package graph; import java.lang.Math; import java.lang.ArithmeticException; /* ************************************************************************** ** ** Class SpecialFunction ** ************************************************************************** ** Copyright (C) 1996 Leigh Brookshaw ** ** 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 ** the Free Software Foundation; either version 2 of the License, or ** (at your option) any later version. ** ** This program is distributed in the hope that it will be useful, ** but WITHOUT ANY WARRANTY; without even the implied warranty of ** MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ** GNU General Public License for more details. ** ** 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., 675 Mass Ave, Cambridge, MA 02139, USA. ************************************************************************** ** ** This class is an extension of java.lang.Math. It includes a number ** of special functions not found in the Math class. ** *************************************************************************/ /** * This class contains physical constants and special functions not found * in the java.lang.Math class. * Like the java.lang.Math class this class is final and cannot be * subclassed. * All physical constants are in cgs units. *

* NOTE: These special functions do not necessarily use the fastest * or most accurate algorithms. * * @version $Revision: 1.7 $, $Date: 1996/08/19 06:04:15 $ * @author Leigh Brookshaw */ public final class SpecialFunction extends Object { /* ** machine constants */ private static final double MACHEP = 1.11022302462515654042E-16; private static final double MAXLOG = 7.09782712893383996732E2; private static final double MINLOG = -7.451332191019412076235E2; private static final double MAXGAM = 171.624376956302725; private static final double SQTPI = 2.50662827463100050242E0; private static final double SQRTH = 7.07106781186547524401E-1; private static final double LOGPI = 1.14472988584940017414; /* ** Physical Constants in cgs Units */ /** * Boltzman Constant. Units erg/deg(K) */ public static final double BOLTZMAN = 1.3807e-16; /** * Elementary Charge. Units statcoulomb */ public static final double ECHARGE = 4.8032e-10; /** * Electron Mass. Units g */ public static final double EMASS = 9.1095e-28; /** * Proton Mass. Units g */ public static final double PMASS = 1.6726e-24; /** * Gravitational Constant. Units dyne-cm^2/g^2 */ public static final double GRAV = 6.6720e-08; /** * Planck constant. Units erg-sec */ public static final double PLANCK = 6.6262e-27; /** * Speed of Light in a Vacuum. Units cm/sec */ public static final double LIGHTSPEED = 2.9979e10; /** * Stefan-Boltzman Constant. Units erg/cm^2-sec-deg^4 */ public static final double STEFANBOLTZ = 5.6703e-5; /** * Avogadro Number. Units 1/mol */ public static final double AVOGADRO = 6.0220e23; /** * Gas Constant. Units erg/deg-mol */ public static final double GASCONSTANT = 8.3144e07; /** * Gravitational Acceleration at the Earths surface. Units cm/sec^2 */ public static final double GRAVACC = 980.67; /** * Solar Mass. Units g */ public static final double SOLARMASS = 1.99e33; /** * Solar Radius. Units cm */ public static final double SOLARRADIUS = 6.96e10; /** * Solar Luminosity. Units erg/sec */ public static final double SOLARLUM = 3.90e33; /** * Solar Flux. Units erg/cm^2-sec */ public static final double SOLARFLUX = 6.41e10; /** * Astronomical Unit (radius of the Earth's orbit). Units cm */ public static final double AU = 1.50e13; /** * Don't let anyone instantiate this class. */ private SpecialFunction() {} /* ** Function Methods */ /** * @param x a double value * @return The log₁₀ */ static public double log10(double x) throws ArithmeticException { if( x <= 0.0 ) throw new ArithmeticException("range exception"); return Math.log(x)/2.30258509299404568401; } /** * @param x a double value * @return the hyperbolic cosine of the argument */ static public double cosh(double x) throws ArithmeticException { double a; a = x; if( a < 0.0 ) a = Math.abs(x); a = Math.exp(a); return 0.5*(a+1/a); } /** * @param x a double value * @return the hyperbolic sine of the argument */ static public double sinh(double x) throws ArithmeticException { double a; if(x == 0.0) return x; a = x; if( a < 0.0 ) a = Math.abs(x); a = Math.exp(a); if( x < 0.0 ) return -0.5*(a-1/a); else return 0.5*(a-1/a); } /** * @param x a double value * @return the hyperbolic tangent of the argument */ static public double tanh(double x) throws ArithmeticException { double a; if( x == 0.0 ) return x; a = x; if( a < 0.0 ) a = Math.abs(x); a = Math.exp(2.0*a); if(x < 0.0 ) return -( 1.0-2.0/(a+1.0) ); else return ( 1.0-2.0/(a+1.0) ); } /** * @param x a double value * @return the hyperbolic arc cosine of the argument */ static public double acosh(double x) throws ArithmeticException { if( x < 1.0 ) throw new ArithmeticException("range exception"); return Math.log( x + Math.sqrt(x*x-1)); } /** * @param x a double value * @return the hyperbolic arc sine of the argument */ static public double asinh(double xx) throws ArithmeticException { double x; int sign; if(xx == 0.0) return xx; if( xx < 0.0 ) { sign = -1; x = -xx; } else { sign = 1; x = xx; } return sign*Math.log( x + Math.sqrt(x*x+1)); } /** * @param x a double value * @return the hyperbolic arc tangent of the argument */ static public double atanh(double x) throws ArithmeticException { if( x > 1.0 || x < -1.0 ) throw new ArithmeticException("range exception"); return 0.5 * Math.log( (1.0+x)/(1.0-x) ); } /** * @param x a double value * @return the Bessel function of order 0 of the argument. */ static public double j0(double x) throws ArithmeticException { double ax; if( (ax=Math.abs(x)) < 8.0 ) { double y=x*x; double ans1=57568490574.0+y*(-13362590354.0+y*(651619640.7 +y*(-11214424.18+y*(77392.33017+y*(-184.9052456))))); double ans2=57568490411.0+y*(1029532985.0+y*(9494680.718 +y*(59272.64853+y*(267.8532712+y*1.0)))); return ans1/ans2; } else { double z=8.0/ax; double y=z*z; double xx=ax-0.785398164; double ans1=1.0+y*(-0.1098628627e-2+y*(0.2734510407e-4 +y*(-0.2073370639e-5+y*0.2093887211e-6))); double ans2 = -0.1562499995e-1+y*(0.1430488765e-3 +y*(-0.6911147651e-5+y*(0.7621095161e-6 -y*0.934935152e-7))); return Math.sqrt(0.636619772/ax)* (Math.cos(xx)*ans1-z*Math.sin(xx)*ans2); } } /** * @param x a double value * @return the Bessel function of order 1 of the argument. */ static public double j1(double x) throws ArithmeticException { double ax; double y; double ans1, ans2; if ((ax=Math.abs(x)) < 8.0) { y=x*x; ans1=x*(72362614232.0+y*(-7895059235.0+y*(242396853.1 +y*(-2972611.439+y*(15704.48260+y*(-30.16036606)))))); ans2=144725228442.0+y*(2300535178.0+y*(18583304.74 +y*(99447.43394+y*(376.9991397+y*1.0)))); return ans1/ans2; } else { double z=8.0/ax; double xx=ax-2.356194491; y=z*z; ans1=1.0+y*(0.183105e-2+y*(-0.3516396496e-4 +y*(0.2457520174e-5+y*(-0.240337019e-6)))); ans2=0.04687499995+y*(-0.2002690873e-3 +y*(0.8449199096e-5+y*(-0.88228987e-6 +y*0.105787412e-6))); double ans=Math.sqrt(0.636619772/ax)* (Math.cos(xx)*ans1-z*Math.sin(xx)*ans2); if (x < 0.0) ans = -ans; return ans; } } /** * @param n integer order * @param x a double value * @return the Bessel function of order n of the argument. */ static public double jn(int n, double x) throws ArithmeticException { int j,m; double ax,bj,bjm,bjp,sum,tox,ans; boolean jsum; double ACC = 40.0; double BIGNO = 1.0e+10; double BIGNI = 1.0e-10; if(n == 0) return j0(x); if(n == 1) return j1(x); ax=Math.abs(x); if(ax == 0.0) return 0.0; else if (ax > (double)n) { tox=2.0/ax; bjm=j0(ax); bj=j1(ax); for (j=1;j0;j--) { bjm=j*tox*bj-bjp; bjp=bj; bj=bjm; if (Math.abs(bj) > BIGNO) { bj *= BIGNI; bjp *= BIGNI; ans *= BIGNI; sum *= BIGNI; } if (jsum) sum += bj; jsum=!jsum; if (j == n) ans=bjp; } sum=2.0*sum-bj; ans /= sum; } return x < 0.0 && n%2 == 1 ? -ans : ans; } /** * @param x a double value * @return the Bessel function of the second kind, * of order 0 of the argument. */ static public double y0(double x) throws ArithmeticException { if (x < 8.0) { double y=x*x; double ans1 = -2957821389.0+y*(7062834065.0+y*(-512359803.6 +y*(10879881.29+y*(-86327.92757+y*228.4622733)))); double ans2=40076544269.0+y*(745249964.8+y*(7189466.438 +y*(47447.26470+y*(226.1030244+y*1.0)))); return (ans1/ans2)+0.636619772*j0(x)*Math.log(x); } else { double z=8.0/x; double y=z*z; double xx=x-0.785398164; double ans1=1.0+y*(-0.1098628627e-2+y*(0.2734510407e-4 +y*(-0.2073370639e-5+y*0.2093887211e-6))); double ans2 = -0.1562499995e-1+y*(0.1430488765e-3 +y*(-0.6911147651e-5+y*(0.7621095161e-6 +y*(-0.934945152e-7)))); return Math.sqrt(0.636619772/x)* (Math.sin(xx)*ans1+z*Math.cos(xx)*ans2); } } /** * @param x a double value * @return the Bessel function of the second kind, * of order 1 of the argument. */ static public double y1(double x) throws ArithmeticException { if (x < 8.0) { double y=x*x; double ans1=x*(-0.4900604943e13+y*(0.1275274390e13 +y*(-0.5153438139e11+y*(0.7349264551e9 +y*(-0.4237922726e7+y*0.8511937935e4))))); double ans2=0.2499580570e14+y*(0.4244419664e12 +y*(0.3733650367e10+y*(0.2245904002e8 +y*(0.1020426050e6+y*(0.3549632885e3+y))))); return (ans1/ans2)+0.636619772*(j1(x)*Math.log(x)-1.0/x); } else { double z=8.0/x; double y=z*z; double xx=x-2.356194491; double ans1=1.0+y*(0.183105e-2+y*(-0.3516396496e-4 +y*(0.2457520174e-5+y*(-0.240337019e-6)))); double ans2=0.04687499995+y*(-0.2002690873e-3 +y*(0.8449199096e-5+y*(-0.88228987e-6 +y*0.105787412e-6))); return Math.sqrt(0.636619772/x)* (Math.sin(xx)*ans1+z*Math.cos(xx)*ans2); } } /** * @param n integer order * @param x a double value * @return the Bessel function of the second kind, * of order n of the argument. */ static public double yn(int n, double x) throws ArithmeticException { double by,bym,byp,tox; if(n == 0) return y0(x); if(n == 1) return y1(x); tox=2.0/x; by=y1(x); bym=y0(x); for (int j=1;j 1) { d *= i--; } if(j < 0) return -d; else return d; } /** * @param x a double value * @return the Gamma function of the value. *

* * Converted to Java from
* Cephes Math Library Release 2.2: July, 1992
* Copyright 1984, 1987, 1989, 1992 by Stephen L. Moshier
* Direct inquiries to 30 Frost Street, Cambridge, MA 02140
**/ static public double gamma(double x) throws ArithmeticException { double P[] = { 1.60119522476751861407E-4, 1.19135147006586384913E-3, 1.04213797561761569935E-2, 4.76367800457137231464E-2, 2.07448227648435975150E-1, 4.94214826801497100753E-1, 9.99999999999999996796E-1 }; double Q[] = { -2.31581873324120129819E-5, 5.39605580493303397842E-4, -4.45641913851797240494E-3, 1.18139785222060435552E-2, 3.58236398605498653373E-2, -2.34591795718243348568E-1, 7.14304917030273074085E-2, 1.00000000000000000320E0 }; double MAXGAM = 171.624376956302725; double LOGPI = 1.14472988584940017414; double p, z; int i; double q = Math.abs(x); if( q > 33.0 ) { if( x < 0.0 ) { p = Math.floor(q); if( p == q ) throw new ArithmeticException("gamma: overflow"); i = (int)p; z = q - p; if( z > 0.5 ) { p += 1.0; z = q - p; } z = q * Math.sin( Math.PI * z ); if( z == 0.0 ) throw new ArithmeticException("gamma: overflow"); z = Math.abs(z); z = Math.PI/(z * stirf(q) ); return -z; } else { return stirf(x); } } z = 1.0; while( x >= 3.0 ) { x -= 1.0; z *= x; } while( x < 0.0 ) { if( x == 0.0 ) { throw new ArithmeticException("gamma: singular"); } else if( x > -1.E-9 ) { return( z/((1.0 + 0.5772156649015329 * x) * x) ); } z /= x; x += 1.0; } while( x < 2.0 ) { if( x == 0.0 ) { throw new ArithmeticException("gamma: singular"); } else if( x < 1.e-9 ) { return( z/((1.0 + 0.5772156649015329 * x) * x) ); } z /= x; x += 1.0; } if( (x == 2.0) || (x == 3.0) ) return z; x -= 2.0; p = polevl( x, P, 6 ); q = polevl( x, Q, 7 ); return z * p / q; } /* Gamma function computed by Stirling's formula. * The polynomial STIR is valid for 33 <= x <= 172. Cephes Math Library Release 2.2: July, 1992 Copyright 1984, 1987, 1989, 1992 by Stephen L. Moshier Direct inquiries to 30 Frost Street, Cambridge, MA 02140 */ static private double stirf(double x) throws ArithmeticException { double STIR[] = { 7.87311395793093628397E-4, -2.29549961613378126380E-4, -2.68132617805781232825E-3, 3.47222221605458667310E-3, 8.33333333333482257126E-2, }; double MAXSTIR = 143.01608; double w = 1.0/x; double y = Math.exp(x); w = 1.0 + w * polevl( w, STIR, 4 ); if( x > MAXSTIR ) { /* Avoid overflow in Math.pow() */ double v = Math.pow( x, 0.5 * x - 0.25 ); y = v * (v / y); } else { y = Math.pow( x, x - 0.5 ) / y; } y = SQTPI * y * w; return y; } /** * @param a double value * @param x double value * @return the Complemented Incomplete Gamma function. *

* * Converted to Java from
* Cephes Math Library Release 2.2: July, 1992
* Copyright 1984, 1987, 1989, 1992 by Stephen L. Moshier
* Direct inquiries to 30 Frost Street, Cambridge, MA 02140
**/ static public double igam(double a, double x) throws ArithmeticException { double ans, ax, c, r; if( x <= 0 || a <= 0 ) return 0.0; if( x > 1.0 && x > a ) return 1.0 - igamc(a,x); /* Compute x**a * exp(-x) / gamma(a) */ ax = a * Math.log(x) - x - lgamma(a); if( ax < -MAXLOG ) return( 0.0 ); ax = Math.exp(ax); /* power series */ r = a; c = 1.0; ans = 1.0; do { r += 1.0; c *= x/r; ans += c; } while( c/ans > MACHEP ); return( ans * ax/a ); } /** * Returns the area under the left hand tail (from 0 to x) * of the Chi square probability density function with * v degrees of freedom. * * @param df degrees of freedom * @param x double value * @return the Chi-Square function. **/ static public double chisq(double df, double x) throws ArithmeticException { if( x < 0.0 || df < 1.0 ) return 0.0; return igam( df/2.0, x/2.0 ); } /** * Returns the area under the right hand tail (from x to * infinity) of the Chi square probability density function * with v degrees of freedom: * * @param df degrees of freedom * @param x double value * @return the Chi-Square function. *

**/ static public double chisqc(double df, double x) throws ArithmeticException { if( x < 0.0 || df < 1.0 ) return 0.0; return igamc( df/2.0, x/2.0 ); } /** * Returns the sum of the first k terms of the Poisson * distribution. * @param k number of terms * @param x double value */ static public double poisson(int k, double x) throws ArithmeticException { if( k < 0 || x < 0 ) return 0.0; return igamc((double)(k+1) ,x); } /** * Returns the sum of the terms k+1 to infinity of the Poisson * distribution. * @param k start * @param x double value */ static public double poissonc(int k, double x) throws ArithmeticException { if( k < 0 || x < 0 ) return 0.0; return igam((double)(k+1),x); } /** * @param a double value * @return The area under the Gaussian probability density * function, integrated from minus infinity to x: */ static public double normal( double a) throws ArithmeticException { double x, y, z; x = a * SQRTH; z = Math.abs(x); if( z < SQRTH ) y = 0.5 + 0.5 * erf(x); else { y = 0.5 * erfc(z); if( x > 0 ) y = 1.0 - y; } return y; } /** * @param a double value * @return The complementary Error function *

* * Converted to Java from
* Cephes Math Library Release 2.2: July, 1992
* Copyright 1984, 1987, 1989, 1992 by Stephen L. Moshier
* Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/ static public double erf(double x) throws ArithmeticException { double y, z; double T[] = { 9.60497373987051638749E0, 9.00260197203842689217E1, 2.23200534594684319226E3, 7.00332514112805075473E3, 5.55923013010394962768E4 }; double U[] = { //1.00000000000000000000E0, 3.35617141647503099647E1, 5.21357949780152679795E2, 4.59432382970980127987E3, 2.26290000613890934246E4, 4.92673942608635921086E4 }; if( Math.abs(x) > 1.0 ) return( 1.0 - erfc(x) ); z = x * x; y = x * polevl( z, T, 4 ) / p1evl( z, U, 5 ); return y; } static private double polevl( double x, double coef[], int N ) throws ArithmeticException { double ans; ans = coef[0]; for(int i=1; i<=N; i++) { ans = ans*x+coef[i]; } return ans; } static private double p1evl( double x, double coef[], int N ) throws ArithmeticException { double ans; ans = x + coef[0]; for(int i=1; i 0.5 ) { p += 1.0; z = p - q; } z = q * Math.sin( Math.PI * z ); if( z == 0.0 ) throw new ArithmeticException("lgamma: Overflow"); z = LOGPI - Math.log( z ) - w; return z; } if( x < 13.0 ) { z = 1.0; while( x >= 3.0 ) { x -= 1.0; z *= x; } while( x < 2.0 ) { if( x == 0.0 ) throw new ArithmeticException("lgamma: Overflow"); z /= x; x += 1.0; } if( z < 0.0 ) z = -z; if( x == 2.0 ) return Math.log(z); x -= 2.0; p = x * polevl( x, B, 5 ) / p1evl( x, C, 6); return( Math.log(z) + p ); } if( x > 2.556348e305 ) throw new ArithmeticException("lgamma: Overflow"); q = ( x - 0.5 ) * Math.log(x) - x + 0.91893853320467274178; if( x > 1.0e8 ) return( q ); p = 1.0/(x*x); if( x >= 1000.0 ) q += (( 7.9365079365079365079365e-4 * p - 2.7777777777777777777778e-3) *p + 0.0833333333333333333333) / x; else q += polevl( p, A, 4 ) / x; return q; } /** * @param aa double value * @param bb double value * @param xx double value * @return The Incomplete Beta Function evaluated from zero to xx. *

* * Converted to Java from
* Cephes Math Library Release 2.3: July, 1995
* Copyright 1984, 1995 by Stephen L. Moshier
* Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/ public static double ibeta( double aa, double bb, double xx ) throws ArithmeticException { double a, b, t, x, xc, w, y; boolean flag; if( aa <= 0.0 || bb <= 0.0 ) throw new ArithmeticException("ibeta: Domain error!"); if( (xx <= 0.0) || ( xx >= 1.0) ) { if( xx == 0.0 ) return 0.0; if( xx == 1.0 ) return 1.0; throw new ArithmeticException("ibeta: Domain error!"); } flag = false; if( (bb * xx) <= 1.0 && xx <= 0.95) { t = pseries(aa, bb, xx); return t; } w = 1.0 - xx; /* Reverse a and b if x is greater than the mean. */ if( xx > (aa/(aa+bb)) ) { flag = true; a = bb; b = aa; xc = xx; x = w; } else { a = aa; b = bb; xc = w; x = xx; } if( flag && (b * x) <= 1.0 && x <= 0.95) { t = pseries(a, b, x); if( t <= MACHEP ) t = 1.0 - MACHEP; else t = 1.0 - t; return t; } /* Choose expansion for better convergence. */ y = x * (a+b-2.0) - (a-1.0); if( y < 0.0 ) w = incbcf( a, b, x ); else w = incbd( a, b, x ) / xc; /* Multiply w by the factor a b _ _ _ x (1-x) | (a+b) / ( a | (a) | (b) ) . */ y = a * Math.log(x); t = b * Math.log(xc); if( (a+b) < MAXGAM && Math.abs(y) < MAXLOG && Math.abs(t) < MAXLOG ) { t = Math.pow(xc,b); t *= Math.pow(x,a); t /= a; t *= w; t *= gamma(a+b) / (gamma(a) * gamma(b)); if( flag ) { if( t <= MACHEP ) t = 1.0 - MACHEP; else t = 1.0 - t; } return t; } /* Resort to logarithms. */ y += t + lgamma(a+b) - lgamma(a) - lgamma(b); y += Math.log(w/a); if( y < MINLOG ) t = 0.0; else t = Math.exp(y); if( flag ) { if( t <= MACHEP ) t = 1.0 - MACHEP; else t = 1.0 - t; } return t; } /* Continued fraction expansion #1 * for incomplete beta integral */ private static double incbcf( double a, double b, double x ) throws ArithmeticException { double xk, pk, pkm1, pkm2, qk, qkm1, qkm2; double k1, k2, k3, k4, k5, k6, k7, k8; double r, t, ans, thresh; int n; double big = 4.503599627370496e15; double biginv = 2.22044604925031308085e-16; k1 = a; k2 = a + b; k3 = a; k4 = a + 1.0; k5 = 1.0; k6 = b - 1.0; k7 = k4; k8 = a + 2.0; pkm2 = 0.0; qkm2 = 1.0; pkm1 = 1.0; qkm1 = 1.0; ans = 1.0; r = 1.0; n = 0; thresh = 3.0 * MACHEP; do { xk = -( x * k1 * k2 )/( k3 * k4 ); pk = pkm1 + pkm2 * xk; qk = qkm1 + qkm2 * xk; pkm2 = pkm1; pkm1 = pk; qkm2 = qkm1; qkm1 = qk; xk = ( x * k5 * k6 )/( k7 * k8 ); pk = pkm1 + pkm2 * xk; qk = qkm1 + qkm2 * xk; pkm2 = pkm1; pkm1 = pk; qkm2 = qkm1; qkm1 = qk; if( qk != 0 ) r = pk/qk; if( r != 0 ) { t = Math.abs( (ans - r)/r ); ans = r; } else t = 1.0; if( t < thresh ) return ans; k1 += 1.0; k2 += 1.0; k3 += 2.0; k4 += 2.0; k5 += 1.0; k6 -= 1.0; k7 += 2.0; k8 += 2.0; if( (Math.abs(qk) + Math.abs(pk)) > big ) { pkm2 *= biginv; pkm1 *= biginv; qkm2 *= biginv; qkm1 *= biginv; } if( (Math.abs(qk) < biginv) || (Math.abs(pk) < biginv) ) { pkm2 *= big; pkm1 *= big; qkm2 *= big; qkm1 *= big; } } while( ++n < 300 ); return ans; } /* Continued fraction expansion #2 * for incomplete beta integral */ static private double incbd( double a, double b, double x ) throws ArithmeticException { double xk, pk, pkm1, pkm2, qk, qkm1, qkm2; double k1, k2, k3, k4, k5, k6, k7, k8; double r, t, ans, z, thresh; int n; double big = 4.503599627370496e15; double biginv = 2.22044604925031308085e-16; k1 = a; k2 = b - 1.0; k3 = a; k4 = a + 1.0; k5 = 1.0; k6 = a + b; k7 = a + 1.0;; k8 = a + 2.0; pkm2 = 0.0; qkm2 = 1.0; pkm1 = 1.0; qkm1 = 1.0; z = x / (1.0-x); ans = 1.0; r = 1.0; n = 0; thresh = 3.0 * MACHEP; do { xk = -( z * k1 * k2 )/( k3 * k4 ); pk = pkm1 + pkm2 * xk; qk = qkm1 + qkm2 * xk; pkm2 = pkm1; pkm1 = pk; qkm2 = qkm1; qkm1 = qk; xk = ( z * k5 * k6 )/( k7 * k8 ); pk = pkm1 + pkm2 * xk; qk = qkm1 + qkm2 * xk; pkm2 = pkm1; pkm1 = pk; qkm2 = qkm1; qkm1 = qk; if( qk != 0 ) r = pk/qk; if( r != 0 ) { t = Math.abs( (ans - r)/r ); ans = r; } else t = 1.0; if( t < thresh ) return ans; k1 += 1.0; k2 -= 1.0; k3 += 2.0; k4 += 2.0; k5 += 1.0; k6 += 1.0; k7 += 2.0; k8 += 2.0; if( (Math.abs(qk) + Math.abs(pk)) > big ) { pkm2 *= biginv; pkm1 *= biginv; qkm2 *= biginv; qkm1 *= biginv; } if( (Math.abs(qk) < biginv) || (Math.abs(pk) < biginv) ) { pkm2 *= big; pkm1 *= big; qkm2 *= big; qkm1 *= big; } } while( ++n < 300 ); return ans; } /* Power series for incomplete beta integral. Use when b*x is small and x not too close to 1. */ static private double pseries( double a, double b, double x ) throws ArithmeticException { double s, t, u, v, n, t1, z, ai; ai = 1.0 / a; u = (1.0 - b) * x; v = u / (a + 1.0); t1 = v; t = u; n = 2.0; s = 0.0; z = MACHEP * ai; while( Math.abs(v) > z ) { u = (n - b) * x / n; t *= u; v = t / (a + n); s += v; n += 1.0; } s += t1; s += ai; u = a * Math.log(x); if( (a+b) < MAXGAM && Math.abs(u) < MAXLOG ) { t = gamma(a+b)/(gamma(a)*gamma(b)); s = s * t * Math.pow(x,a); } else { t = lgamma(a+b) - lgamma(a) - lgamma(b) + u + Math.log(s); if( t < MINLOG ) s = 0.0; else s = Math.exp(t); } return s; } }