- speedup by declaring x_pt and so on const

[ginac.git] / ginac / inifcns_gamma.cpp

diff --git a/ginac/inifcns_gamma.cpp b/ginac/inifcns_gamma.cpp
index 7bd6669bb8b5eb8c817fd016718c5e3bfdaa2546..8ed385150e8175d15df02931ecd5c8fd8796c5cc 100644 (file)
--- a/ginac/inifcns_gamma.cpp
+++ b/ginac/inifcns_gamma.cpp
@@ -1,6 +1,7 @@
 /** @file inifcns_gamma.cpp
  *
 /** @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 Johannes Gutenberg University Mainz, Germany


@@ -26,99 +27,297 @@
 #include "inifcns.h"
 #include "ex.h"
 #include "constant.h"
 #include "inifcns.h"
 #include "ex.h"
 #include "constant.h"

+#include "series.h"

 #include "numeric.h"
 #include "power.h"
 #include "numeric.h"
 #include "power.h"

+#include "relational.h"

 #include "symbol.h"
 #include "symbol.h"

+#include "utils.h"

 
 

+#ifndef NO_GINAC_NAMESPACE

 namespace GiNaC {
 namespace GiNaC {

+#endif // ndef NO_GINAC_NAMESPACE

 
 //////////
 
 //////////

-// gamma function
+// 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...
  *
 /** 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 fail_numeric("complex_infinity") or something similar... */
-static ex gamma_eval(ex const & x)
+ *  @exception std::domain_error("gamma_eval(): simple pole") */
+static ex gamma_eval(const ex & x)

 {
     if (x.info(info_flags::numeric)) {
 {
     if (x.info(info_flags::numeric)) {

-

         // trap integer arguments:
         // trap integer arguments:

-        if ( x.info(info_flags::integer) ) {
+        if (x.info(info_flags::integer)) {

             // gamma(n+1) -> n! for postitive n
             // gamma(n+1) -> n! for postitive n

-            if ( x.info(info_flags::posint) ) {
-                return factorial(ex_to_numeric(x).sub(numONE()));
+            if (x.info(info_flags::posint)) {
+                return factorial(ex_to_numeric(x).sub(_num1()));

             } else {
             } else {

-                return numZERO();  // Infinity. Throw? What?
+                throw (std::domain_error("gamma_eval(): simple pole"));

             }
         }
         // trap half integer arguments:
             }
         }
         // trap half integer arguments:

-        if ( (x*2).info(info_flags::integer) ) {
-            // trap positive x=(n+1/2)
+        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)
             // 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 {
             } else {

-                // trap negative x=(-n+1/2)
+                // trap negative x==(-n+1/2)

                 // gamma(-n+1/2) -> Pi^(1/2)*(-2)^n/(1*3*..*(2*n-1))
                 // 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();
 }    
     return gamma(x).hold();
 }    

+
+static ex gamma_diff(const ex & x, unsigned diff_param)
+{
+    GINAC_ASSERT(diff_param==0);

     
     

-static ex gamma_evalf(ex const & 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(const ex & x, const symbol & s, const ex & pt, int order)
+{
+    // method:
+    // 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, pt, order+1);
+}
+
+REGISTER_FUNCTION(gamma, gamma_eval, gamma_evalf, gamma_diff, gamma_series);
+
+//////////
+// Beta-function
+//////////
+
+static ex beta_evalf(const ex & x, const ex & y)

 {
     BEGIN_TYPECHECK
         TYPECHECK(x,numeric)
 {
     BEGIN_TYPECHECK
         TYPECHECK(x,numeric)

-    END_TYPECHECK(gamma(x))
+        TYPECHECK(y,numeric)
+    END_TYPECHECK(beta(x,y))

     
     

-    return gamma(ex_to_numeric(x));
+    return gamma(ex_to_numeric(x))*gamma(ex_to_numeric(y))
+        / gamma(ex_to_numeric(x+y));

 }
 
 }
 

-static ex gamma_diff(ex const & x, unsigned diff_param)
+static ex beta_eval(const ex & x, const ex & y)

 {
 {

-    ASSERT(diff_param==0);
+    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));
+        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 (std::domain_error("beta_eval(): simple pole"));
+            }
+            if (ny.is_negative()) {
+                if (ny<=-nx)
+                    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);
+        }
+        // no problem in numerator, but denominator has pole:
+        if ((nx+ny).is_real() &&
+            (nx+ny).is_integer() &&
+            !(nx+ny).is_positive())
+             return _ex0();
+        // everything is ok:
+        return gamma(x)*gamma(y)/gamma(x+y);
+    }
+    
+    return beta(x,y).hold();
+}

 
 

-    return psi(exZERO(),x)*gamma(x);
+static ex beta_diff(const ex & x, const ex & y, unsigned diff_param)
+{
+    GINAC_ASSERT(diff_param<2);
+    ex retval;
+    
+    // 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);
+    // 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;

 }
 
 }
 

-static ex gamma_series(ex const & x, symbol const & s, ex const & point, int order)
+static ex beta_series(const ex & x, const ex & y, const symbol & s, const ex & pt, int order)

 {
 {

-       // 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"));
+    // 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(gamma, gamma_eval, gamma_evalf, gamma_diff, gamma_series);
+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(n,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... */
  *  Somebody ought to provide some good numerical evaluation some day... */

-static ex psi_eval(ex const & n, ex const & x)
+static ex psi1_eval(const ex & x)

 {
 {

-    if (n.info(info_flags::numeric) && x.info(info_flags::numeric)) {
-        // do some stuff...
+    if (x.info(info_flags::numeric)) {
+        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(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"));
+            }
+        }
+        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((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(n, x).hold();
-}    

     
     

-static ex psi_evalf(ex const & n, ex const & x)
+    return psi(x).hold();
+}
+
+static ex psi1_diff(const ex & x, unsigned diff_param)
+{
+    GINAC_ASSERT(diff_param==0);
+    
+    // d/dx psi(x) -> psi(1,x)
+    return psi(_ex1(), x);
+}
+
+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(const ex & n, const ex & x)

 {
     BEGIN_TYPECHECK
         TYPECHECK(n,numeric)
 {
     BEGIN_TYPECHECK
         TYPECHECK(n,numeric)


@@ -128,18 +327,108 @@ static ex psi_evalf(ex const & n, ex const & x)
     return psi(ex_to_numeric(n), ex_to_numeric(x));
 }
 
     return psi(ex_to_numeric(n), ex_to_numeric(x));
 }
 

-static ex psi_diff(ex const & n, ex const & x, unsigned diff_param)
+/** 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)

 {
 {

-    ASSERT(diff_param==0);
+    // psi(0,x) -> psi(x)
+    if (n.is_zero())
+        return psi(x);
+    // psi(-1,x) -> log(gamma(x))
+    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 (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+1, x);
+    return psi(n, x).hold();
+}    
+
+static ex psi2_diff(const ex & n, const ex & x, unsigned diff_param)
+{
+    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) -> psi(n+1,x)
+    return psi(n+_ex1(), x);

 }
 
 }
 

-static ex psi_series(ex const & n, ex const & x, symbol const & s, ex const & point, int order)
+static ex psi2_series(const ex & n, const ex & x, const symbol & s, const ex & pt, int order)

 {
 {

-    throw(std::logic_error("Nobody told me how to series expand the psi 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(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);

 }
 
 }
 

-REGISTER_FUNCTION(psi, psi_eval, psi_evalf, psi_diff, psi_series);
+const unsigned function_index_psi2 = function::register_new("psi", psi2_eval, psi2_evalf, psi2_diff, psi2_series);

 
 

+#ifndef NO_GINAC_NAMESPACE

 } // namespace GiNaC
 } // namespace GiNaC

+#endif // ndef NO_GINAC_NAMESPACE