X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fpseries.cpp;h=dcc25a0493a767d45980c2d1bc0be27b3f6d1ce4;hp=720c4eca856f1de8315826c056f436ab4c29a24d;hb=40234423820294740aa535a713c2784d1bb23351;hpb=3a63743e24046766b37c3d1bd38605542ee0a536 diff --git a/ginac/pseries.cpp b/ginac/pseries.cpp index 720c4eca..dcc25a04 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) 1999-2018 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 @@ -18,12 +18,9 @@ * * You should have received a copy of the GNU General Public License * along with this program; if not, write to the Free Software - * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA + * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA */ -#include -#include - #include "pseries.h" #include "add.h" #include "inifcns.h" // for Order function @@ -31,31 +28,31 @@ #include "mul.h" #include "power.h" #include "relational.h" +#include "operators.h" #include "symbol.h" -#include "print.h" +#include "integral.h" #include "archive.h" #include "utils.h" +#include +#include +#include + 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() : inherited(TINFO_pseries) { } - -void pseries::copy(const pseries &other) -{ - inherited::copy(other); - seq = other.seq; - var = other.var; - point = other.point; -} - -DEFAULT_DESTROY(pseries) +pseries::pseries() { } /* @@ -71,10 +68,45 @@ DEFAULT_DESTROY(pseries) * @param rel_ expansion variable and point (must hold a relational) * @param ops_ vector of {coefficient, power} pairs (coefficient must not be zero) * @return newly constructed pseries */ -pseries::pseries(const ex &rel_, const epvector &ops_) : basic(TINFO_pseries), seq(ops_) +pseries::pseries(const ex &rel_, const epvector &ops_) + : seq(ops_) { - GINAC_ASSERT(is_exactly_a(rel_)); - GINAC_ASSERT(is_exactly_a(rel_.lhs())); +#ifdef DO_GINAC_ASSERT + auto i = seq.begin(); + while (i != seq.end()) { + auto ip1 = i+1; + if (ip1 != seq.end()) + GINAC_ASSERT(!is_order_function(i->rest)); + else + break; + GINAC_ASSERT(is_a(i->coeff)); + GINAC_ASSERT(ex_to(i->coeff) < ex_to(ip1->coeff)); + ++i; + } +#endif // def DO_GINAC_ASSERT + GINAC_ASSERT(is_a(rel_)); + GINAC_ASSERT(is_a(rel_.lhs())); + point = rel_.rhs(); + var = rel_.lhs(); +} +pseries::pseries(const ex &rel_, epvector &&ops_) + : seq(std::move(ops_)) +{ +#ifdef DO_GINAC_ASSERT + auto i = seq.begin(); + while (i != seq.end()) { + auto ip1 = i+1; + if (ip1 != seq.end()) + GINAC_ASSERT(!is_order_function(i->rest)); + else + break; + GINAC_ASSERT(is_a(i->coeff)); + GINAC_ASSERT(ex_to(i->coeff) < ex_to(ip1->coeff)); + ++i; + } +#endif // def DO_GINAC_ASSERT + GINAC_ASSERT(is_a(rel_)); + GINAC_ASSERT(is_a(rel_.lhs())); point = rel_.rhs(); var = rel_.lhs(); } @@ -84,16 +116,22 @@ 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) +void pseries::read_archive(const archive_node &n, lst &sym_lst) { - for (unsigned int i=0; true; ++i) { + inherited::read_archive(n, sym_lst); + auto first = n.find_first("coeff"); + auto last = n.find_last("power"); + ++last; + seq.reserve((last-first)/2); + + for (auto loc = first; loc < last;) { ex rest; ex coeff; - if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i)) - seq.push_back(expair(rest, coeff)); - else - break; + n.find_ex_by_loc(loc++, rest, sym_lst); + n.find_ex_by_loc(loc++, coeff, sym_lst); + seq.push_back(expair(rest, coeff)); } + n.find_ex("var", var, sym_lst); n.find_ex("point", point, sym_lst); } @@ -101,122 +139,126 @@ pseries::pseries(const archive_node &n, const lst &sym_lst) : inherited(n, sym_l void pseries::archive(archive_node &n) const { inherited::archive(n); - epvector::const_iterator i = seq.begin(), iend = seq.end(); - while (i != iend) { - n.add_ex("coeff", i->rest); - n.add_ex("power", i->coeff); - ++i; + for (auto & it : seq) { + n.add_ex("coeff", it.rest); + n.add_ex("power", it.coeff); } n.add_ex("var", var); n.add_ex("point", point); } -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 { - if (is_a(c)) { + 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'; - 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; i(c)) { - c.s << class_name() << "(relational("; - var.print(c); - c.s << ','; - point.print(c); - c.s << "),["; - unsigned num = seq.size(); - for (unsigned i=0; i(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; + 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 << 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)) { - if (is_a(c)) - c.s << "**"; - else - 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(pow(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; icompare(*o_it); if (cmpval) @@ -253,22 +295,20 @@ 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")); + if (is_order_function(seq[i].rest)) + return Order(pow(var-point, seq[i].coeff)); + return seq[i].rest * pow(var - point, seq[i].coeff); } /** Return degree of highest power of the series. This is usually the exponent @@ -276,25 +316,17 @@ ex &pseries::let_op(int i) * series is examined termwise. */ 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(); - else - return 0; - } else { - epvector::const_iterator it = seq.begin(), itend = seq.end(); - if (it == itend) - return 0; - int max_pow = INT_MIN; - while (it != itend) { - int pow = it->rest.degree(s); - if (pow > max_pow) - max_pow = pow; - ++it; - } - return max_pow; - } + if (seq.empty()) + return 0; + + if (var.is_equal(s)) + // Return last/greatest exponent + return ex_to((seq.end()-1)->coeff).to_int(); + + int max_pow = std::numeric_limits::min(); + for (auto & it : seq) + max_pow = std::max(max_pow, it.rest.degree(s)); + return max_pow; } /** Return degree of lowest power of the series. This is usually the exponent @@ -304,25 +336,17 @@ int pseries::degree(const ex &s) const * I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */ 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(); - else - return 0; - } else { - epvector::const_iterator it = seq.begin(), itend = seq.end(); - if (it == itend) - return 0; - int min_pow = INT_MAX; - while (it != itend) { - int pow = it->rest.ldegree(s); - if (pow < min_pow) - min_pow = pow; - ++it; - } - return min_pow; - } + if (seq.empty()) + return 0; + + if (var.is_equal(s)) + // Return first/smallest exponent + return ex_to((seq.begin())->coeff).to_int(); + + int min_pow = std::numeric_limits::max(); + for (auto & it : seq) + min_pow = std::min(min_pow, it.rest.degree(s)); + return min_pow; } /** Return coefficient of degree n in power series if s is the expansion @@ -370,63 +394,144 @@ ex pseries::collect(const ex &s, bool distributed) const } /** Perform coefficient-wise automatic term rewriting rules in this class. */ -ex pseries::eval(int level) const +ex pseries::eval() const { - if (level == 1) - return this->hold(); - - if (level == -max_recursion_level) - throw (std::runtime_error("pseries::eval(): recursion limit exceeded")); - + if (flags & status_flags::evaluated) { + return *this; + } + // Construct a new series with evaluated coefficients epvector new_seq; new_seq.reserve(seq.size()); - epvector::const_iterator it = seq.begin(), itend = seq.end(); - while (it != itend) { - new_seq.push_back(expair(it->rest.eval(level-1), it->coeff)); - ++it; - } - return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated); + for (auto & it : seq) + new_seq.push_back(expair(it.rest, it.coeff)); + + return dynallocate(relational(var,point), std::move(new_seq)).setflag(status_flags::evaluated); } /** Evaluate coefficients numerically. */ -ex pseries::evalf(int level) const +ex pseries::evalf() const { - if (level == 1) - return *this; - - if (level == -max_recursion_level) - throw (std::runtime_error("pseries::evalf(): recursion limit exceeded")); - // Construct a new series with evaluated coefficients epvector new_seq; new_seq.reserve(seq.size()); - epvector::const_iterator it = seq.begin(), itend = seq.end(); - while (it != itend) { - new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff)); - ++it; + for (auto & it : seq) + new_seq.push_back(expair(it.rest, it.coeff)); + + return dynallocate(relational(var,point), std::move(new_seq)).setflag(status_flags::evaluated); +} + +ex pseries::conjugate() const +{ + if(!var.info(info_flags::real)) + return conjugate_function(*this).hold(); + + std::unique_ptr newseq(conjugateepvector(seq)); + ex newpoint = point.conjugate(); + + if (!newseq && are_ex_trivially_equal(point, newpoint)) { + return *this; + } + + return dynallocate(var==newpoint, newseq ? std::move(*newseq) : seq); +} + +ex pseries::real_part() const +{ + if(!var.info(info_flags::real)) + return real_part_function(*this).hold(); + ex newpoint = point.real_part(); + if(newpoint != point) + return real_part_function(*this).hold(); + + epvector v; + v.reserve(seq.size()); + for (auto & it : seq) + v.push_back(expair((it.rest).real_part(), it.coeff)); + return dynallocate(var==point, std::move(v)); +} + +ex pseries::imag_part() const +{ + if(!var.info(info_flags::real)) + return imag_part_function(*this).hold(); + ex newpoint = point.real_part(); + if(newpoint != point) + return imag_part_function(*this).hold(); + + epvector v; + v.reserve(seq.size()); + for (auto & it : seq) + v.push_back(expair((it.rest).imag_part(), it.coeff)); + return dynallocate(var==point, std::move(v)); +} + +ex pseries::eval_integ() const +{ + std::unique_ptr newseq(nullptr); + for (auto i=seq.begin(); i!=seq.end(); ++i) { + if (newseq) { + newseq->push_back(expair(i->rest.eval_integ(), i->coeff)); + continue; + } + ex newterm = i->rest.eval_integ(); + if (!are_ex_trivially_equal(newterm, i->rest)) { + newseq.reset(new epvector); + newseq->reserve(seq.size()); + for (auto j=seq.begin(); j!=i; ++j) + newseq->push_back(*j); + newseq->push_back(expair(newterm, i->coeff)); + } + } + + ex newpoint = point.eval_integ(); + if (newseq || !are_ex_trivially_equal(newpoint, point)) + return dynallocate(var==newpoint, std::move(*newseq)); + return *this; +} + +ex pseries::evalm() const +{ + // evalm each coefficient + epvector newseq; + bool something_changed = false; + for (auto i=seq.begin(); i!=seq.end(); ++i) { + if (something_changed) { + ex newcoeff = i->rest.evalm(); + if (!newcoeff.is_zero()) + newseq.push_back(expair(newcoeff, i->coeff)); + } else { + ex newcoeff = i->rest.evalm(); + if (!are_ex_trivially_equal(newcoeff, i->rest)) { + something_changed = true; + newseq.reserve(seq.size()); + std::copy(seq.begin(), i, std::back_inserter(newseq)); + if (!newcoeff.is_zero()) + newseq.push_back(expair(newcoeff, i->coeff)); + } + } } - return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated); + if (something_changed) + return dynallocate(var==point, std::move(newseq)); + else + return *this; } -ex pseries::subs(const lst & ls, const lst & lr, bool no_pattern) 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, 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 epvector newseq; 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)); - ++it; - } - return (new pseries(relational(var,point.subs(ls, lr, no_pattern)), newseq))->setflag(status_flags::dynallocated); + for (auto & it : seq) + newseq.push_back(expair(it.rest.subs(m, options), it.coeff)); + return dynallocate(relational(var,point.subs(m, options)), std::move(newseq)); } /** Implementation of ex::expand() for a power series. It expands all the @@ -434,64 +539,81 @@ ex pseries::subs(const lst & ls, const lst & lr, bool no_pattern) const ex pseries::expand(unsigned options) const { epvector newseq; - epvector::const_iterator i = seq.begin(), end = seq.end(); - while (i != end) { - ex restexp = i->rest.expand(); + for (auto & it : seq) { + ex restexp = it.rest.expand(); if (!restexp.is_zero()) - newseq.push_back(expair(restexp, i->coeff)); - ++i; + newseq.push_back(expair(restexp, it.coeff)); } - return (new pseries(relational(var,point), newseq)) - ->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0)); + return dynallocate(relational(var,point), std::move(newseq)).setflag(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; + if (s == var) { - epvector new_seq; - epvector::const_iterator it = seq.begin(), itend = seq.end(); // FIXME: coeff might depend on var - while (it != itend) { - if (is_order_function(it->rest)) { - new_seq.push_back(expair(it->rest, it->coeff - 1)); + for (auto & it : seq) { + if (is_order_function(it.rest)) { + new_seq.push_back(expair(it.rest, it.coeff - 1)); } else { - ex c = it->rest * it->coeff; + ex c = it.rest * it.coeff; if (!c.is_zero()) - new_seq.push_back(expair(c, it->coeff - 1)); + new_seq.push_back(expair(c, it.coeff - 1)); } - ++it; } - return pseries(relational(var,point), new_seq); + } else { - return *this; + + for (auto & it : seq) { + 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)); + } + } } + + return pseries(relational(var,point), std::move(new_seq)); } ex pseries::convert_to_poly(bool no_order) const { ex e; - epvector::const_iterator it = seq.begin(), itend = seq.end(); - - while (it != itend) { - if (is_order_function(it->rest)) { + for (auto & it : seq) { + if (is_order_function(it.rest)) { if (!no_order) - e += Order(power(var - point, it->coeff)); + e += Order(pow(var - point, it.coeff)); } else - e += it->rest * power(var - point, it->coeff); - ++it; + e += it.rest * pow(var - point, it.coeff); } return e; } -bool pseries::is_terminating(void) const +bool pseries::is_terminating() const { return seq.empty() || !is_order_function((seq.end()-1)->rest); } +ex pseries::coeffop(size_t i) const +{ + if (i >= nops()) + throw (std::out_of_range("coeffop() out of range")); + return seq[i].rest; +} + +ex pseries::exponop(size_t i) const +{ + if (i >= nops()) + throw (std::out_of_range("exponop() out of range")); + return seq[i].coeff; +} + /* * Implementations of series expansion @@ -502,34 +624,43 @@ bool pseries::is_terminating(void) const ex basic::series(const relational & r, int order, unsigned options) const { epvector seq; + 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, std::move(seq)); + } + + // do Taylor expansion numeric fac = 1; ex deriv = *this; - ex coeff = deriv.subs(r); - const symbol &s = ex_to(r.lhs()); - - if (!coeff.is_zero()) + 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())); + GINAC_ASSERT(is_a(r.lhs())); if (this->is_equal_same_type(ex_to(r.lhs()))) { if (order > 0 && !point.is_zero()) @@ -550,7 +681,7 @@ ex symbol::series(const relational & r, int order, unsigned options) const seq.push_back(expair(Order(_ex1), numeric(order))); } else seq.push_back(expair(*this, _ex0)); - return pseries(r, seq); + return pseries(r, std::move(seq)); } @@ -564,18 +695,15 @@ ex pseries::add_series(const pseries &other) const // Adding two series with different variables or expansion points // results in an empty (constant) series if (!is_compatible_to(other)) { - epvector nul; - nul.push_back(expair(Order(_ex1), _ex0)); - return pseries(relational(var,point), nul); + epvector nul { expair(Order(_ex1), _ex0) }; + return pseries(relational(var,point), std::move(nul)); } // Series addition epvector new_seq; - epvector::const_iterator a = seq.begin(); - epvector::const_iterator b = other.seq.begin(); - epvector::const_iterator a_end = seq.end(); - epvector::const_iterator b_end = other.seq.end(); - int pow_a = INT_MAX, pow_b = INT_MAX; + auto a = seq.begin(), a_end = seq.end(); + auto b = other.seq.begin(), b_end = other.seq.end(); + int pow_a = std::numeric_limits::max(), pow_b = std::numeric_limits::max(); for (;;) { // If a is empty, fill up with elements from b and stop if (a == a_end) { @@ -624,7 +752,7 @@ ex pseries::add_series(const pseries &other) const } } } - return pseries(relational(var,point), new_seq); + return pseries(relational(var,point), std::move(new_seq)); } @@ -639,16 +767,14 @@ ex add::series(const relational & r, int order, unsigned options) const acc = overall_coeff.series(r, order, options); // Add remaining terms - epvector::const_iterator it = seq.begin(); - epvector::const_iterator itend = seq.end(); - for (; it!=itend; ++it) { + for (auto & it : seq) { ex op; - if (is_ex_exactly_of_type(it->rest, pseries)) - op = it->rest; + 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(op).mul_const(ex_to(it->coeff)); + op = it.rest.series(r, order, options); + if (!it.coeff.is_equal(_ex1)) + op = ex_to(op).mul_const(ex_to(it.coeff)); // Series addition acc = ex_to(acc).add_series(ex_to(op)); @@ -667,15 +793,13 @@ ex pseries::mul_const(const numeric &other) const epvector new_seq; new_seq.reserve(seq.size()); - epvector::const_iterator it = seq.begin(), itend = seq.end(); - while (it != itend) { - if (!is_order_function(it->rest)) - new_seq.push_back(expair(it->rest * other, it->coeff)); + for (auto & it : seq) { + if (!is_order_function(it.rest)) + new_seq.push_back(expair(it.rest * other, it.coeff)); else - new_seq.push_back(*it); - ++it; + new_seq.push_back(it); } - return pseries(relational(var,point), new_seq); + return pseries(relational(var,point), std::move(new_seq)); } @@ -689,45 +813,60 @@ ex pseries::mul_series(const pseries &other) const // Multiplying two series with different variables or expansion points // results in an empty (constant) series if (!is_compatible_to(other)) { - epvector nul; - nul.push_back(expair(Order(_ex1), _ex0)); - return pseries(relational(var,point), nul); + epvector nul { expair(Order(_ex1), _ex0) }; + return pseries(relational(var,point), std::move(nul)); + } + + if (seq.empty() || other.seq.empty()) { + return dynallocate(var==point, epvector()); } // Series multiplication epvector new_seq; - 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; + const int a_max = degree(var); + const int b_max = other.degree(var); + const int a_min = ldegree(var); + const int b_min = other.ldegree(var); + const 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; + int higher_order_a = std::numeric_limits::max(); + int higher_order_b = std::numeric_limits::max(); if (is_order_function(coeff(var, a_max))) higher_order_a = a_max + b_min; 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); + const int higher_order_c = std::min(higher_order_a, higher_order_b); if (cdeg_max >= higher_order_c) cdeg_max = higher_order_c - 1; - + + std::map rest_map_a, rest_map_b; + for (const auto& it : seq) + rest_map_a[ex_to(it.coeff).to_int()] = it.rest; + + if (other.var.is_equal(var)) + for (const auto& it : other.seq) + rest_map_b[ex_to(it.coeff).to_int()] = it.rest; + for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) { 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); - ex b_coeff = other.coeff(var, cdeg-i); - if (!is_order_function(a_coeff) && !is_order_function(b_coeff)) - co += a_coeff * b_coeff; + const auto& ita = rest_map_a.find(i); + if (ita == rest_map_a.end()) + continue; + const auto& itb = rest_map_b.find(cdeg-i); + if (itb == rest_map_b.end()) + continue; + if (!is_order_function(ita->second) && !is_order_function(itb->second)) + co += ita->second * itb->second; } if (!co.is_zero()) new_seq.push_back(expair(co, numeric(cdeg))); } - if (higher_order_c < INT_MAX) + if (higher_order_c < std::numeric_limits::max()) new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c))); - return pseries(relational(var, point), new_seq); + return pseries(relational(var, point), std::move(new_seq)); } @@ -738,18 +877,107 @@ ex mul::series(const relational & r, int order, unsigned options) const { 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; + std::vector ldegree_redo; + + // find minimal degrees + // first round: obtain a bound up to which minimal degrees have to be + // considered + for (auto & it : seq) { + + 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; + bool flag_redo = false; + try { + real_ldegree = buf.expand().ldegree(sym-r.rhs()); + } catch (std::runtime_error) {} + + if (real_ldegree == 0) { + if ( factor < 0 ) { + // This case must terminate, otherwise we would have division by + // zero. + int orderloop = 0; + do { + orderloop++; + real_ldegree = buf.series(r, orderloop, options).ldegree(sym); + } while (real_ldegree == orderloop); + } else { + // Here it is possible that buf does not have a ldegree, therefore + // check only if ldegree is negative, otherwise reconsider the case + // in the second round. + real_ldegree = buf.series(r, 0, options).ldegree(sym); + if (real_ldegree == 0) + flag_redo = true; + } + } + + ldegrees.push_back(factor * real_ldegree); + ldegree_redo.push_back(flag_redo); + } + + int degbound = order-std::accumulate(ldegrees.begin(), ldegrees.end(), 0); + // Second round: determine the remaining positive ldegrees by the series + // method. + // here we can ignore ldegrees larger than degbound + size_t j = 0; + for (auto & it : seq) { + if ( ldegree_redo[j] ) { + 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; + int orderloop = 0; + do { + orderloop++; + real_ldegree = buf.series(r, orderloop, options).ldegree(sym); + } while ((real_ldegree == orderloop) + && (factor*real_ldegree < degbound)); + ldegrees[j] = factor * real_ldegree; + degbound -= factor * real_ldegree; + } + j++; + } + + int degsum = std::accumulate(ldegrees.begin(), ldegrees.end(), 0); + + if (degsum >= order) { + epvector epv { expair(Order(_ex1), order) }; + return dynallocate(r, std::move(epv)); + } + // Multiply with remaining terms - const epvector::const_iterator itbeg = seq.begin(); - const epvector::const_iterator itend = seq.end(); - for (epvector::const_iterator it=itbeg; it!=itend; ++it) { - ex op = recombine_pair_to_ex(*it).series(r, order, options); + auto itd = ldegrees.begin(); + for (auto it=seq.begin(), itend=seq.end(); 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 - if (it==itbeg) + if (it == seq.begin()) acc = ex_to(op); else acc = ex_to(acc.mul_series(ex_to(op))); } + return acc.mul_const(ex_to(overall_coeff)); } @@ -796,16 +1024,22 @@ ex pseries::power_const(const numeric &p, int deg) const if (!(p*ldeg).is_integer()) throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series"); + // adjust number of coefficients + int numcoeff = deg - (p*ldeg).to_int(); + if (numcoeff <= 0) { + epvector epv { expair(Order(_ex1), deg) }; + return dynallocate(relational(var,point), std::move(epv)); + } + // 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; icoeff += deg; - ++i; - } - return pseries(relational(var, point), newseq); + for (auto & it : newseq) + it.coeff += deg; + return pseries(relational(var, point), std::move(newseq)); } @@ -856,23 +1086,48 @@ pseries pseries::shift_exponents(int deg) const ex power::series(const relational & r, int order, unsigned options) const { // If basis is already a series, just power it - if (is_ex_exactly_of_type(basis, pseries)) + 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); + basis.subs(r, subs_options::no_pattern); } catch (pole_error) { must_expand_basis = true; } - + + bool exponent_is_regular = true; + try { + exponent.subs(r, subs_options::no_pattern); + } catch (pole_error) { + exponent_is_regular = false; + } + + if (!exponent_is_regular) { + ex l = exponent*log(basis); + // this == exp(l); + ex le = l.series(r, order, options); + // Note: expanding exp(l) won't help, since that will attempt + // Taylor expansion, and fail (because exponent is "singular") + // Still l itself might be expanded in Taylor series. + // Examples: + // sin(x)/x*log(cos(x)) + // 1/x*log(1 + x) + return exp(le).series(r, order, options); + // Note: if l happens to have a Laurent expansion (with + // negative powers of (var - point)), expanding exp(le) + // will barf (which is The Right Thing). + } + // Is the expression of type something^(-int)? - if (!must_expand_basis && !exponent.info(info_flags::negint)) + if (!must_expand_basis && !exponent.info(info_flags::negint) + && (!is_a(basis) || !is_a(exponent))) return basic::series(r, order, options); - + // Is the expression of type 0^something? - if (!must_expand_basis && !basis.subs(r).is_zero()) + if (!must_expand_basis && !basis.subs(r, subs_options::no_pattern).is_zero() + && (!is_a(basis) || !is_a(exponent))) return basic::series(r, order, options); // Singularity encountered, is the basis equal to (var - point)? @@ -882,12 +1137,44 @@ ex power::series(const relational & r, int order, unsigned options) const new_seq.push_back(expair(_ex1, exponent)); else new_seq.push_back(expair(Order(_ex1), exponent)); - return pseries(r, new_seq); + return pseries(r, std::move(new_seq)); } // No, expand basis into series - ex e = basis.series(r, order, options); - return ex_to(e).power_const(ex_to(exponent), order); + + numeric numexp; + if (is_a(exponent)) { + numexp = ex_to(exponent); + } else { + numexp = 0; + } + const ex& sym = r.lhs(); + // find existing minimal degree + ex eb = basis.expand(); + int real_ldegree = 0; + if (eb.info(info_flags::rational_function)) + real_ldegree = eb.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); + + ex result; + try { + result = ex_to(e).power_const(numexp, order); + } catch (pole_error) { + epvector ser { expair(Order(_ex1), order) }; + result = pseries(r, std::move(ser)); + } + + return result; } @@ -895,7 +1182,7 @@ 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_exactly_a(r.lhs())); + GINAC_ASSERT(is_a(r.lhs())); const symbol &s = ex_to(r.lhs()); if (var.is_equal(s) && point.is_equal(p)) { @@ -903,22 +1190,75 @@ ex pseries::series(const relational & r, int order, unsigned options) const return *this; else { epvector new_seq; - epvector::const_iterator it = seq.begin(), itend = seq.end(); - while (it != itend) { - int o = ex_to(it->coeff).to_int(); + for (auto & it : seq) { + int o = ex_to(it.coeff).to_int(); if (o >= order) { new_seq.push_back(expair(Order(_ex1), o)); break; } - new_seq.push_back(*it); - ++it; + new_seq.push_back(it); } - return pseries(r, new_seq); + return pseries(r, std::move(new_seq)); } } else return convert_to_poly().series(r, order, options); } +ex integral::series(const relational & r, int order, unsigned options) const +{ + if (x.subs(r) != x) + throw std::logic_error("Cannot series expand wrt dummy variable"); + + // Expanding integrand with r substituted taken in boundaries. + ex fseries = f.series(r, order, options); + epvector fexpansion; + fexpansion.reserve(fseries.nops()); + for (size_t i=0; i(fseries).coeffop(i); + currcoeff = (currcoeff == Order(_ex1)) + ? currcoeff + : integral(x, a.subs(r), b.subs(r), currcoeff); + if (currcoeff != 0) + fexpansion.push_back( + expair(currcoeff, ex_to(fseries).exponop(i))); + } + + // Expanding lower boundary + ex result = dynallocate(r, std::move(fexpansion)); + ex aseries = (a-a.subs(r)).series(r, order, options); + fseries = f.series(x == (a.subs(r)), order, options); + for (size_t i=0; i(fseries).coeffop(i); + if (is_order_function(currcoeff)) + break; + ex currexpon = ex_to(fseries).exponop(i); + int orderforf = order-ex_to(currexpon).to_int()-1; + currcoeff = currcoeff.series(r, orderforf); + ex term = ex_to(aseries).power_const(ex_to(currexpon+1),order); + term = ex_to(term).mul_const(ex_to(-1/(currexpon+1))); + term = ex_to(term).mul_series(ex_to(currcoeff)); + result = ex_to(result).add_series(ex_to(term)); + } + + // Expanding upper boundary + ex bseries = (b-b.subs(r)).series(r, order, options); + fseries = f.series(x == (b.subs(r)), order, options); + for (size_t i=0; i(fseries).coeffop(i); + if (is_order_function(currcoeff)) + break; + ex currexpon = ex_to(fseries).exponop(i); + int orderforf = order-ex_to(currexpon).to_int()-1; + currcoeff = currcoeff.series(r, orderforf); + ex term = ex_to(bseries).power_const(ex_to(currexpon+1),order); + term = ex_to(term).mul_const(ex_to(1/(currexpon+1))); + term = ex_to(term).mul_series(ex_to(currcoeff)); + result = ex_to(result).add_series(ex_to(term)); + } + + return result; +} + /** Compute the truncated series expansion of an expression. * This function returns an expression containing an object of class pseries @@ -931,23 +1271,20 @@ 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)) + else if (is_a(r)) rel_ = relational(r,_ex0); else throw (std::logic_error("ex::series(): expansion point has unknown type")); - try { - e = bp->series(rel_, order, options); - } catch (std::exception &x) { - throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")")); - } + e = bp->series(rel_, order, options); return e; } +GINAC_BIND_UNARCHIVER(pseries); + } // namespace GiNaC