      SUBROUTINE NBCDF(X,P,N,CDF)
C
C     PURPOSE--THIS SUBROUTINE COMPUTES THE CUMULATIVE DISTRIBUTION
C              FUNCTION VALUE AT THE SINGLE PRECISION VALUE X
C              FOR THE NEGATIVE BINOMIAL DISTRIBUTION
C              WITH SINGLE PRECISION 'BERNOULLI PROBABILITY'
C              PARAMETER = P, 
C              AND INTEGER 'NUMBER OF SUCCESSES IN BERNOULLI TRIALS'
C              PARAMETER = N. 
C              THE NEGATIVE BINOMIAL DISTRIBUTION USED
C              HEREIN HAS MEAN = N*(1-P)/P
C              AND STANDARD DEVIATION = SQRT(N*(1-P)/(P*P))).
C              THIS DISTRIBUTION IS DEFINED FOR
C              ALL NON-NEGATIVE INTEGER X--X = 0, 1, 2, ... .
C              THIS DISTRIBUTION HAS THE PROBABILITY FUNCTION
C              F(X) = C(N+X-1,N) * P**N * (1-P)**X.
C              WHERE C(N+X-1,N) IS THE COMBINATORIAL FUNCTION
C              EQUALING THE NUMBER OF COMBINATIONS OF N+X-1 ITEMS
C              TAKEN N AT A TIME.
C              THE NEGATIVE BINOMIAL DISTRIBUTION IS THE
C              DISTRIBUTION OF THE NUMBER OF FAILURES
C              BEFORE OBTAINING N SUCCESSES IN AN 
C              INDEFINITE SEQUENCE OF BERNOULLI (0,1)
C              TRIALS WHERE THE PROBABILITY OF SUCCESS
C              IN A SINGLE TRIAL = P.
C     INPUT  ARGUMENTS--X      = THE SINGLE PRECISION VALUE 
C                                AT WHICH THE CUMULATIVE DISTRIBUTION 
C                                FUNCTION IS TO BE EVALUATED.
C                                X SHOULD BE NON-NEGATIVE AND
C                                INTEGRAL-VALUED. 
C                     --P      = THE SINGLE PRECISION VALUE 
C                                OF THE 'BERNOULLI PROBABILITY'
C                                PARAMETER FOR THE NEGATIVE BINOMIAL
C                                DISTRIBUTION.
C                                P SHOULD BE BETWEEN
C                                0.0 (EXCLUSIVELY) AND
C                                1.0 (EXCLUSIVELY).
C                     --N      = THE INTEGER VALUE
C                                OF THE 'NUMBER OF SUCCESSES
C                                IN BERNOULLI TRIALS' PARAMETER.
C                                N SHOULD BE A POSITIVE INTEGER.
C     OUTPUT ARGUMENTS--CDF    = THE SINGLE PRECISION CUMULATIVE
C                                DISTRIBUTION FUNCTION VALUE.
C     OUTPUT--THE SINGLE PRECISION CUMULATIVE DISTRIBUTION
C             FUNCTION VALUE CDF
C             FOR THE NEGATIVE BINOMIAL DISTRIBUTION
C             WITH 'BERNOULLI PROBABILITY' PARAMETER = P
C             AND 'NUMBER OF SUCCESSES IN BERNOULLI TRIALS' 
C             PARAMETER = N.
C     PRINTING--NONE UNLESS AN INPUT ARGUMENT ERROR CONDITION EXISTS. 
C     RESTRICTIONS--X SHOULD BE NON-NEGATIVE AND INTEGRAL-VALUED.
C                 --P SHOULD BE BETWEEN 0.0 (EXCLUSIVELY)
C                   AND 1.0 (EXCLUSIVELY).
C                 --N SHOULD BE A POSITIVE INTEGER.
C     OTHER DATAPAC   SUBROUTINES NEEDED--NONE.
C     FORTRAN LIBRARY SUBROUTINES NEEDED--DSQRT, DATAN.
C     MODE OF INTERNAL OPERATIONS--DOUBLE PRECISION.
C     LANGUAGE--ANSI FORTRAN. 
C     COMMENT--NOTE THAT EVEN THOUGH THE INPUT
C              TO THIS CUMULATIVE
C              DISTRIBUTION FUNCTION SUBROUTINE
C              FOR THIS DISCRETE DISTRIBUTION
C              SHOULD (UNDER NORMAL CIRCUMSTANCES) BE A
C              DISCRETE INTEGER VALUE,
C              THE INPUT VARIABLE X IS SINGLE
C              PRECISION IN MODE.
C              X HAS BEEN SPECIFIED AS SINGLE
C              PRECISION SO AS TO CONFORM WITH THE DATAPAC
C              CONVENTION THAT ALL INPUT ****DATA****
C              (AS OPPOSED TO SAMPLE SIZE, FOR EXAMPLE)
C              VARIABLES TO ALL
C              DATAPAC SUBROUTINES ARE SINGLE PRECISION.
C              THIS CONVENTION IS BASED ON THE BELIEF THAT
C              1) A MIXTURE OF MODES (FLOATING POINT
C              VERSUS INTEGER) IS INCONSISTENT AND
C              AN UNNECESSARY COMPLICATION
C              IN A DATA ANALYSIS; AND
C              2) FLOATING POINT MACHINE ARITHMETIC
C              (AS OPPOSED TO INTEGER ARITHMETIC) 
C              IS THE MORE NATURAL MODE FOR DOING 
C              DATA ANALYSIS. 
C     REFERENCES--NATIONAL BUREAU OF STANDARDS APPLIED MATHEMATICS
C                 SERIES 55, 1964, PAGE 945, FORMULAE 26.5.24 AND
C                 26.5.28, AND PAGE 929.
C               --JOHNSON AND KOTZ, DISCRETE
C                 DISTRIBUTIONS, 1969, PAGES 122-142,
C                 ESPECIALLY PAGE 127.
C               --HASTINGS AND PEACOCK, STATISTICAL
C                 DISTRIBUTIONS--A HANDBOOK FOR
C                 STUDENTS AND PRACTITIONERS, 1975,
C                 PAGES 92-95.
C               --FELLER, AN INTRODUCTION TO PROBABILITY
C                 THEORY AND ITS APPLICATIONS, VOLUME 1,
C                 EDITION 2, 1957, PAGES 155-157, 210.
C               --KENDALL AND STUART, THE ADVANCED THEORY OF
C                 STATISTICS, VOLUME 1, EDITION 2, 1963, PAGES 130-131.
C               --WILLIAMSON AND BRETHERTON, TABLES OF
C                 THE NEGATIVE BINOMIAL PROBABILITY
C                 DISTRIBUTION, 1963.
C               --OWEN, HANDBOOK OF STATISTICAL
C                 TABLES, 1962, PAGE 304.
C     WRITTEN BY--JAMES J. FILLIBEN
C                 STATISTICAL ENGINEERING LABORATORY (205.03)
C                 NATIONAL BUREAU OF STANDARDS
C                 WASHINGTON, D. C. 20234
C                 PHONE:  301-921-2315
C     ORIGINAL VERSION--NOVEMBER  1975. 
C
C---------------------------------------------------------------------
C
      DOUBLE PRECISION DX2,PI,ANU1,ANU2,Z,SUM,TERM,AI,COEF1,COEF2,ARG 
      DOUBLE PRECISION COEF
      DOUBLE PRECISION THETA,SINTH,COSTH,A,B
      DOUBLE PRECISION DSQRT,DATAN
      DATA PI/3.14159265358979D0/
C
      IPR=6
C
C     CHECK THE INPUT ARGUMENTS FOR ERRORS
C
      AN=N
      IF(P.LE.0.0.OR.P.GE.1.0)GOTO50
      IF(N.LT.1)GOTO55
      IF(X.LT.0.0)GOTO60
      INTX=X+0.0001 
      FINTX=INTX
      DEL=X-FINTX
      IF(DEL.LT.0.0)DEL=-DEL
      IF(DEL.GT.0.001)GOTO65
      GOTO90
   50 WRITE(IPR,11) 
      WRITE(IPR,46)P
      CDF=0.0
      RETURN
   55 WRITE(IPR,25) 
      WRITE(IPR,47)N
      CDF=0.0
      RETURN
   60 WRITE(IPR,4)
      WRITE(IPR,46)X
      IF(X.LT.0.0)CDF=0.0
      RETURN
   65 WRITE(IPR,5)
      WRITE(IPR,46)X
   90 CONTINUE
    4 FORMAT(1H , 96H***** NON-FATAL DIAGNOSTIC--THE FIRST  INPUT ARGUME
     1NT TO THE NBCDF  SUBROUTINE IS NEGATIVE *****)
    5 FORMAT(1H ,100H***** NON-FATAL DIAGNOSTIC--THE FIRST  INPUT ARGUME
     1NT TO THE NBCDF  SUBROUTINE IS NON-INTEGRAL *****)
   11 FORMAT(1H ,115H***** FATAL ERROR--THE SECOND INPUT ARGUMENT TO THE
     1 NBCDF  SUBROUTINE IS OUTSIDE THE ALLOWABLE (0,1) INTERVAL *****)
   25 FORMAT(1H , 91H***** FATAL ERROR--THE THIRD  INPUT ARGUMENT TO THE
     1 NBCDF  SUBROUTINE IS NON-POSITIVE *****)
   46 FORMAT(1H , 35H***** THE VALUE OF THE ARGUMENT IS ,E15.8,6H *****)
   47 FORMAT(1H , 35H***** THE VALUE OF THE ARGUMENT IS ,I8   ,6H *****)
C
C-----START POINT-----------------------------------------------------
C
C     EXPRESS THE NEGATIVE BINOMIAL CUMULATIVE DISTRIBUTION 
C     FUNCTION IN TERMS OF THE EQUIVALENT BINOMIAL
C     CUMULATIVE DISTRIBUTION FUNCTION, 
C     AND THEN OPERATE ON THE LATTER.
C
      INTX=X+0.0001 
      K=N-1
      N2=N+INTX
C
C     EXPRESS THE BINOMIAL CUMULATIVE DISTRIBUTION
C     FUNCTION IN TERMS OF THE EQUIVALENT F
C     CUMULATIVE DISTRIBUTION FUNCTION, 
C     AND THEN EVALUATE THE LATTER.
C
      AK=K
      AN2=N2
      DX2=(P/(1.0-P))*((AN2-AK)/(AK+1.0))
      NU1=2*(K+1)
      NU2=2*(N2-K)
      ANU1=NU1
      ANU2=NU2
      Z=ANU2/(ANU2+ANU1*DX2)
C
C     DETERMINE IF NU1 AND NU2 ARE EVEN OR ODD
C
      IFLAG1=NU1-2*(NU1/2)
      IFLAG2=NU2-2*(NU2/2)
      IF(IFLAG1.EQ.0)GOTO120
      IF(IFLAG2.EQ.0)GOTO150
      GOTO250
C
C     DO THE NU1 EVEN AND NU2 EVEN OR ODD CASE
C
  120 SUM=0.0D0
      TERM=1.0D0
      IMAX=(NU1-2)/2
      IF(IMAX.LE.0)GOTO110
      DO100I=1,IMAX 
      AI=I
      COEF1=2.0D0*(AI-1.0D0)
      COEF2=2.0D0*AI
      TERM=TERM*((ANU2+COEF1)/COEF2)*(1.0D0-Z)
      SUM=SUM+TERM
  100 CONTINUE
C
  110 SUM=SUM+1.0D0 
      SUM=(Z**(ANU2/2.0D0))*SUM
      CDF=1.0D0-SUM 
      RETURN
C
C     DO THE NU1 ODD AND NU2 EVEN CASE
C
  150 SUM=0.0D0
      TERM=1.0D0
      IMAX=(NU2-2)/2
      IF(IMAX.LE.0)GOTO210
      DO200I=1,IMAX 
      AI=I
      COEF1=2.0D0*(AI-1.0D0)
      COEF2=2.0D0*AI
      TERM=TERM*((ANU1+COEF1)/COEF2)*Z
      SUM=SUM+TERM
  200 CONTINUE
C
  210 SUM=SUM+1.0D0 
      CDF=((1.0D0-Z)**(ANU1/2.0D0))*SUM 
      RETURN
C
C     DO THE NU1 ODD AND NU2 ODD CASE
C
  250 SUM=0.0D0
      TERM=1.0D0
      ARG=DSQRT((ANU1/ANU2)*DX2)
      THETA=DATAN(ARG)
      SINTH=ARG/DSQRT(1.0D0+ARG*ARG)
      COSTH=1.0D0/DSQRT(1.0D0+ARG*ARG)
      IF(NU2.EQ.1)GOTO320
      IF(NU2.EQ.3)GOTO310
      IMAX=NU2-2
      DO300I=3,IMAX,2
      AI=I
      COEF1=AI-1.0D0
      COEF2=AI
      TERM=TERM*(COEF1/COEF2)*(COSTH*COSTH)
      SUM=SUM+TERM
  300 CONTINUE
C
  310 SUM=SUM+1.0D0 
      SUM=SUM*SINTH*COSTH
C
  320 A=(2.0D0/PI)*(THETA+SUM)
  350 SUM=0.0D0
      TERM=1.0D0
      IF(NU1.EQ.1)B=0.0D0
      IF(NU1.EQ.1)GOTO450
      IF(NU1.EQ.3)GOTO410
      IMAX=NU1-3
      DO400I=1,IMAX,2
      AI=I
      COEF1=AI
      COEF2=AI+2.0D0
      TERM=TERM*((ANU2+COEF1)/COEF2)*(SINTH*SINTH)
      SUM=SUM+TERM
  400 CONTINUE
C
  410 SUM=SUM+1.0D0 
      SUM=SUM*SINTH*(COSTH**N)
      COEF=1.0D0
      IEVODD=NU2-2*(NU2/2)
      IMIN=3
      IF(IEVODD.EQ.0)IMIN=2
      IF(IMIN.GT.NU2)GOTO420
      DO430I=IMIN,NU2,2
      AI=I
      COEF=((AI-1.0D0)/AI)*COEF
  430 CONTINUE
C
  420 COEF=COEF*ANU2
      IF(IEVODD.EQ.0)GOTO440
      COEF=COEF*(2.0D0/PI)
C
  440 B=COEF*SUM
C
  450 CDF=A-B
      RETURN
C
      END 
