -#include "inifcns.h" +#include "inifcns.h" // for Order function #include "lst.h" #include "mul.h" #include "power.h" #include "relational.h" +#include "operators.h" #include "symbol.h" #include "archive.h" #include "utils.h" -#include "debugmsg.h" -#ifndef NO_NAMESPACE_GINAC namespace GiNaC { -#endif // ndef NO_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 constructor, destructor, copy constructor, assignment operator and helpers + * Default constructor */ -pseries::pseries() : basic(TINFO_pseries) -{ - debugmsg("pseries default constructor", LOGLEVEL_CONSTRUCT); -} - -pseries::~pseries() -{ - debugmsg("pseries destructor", LOGLEVEL_DESTRUCT); - destroy(false); -} - -pseries::pseries(const pseries &other) -{ - debugmsg("pseries copy constructor", LOGLEVEL_CONSTRUCT); - copy(other); -} - -const pseries &pseries::operator=(const pseries & other) -{ - debugmsg("pseries operator=", LOGLEVEL_ASSIGNMENT); - if (this != &other) { - destroy(true); - copy(other); - } - return *this; -} - -void pseries::copy(const pseries &other) -{ - inherited::copy(other); - seq = other.seq; - var = other.var; - point = other.point; -} - -void pseries::destroy(bool call_parent) -{ - if (call_parent) - inherited::destroy(call_parent); -} +pseries::pseries() : inherited(TINFO_pseries) { } /* - * Other constructors + * Other ctors */ /** 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) @@ -102,11 +68,10 @@ void pseries::destroy(bool call_parent) * @return newly constructed pseries */ pseries::pseries(const ex &rel_, const epvector &ops_) : basic(TINFO_pseries), seq(ops_) { - debugmsg("pseries constructor 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(); } @@ -114,10 +79,8 @@ pseries::pseries(const ex &rel_, const epvector &ops_) : basic(TINFO_pseries), s * Archiving */ -/** Construct object from archive_node. */ -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 constructor from archive_node", LOGLEVEL_CONSTRUCT); for (unsigned int i=0; true; ++i) { ex rest; ex coeff; @@ -130,13 +93,6 @@ pseries::pseries(const archive_node &n, const lst &sym_lst) : inherited(n, sym_l n.find_ex("point", point, sym_lst); } -/** Unarchive the object. */ -ex pseries::unarchive(const archive_node &n, const lst &sym_lst) -{ - return (new pseries(n, sym_lst))->setflag(status_flags::dynallocated); -} - -/** Archive the object. */ void pseries::archive(archive_node &n) const { inherited::archive(n); @@ -150,116 +106,178 @@ void pseries::archive(archive_node &n) const n.add_ex("point", point); } +DEFAULT_UNARCHIVE(pseries) + ////////// -// functions overriding virtual functions from bases classes +// functions overriding virtual functions from base classes ////////// -basic *pseries::duplicate() 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 duplicate", LOGLEVEL_DUPLICATE); - return new pseries(*this); -} + 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'; + + epvector::const_iterator i = seq.begin(), end = seq.end(); + while (i != end) { -void pseries::print(std::ostream &os, unsigned upper_precedence) const -{ - debugmsg("pseries print", LOGLEVEL_PRINT); - if (precedence<=upper_precedence) os << "("; - for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) { - // omit zero terms - if (i->rest.is_zero()) - continue; // print a sign, if needed - if (i!=seq.begin()) - os << '+'; + 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)) { - os << i->rest; - } else - os << "(" << i->rest << ')'; + 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()) { - os << '*'; - if (!point.is_zero()) - os << '(' << var-point << ')'; - else - os << var; - if (i->coeff.compare(_ex1())) { - os << '^'; - if (i->coeff.info(info_flags::negative)) - os << '(' << i->coeff << ')'; - else - os << i->coeff; + 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 + i->coeff.print(c); + c.s << closebrace; } } - } else { - os << Order(power(var-point,i->coeff)); - } + } else + Order(power(var-point,i->coeff)).print(c); + ++i; } - if (precedence<=upper_precedence) os << ")"; + + if (precedence() <= level) + c.s << ')'; } +void pseries::do_print(const print_context & c, unsigned level) const +{ + print_series(c, "", "", "*", "^", level); +} -void pseries::printraw(std::ostream &os) const +void pseries::do_print_latex(const print_latex & c, unsigned level) const { - debugmsg("pseries printraw", LOGLEVEL_PRINT); - os << "pseries(" << var << ";" << point << ";"; - for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) { - os << "(" << (*i).rest << "," << (*i).coeff << "),"; + print_series(c, "{", "}", " ", "^", level); +} + +void pseries::do_print_python(const print_python & c, unsigned level) const +{ + print_series(c, "", "", "*", "**", level); +} + +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... + if (seq.size()>o.seq.size()) + return 1; + if (seq.size()compare(*o_it); + if (cmpval) + return cmpval; + ++it; + ++o_it; } - var.printtree(os, indent+delta_indent); - point.printtree(os, indent+delta_indent); + + // so they are equal. + return 0; } /** 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 * of the Order term. If s is not the expansion variable of the series, the * series is examined termwise. */ -int pseries::degree(const symbol &s) const +int pseries::degree(const ex &s) const { if (var.is_equal(s)) { // Return last exponent if (seq.size()) - return ex_to_numeric((*(seq.end() - 1)).coeff).to_int(); + return ex_to((seq.end()-1)->coeff).to_int(); else return 0; } else { @@ -282,12 +300,12 @@ int pseries::degree(const symbol &s) const * series is examined termwise. If s is the expansion variable but the * expansion point is not zero the series is not expanded to find the degree. * I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */ -int pseries::ldegree(const symbol &s) const +int pseries::ldegree(const ex &s) const { if (var.is_equal(s)) { // Return first exponent if (seq.size()) - return ex_to_numeric((*(seq.begin())).coeff).to_int(); + return ex_to((seq.begin())->coeff).to_int(); else return 0; } else { @@ -305,19 +323,26 @@ int pseries::ldegree(const symbol &s) const } } -ex pseries::coeff(const symbol &s, int n) const +/** Return coefficient of degree n in power series if s is the expansion + * variable. If the expansion point is nonzero, by definition the n=1 + * coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming + * the expansion took place in the s in the first place). + * If s is not the expansion variable, an attempt is made to convert the + * series to a polynomial and return the corresponding coefficient from + * there. */ +ex pseries::coeff(const ex &s, int n) const { if (var.is_equal(s)) { - if (seq.size() == 0) - return _ex0(); + if (seq.empty()) + 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)); - int cmp = ex_to_numeric(seq[mid].coeff).compare(looking_for); + GINAC_ASSERT(is_exactly_a(seq[mid].coeff)); + int cmp = ex_to(seq[mid].coeff).compare(looking_for); switch (cmp) { case -1: lo = mid + 1; @@ -331,19 +356,18 @@ ex pseries::coeff(const symbol &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); } - -ex pseries::collect(const symbol &s) const +/** Does nothing. */ +ex pseries::collect(const ex &s, bool distributed) const { return *this; } - -/** Evaluate coefficients. */ +/** Perform coefficient-wise automatic term rewriting rules in this class. */ ex pseries::eval(int level) const { if (level == 1) @@ -363,7 +387,6 @@ ex pseries::eval(int level) const return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated); } - /** Evaluate coefficients numerically. */ ex pseries::evalf(int level) const { @@ -384,14 +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::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 lst & ls, const lst & lr) const +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); + 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 @@ -399,35 +438,36 @@ ex pseries::subs(const lst & ls, const lst & lr) 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), it->coeff)); + newseq.push_back(expair(it->rest.subs(m, options), it->coeff)); ++it; } - return (new pseries(relational(var,point.subs(ls, lr)), 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 - * terms individually and returns the resulting series as a new pseries. - * @see ex::diff */ + * terms individually and returns the resulting series as a new pseries. */ ex pseries::expand(unsigned options) const { epvector newseq; - newseq.reserve(seq.size()); - for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) - newseq.push_back(expair(i->rest.expand(), i->coeff)); + epvector::const_iterator i = seq.begin(), end = seq.end(); + while (i != end) { + ex restexp = i->rest.expand(); + if (!restexp.is_zero()) + newseq.push_back(expair(restexp, i->coeff)); + ++i; + } return (new pseries(relational(var,point), newseq)) - ->setflag(status_flags::dynallocated | status_flags::expanded); + ->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) { @@ -440,20 +480,24 @@ 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; + } + } -/* - * Construct ordinary polynomial out of series - */ + return pseries(relational(var,point), new_seq); +} -/** Convert a pseries object to an ordinary polynomial. - * - * @param no_order flag: discard higher order terms */ ex pseries::convert_to_poly(bool no_order) const { ex e; @@ -470,16 +514,14 @@ ex pseries::convert_to_poly(bool no_order) const return e; } -/** Returns true if there is no order term, i.e. the series terminates and - * false otherwise. */ -bool pseries::is_terminating(void) const +bool pseries::is_terminating() const { - return seq.size() == 0 || !is_order_function((seq.end()-1)->rest); + return seq.empty() || !is_order_function((seq.end()-1)->rest); } /* - * Implementation of series expansion + * Implementations of series expansion */ /** Default implementation of ex::series(). This performs Taylor expansion. @@ -487,31 +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, numeric(0))); - + ex coeff = deriv.subs(r, subs_options::no_pattern); + + if (!coeff.is_zero()) { + seq.push_back(expair(coeff, _ex0)); + } + int n; for (n=1; n(r.lhs().bp); - - if (this->is_equal(*s)) { + 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); } @@ -549,7 +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); } @@ -569,7 +621,7 @@ ex pseries::add_series(const pseries &other) const } break; } else - pow_a = ex_to_numeric((*a).coeff).to_int(); + pow_a = ex_to((*a).coeff).to_int(); // If b is empty, fill up with elements from a and stop if (b == b_end) { @@ -579,7 +631,7 @@ ex pseries::add_series(const pseries &other) const } break; } else - pow_b = ex_to_numeric((*b).coeff).to_int(); + pow_b = ex_to((*b).coeff).to_int(); // a and b are non-empty, compare powers if (pow_a < pow_b) { @@ -597,7 +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; @@ -627,15 +679,15 @@ 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())) - op = ex_to_pseries(op).mul_const(ex_to_numeric(it->coeff)); + if (!it->coeff.is_equal(_ex1)) + op = ex_to(op).mul_const(ex_to(it->coeff)); // Series addition - acc = ex_to_pseries(acc).add_series(ex_to_pseries(op)); + acc = ex_to(acc).add_series(ex_to(op)); } return acc; } @@ -674,37 +726,35 @@ 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; - - const symbol *s = static_cast(var.bp); - int a_max = degree(*s); - int b_max = other.degree(*s); - int a_min = ldegree(*s); - int b_min = other.ldegree(*s); + int a_max = degree(var); + int b_max = other.degree(var); + int a_min = ldegree(var); + int b_min = other.ldegree(var); int cdeg_min = a_min + b_min; int cdeg_max = a_max + b_max; int higher_order_a = INT_MAX; int higher_order_b = INT_MAX; - if (is_order_function(coeff(*s, a_max))) + if (is_order_function(coeff(var, a_max))) higher_order_a = a_max + b_min; - if (is_order_function(other.coeff(*s, b_max))) + if (is_order_function(other.coeff(var, b_max))) higher_order_b = b_max + a_min; int higher_order_c = std::min(higher_order_a, higher_order_b); if (cdeg_max >= higher_order_c) 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(*s, i); - ex b_coeff = other.coeff(*s, cdeg-i); + ex a_coeff = coeff(var, i); + ex b_coeff = other.coeff(var, cdeg-i); if (!is_order_function(a_coeff) && !is_order_function(b_coeff)) co += a_coeff * b_coeff; } @@ -712,8 +762,8 @@ 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))); - return pseries(relational(var,point), new_seq); + new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c))); + return pseries(relational(var, point), new_seq); } @@ -722,30 +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_pseries(acc).mul_const(ex_to_numeric(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_pseries(op).power_const(ex_to_numeric(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_pseries(acc).mul_series(ex_to_pseries(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)); } @@ -755,44 +843,80 @@ ex mul::series(const relational & r, int order, unsigned options) const * @param deg truncation order of series calculation */ ex pseries::power_const(const numeric &p, int deg) const { - int i; - const symbol *s = static_cast(var.bp); - int ldeg = ldegree(*s); + // method: + // (due to Leonhard Euler) + // let A(x) be this series and for the time being let it start with a + // constant (later we'll generalize): + // A(x) = a_0 + a_1*x + a_2*x^2 + ... + // We want to compute + // C(x) = A(x)^p + // C(x) = c_0 + c_1*x + c_2*x^2 + ... + // Taking the derivative on both sides and multiplying with A(x) one + // immediately arrives at + // C'(x)*A(x) = p*C(x)*A'(x) + // Multiplying this out and comparing coefficients we get the recurrence + // formula + // c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ... + // ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i) + // which can easily be solved given the starting value c_0 = (a_0)^p. + // For the more general case where the leading coefficient of A(x) is not + // a constant, just consider A2(x) = A(x)*x^m, with some integer m and + // repeat the above derivation. The leading power of C2(x) = A2(x)^2 is + // then of course x^(p*m) but the recurrence formula still holds. + + if (seq.empty()) { + // 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"); + else if (p.real().is_negative()) + throw pole_error("pseries::power_const(): division by zero",1); + else + return *this; + } + + 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); - // Calculate coefficients of powered series + // Compute coefficients of the powered series exvector co; co.reserve(deg); - ex co0; - co.push_back(co0 = power(coeff(*s, ldeg), p)); - bool all_sums_zero = true; - for (i=1; icoeff = i->coeff + deg; + epvector newseq = seq; + epvector::iterator i = newseq.begin(), end = newseq.end(); + while (i != end) { + i->coeff += deg; + ++i; + } return pseries(relational(var, point), newseq); } @@ -812,25 +939,69 @@ pseries pseries::shift_exponents(int deg) const * @see ex::series */ ex power::series(const relational & r, int order, unsigned options) const { - ex e; - if (!is_ex_exactly_of_type(basis, pseries)) { - // Basis is not a series, may there be a singulary? - if (!exponent.info(info_flags::negint)) - return basic::series(r, order, options); - - // Expression is of type something^(-int), check for singularity - if (!basis.subs(r).is_zero()) - return basic::series(r, order, options); - - // Singularity encountered, expand basis into series - e = basis.series(r, order, options); + // If basis is already a series, just power it + if (is_exactly_a(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; + if (is_a(exponent)) { + numexp = ex_to(exponent); } else { - // Basis is a series - e = basis; + numexp = 0; + } + 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_pseries(e).power_const(ex_to_numeric(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; } @@ -838,19 +1009,19 @@ 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)) + if (var.is_equal(s) && point.is_equal(p)) { + if (order > degree(s)) return *this; else { epvector new_seq; epvector::const_iterator it = seq.begin(), itend = seq.end(); while (it != itend) { - int o = ex_to_numeric(it->coeff).to_int(); + 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); @@ -874,14 +1045,13 @@ 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)) - rel_ = ex_to_relational(r); - else if (is_ex_exactly_of_type(r,symbol)) - rel_ = relational(r,_ex0()); + if (is_a(r)) + rel_ = ex_to(r); + else if (is_a(r)) + rel_ = relational(r,_ex0); else throw (std::logic_error("ex::series(): expansion point has unknown type")); @@ -893,21 +1063,4 @@ ex ex::series(const ex & r, int order, unsigned options) const return e; } -////////// -// static member variables -////////// - -// protected - -unsigned pseries::precedence = 38; // for clarity just below add::precedence - -////////// -// global constants -////////// - -const pseries some_pseries; -const std::type_info & typeid_pseries = typeid(some_pseries); - -#ifndef NO_NAMESPACE_GINAC } // namespace GiNaC -#endif // ndef NO_NAMESPACE_GINAC