normal.java

一个用于排队系统仿真的开源软件,有非常形象的图象仿真过程!

/**
 * Class contains the implementation of:
 * - Inverse Normal Cummulative Distribution Function Algorythm
 * - Error Function Algorythm
 * - Complimentary Error Function Algorythm
 *
 * @author Sherali Karimov ([e-mail])
 */
package jmt.engine.math;

public class Normal {
/* ********************************************
 * Original algorythm and Perl implementation can
 * be found at:
 * http://www.math.uio.no/~jacklam/notes/invnorm/index.html
 * Author:
 *  Peter J. Acklam
 *  [e-mail]
 * ****************************************** */
	private static final double P_LOW = 0.02425D;
	private static final double P_HIGH = 1.0D - P_LOW;

	// Coefficients in rational approximations.
	private static final double ICDF_A[] = {-3.969683028665376e+01, 2.209460984245205e+02, -2.759285104469687e+02, 1.383577518672690e+02, -3.066479806614716e+01, 2.506628277459239e+00};

	private static final double ICDF_B[] = {-5.447609879822406e+01, 1.615858368580409e+02, -1.556989798598866e+02, 6.680131188771972e+01, -1.328068155288572e+01};

	private static final double ICDF_C[] = {-7.784894002430293e-03, -3.223964580411365e-01, -2.400758277161838e+00, -2.549732539343734e+00, 4.374664141464968e+00, 2.938163982698783e+00};

	private static final double ICDF_D[] = {7.784695709041462e-03, 3.224671290700398e-01, 2.445134137142996e+00, 3.754408661907416e+00};

	/* computs the inverse Comulative Distribution function for the normal
	 * distribution.
	 * @param d
	 * @param highPrecision set false
	 */
	public static double getInvCDF(double d, boolean highPrecision) {
		// Define break-points.
		// variable for result
		double z = 0;

		if (d == 0)
			z = Double.NEGATIVE_INFINITY;
		else if (d == 1)
			z = Double.POSITIVE_INFINITY;
		else if (Double.isNaN(d) || d < 0 || d > 1)
			z = Double.NaN;

		// Rational approximation for lower region:
		else if (d < P_LOW) {
			double q = Math.sqrt(-2 * Math.log(d));
			z = (((((ICDF_C[0] * q + ICDF_C[1]) * q + ICDF_C[2]) * q + ICDF_C[3]) * q + ICDF_C[4]) * q + ICDF_C[5]) / ((((ICDF_D[0] * q + ICDF_D[1]) * q + ICDF_D[2]) * q + ICDF_D[3]) * q + 1);
		}

		// Rational approximation for upper region:
		else if (P_HIGH < d) {
			double q = Math.sqrt(-2 * Math.log(1 - d));
			z = -(((((ICDF_C[0] * q + ICDF_C[1]) * q + ICDF_C[2]) * q + ICDF_C[3]) * q + ICDF_C[4]) * q + ICDF_C[5]) / ((((ICDF_D[0] * q + ICDF_D[1]) * q + ICDF_D[2]) * q + ICDF_D[3]) * q + 1);
		}
		// Rational approximation for central region:
		else {
			double q = d - 0.5D;
			double r = q * q;
			z = (((((ICDF_A[0] * r + ICDF_A[1]) * r + ICDF_A[2]) * r + ICDF_A[3]) * r + ICDF_A[4]) * r + ICDF_A[5]) * q / (((((ICDF_B[0] * r + ICDF_B[1]) * r + ICDF_B[2]) * r + ICDF_B[3]) * r + ICDF_B[4]) * r + 1);
		}
		if (highPrecision) z = refine(z, d);
		return z;
	}

//C------------------------------------------------------------------
//C  Coefficients for approximation to  erf  in first interval
//C------------------------------------------------------------------
	private static final double ERF_A[] = {3.16112374387056560E00, 1.13864154151050156E02, 3.77485237685302021E02, 3.20937758913846947E03, 1.85777706184603153E-1};

	private static final double ERF_B[] = {2.36012909523441209E01, 2.44024637934444173E02, 1.28261652607737228E03, 2.84423683343917062E03};

//C------------------------------------------------------------------
//C  Coefficients for approximation to  erfc  in second interval
//C------------------------------------------------------------------
	private static final double ERF_C[] = {5.64188496988670089E-1, 8.88314979438837594E0, 6.61191906371416295E01, 2.98635138197400131E02, 8.81952221241769090E02, 1.71204761263407058E03, 2.05107837782607147E03, 1.23033935479799725E03, 2.15311535474403846E-8};

	private static final double ERF_D[] = {1.57449261107098347E01, 1.17693950891312499E02, 5.37181101862009858E02, 1.62138957456669019E03, 3.29079923573345963E03, 4.36261909014324716E03, 3.43936767414372164E03, 1.23033935480374942E03};

//C------------------------------------------------------------------
//C  Coefficients for approximation to  erfc  in third interval
//C------------------------------------------------------------------
	private static final double ERF_P[] = {3.05326634961232344E-1, 3.60344899949804439E-1, 1.25781726111229246E-1, 1.60837851487422766E-2, 6.58749161529837803E-4, 1.63153871373020978E-2};

	private static final double ERF_Q[] = {2.56852019228982242E00, 1.87295284992346047E00, 5.27905102951428412E-1, 6.05183413124413191E-2, 2.33520497626869185E-3};

	private static final double PI_SQRT = Math.sqrt(Math.PI);
	private static final double THRESHOLD = 0.46875D;

/* **************************************
 * Hardware dependant constants were calculated
 * on Dell "Dimension 4100":
 * - Pentium III 800 MHz
 * running Microsoft Windows 2000
 * ************************************* */
	private static final double X_MIN = Double.MIN_VALUE;
	private static final double X_INF = Double.MAX_VALUE;
	private static final double X_NEG = -9.38241396824444;
	private static final double X_SMALL = 1.110223024625156663E-16;
	private static final double X_BIG = 9.194E0;
	private static final double X_HUGE = 1.0D / (2.0D * Math.sqrt(X_SMALL));
	private static final double X_MAX = Math.min(X_INF, (1 / (Math.sqrt(Math.PI) * X_MIN)));

	private static double calerf(double X, int jint) {
/* ******************************************
 * ORIGINAL FORTRAN version can be found at:
 * http://www.netlib.org/specfun/erf
 ********************************************
C------------------------------------------------------------------
C
C THIS PACKET COMPUTES THE ERROR AND COMPLEMENTARY ERROR FUNCTIONS
C   FOR REAL ARGUMENTS  ARG.  IT CONTAINS TWO FUNCTION TYPE
C   SUBPROGRAMS,  ERF  AND  ERFC  (OR  DERF  AND  DERFC),  AND ONE
C   SUBROUTINE TYPE SUBPROGRAM,  CALERF.  THE CALLING STATEMENTS
C   FOR THE PRIMARY ENTRIES ARE
C
C                   Y=ERF(X)     (OR   Y=DERF(X) )
C   AND
C                   Y=ERFC(X)    (OR   Y=DERFC(X) ).
C
C   THE ROUTINE  CALERF  IS INTENDED FOR INTERNAL PACKET USE ONLY,
C   LEVEL_ALL COMPUTATIONS WITHIN THE PACKET BEING CONCENTRATED IN THIS
C   ROUTINE.  THE FUNCTION SUBPROGRAMS INVOKE  CALERF  WITH THE
C   STATEMENT
C          CALL CALERF(ARG,RESULT,JINT)
C   WHERE THE PARAMETER USAGE IS AS FOLLOWS
C
C      FUNCTION                     PARAMETERS FOR CALERF
C       CALL              ARG                  RESULT          JINT
C     ERF(ARG)      ANY REAL ARGUMENT         ERF(ARG)          0
C     ERFC(ARG)     ABS(ARG) .LT. XMAX        ERFC(ARG)         1
C
C   THE MAIN COMPUTATION EVALUATES NEAR MINIMAX APPROXIMATIONS
C   FROM "RATIONAL CHEBYSHEV APPROXIMATIONS FOR THE ERROR FUNCTION"
C   BY W. J. CODY, MATH. COMP., 1969, PP. 631-638.  THIS
C   TRANSPORTABLE PROGRAM USES RATIONAL FUNCTIONS THAT THEORETICALLY
C       APPROXIMATE  ERF(X)  AND  ERFC(X)  TO AT LEAST 18 SIGNIFICANT
C   DECIMAL DIGITS.  THE ACCURACY ACHIEVED DEPENDS ON THE ARITHMETIC
C   SYSTEM, THE COMPILER, THE INTRINSIC FUNCTIONS, AND PROPER
C   SELECTION OF THE MACHINE-DEPENDENT CONSTANTS.
C
C  AUTHOR: W. J. CODY
C          MATHEMATICS AND COMPUTER SCIENCE DIVISION
C          ARGONNE NATIONAL LABORATORY
C          ARGONNE, IL 60439
C
C  LATEST MODIFICATION: JANUARY 8, 1985
C
C------------------------------------------------------------------
*/
		double result = 0;
		double Y = Math.abs(X);
		double Y_SQ, X_NUM, X_DEN;

		if (Y <= THRESHOLD) {
			Y_SQ = 0.0D;
			if (Y > X_SMALL) Y_SQ = Y * Y;
			X_NUM = ERF_A[4] * Y_SQ;
			X_DEN = Y_SQ;
			for (int i = 0; i < 3; i++) {
				X_NUM = (X_NUM + ERF_A[i]) * Y_SQ;
				X_DEN = (X_DEN + ERF_B[i]) * Y_SQ;
			}
			result = X * (X_NUM + ERF_A[3]) / (X_DEN + ERF_B[3]);
			if (jint != 0) result = 1 - result;
			if (jint == 2) result = Math.exp(Y_SQ) * result;
			return result;
		} else if (Y <= 4.0D) {
			X_NUM = ERF_C[8] * Y;
			X_DEN = Y;
			for (int i = 0; i < 7; i++) {
				X_NUM = (X_NUM + ERF_C[i]) * Y;
				X_DEN = (X_DEN + ERF_D[i]) * Y;
			}
			result = (X_NUM + ERF_C[7]) / (X_DEN + ERF_D[7]);
			if (jint != 2) {
				Y_SQ = Math.round(Y * 16.0D) / 16.0D;
				double del = (Y - Y_SQ) * (Y + Y_SQ);
				result = Math.exp(-Y_SQ * Y_SQ) * Math.exp(-del) * result;
			}
		} else {
			result = 0.0D;
			if (Y >= X_BIG && (jint != 2 || Y >= X_MAX))
				;
			else if (Y >= X_BIG && Y >= X_HUGE)
				result = PI_SQRT / Y;
			else {
				Y_SQ = 1.0D / (Y * Y);
				X_NUM = ERF_P[5] * Y_SQ;
				X_DEN = Y_SQ;
				for (int i = 0; i < 4; i++) {
					X_NUM = (X_NUM + ERF_P[i]) * Y_SQ;
					X_DEN = (X_DEN + ERF_Q[i]) * Y_SQ;
				}
				result = Y_SQ * (X_NUM + ERF_P[4]) / (X_DEN + ERF_Q[4]);
				result = (PI_SQRT - result) / Y;
				if (jint != 2) {
					Y_SQ = Math.round(Y * 16.0D) / 16.0D;
					double del = (Y - Y_SQ) * (Y + Y_SQ);
					result = Math.exp(-Y_SQ * Y_SQ) * Math.exp(-del) * result;
				}
			}
		}

		if (jint == 0) {
			result = (0.5D - result) + 0.5D;
			if (X < 0) result = -result;
		} else if (jint == 1) {
			if (X < 0) result = 2.0D - result;
		} else {
			if (X < 0) {
				if (X < X_NEG)
					result = X_INF;
				else {
					Y_SQ = Math.round(X * 16.0D) / 16.0D;
					double del = (X - Y_SQ) * (X + Y_SQ);
					Y = Math.exp(Y_SQ * Y_SQ) * Math.exp(del);
					result = (Y + Y) - result;
				}
			}
		}
		return result;
	}

	public static double erf(double d) {
		return calerf(d, 0);
	}

	public static double erfc(double d) {
		return calerf(d, 1);
	}

	public static double erfcx(double d) {
		return calerf(d, 2);
	}

/* ****************************************************
 * Refining algorytm is based on Halley rational method
 * for finding roots of equations as described at:
 * http://www.math.uio.no/~jacklam/notes/invnorm/index.html
 * by:
 *  Peter J. Acklam
 *  [e-mail]
 * ************************************************** */
	public static double refine(double x, double d) {
		if (d > 0 && d < 1) {
			double e = 0.5D * erfc(-x / Math.sqrt(2.0D)) - d;
			double u = e * Math.sqrt(2.0D * Math.PI) * Math.exp((x * x) / 2.0D);
			x = x - u / (1.0D + x * u / 2.0D);
		}
		return x;
	}
}