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=ecec37a603dbc208f12aeacb790a2daac18b6309;hb=a6bb52b00bf185271774e7d56215923700a3ec40;hpb=26741891dadf23162799009b6fd57b4984bd4ce5;ds=sidebyside diff --git a/ginac/inifcns_gamma.cpp b/ginac/inifcns_gamma.cpp index ecec37a6..8ed38515 100644 --- a/ginac/inifcns_gamma.cpp +++ b/ginac/inifcns_gamma.cpp @@ -42,7 +42,7 @@ namespace GiNaC { // Gamma-function ////////// -static ex gamma_evalf(ex const & x) +static ex gamma_evalf(const ex & x) { BEGIN_TYPECHECK TYPECHECK(x,numeric) @@ -56,14 +56,14 @@ static ex gamma_evalf(ex const & x) * evaluation some day... * * @exception std::domain_error("gamma_eval(): simple pole") */ -static ex gamma_eval(ex const & x) +static ex gamma_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 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")); } @@ -72,48 +72,55 @@ static ex gamma_eval(ex const & x) 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) // 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(ex const & x, unsigned diff_param) +static ex gamma_diff(const ex & x, unsigned diff_param) { GINAC_ASSERT(diff_param==0); - return psi(x)*gamma(x); // diff(log(gamma(x)),x)==psi(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(ex const & x, symbol const & s, ex const & point, int order) +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 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 + // 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, point, order+1); + return (ser_numer/ser_denom).series(s, pt, order+1); } REGISTER_FUNCTION(gamma, gamma_eval, gamma_evalf, gamma_diff, gamma_series); @@ -122,7 +129,7 @@ REGISTER_FUNCTION(gamma, gamma_eval, gamma_evalf, gamma_diff, gamma_series); // 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) @@ -133,24 +140,25 @@ static ex beta_evalf(ex const & x, ex const & y) / gamma(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 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)); - // 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")); } @@ -160,31 +168,67 @@ static ex beta_eval(ex const & x, ex const & y) if ((nx+ny).is_real() && (nx+ny).is_integer() && !(nx+ny).is_positive()) - return exZERO(); + return _ex0(); + // everything is ok: return gamma(x)*gamma(y)/gamma(x+y); } + return beta(x,y).hold(); } -static ex beta_diff(ex const & x, ex const & y, unsigned diff_param) +static ex beta_diff(const ex & x, const ex & y, unsigned diff_param) { GINAC_ASSERT(diff_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 (diff_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 (diff_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 symbol & s, const ex & pt, int order) +{ + // 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(beta, beta_eval, beta_evalf, beta_diff, 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 +237,87 @@ 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) - EulerGamma 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; + } 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 - EulerGamma - 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-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(x).hold(); } -static ex psi1_diff(ex const & x, unsigned diff_param) +static ex psi1_diff(const ex & x, unsigned diff_param) { GINAC_ASSERT(diff_param==0); - return psi(exONE(), x); + // d/dx psi(x) -> psi(1,x) + return psi(_ex1(), x); } -const unsigned function_index_psi1 = function::register_new("psi", psi1_eval, psi1_evalf, psi1_diff, NULL); +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(ex const & n, ex const & x) +static ex psi2_evalf(const ex & n, const ex & x) { BEGIN_TYPECHECK TYPECHECK(n,numeric) @@ -248,25 +329,70 @@ 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())) + 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 (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); +} + +static ex psi2_series(const ex & n, 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(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); } -const unsigned function_index_psi2 = function::register_new("psi", psi2_eval, psi2_evalf, psi2_diff, NULL); +const unsigned function_index_psi2 = function::register_new("psi", psi2_eval, psi2_evalf, psi2_diff, psi2_series); #ifndef NO_GINAC_NAMESPACE } // namespace GiNaC