// 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 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"));
}
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_evalf(ex const & x)
-{
- BEGIN_TYPECHECK
- TYPECHECK(x,numeric)
- END_TYPECHECK(gamma(x))
-
- return gamma(ex_to_numeric(x));
-}
-
-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);
// Beta-function
//////////
-static ex beta_eval(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));
+}
+
+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"));
}
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_evalf(ex const & x, ex const & 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 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)
//////////
-/** Evaluation of polygamma-function psi(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 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_evalf(ex const & x)
+static ex psi1_diff(const ex & x, unsigned diff_param)
{
- BEGIN_TYPECHECK
- TYPECHECK(x,numeric)
- END_TYPECHECK(psi(x))
+ GINAC_ASSERT(diff_param==0);
- return psi(ex_to_numeric(x));
+ // d/dx psi(x) -> psi(1,x)
+ return psi(_ex1(), x);
}
-static ex psi1_diff(ex const & x, unsigned diff_param)
+static ex psi1_series(const ex & x, const symbol & s, const ex & pt, int order)
{
- GINAC_ASSERT(diff_param==0);
-
- return psi(exONE(), 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(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, NULL);
+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)
+ TYPECHECK(x,numeric)
+ END_TYPECHECK(psi(n,x))
+
+ return psi(ex_to_numeric(n), ex_to_numeric(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<nx; ++p)
+ recur += pow(p, -nn+_num_1());
+ recur *= factorial(nn)*pow(_num_1(), nn);
+ return recur+psi(n,_ex1());
+ } else {
+ // for non-positive integers there is a pole:
+ throw (std::domain_error("psi2_eval(): pole"));
+ }
+ }
+ if ((_num2()*nx).is_integer()) {
+ // half integer case
+ if (nx.is_equal(_num1_2()))
+ // use psi(n,1/2) == (-)^(n+1) * n! * (2^(n+1)-1) * zeta(n+1)
+ return pow(_num_1(),nn+_num1())*factorial(nn)*(pow(_num2(),nn+_num1()) + _num_1())*zeta(ex(nn+_num1()));
+ if (nx.is_positive()) {
+ numeric m = nx - _num1_2();
+ // use the multiplication formula
+ // psi(n,2*m) == (psi(n,m) + psi(n,m+1/2)) / 2^(n+1)
+ // to revert to positive integer case
+ return psi(n,_num2()*m)*pow(_num2(),nn+_num1())-psi(n,m);
+ } else {
+ // use the recurrence relation
+ // psi(n,-m-1/2) == psi(n,-m-1/2+1) - (-)^n * n! / (-m-1/2)^(n+1)
+ // to relate psi(n,-m-1/2) to psi(n,1/2):
+ // psi(n,-m-1/2) == psi(n,1/2) + r
+ // where r == (-)^(n+1) * n! * ((-1/2)^(-n-1) + ... + (-m-1/2)^(-n-1))
+ numeric recur(0);
+ for (numeric p(nx); p<0; ++p)
+ recur += pow(p, -nn+_num_1());
+ recur *= factorial(nn)*pow(_num_1(), nn+_num_1());
+ return recur+psi(n,_ex1_2());
+ }
+ }
+ // psi2_evalf should be called here once it becomes available
}
+
return psi(n, x).hold();
}
-static ex psi2_evalf(ex const & n, ex const & x)
-{
- BEGIN_TYPECHECK
- TYPECHECK(n,numeric)
- TYPECHECK(x,numeric)
- END_TYPECHECK(psi(n,x))
-
- return psi(ex_to_numeric(n), ex_to_numeric(x));
-}
-
-static ex psi2_diff(ex const & n, ex const & x, unsigned diff_param)
+static ex psi2_diff(const ex & n, const ex & x, unsigned diff_param)
{
GINAC_ASSERT(diff_param<2);
// 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);
+ // d/dx psi(n,x) -> 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
git://www.ginac.de / ginac.git / blobdiff