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eva2/src/wsi/ra/math/SpecialFunction.java
Marcel Kronfeld 7ae15be788 Importing release version 322 from old repos
2007-12-11 16:38:11 +00:00

1365 lines
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Java
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package wsi.ra.math;
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.
* <P>
* <B>NOTE:</B> These special functions do not necessarily use the fastest
* or most accurate algorithms.
*
* @version $Revision: 1.1.1.1 $, $Date: 2003/07/03 14:59:40 $
* @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 log<sub>10</sub>
*/
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;j<n;j++) {
bjp=j*tox*bj-bjm;
bjm=bj;
bj=bjp;
}
ans=bj;
} else {
tox=2.0/ax;
m=2*((n+(int)Math.sqrt(ACC*n))/2);
jsum=false;
bjp=ans=sum=0.0;
bj=1.0;
for (j=m;j>0;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<n;j++) {
byp=j*tox*by-bym;
bym=by;
by=byp;
}
return by;
}
/**
* @param x a double value
* @return the factorial of the argument
*/
static public double fac(double x) throws ArithmeticException {
double d = Math.abs(x);
if(Math.floor(d) == d) return (double)fac( (int)x );
else return gamma(x+1.0);
}
/**
* @param x an integer value
* @return the factorial of the argument
*/
static public int fac(int j) throws ArithmeticException {
int i = j;
int d = 1;
if(j < 0) i = Math.abs(j);
while( i > 1) { d *= i--; }
if(j < 0) return -d;
else return d;
}
/**
* @param x a double value
* @return the Gamma function of the value.
* <P>
* <FONT size=2>
* Converted to Java from<BR>
* Cephes Math Library Release 2.2: July, 1992<BR>
* Copyright 1984, 1987, 1989, 1992 by Stephen L. Moshier<BR>
* Direct inquiries to 30 Frost Street, Cambridge, MA 02140<BR>
**/
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.
* <P>
* <FONT size=2>
* Converted to Java from<BR>
* Cephes Math Library Release 2.2: July, 1992<BR>
* Copyright 1984, 1987, 1989, 1992 by Stephen L. Moshier<BR>
* Direct inquiries to 30 Frost Street, Cambridge, MA 02140<BR>
**/
static public double igamc( double a, double x )
throws ArithmeticException {
double big = 4.503599627370496e15;
double biginv = 2.22044604925031308085e-16;
double ans, ax, c, yc, r, t, y, z;
double pk, pkm1, pkm2, qk, qkm1, qkm2;
if( x <= 0 || a <= 0 ) return 1.0;
if( x < 1.0 || x < a ) return 1.0 - igam(a,x);
ax = a * Math.log(x) - x - lgamma(a);
if( ax < -MAXLOG ) return 0.0;
ax = Math.exp(ax);
/* continued fraction */
y = 1.0 - a;
z = x + y + 1.0;
c = 0.0;
pkm2 = 1.0;
qkm2 = x;
pkm1 = x + 1.0;
qkm1 = z * x;
ans = pkm1/qkm1;
do {
c += 1.0;
y += 1.0;
z += 2.0;
yc = y * c;
pk = pkm1 * z - pkm2 * yc;
qk = qkm1 * z - qkm2 * yc;
if( qk != 0 ) {
r = pk/qk;
t = Math.abs( (ans - r)/r );
ans = r;
} else
t = 1.0;
pkm2 = pkm1;
pkm1 = pk;
qkm2 = qkm1;
qkm1 = qk;
if( Math.abs(pk) > big ) {
pkm2 *= biginv;
pkm1 *= biginv;
qkm2 *= biginv;
qkm1 *= biginv;
}
} while( t > MACHEP );
return ans * ax;
}
/**
* @param a double value
* @param x double value
* @return the Incomplete Gamma function.
* <P>
* <FONT size=2>
* Converted to Java from<BR>
* Cephes Math Library Release 2.2: July, 1992<BR>
* Copyright 1984, 1987, 1989, 1992 by Stephen L. Moshier<BR>
* Direct inquiries to 30 Frost Street, Cambridge, MA 02140<BR>
**/
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.
* <P>
**/
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
* <P>
* <FONT size=2>
* Converted to Java from<BR>
* Cephes Math Library Release 2.2: July, 1992<BR>
* Copyright 1984, 1987, 1989, 1992 by Stephen L. Moshier<BR>
* Direct inquiries to 30 Frost Street, Cambridge, MA 02140<BR>
*/
static public double erfc(double a)
throws ArithmeticException {
double x,y,z,p,q;
double P[] = {
2.46196981473530512524E-10,
5.64189564831068821977E-1,
7.46321056442269912687E0,
4.86371970985681366614E1,
1.96520832956077098242E2,
5.26445194995477358631E2,
9.34528527171957607540E2,
1.02755188689515710272E3,
5.57535335369399327526E2
};
double Q[] = {
//1.0
1.32281951154744992508E1,
8.67072140885989742329E1,
3.54937778887819891062E2,
9.75708501743205489753E2,
1.82390916687909736289E3,
2.24633760818710981792E3,
1.65666309194161350182E3,
5.57535340817727675546E2
};
double R[] = {
5.64189583547755073984E-1,
1.27536670759978104416E0,
5.01905042251180477414E0,
6.16021097993053585195E0,
7.40974269950448939160E0,
2.97886665372100240670E0
};
double S[] = {
//1.00000000000000000000E0,
2.26052863220117276590E0,
9.39603524938001434673E0,
1.20489539808096656605E1,
1.70814450747565897222E1,
9.60896809063285878198E0,
3.36907645100081516050E0
};
if( a < 0.0 ) x = -a;
else x = a;
if( x < 1.0 ) return 1.0 - erf(a);
z = -a * a;
if( z < -MAXLOG ) {
if( a < 0 ) return( 2.0 );
else return( 0.0 );
}
z = Math.exp(z);
if( x < 8.0 ) {
p = polevl( x, P, 8 );
q = p1evl( x, Q, 8 );
} else {
p = polevl( x, R, 5 );
q = p1evl( x, S, 6 );
}
y = (z * p)/q;
if( a < 0 ) y = 2.0 - y;
if( y == 0.0 ) {
if( a < 0 ) return 2.0;
else return( 0.0 );
}
return y;
}
/**
* @param a double value
* @return The Error function
* <P>
* <FONT size=2>
* Converted to Java from<BR>
* Cephes Math Library Release 2.2: July, 1992<BR>
* Copyright 1984, 1987, 1989, 1992 by Stephen L. Moshier<BR>
* Direct inquiries to 30 Frost Street, Cambridge, MA 02140<BR>
*/
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<N; i++) { ans = ans*x+coef[i]; }
return ans;
}
/*
*
* Natural logarithm of gamma function
*
*/
/*
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 lgamma(double x)
throws ArithmeticException {
double p, q, w, z;
double A[] = {
8.11614167470508450300E-4,
-5.95061904284301438324E-4,
7.93650340457716943945E-4,
-2.77777777730099687205E-3,
8.33333333333331927722E-2
};
double B[] = {
-1.37825152569120859100E3,
-3.88016315134637840924E4,
-3.31612992738871184744E5,
-1.16237097492762307383E6,
-1.72173700820839662146E6,
-8.53555664245765465627E5
};
double C[] = {
/* 1.00000000000000000000E0, */
-3.51815701436523470549E2,
-1.70642106651881159223E4,
-2.20528590553854454839E5,
-1.13933444367982507207E6,
-2.53252307177582951285E6,
-2.01889141433532773231E6
};
if( x < -34.0 ) {
q = -x;
w = lgamma(q);
p = Math.floor(q);
if( p == q ) throw new ArithmeticException("lgam: Overflow");
z = q - p;
if( z > 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.
* <P>
* <FONT size=2>
* Converted to Java from<BR>
* Cephes Math Library Release 2.3: July, 1995<BR>
* Copyright 1984, 1995 by Stephen L. Moshier<BR>
* Direct inquiries to 30 Frost Street, Cambridge, MA 02140<BR>
*/
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;
}
public static double getnormpdf(double x) {
// k = find(sigma > 0);
// if any(k)
// xn = (x(k) - mu(k)) ./ sigma(k);
// y(k) = exp(-0.5 * xn .^2) ./ (sqrt(2*pi) .* sigma(k));
// end
double y = Math.exp(-0.5 *x *x) / Math.sqrt(2.0*Math.PI);
return y;
}
public static double getnormcdf(double x) {
//p = 0.5 * erfc(-(x-mu)./(sqrt(2)*sigma));
double p = 0.5 * erfc(-x/Math.sqrt(2));
return p;
}
}
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