X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Finifcns_gamma.cpp;h=f3d44a81e462325f5c6818ac29ba07a641903fab;hp=34a390ed23e23e1b412609399f926328596570cf;hb=073bf40a73e419a3dbcb6dfa190947ce2cc3bdce;hpb=487e5659efe401683eee0381b0d23f967ffffc3c diff --git a/ginac/inifcns_gamma.cpp b/ginac/inifcns_gamma.cpp index 34a390ed..f3d44a81 100644 --- a/ginac/inifcns_gamma.cpp +++ b/ginac/inifcns_gamma.cpp @@ -1,9 +1,10 @@ /** @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 + * GiNaC Copyright (C) 1999-2011 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 @@ -17,87 +18,572 @@ * * 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 "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" + +#include +#include + +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. + * Handles integer arguments as a special case. * - * @exception fail_numeric("complex_infinity") or something similar... */ -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 * power(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 * power(Pi,numHALF()); - } - } - } - return gamma(x).hold(); -} - -ex gamma_evalf(ex const & x) + * @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)); + } + if (!ex_to(x).is_rational()) + return lgamma(ex_to(x)); + } + + return lgamma(x).hold(); +} + + +static ex lgamma_deriv(const ex & x, unsigned deriv_param) +{ + GINAC_ASSERT(deriv_param==0); + + // d/dx lgamma(x) -> psi(x) + return psi(x); +} + + +static ex lgamma_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 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 lgamma_conjugate(const ex & x) +{ + // conjugate(lgamma(x))==lgamma(conjugate(x)) unless on the branch cut + // which runs along the negative real axis. + if (x.info(info_flags::positive)) { + return lgamma(x); + } + if (is_exactly_a(x) && + !x.imag_part().is_zero()) { + return lgamma(x.conjugate()); + } + return conjugate_function(lgamma(x)).hold(); +} + + +REGISTER_FUNCTION(lgamma, eval_func(lgamma_eval). + evalf_func(lgamma_evalf). + derivative_func(lgamma_deriv). + series_func(lgamma_series). + conjugate_func(lgamma_conjugate). + latex_name("\\log \\Gamma")); + + +////////// +// true Gamma function +////////// + +static ex tgamma_evalf(const ex & x) +{ + if (is_exactly_a(x)) { + try { + return tgamma(ex_to(x)); + } catch (const dunno &e) { } + } + + return tgamma(x).hold(); +} + + +/** 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_p)*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_p)); + } 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_p); + return (doublefactorial(n.mul(*_num2_p).sub(*_num1_p)).div(pow(*_num2_p,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_p)); + return (pow(*_num_2_p, n).div(doublefactorial(n.mul(*_num2_p).sub(*_num1_p))))*sqrt(Pi); + } + } + if (!ex_to(x).is_rational()) + return tgamma(ex_to(x)); + } + + 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); + 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, options); +} + + +static ex tgamma_conjugate(const ex & x) { - BEGIN_TYPECHECK - TYPECHECK(x,numeric) - END_TYPECHECK(gamma(x)) - - return gamma(ex_to_numeric(x)); + // conjugate(tgamma(x))==tgamma(conjugate(x)) + return tgamma(x.conjugate()); } -ex gamma_diff(ex const & x, unsigned diff_param) + +REGISTER_FUNCTION(tgamma, eval_func(tgamma_eval). + evalf_func(tgamma_evalf). + derivative_func(tgamma_deriv). + series_func(tgamma_series). + conjugate_func(tgamma_conjugate). + latex_name("\\Gamma")); + + +////////// +// beta-function +////////// + +static ex beta_evalf(const ex & x, const ex & y) +{ + if (is_exactly_a(x) && is_exactly_a(y)) { + try { + return exp(lgamma(ex_to(x))+lgamma(ex_to(y))-lgamma(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_p, 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_p, 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; + if (!ex_to(x).is_rational() || !ex_to(x).is_rational()) + return evalf(beta(x, y).hold()); + } + + 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); + else + arg1_ser = tgamma(arg1); + // 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); + else + arg2_ser = tgamma(arg2); + // 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); + else + arg1arg2_ser = tgamma(arg2+arg1); + // compose the result (expanding all the terms): + return (arg1_ser*arg2_ser/arg1arg2_ser).series(rel, order, options).expand(); +} + + +REGISTER_FUNCTION(beta, eval_func(beta_eval). + evalf_func(beta_evalf). + derivative_func(beta_deriv). + series_func(beta_series). + latex_name("\\mathrm{B}"). + set_symmetry(sy_symm(0, 1))); + + +////////// +// Psi-function (aka digamma-function) +////////// + +static ex psi1_evalf(const ex & x) { - ASSERT(diff_param==0); + if (is_exactly_a(x)) { + try { + return psi(ex_to(x)); + } catch (const dunno &e) { } + } + + return psi(x).hold(); +} - return power(x, -1); //!! +/** Evaluation of digamma-function psi(x). + * Somebody ought to provide some good numerical evaluation some day... */ +static ex psi1_eval(const ex & 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_p)); 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_p)*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_p))*(*_num2_p); i>0; i-=(*_num2_p)) + rat += (*_num2_p)*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_p); + return recur+psi(_ex1_2); + } + } + // psi1_evalf should be called here once it becomes available + } + + return psi(x).hold(); +} + +static ex psi1_deriv(const ex & x, unsigned deriv_param) +{ + GINAC_ASSERT(deriv_param==0); + + // d/dx psi(x) -> psi(1,x) + return psi(_ex1, x); } -ex gamma_series(ex const & x, symbol const & s, ex const & point, int order) +static ex psi1_series(const ex & arg, + const relational & rel, + int order, + unsigned options) { - //!! Only handle one special case for now... - if (x.is_equal(s) && point.is_zero()) { - ex e = 1 / s - EulerGamma + s * (power(Pi, 2) / 12 + power(EulerGamma, 2) / 2) + Order(power(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 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); } -REGISTER_FUNCTION(gamma, gamma_eval, gamma_evalf, gamma_diff, gamma_series); +unsigned psi1_SERIAL::serial = + function::register_new(function_options("psi", 1). + 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(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_p)) + // use psi(n,1) == (-)^(n+1) * n! * zeta(n+1) + return pow(*_num_1_p,nn+(*_num1_p))*factorial(nn)*zeta(ex(nn+(*_num1_p))); + 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_series(const ex & n, + 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(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); +} + +unsigned psi2_SERIAL::serial = + function::register_new(function_options("psi", 2). + eval_func(psi2_eval). + evalf_func(psi2_evalf). + derivative_func(psi2_deriv). + series_func(psi2_series). + latex_name("\\psi"). + overloaded(2)); + + +} // namespace GiNaC