X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Finifcns_gamma.cpp;h=8ed385150e8175d15df02931ecd5c8fd8796c5cc;hp=7bd6669bb8b5eb8c817fd016718c5e3bfdaa2546;hb=a6bb52b00bf185271774e7d56215923700a3ec40;hpb=8d16780dbd9f4da9397e638aca213745589818c0 diff --git a/ginac/inifcns_gamma.cpp b/ginac/inifcns_gamma.cpp index 7bd6669b..8ed38515 100644 --- a/ginac/inifcns_gamma.cpp +++ b/ginac/inifcns_gamma.cpp @@ -1,6 +1,7 @@ /** @file inifcns_gamma.cpp * - * Implementation of Gamma function and some related stuff. */ + * Implementation of Gamma-function, Beta-function, Polygamma-functions, and + * some related stuff. */ /* * GiNaC Copyright (C) 1999 Johannes Gutenberg University Mainz, Germany @@ -26,99 +27,297 @@ #include "inifcns.h" #include "ex.h" #include "constant.h" +#include "series.h" #include "numeric.h" #include "power.h" +#include "relational.h" #include "symbol.h" +#include "utils.h" +#ifndef NO_GINAC_NAMESPACE namespace GiNaC { +#endif // ndef NO_GINAC_NAMESPACE ////////// -// gamma function +// Gamma-function ////////// +static ex gamma_evalf(const ex & x) +{ + BEGIN_TYPECHECK + TYPECHECK(x,numeric) + END_TYPECHECK(gamma(x)) + + return gamma(ex_to_numeric(x)); +} + /** Evaluation of gamma(x). Knows about integer arguments, half-integer * arguments and that's it. Somebody ought to provide some good numerical * evaluation some day... * - * @exception fail_numeric("complex_infinity") or something similar... */ -static ex gamma_eval(ex const & x) + * @exception std::domain_error("gamma_eval(): simple pole") */ +static ex gamma_eval(const ex & x) { if (x.info(info_flags::numeric)) { - // trap integer arguments: - if ( x.info(info_flags::integer) ) { + if (x.info(info_flags::integer)) { // gamma(n+1) -> n! for postitive n - if ( x.info(info_flags::posint) ) { - return factorial(ex_to_numeric(x).sub(numONE())); + if (x.info(info_flags::posint)) { + return factorial(ex_to_numeric(x).sub(_num1())); } else { - return numZERO(); // Infinity. Throw? What? + throw (std::domain_error("gamma_eval(): simple pole")); } } // trap half integer arguments: - if ( (x*2).info(info_flags::integer) ) { - // trap positive x=(n+1/2) + if ((x*2).info(info_flags::integer)) { + // trap positive x==(n+1/2) // gamma(n+1/2) -> Pi^(1/2)*(1*3*..*(2*n-1))/(2^n) - if ( (x*2).info(info_flags::posint) ) { - numeric n = ex_to_numeric(x).sub(numHALF()); - numeric coefficient = doublefactorial(n.mul(numTWO()).sub(numONE())); - coefficient = coefficient.div(numTWO().power(n)); - return coefficient * pow(Pi,numHALF()); + if ((x*_ex2()).info(info_flags::posint)) { + numeric n = ex_to_numeric(x).sub(_num1_2()); + numeric coefficient = doublefactorial(n.mul(_num2()).sub(_num1())); + coefficient = coefficient.div(pow(_num2(),n)); + return coefficient * pow(Pi,_ex1_2()); } else { - // trap negative x=(-n+1/2) + // trap negative x==(-n+1/2) // gamma(-n+1/2) -> Pi^(1/2)*(-2)^n/(1*3*..*(2*n-1)) - numeric n = abs(ex_to_numeric(x).sub(numHALF())); - numeric coefficient = numeric(-2).power(n); - coefficient = coefficient.div(doublefactorial(n.mul(numTWO()).sub(numONE())));; - return coefficient*sqrt(Pi); + numeric n = abs(ex_to_numeric(x).sub(_num1_2())); + numeric coefficient = pow(_num_2(), n); + coefficient = coefficient.div(doublefactorial(n.mul(_num2()).sub(_num1())));; + return coefficient*power(Pi,_ex1_2()); } } + // gamma_evalf should be called here once it becomes available } + return gamma(x).hold(); } + +static ex gamma_diff(const ex & x, unsigned diff_param) +{ + GINAC_ASSERT(diff_param==0); -static ex gamma_evalf(ex const & x) + // d/dx log(gamma(x)) -> psi(x) + // d/dx gamma(x) -> psi(x)*gamma(x) + return psi(x)*gamma(x); +} + +static ex gamma_series(const ex & x, const symbol & s, const ex & pt, int order) +{ + // method: + // Taylor series where there is no pole falls back to psi function + // evaluation. + // On a pole at -m use the recurrence relation + // gamma(x) == gamma(x+1) / x + // from which follows + // series(gamma(x),x,-m,order) == + // series(gamma(x+m+1)/(x*(x+1)*...*(x+m)),x,-m,order+1); + const ex x_pt = x.subs(s==pt); + if (!x_pt.info(info_flags::integer) || x_pt.info(info_flags::positive)) + throw do_taylor(); // caught by function::series() + // if we got here we have to care for a simple pole at -m: + numeric m = -ex_to_numeric(x_pt); + ex ser_numer = gamma(x+m+_ex1()); + ex ser_denom = _ex1(); + for (numeric p; p<=m; ++p) + ser_denom *= x+p; + return (ser_numer/ser_denom).series(s, pt, order+1); +} + +REGISTER_FUNCTION(gamma, gamma_eval, gamma_evalf, gamma_diff, gamma_series); + +////////// +// Beta-function +////////// + +static ex beta_evalf(const ex & x, const ex & y) { BEGIN_TYPECHECK TYPECHECK(x,numeric) - END_TYPECHECK(gamma(x)) + TYPECHECK(y,numeric) + END_TYPECHECK(beta(x,y)) - return gamma(ex_to_numeric(x)); + return gamma(ex_to_numeric(x))*gamma(ex_to_numeric(y)) + / gamma(ex_to_numeric(x+y)); } -static ex gamma_diff(ex const & x, unsigned diff_param) +static ex beta_eval(const ex & x, const ex & y) { - ASSERT(diff_param==0); + if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) { + // treat all problematic x and y that may not be passed into gamma, + // because they would throw there although beta(x,y) is well-defined + // using the formula beta(x,y) == (-1)^y * beta(1-x-y, y) + numeric nx(ex_to_numeric(x)); + numeric ny(ex_to_numeric(y)); + if (nx.is_real() && nx.is_integer() && + ny.is_real() && ny.is_integer()) { + if (nx.is_negative()) { + if (nx<=-ny) + return pow(_num_1(), ny)*beta(1-x-y, y); + else + throw (std::domain_error("beta_eval(): simple pole")); + } + if (ny.is_negative()) { + if (ny<=-nx) + return pow(_num_1(), nx)*beta(1-y-x, x); + else + throw (std::domain_error("beta_eval(): simple pole")); + } + return gamma(x)*gamma(y)/gamma(x+y); + } + // no problem in numerator, but denominator has pole: + if ((nx+ny).is_real() && + (nx+ny).is_integer() && + !(nx+ny).is_positive()) + return _ex0(); + // everything is ok: + return gamma(x)*gamma(y)/gamma(x+y); + } + + return beta(x,y).hold(); +} - return psi(exZERO(),x)*gamma(x); +static ex beta_diff(const ex & x, const ex & y, unsigned diff_param) +{ + GINAC_ASSERT(diff_param<2); + ex retval; + + // d/dx beta(x,y) -> (psi(x)-psi(x+y)) * beta(x,y) + if (diff_param==0) + retval = (psi(x)-psi(x+y))*beta(x,y); + // d/dy beta(x,y) -> (psi(y)-psi(x+y)) * beta(x,y) + if (diff_param==1) + retval = (psi(y)-psi(x+y))*beta(x,y); + return retval; } -static ex gamma_series(ex const & x, symbol const & s, ex const & point, int order) +static ex beta_series(const ex & x, const ex & y, const symbol & s, const ex & pt, int order) { - // FIXME: Only handle one special case for now... - if (x.is_equal(s) && point.is_zero()) { - ex e = 1 / s - EulerGamma + s * (pow(Pi, 2) / 12 + pow(EulerGamma, 2) / 2) + Order(pow(s, 2)); - return e.series(s, point, order); - } else - throw(std::logic_error("don't know the series expansion of this particular gamma function")); + // method: + // Taylor series where there is no pole of one of the gamma functions + // falls back to beta function evaluation. Otherwise, fall back to + // gamma series directly. + // FIXME: this could need some testing, maybe it's wrong in some cases? + const ex x_pt = x.subs(s==pt); + const ex y_pt = y.subs(s==pt); + ex x_ser, y_ser, xy_ser; + if ((!x_pt.info(info_flags::integer) || x_pt.info(info_flags::positive)) && + (!y_pt.info(info_flags::integer) || y_pt.info(info_flags::positive))) + throw do_taylor(); // caught by function::series() + // trap the case where x is on a pole directly: + if (x.info(info_flags::integer) && !x.info(info_flags::positive)) + x_ser = gamma(x+s).series(s,pt,order); + else + x_ser = gamma(x).series(s,pt,order); + // trap the case where y is on a pole directly: + if (y.info(info_flags::integer) && !y.info(info_flags::positive)) + y_ser = gamma(y+s).series(s,pt,order); + else + y_ser = gamma(y).series(s,pt,order); + // trap the case where y is on a pole directly: + if ((x+y).info(info_flags::integer) && !(x+y).info(info_flags::positive)) + xy_ser = gamma(y+x+s).series(s,pt,order); + else + xy_ser = gamma(y+x).series(s,pt,order); + // compose the result: + return (x_ser*y_ser/xy_ser).series(s,pt,order); } -REGISTER_FUNCTION(gamma, gamma_eval, gamma_evalf, gamma_diff, gamma_series); +REGISTER_FUNCTION(beta, beta_eval, beta_evalf, beta_diff, beta_series); ////////// -// psi function (aka polygamma function) +// Psi-function (aka digamma-function) ////////// -/** Evaluation of polygamma-function psi(n,x). +static ex psi1_evalf(const ex & x) +{ + BEGIN_TYPECHECK + TYPECHECK(x,numeric) + END_TYPECHECK(psi(x)) + + return psi(ex_to_numeric(x)); +} + +/** Evaluation of digamma-function psi(x). * Somebody ought to provide some good numerical evaluation some day... */ -static ex psi_eval(ex const & n, ex const & x) +static ex psi1_eval(const ex & x) { - if (n.info(info_flags::numeric) && x.info(info_flags::numeric)) { - // do some stuff... + if (x.info(info_flags::numeric)) { + numeric nx = ex_to_numeric(x); + if (nx.is_integer()) { + // integer case + if (nx.is_positive()) { + // psi(n) -> 1 + 1/2 +...+ 1/(n-1) - EulerGamma + numeric rat(0); + for (numeric i(nx+_num_1()); i.is_positive(); --i) + rat += i.inverse(); + return rat-EulerGamma; + } else { + // for non-positive integers there is a pole: + throw (std::domain_error("psi_eval(): simple pole")); + } + } + if ((_num2()*nx).is_integer()) { + // half integer case + if (nx.is_positive()) { + // psi((2m+1)/2) -> 2/(2m+1) + 2/2m +...+ 2/1 - EulerGamma - 2log(2) + numeric rat(0); + for (numeric i((nx+_num_1())*_num2()); i.is_positive(); i-=_num2()) + rat += _num2()*i.inverse(); + return rat-EulerGamma-_ex2()*log(_ex2()); + } else { + // use the recurrence relation + // psi(-m-1/2) == psi(-m-1/2+1) - 1 / (-m-1/2) + // to relate psi(-m-1/2) to psi(1/2): + // psi(-m-1/2) == psi(1/2) + r + // where r == ((-1/2)^(-1) + ... + (-m-1/2)^(-1)) + numeric recur(0); + for (numeric p(nx); p<0; ++p) + recur -= pow(p, _num_1()); + return recur+psi(_ex1_2()); + } + } + // psi1_evalf should be called here once it becomes available } - return psi(n, x).hold(); -} -static ex psi_evalf(ex const & n, ex const & x) + return psi(x).hold(); +} + +static ex psi1_diff(const ex & x, unsigned diff_param) +{ + GINAC_ASSERT(diff_param==0); + + // d/dx psi(x) -> psi(1,x) + return psi(_ex1(), x); +} + +static ex psi1_series(const ex & x, const symbol & s, const ex & pt, int order) +{ + // method: + // Taylor series where there is no pole falls back to polygamma function + // evaluation. + // On a pole at -m use the recurrence relation + // psi(x) == psi(x+1) - 1/z + // from which follows + // series(psi(x),x,-m,order) == + // series(psi(x+m+1) - 1/x - 1/(x+1) - 1/(x+m)),x,-m,order); + const ex x_pt = x.subs(s==pt); + if (!x_pt.info(info_flags::integer) || x_pt.info(info_flags::positive)) + throw do_taylor(); // caught by function::series() + // if we got here we have to care for a simple pole at -m: + numeric m = -ex_to_numeric(x_pt); + ex recur; + for (numeric p; p<=m; ++p) + recur += power(x+p,_ex_1()); + return (psi(x+m+_ex1())-recur).series(s, pt, order); +} + +const unsigned function_index_psi1 = function::register_new("psi", psi1_eval, psi1_evalf, psi1_diff, psi1_series); + +////////// +// Psi-functions (aka polygamma-functions) psi(0,x)==psi(x) +////////// + +static ex psi2_evalf(const ex & n, const ex & x) { BEGIN_TYPECHECK TYPECHECK(n,numeric) @@ -128,18 +327,108 @@ static ex psi_evalf(ex const & n, ex const & x) return psi(ex_to_numeric(n), ex_to_numeric(x)); } -static ex psi_diff(ex const & n, ex const & x, unsigned diff_param) +/** Evaluation of polygamma-function psi(n,x). + * Somebody ought to provide some good numerical evaluation some day... */ +static ex psi2_eval(const ex & n, const ex & x) { - ASSERT(diff_param==0); + // psi(0,x) -> psi(x) + if (n.is_zero()) + return psi(x); + // psi(-1,x) -> log(gamma(x)) + if (n.is_equal(_ex_1())) + return log(gamma(x)); + if (n.info(info_flags::numeric) && n.info(info_flags::posint) && + x.info(info_flags::numeric)) { + numeric nn = ex_to_numeric(n); + numeric nx = ex_to_numeric(x); + if (nx.is_integer()) { + // integer case + if (nx.is_equal(_num1())) + // use psi(n,1) == (-)^(n+1) * n! * zeta(n+1) + return pow(_num_1(),nn+_num1())*factorial(nn)*zeta(ex(nn+_num1())); + if (nx.is_positive()) { + // use the recurrence relation + // psi(n,m) == psi(n,m+1) - (-)^n * n! / m^(n+1) + // to relate psi(n,m) to psi(n,1): + // psi(n,m) == psi(n,1) + r + // where r == (-)^n * n! * (1^(-n-1) + ... + (m-1)^(-n-1)) + numeric recur(0); + for (numeric p(1); p psi(n+1,x) + return psi(n+_ex1(), x); } -static ex psi_series(ex const & n, ex const & x, symbol const & s, ex const & point, int order) +static ex psi2_series(const ex & n, const ex & x, const symbol & s, const ex & pt, int order) { - throw(std::logic_error("Nobody told me how to series expand the psi function. :-(")); + // method: + // Taylor series where there is no pole falls back to polygamma function + // evaluation. + // On a pole at -m use the recurrence relation + // psi(n,x) == psi(n,x+1) - (-)^n * n! / x^(n+1) + // from which follows + // series(psi(x),x,-m,order) == + // series(psi(x+m+1) - (-1)^n * n! * ((x)^(-n-1) + (x+1)^(-n-1) + ... + // ... + (x+m)^(-n-1))),x,-m,order); + const ex x_pt = x.subs(s==pt); + if (!x_pt.info(info_flags::integer) || x_pt.info(info_flags::positive)) + throw do_taylor(); // caught by function::series() + // if we got here we have to care for a pole of order n+1 at -m: + numeric m = -ex_to_numeric(x_pt); + ex recur; + for (numeric p; p<=m; ++p) + recur += power(x+p,-n+_ex_1()); + recur *= factorial(n)*power(_ex_1(),n); + return (psi(n, x+m+_ex1())-recur).series(s, pt, order); } -REGISTER_FUNCTION(psi, psi_eval, psi_evalf, psi_diff, psi_series); +const unsigned function_index_psi2 = function::register_new("psi", psi2_eval, psi2_evalf, psi2_diff, psi2_series); +#ifndef NO_GINAC_NAMESPACE } // namespace GiNaC +#endif // ndef NO_GINAC_NAMESPACE