X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Finifcns_gamma.cpp;h=6bf85621d8e6f8f216edcb182b44fa4b7d2ddeb0;hp=a208592f82eab47de23bcfde1d83aeaea4095297;hb=eefedc70f63222beca918a3df89cabac700df1eb;hpb=be0485a03e9886496eeb7e8cdc2cc5c95b848632 diff --git a/ginac/inifcns_gamma.cpp b/ginac/inifcns_gamma.cpp index a208592f..6bf85621 100644 --- a/ginac/inifcns_gamma.cpp +++ b/ginac/inifcns_gamma.cpp @@ -1,10 +1,10 @@ /** @file inifcns_gamma.cpp * - * Implementation of Gamma-function, Polygamma-functions, 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 + * GiNaC Copyright (C) 1999-2003 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 @@ -25,166 +25,539 @@ #include #include "inifcns.h" -#include "ex.h" #include "constant.h" +#include "pseries.h" #include "numeric.h" #include "power.h" +#include "relational.h" +#include "operators.h" #include "symbol.h" +#include "symmetry.h" +#include "utils.h" namespace GiNaC { ////////// -// Gamma-function +// Logarithm of Gamma function ////////// -/** 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... +static ex lgamma_evalf(const ex & x) +{ + if (is_exactly_a(x)) { + try { + return lgamma(ex_to(x)); + } catch (const dunno &e) { } + } + + return lgamma(x).hold(); +} + + +/** Evaluation of lgamma(x), the natural logarithm of the Gamma function. + * Knows about 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) -{ - if (x.info(info_flags::numeric)) { - // trap integer arguments: - 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())); - } else { - return numZERO(); // Infinity. Throw? What? - } - } - // trap half integer arguments: - 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()); - } else { - // 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); - } - } - } - return gamma(x).hold(); -} + * @exception GiNaC::pole_error("lgamma_eval(): logarithmic pole",0) */ +static ex lgamma_eval(const ex & x) +{ + if (x.info(info_flags::numeric)) { + // trap integer arguments: + if (x.info(info_flags::integer)) { + // lgamma(n) -> log((n-1)!) for postitive n + if (x.info(info_flags::posint)) + return log(factorial(x + _ex_1)); + else + throw (pole_error("lgamma_eval(): logarithmic pole",0)); + } + // lgamma_evalf should be called here once it becomes available + } + + return lgamma(x).hold(); +} + -static ex gamma_evalf(ex const & x) +static ex lgamma_deriv(const ex & x, unsigned deriv_param) { - BEGIN_TYPECHECK - TYPECHECK(x,numeric) - END_TYPECHECK(gamma(x)) - - return gamma(ex_to_numeric(x)); + GINAC_ASSERT(deriv_param==0); + + // d/dx lgamma(x) -> psi(x) + return psi(x); } -static ex gamma_diff(ex const & x, unsigned diff_param) + +static ex lgamma_series(const ex & arg, + const relational & rel, + int order, + unsigned options) { - GINAC_ASSERT(diff_param==0); - - return psi(x)*gamma(x); // diff(log(gamma(x)),x)==psi(x) + // method: + // Taylor series where there is no pole falls back to psi function + // evaluation. + // On a pole at -m we could use the recurrence relation + // lgamma(x) == lgamma(x+1)-log(x) + // from which follows + // series(lgamma(x),x==-m,order) == + // series(lgamma(x+m+1)-log(x)...-log(x+m)),x==-m,order); + const ex arg_pt = arg.subs(rel, subs_options::no_pattern); + if (!arg_pt.info(info_flags::integer) || arg_pt.info(info_flags::positive)) + throw do_taylor(); // caught by function::series() + // if we got here we have to care for a simple pole of tgamma(-m): + numeric m = -ex_to(arg_pt); + ex recur; + for (numeric p = 0; p<=m; ++p) + recur += log(arg+p); + return (lgamma(arg+m+_ex1)-recur).series(rel, order, options); } -static ex gamma_series(ex const & x, symbol const & s, ex const & point, int order) + +REGISTER_FUNCTION(lgamma, eval_func(lgamma_eval). + evalf_func(lgamma_evalf). + derivative_func(lgamma_deriv). + series_func(lgamma_series). + latex_name("\\log \\Gamma")); + + +////////// +// true Gamma function +////////// + +static ex tgamma_evalf(const ex & x) { - // 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")); + if (is_exactly_a(x)) { + try { + return tgamma(ex_to(x)); + } catch (const dunno &e) { } + } + + return tgamma(x).hold(); } -REGISTER_FUNCTION(gamma, gamma_eval, gamma_evalf, gamma_diff, gamma_series); + +/** Evaluation of tgamma(x), the true Gamma function. Knows about integer + * arguments, half-integer arguments and that's it. Somebody ought to provide + * some good numerical evaluation some day... + * + * @exception pole_error("tgamma_eval(): simple pole",0) */ +static ex tgamma_eval(const ex & x) +{ + if (x.info(info_flags::numeric)) { + // trap integer arguments: + const numeric two_x = _num2*ex_to(x); + if (two_x.is_even()) { + // tgamma(n) -> (n-1)! for postitive n + if (two_x.is_positive()) { + return factorial(ex_to(x).sub(_num1)); + } else { + throw (pole_error("tgamma_eval(): simple pole",1)); + } + } + // trap half integer arguments: + if (two_x.is_integer()) { + // trap positive x==(n+1/2) + // tgamma(n+1/2) -> Pi^(1/2)*(1*3*..*(2*n-1))/(2^n) + if (two_x.is_positive()) { + const numeric n = ex_to(x).sub(_num1_2); + return (doublefactorial(n.mul(_num2).sub(_num1)).div(pow(_num2,n))) * sqrt(Pi); + } else { + // trap negative x==(-n+1/2) + // tgamma(-n+1/2) -> Pi^(1/2)*(-2)^n/(1*3*..*(2*n-1)) + const numeric n = abs(ex_to(x).sub(_num1_2)); + return (pow(_num_2, n).div(doublefactorial(n.mul(_num2).sub(_num1))))*sqrt(Pi); + } + } + // tgamma_evalf should be called here once it becomes available + } + + return tgamma(x).hold(); +} + + +static ex tgamma_deriv(const ex & x, unsigned deriv_param) +{ + GINAC_ASSERT(deriv_param==0); + + // d/dx tgamma(x) -> psi(x)*tgamma(x) + return psi(x)*tgamma(x); +} + + +static ex tgamma_series(const ex & arg, + const relational & rel, + int order, + unsigned options) +{ + // method: + // Taylor series where there is no pole falls back to psi function + // evaluation. + // On a pole at -m use the recurrence relation + // tgamma(x) == tgamma(x+1) / x + // from which follows + // series(tgamma(x),x==-m,order) == + // series(tgamma(x+m+1)/(x*(x+1)*...*(x+m)),x==-m,order+1); + const ex arg_pt = arg.subs(rel, subs_options::no_pattern); + if (!arg_pt.info(info_flags::integer) || arg_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: + const numeric m = -ex_to(arg_pt); + ex ser_denom = _ex1; + for (numeric p; p<=m; ++p) + ser_denom *= arg+p; + return (tgamma(arg+m+_ex1)/ser_denom).series(rel, order+1, options); +} + + +REGISTER_FUNCTION(tgamma, eval_func(tgamma_eval). + evalf_func(tgamma_evalf). + derivative_func(tgamma_deriv). + series_func(tgamma_series). + latex_name("\\Gamma")); + ////////// -// Psi-function (aka polygamma-function) +// beta-function ////////// -/** Evaluation of polygamma-function psi(x). - * Somebody ought to provide some good numerical evaluation some day... */ -static ex psi1_eval(ex const & x) +static ex beta_evalf(const ex & x, const ex & y) { - if (x.info(info_flags::numeric)) { - // do some stuff... - } - return psi(x).hold(); -} + if (is_exactly_a(x) && is_exactly_a(y)) { + try { + return tgamma(ex_to(x))*tgamma(ex_to(y))/tgamma(ex_to(x+y)); + } catch (const dunno &e) { } + } + + return beta(x,y).hold(); +} + + +static ex beta_eval(const ex & x, const ex & y) +{ + if (x.is_equal(_ex1)) + return 1/y; + if (y.is_equal(_ex1)) + return 1/x; + if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) { + // treat all problematic x and y that may not be passed into tgamma, + // 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) + const numeric &nx = ex_to(x); + const numeric &ny = ex_to(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 (pole_error("beta_eval(): simple pole",1)); + } + if (ny.is_negative()) { + if (ny<=-nx) + return pow(_num_1, nx)*beta(1-y-x, x); + else + throw (pole_error("beta_eval(): simple pole",1)); + } + return tgamma(x)*tgamma(y)/tgamma(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; + // beta_evalf should be called here once it becomes available + } + + return beta(x,y).hold(); +} + + +static ex beta_deriv(const ex & x, const ex & y, unsigned deriv_param) +{ + GINAC_ASSERT(deriv_param<2); + ex retval; + + // d/dx beta(x,y) -> (psi(x)-psi(x+y)) * beta(x,y) + if (deriv_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 (deriv_param==1) + retval = (psi(y)-psi(x+y))*beta(x,y); + return retval; +} + + +static ex beta_series(const ex & arg1, + const ex & arg2, + const relational & rel, + int order, + unsigned options) +{ + // method: + // Taylor series where there is no pole of one of the tgamma functions + // falls back to beta function evaluation. Otherwise, fall back to + // tgamma series directly. + const ex arg1_pt = arg1.subs(rel, subs_options::no_pattern); + const ex arg2_pt = arg2.subs(rel, subs_options::no_pattern); + GINAC_ASSERT(is_a(rel.lhs())); + const symbol &s = ex_to(rel.lhs()); + ex arg1_ser, arg2_ser, arg1arg2_ser; + if ((!arg1_pt.info(info_flags::integer) || arg1_pt.info(info_flags::positive)) && + (!arg2_pt.info(info_flags::integer) || arg2_pt.info(info_flags::positive))) + throw do_taylor(); // caught by function::series() + // trap the case where arg1 is on a pole: + if (arg1.info(info_flags::integer) && !arg1.info(info_flags::positive)) + arg1_ser = tgamma(arg1+s).series(rel, order, options); + else + arg1_ser = tgamma(arg1).series(rel,order); + // trap the case where arg2 is on a pole: + if (arg2.info(info_flags::integer) && !arg2.info(info_flags::positive)) + arg2_ser = tgamma(arg2+s).series(rel, order, options); + else + arg2_ser = tgamma(arg2).series(rel,order); + // trap the case where arg1+arg2 is on a pole: + if ((arg1+arg2).info(info_flags::integer) && !(arg1+arg2).info(info_flags::positive)) + arg1arg2_ser = tgamma(arg2+arg1+s).series(rel, order, options); + else + arg1arg2_ser = tgamma(arg2+arg1).series(rel,order); + // compose the result (expanding all the terms): + return (arg1_ser*arg2_ser/arg1arg2_ser).series(rel, order, options).expand(); +} + -static ex psi1_evalf(ex const & x) +REGISTER_FUNCTION(beta, eval_func(beta_eval). + evalf_func(beta_evalf). + derivative_func(beta_deriv). + series_func(beta_series). + latex_name("\\mbox{B}"). + set_symmetry(sy_symm(0, 1))); + + +////////// +// Psi-function (aka digamma-function) +////////// + +static ex psi1_evalf(const ex & x) { - BEGIN_TYPECHECK - TYPECHECK(x,numeric) - END_TYPECHECK(psi(x)) - - return psi(ex_to_numeric(x)); + if (is_exactly_a(x)) { + try { + return psi(ex_to(x)); + } catch (const dunno &e) { } + } + + return psi(x).hold(); } -static ex psi1_diff(ex const & x, unsigned diff_param) +/** Evaluation of digamma-function psi(x). + * Somebody ought to provide some good numerical evaluation some day... */ +static ex psi1_eval(const ex & x) { - GINAC_ASSERT(diff_param==0); - - return psi(exONE(), x); + if (x.info(info_flags::numeric)) { + const numeric &nx = ex_to(x); + if (nx.is_integer()) { + // integer case + if (nx.is_positive()) { + // psi(n) -> 1 + 1/2 +...+ 1/(n-1) - Euler + numeric rat = 0; + for (numeric i(nx+_num_1); i>0; --i) + rat += i.inverse(); + return rat-Euler; + } else { + // for non-positive integers there is a pole: + throw (pole_error("psi_eval(): simple pole",1)); + } + } + if ((_num2*nx).is_integer()) { + // half integer case + if (nx.is_positive()) { + // psi((2m+1)/2) -> 2/(2m+1) + 2/2m +...+ 2/1 - Euler - 2log(2) + numeric rat = 0; + for (numeric i = (nx+_num_1)*_num2; i>0; i-=_num2) + rat += _num2*i.inverse(); + return rat-Euler-_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(x).hold(); } -static ex psi1_series(ex const & x, symbol const & s, ex const & point, int order) +static ex psi1_deriv(const ex & x, unsigned deriv_param) { - throw(std::logic_error("Nobody told me how to series expand the psi function. :-(")); + GINAC_ASSERT(deriv_param==0); + + // d/dx psi(x) -> psi(1,x) + return psi(_ex1, x); } -unsigned function_index_psi1 = function::register_new("psi", psi1_eval, psi1_evalf, psi1_diff, psi1_series); +static ex psi1_series(const ex & arg, + const relational & rel, + int order, + unsigned options) +{ + // 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 arg_pt = arg.subs(rel, subs_options::no_pattern); + if (!arg_pt.info(info_flags::integer) || arg_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: + const numeric m = -ex_to(arg_pt); + ex recur; + for (numeric p; p<=m; ++p) + recur += power(arg+p,_ex_1); + return (psi(arg+m+_ex1)-recur).series(rel, order, options); +} + +unsigned psi1_SERIAL::serial = + function::register_new(function_options("psi"). + eval_func(psi1_eval). + evalf_func(psi1_evalf). + derivative_func(psi1_deriv). + series_func(psi1_series). + latex_name("\\psi"). + overloaded(2)); ////////// // Psi-functions (aka polygamma-functions) psi(0,x)==psi(x) ////////// +static ex psi2_evalf(const ex & n, const ex & x) +{ + if (is_exactly_a(n) && is_exactly_a(x)) { + try { + return psi(ex_to(n),ex_to(x)); + } catch (const dunno &e) { } + } + + return psi(n,x).hold(); +} + /** Evaluation of polygamma-function psi(n,x). * Somebody ought to provide some good numerical evaluation some day... */ -static ex psi2_eval(ex const & n, ex const & x) -{ - // psi(0,x) -> psi(x) - if (n.is_zero()) - return psi(x).hold(); - if (n.info(info_flags::numeric) && x.info(info_flags::numeric)) { - // do some stuff... - } - return psi(n, x).hold(); +static ex psi2_eval(const ex & n, const ex & x) +{ + // psi(0,x) -> psi(x) + if (n.is_zero()) + return psi(x); + // psi(-1,x) -> log(tgamma(x)) + if (n.is_equal(_ex_1)) + return log(tgamma(x)); + if (n.info(info_flags::numeric) && n.info(info_flags::posint) && + x.info(info_flags::numeric)) { + const numeric &nn = ex_to(n); + const numeric &nx = ex_to(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 psi2_diff(ex const & n, ex const & x, unsigned diff_param) +static ex psi2_series(const ex & n, + const ex & arg, + const relational & rel, + int order, + unsigned options) { - GINAC_ASSERT(diff_param<2); - - if (diff_param==0) { - // d/dn psi(n,x) - throw(std::logic_error("cannot diff psi(n,x) with respect to n")); - } - // d/dx psi(n,x) - return psi(n+1, x); + // 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 arg_pt = arg.subs(rel, subs_options::no_pattern); + if (!arg_pt.info(info_flags::integer) || arg_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: + const numeric m = -ex_to(arg_pt); + ex recur; + for (numeric p; p<=m; ++p) + recur += power(arg+p,-n+_ex_1); + recur *= factorial(n)*power(_ex_1,n); + return (psi(n, arg+m+_ex1)-recur).series(rel, order, options); } -static ex psi2_series(ex const & n, ex const & x, symbol const & s, ex const & point, int order) -{ - throw(std::logic_error("Nobody told me how to series expand the psi functions. :-(")); -} +unsigned psi2_SERIAL::serial = + function::register_new(function_options("psi"). + eval_func(psi2_eval). + evalf_func(psi2_evalf). + derivative_func(psi2_deriv). + series_func(psi2_series). + latex_name("\\psi"). + overloaded(2)); -unsigned function_index_psi2 = function::register_new("psi", psi2_eval, psi2_evalf, psi2_diff, psi2_series); } // namespace GiNaC