X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Finifcns_gamma.cpp;h=213d9d7ff04ea70fd902b39e9bd4b4a282256209;hp=ecec37a603dbc208f12aeacb790a2daac18b6309;hb=b7c15cf3731d4b3d4f9bcd7b95b42982e91a69bd;hpb=26741891dadf23162799009b6fd57b4984bd4ce5 diff --git a/ginac/inifcns_gamma.cpp b/ginac/inifcns_gamma.cpp index ecec37a6..213d9d7f 100644 --- a/ginac/inifcns_gamma.cpp +++ b/ginac/inifcns_gamma.cpp @@ -4,7 +4,7 @@ * some related stuff. */ /* - * GiNaC Copyright (C) 1999 Johannes Gutenberg University Mainz, Germany + * GiNaC Copyright (C) 1999-2000 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 @@ -27,164 +27,297 @@ #include "inifcns.h" #include "ex.h" #include "constant.h" -#include "series.h" +#include "pseries.h" #include "numeric.h" #include "power.h" #include "relational.h" #include "symbol.h" #include "utils.h" -#ifndef NO_GINAC_NAMESPACE +#ifndef NO_NAMESPACE_GINAC namespace GiNaC { -#endif // ndef NO_GINAC_NAMESPACE +#endif // ndef NO_NAMESPACE_GINAC ////////// -// Gamma-function +// Logarithm of Gamma function ////////// -static ex gamma_evalf(ex const & x) +static ex lgamma_evalf(const ex & x) { BEGIN_TYPECHECK TYPECHECK(x,numeric) - END_TYPECHECK(gamma(x)) + END_TYPECHECK(lgamma(x)) - return gamma(ex_to_numeric(x)); + return lgamma(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... + +/** 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 std::domain_error("lgamma_eval(): logarithmic pole") */ +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.exadd(_ex_1()))); + } else { + throw (std::domain_error("lgamma_eval(): logarithmic pole")); + } + } + // lgamma_evalf should be called here once it becomes available + } + + 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 & x, const relational & rel, int order) +{ + // 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); + // This, however, seems to fail utterly because you run into branch-cut + // problems. Somebody ought to implement it some day using an asymptotic + // series for tgamma: + const ex x_pt = x.subs(rel); + 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 of tgamma(-m): + throw (std::domain_error("lgamma_series: please implemnt my at the poles")); + return _ex0(); // not reached +} + + +REGISTER_FUNCTION(lgamma, eval_func(lgamma_eval). + evalf_func(lgamma_evalf). + derivative_func(lgamma_deriv). + series_func(lgamma_series)); + + +////////// +// true Gamma function +////////// + +static ex tgamma_evalf(const ex & x) +{ + BEGIN_TYPECHECK + TYPECHECK(x,numeric) + END_TYPECHECK(tgamma(x)) + + return tgamma(ex_to_numeric(x)); +} + + +/** 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 std::domain_error("gamma_eval(): simple pole") */ -static ex gamma_eval(ex const & x) + * @exception std::domain_error("tgamma_eval(): simple pole") */ +static ex tgamma_eval(const ex & x) { if (x.info(info_flags::numeric)) { // trap integer arguments: if (x.info(info_flags::integer)) { - // gamma(n+1) -> n! for postitive n + // tgamma(n) -> (n-1)! for postitive n if (x.info(info_flags::posint)) { - return factorial(ex_to_numeric(x).sub(numONE())); + return factorial(ex_to_numeric(x).sub(_num1())); } else { - throw (std::domain_error("gamma_eval(): simple pole")); + throw (std::domain_error("tgamma_eval(): simple pole")); } } // 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()); + // tgamma(n+1/2) -> Pi^(1/2)*(1*3*..*(2*n-1))/(2^n) + 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) - // 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); + // tgamma(-n+1/2) -> Pi^(1/2)*(-2)^n/(1*3*..*(2*n-1)) + 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()); } } + // tgamma_evalf should be called here once it becomes available } - return gamma(x).hold(); -} + + return tgamma(x).hold(); +} + -static ex gamma_diff(ex const & x, unsigned diff_param) +static ex tgamma_deriv(const ex & x, unsigned deriv_param) { - GINAC_ASSERT(diff_param==0); + GINAC_ASSERT(deriv_param==0); - return psi(x)*gamma(x); // diff(log(gamma(x)),x)==psi(x) + // d/dx tgamma(x) -> psi(x)*tgamma(x) + return psi(x)*tgamma(x); } -static ex gamma_series(ex const & x, symbol const & s, ex const & point, int order) + +static ex tgamma_series(const ex & x, const relational & rel, int order) { // method: - // Taylor series where there is no pole falls back to psi functions. - // On a pole at -n use the identity - // series(GAMMA(x),x=-n,order) == - // series(GAMMA(x+n+1)/(x*(x+1)...*(x+n)),x=-n,order+1); - ex xpoint = x.subs(s==point); - if (!xpoint.info(info_flags::integer) || xpoint.info(info_flags::positive)) - throw do_taylor(); - // if we got here we have to care for a simple pole at -n: - numeric n = -ex_to_numeric(xpoint); - ex ser_numer = gamma(x+n+exONE()); - ex ser_denom = exONE(); - for (numeric p; p<=n; ++p) + // 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 x_pt = x.subs(rel); + 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_denom = _ex1(); + for (numeric p; p<=m; ++p) ser_denom *= x+p; - return (ser_numer/ser_denom).series(s, point, order+1); + return (tgamma(x+m+_ex1())/ser_denom).series(rel, order+1); } -REGISTER_FUNCTION(gamma, gamma_eval, gamma_evalf, gamma_diff, gamma_series); + +REGISTER_FUNCTION(tgamma, eval_func(tgamma_eval). + evalf_func(tgamma_evalf). + derivative_func(tgamma_deriv). + series_func(tgamma_series)); + ////////// -// Beta-function +// beta-function ////////// -static ex beta_evalf(ex const & x, ex const & y) +static ex beta_evalf(const ex & x, const ex & y) { BEGIN_TYPECHECK TYPECHECK(x,numeric) TYPECHECK(y,numeric) END_TYPECHECK(beta(x,y)) - return gamma(ex_to_numeric(x))*gamma(ex_to_numeric(y)) - / gamma(ex_to_numeric(x+y)); + return tgamma(ex_to_numeric(x))*tgamma(ex_to_numeric(y))/tgamma(ex_to_numeric(x+y)); } -static ex beta_eval(ex const & x, ex const & y) + +static ex beta_eval(const ex & x, const ex & y) { 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) numeric nx(ex_to_numeric(x)); numeric ny(ex_to_numeric(y)); - // 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: if (nx.is_real() && nx.is_integer() && ny.is_real() && ny.is_integer()) { if (nx.is_negative()) { if (nx<=-ny) - return numMINUSONE().power(ny)*beta(1-x-y, y); + 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 numMINUSONE().power(nx)*beta(1-y-x, x); + 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); + 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 exZERO(); - return gamma(x)*gamma(y)/gamma(x+y); + return _ex0(); + // everything is ok: + return tgamma(x)*tgamma(y)/tgamma(x+y); } + return beta(x,y).hold(); } -static ex beta_diff(ex const & x, ex const & y, unsigned diff_param) + +static ex beta_deriv(const ex & x, const ex & y, unsigned deriv_param) { - GINAC_ASSERT(diff_param<2); + GINAC_ASSERT(deriv_param<2); ex retval; - if (diff_param==0) // d/dx beta(x,y) + // 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); - if (diff_param==1) // d/dy 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; } -REGISTER_FUNCTION(beta, beta_eval, beta_evalf, beta_diff, NULL); + +static ex beta_series(const ex & x, const ex & y, const relational & rel, int order) +{ + // 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 x_pt = x.subs(rel); + const ex y_pt = y.subs(rel); + GINAC_ASSERT(is_ex_exactly_of_type(rel.lhs(),symbol)); + const symbol *s = static_cast(rel.lhs().bp); + 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 = tgamma(x+*s).series(rel,order); + else + x_ser = tgamma(x).series(rel,order); + // trap the case where y is on a pole directly: + if (y.info(info_flags::integer) && !y.info(info_flags::positive)) + y_ser = tgamma(y+*s).series(rel,order); + else + y_ser = tgamma(y).series(rel,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 = tgamma(y+x+*s).series(rel,order); + else + xy_ser = tgamma(y+x).series(rel,order); + // compose the result: + return (x_ser*y_ser/xy_ser).series(rel,order); +} + + +REGISTER_FUNCTION(beta, eval_func(beta_eval). + evalf_func(beta_evalf). + derivative_func(beta_deriv). + series_func(beta_series)); + ////////// -// Psi-function (aka polygamma-function) +// Psi-function (aka digamma-function) ////////// -static ex psi1_evalf(ex const & x) +static ex psi1_evalf(const ex & x) { BEGIN_TYPECHECK TYPECHECK(x,numeric) @@ -193,50 +326,93 @@ static ex psi1_evalf(ex const & x) return psi(ex_to_numeric(x)); } -/** Evaluation of polygamma-function psi(x). +/** Evaluation of digamma-function psi(x). * Somebody ought to provide some good numerical evaluation some day... */ -static ex psi1_eval(ex const & x) +static ex psi1_eval(const ex & x) { if (x.info(info_flags::numeric)) { - if (x.info(info_flags::integer) && !x.info(info_flags::positive)) - throw (std::domain_error("psi_eval(): simple pole")); - if (x.info(info_flags::positive)) { - // psi(n) -> 1 + 1/2 +...+ 1/(n-1) - EulerGamma - if (x.info(info_flags::integer)) { + numeric nx = ex_to_numeric(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(ex_to_numeric(x)-numONE()); i.is_positive(); --i) + for (numeric i(nx+_num_1()); i.is_positive(); --i) rat += i.inverse(); - return rat-EulerGamma; + return rat-Euler; + } else { + // for non-positive integers there is a pole: + throw (std::domain_error("psi_eval(): simple pole")); } - // psi((2m+1)/2) -> 2/(2m+1) + 2/2m +...+ 2/1 - EulerGamma - 2log(2) - if ((exTWO()*x).info(info_flags::integer)) { + } + 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((ex_to_numeric(x)-numONE())*numTWO()); i.is_positive(); i-=numTWO()) - rat += numTWO()*i.inverse(); - return rat-EulerGamma-exTWO()*log(exTWO()); - } - if (x.compare(exONE())==1) { - // should call numeric, since >1 + for (numeric i((nx+_num_1())*_num2()); i.is_positive(); 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_diff(ex const & x, unsigned diff_param) +static ex psi1_deriv(const ex & x, unsigned deriv_param) { - GINAC_ASSERT(diff_param==0); + GINAC_ASSERT(deriv_param==0); - return psi(exONE(), x); + // d/dx psi(x) -> psi(1,x) + return psi(_ex1(), x); +} + +static ex psi1_series(const ex & x, const relational & rel, 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(rel); + 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(rel, order); } -const unsigned function_index_psi1 = function::register_new("psi", psi1_eval, psi1_evalf, psi1_diff, NULL); +const unsigned function_index_psi1 = + function::register_new(function_options("psi"). + eval_func(psi1_eval). + evalf_func(psi1_evalf). + derivative_func(psi1_deriv). + series_func(psi1_series). + overloaded(2)); ////////// // Psi-functions (aka polygamma-functions) psi(0,x)==psi(x) ////////// -static ex psi2_evalf(ex const & n, ex const & x) +static ex psi2_evalf(const ex & n, const ex & x) { BEGIN_TYPECHECK TYPECHECK(n,numeric) @@ -248,38 +424,113 @@ static ex psi2_evalf(ex const & n, ex const & x) /** 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) +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(gamma(x)) - if (n.is_equal(exMINUSONE())) - return log(gamma(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)) { numeric nn = ex_to_numeric(n); numeric nx = ex_to_numeric(x); - if (x.is_equal(exONE())) - return numMINUSONE().power(nn+numONE())*factorial(nn)*zeta(ex(nn+numONE())); + 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); } -const unsigned function_index_psi2 = function::register_new("psi", psi2_eval, psi2_evalf, psi2_diff, NULL); +static ex psi2_series(const ex & n, const ex & x, const relational & rel, 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(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(rel); + 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(rel, order); +} + +const unsigned function_index_psi2 = + function::register_new(function_options("psi"). + eval_func(psi2_eval). + evalf_func(psi2_evalf). + derivative_func(psi2_deriv). + series_func(psi2_series). + overloaded(2)); + -#ifndef NO_GINAC_NAMESPACE +#ifndef NO_NAMESPACE_GINAC } // namespace GiNaC -#endif // ndef NO_GINAC_NAMESPACE +#endif // ndef NO_NAMESPACE_GINAC