CDFLIB
Cumulative Density Functions

CDFLIB is a FORTRAN90 library which evaluates the cumulative density function (CDF) associated with common probability distributions, by Barry Brown, James Lovato, Kathy Russell.

CDFLIB includes routines for evaluating the cumulative density functions of a variety of standard probability distributions. An unusual feature of this library is its ability to easily compute any one parameter of the CDF given the others. This means that a single routine can evaluate the CDF given the usual parameters, or determine the value of a parameter that produced a given CDF value.

The probability distributions covered include:

• the Beta distribution;
• the binomial distribution;
• the chi-square distribution;
• the noncentral chi-square distribution;
• the F distribution;
• the noncentral F distribution;
• the Gamma distribution;
• the negative binomial distribution;
• the normal distribution;
• the Poisson distribution;
• the T distribution;

Note that the F and noncentral F distributions are not necessarily monotone in either degree of freedom argument. Consequently, there may be two degree of freedom arguments that satisfy the specified condition. An arbitrary one of these will be found by the routines.

The amount of computation required for the noncentral chisquare and noncentral F distribution is proportional to the value of the noncentrality parameter. Very large values of this parameter can require immense numbers of computation. Consequently, when the noncentrality parameter is to be calculated, the upper limit searched is 10,000.

http://www.netlib.org/random
the NETLIB random number web site, distributes a TAR file of the source code for the original CDFLIB library in C and FORTRAN;

http://biostatistics.mdanderson.org/SoftwareDownload/SingleSoftware.aspx?Software_Id=21 is a site at the University of Texas Department of Biostatistics and Applied Mathematics which makes available a more up-to-date FORTRAN90 version of the software, known as CDFLIB90.

Languages:

CDFLIB is available in a C version and a C++ version and a FORTRAN90 version.

Related Data and Programs:

ASA310, a FORTRAN90 library which computes the CDF of the noncentral Beta distribution.

BETA_NC, a FORTRAN90 library which evaluates the CDF of the noncentral Beta distribution.

GSL, a C++ library which contains routines for evaluating, sampling and inverting various probability distributions.

NORMAL, a FORTRAN90 library which contains routines for sampling the normal distribution.

PROB, a FORTRAN90 library which contains routines for evaluating and inverting the normal CDF, and many other distributions.

TEST_VALUES, a FORTRAN90 library which contains routines that store selected values of the normal PDF, and many other statistical distributions.

UNIFORM, a FORTRAN90 library which contains routines for sampling the uniform distribution.

WALKER_SAMPLE, a FORTRAN90 library which efficiently samples a discrete probability vector using Walker sampling.

Author:

Barry Brown, James Lovato, Kathy Russell,
Department of Biomathematics,
University of Texas,
Houston, Texas.

Reference:

1. Milton Abramowitz, Irene Stegun,
   Handbook of Mathematical Functions,
   National Bureau of Standards, 1964,
   ISBN: 0-486-61272-4,
   LC: QA47.A34.
2. Jacobus Bus, Jacob Dekker,
   Two Efficient Algorithms with Guaranteed Convergence for Finding a Zero of a Function,
   ACM Transactions on Mathematical Software,
   Volume 1, Number 4, December 1975, pages 330-345.
3. William Cody,
   Algorithm 715: SPECFUN - A Portable FORTRAN Package of Special Function Routines and Test Drivers,
   ACM Transactions on Mathematical Software,
   Volume 19, Number 1, March 1993, pages 22-32.
4. William Cody,
   Rational Chebyshev Approximations for the Error Function,
   Mathematics of Computation,
   Volume 23, Number 107, July 1969, pages 631-638.
5. William Cody, Anthony Strecok, Henry Thacher,
   Chebyshev Approximations for the Psi Function,
   Mathematics of Computation,
   Volume 27, Number 121, January 1973, pages 123-127.
6. Armido DiDinato, Alfred Morris,
   Algorithm 708: Significant Digit Computation of the Incomplete Beta Function Ratios,
   ACM Transactions on Mathematical Software,
   Volume 18, Number 3, September 1993, pages 360-373.
7. Armido DiDinato, Alfred Morris,
   Algorithm 654: Computation of the Incomplete Gamma Function Ratios and their Inverse,
   ACM Transactions on Mathematical Software,
   Volume 12, Number 4, December 1986, pages 377-393.
8. Phyllis Fox, Andrew Hall, Norman Schryer,
   Algorithm 528: Framework for a Portable Library,
   ACM Transactions on Mathematical Software,
   Volume 4, Number 2, June 1978, page 176-188.
9. William Kennedy, James Gentle,
   Statistical Computing,
   Marcel Dekker, 1980,
   ISBN: 0824768981,
   LC: QA276.4 K46.
10. Karl Pearson,
    Tables of the Incomplete Beta Function,
    Cambridge University Press, 1968,
    ISBN: 0521059224,
    LC: QA351.P38.
11. Frank Powell,
    Statistical Tables for Sociology, Biology and Physical Sciences,
    Cambridge University Press, 1982,
    ISBN: 0521284732,
    LC: QA276.25.S73.
12. Stephen Wolfram,
    The Mathematica Book,
    Fourth Edition,
    Cambridge University Press, 1999,
    ISBN: 0-521-64314-7,
    LC: QA76.95.W65.
13. Daniel Zwillinger, editor,
    CRC Standard Mathematical Tables and Formulae,
    30th Edition,
    CRC Press, 1996,
    ISBN: 0-8493-2479-3,
    LC: QA47.M315.
14. http://www.netlib.org/random, the web site.

Source Code:

• cdflib.f90, the source code;

Examples and Tests:

• cdflib_test.f90, a sample calling program;
• cdflib_test.txt, the output file.

List of Routines:

• ALGDIV computes ln ( Gamma ( B ) / Gamma ( A + B ) ) when 8 <= B.
• ALNREL evaluates the function ln ( 1 + A ).
• APSER computes the incomplete beta ratio I(SUB(1-X))(B,A).
• BCORR evaluates DEL(A0) + DEL(B0) - DEL(A0 + B0).
• BETA evaluates the beta function.
• BETA_ASYM computes an asymptotic expansion for IX(A,B), for large A and B.
• BETA_FRAC evaluates a continued fraction expansion for IX(A,B).
• BETA_GRAT evaluates an asymptotic expansion for IX(A,B).
• BETA_INC evaluates the incomplete beta function IX(A,B).
• BETA_INC_VALUES returns some values of the incomplete Beta function.
• BETA_LOG evaluates the logarithm of the beta function.
• BETA_PSER uses a power series expansion to evaluate IX(A,B)(X).
• BETA_RCOMP evaluates X**A * Y**B / Beta(A,B).
• BETA_RCOMP1 evaluates exp(MU) * X**A * Y**B / Beta(A,B).
• BETA_UP evaluates IX(A,B) - IX(A+N,B) where N is a positive integer.
• BINOMIAL_CDF_VALUES returns some values of the binomial CDF.
• CDFBET evaluates the CDF of the Beta Distribution.
• CDFBIN evaluates the CDF of the Binomial distribution.
• CDFCHI evaluates the CDF of the chi square distribution.
• CDFCHN evaluates the CDF of the Noncentral Chi-Square.
• CDFF evaluates the CDF of the F distribution.
• CDFFNC evaluates the CDF of the Noncentral F distribution.
• CDFGAM evaluates the CDF of the Gamma Distribution.
• CDFNBN evaluates the CDF of the Negative Binomial distribution
• CDFNOR evaluates the CDF of the Normal distribution.
• CDFPOI evaluates the CDF of the Poisson distribution.
• CDFT evaluates the CDF of the T distribution.
• CHI_NONCENTRAL_CDF_VALUES returns values of the noncentral chi CDF.
• CHI_SQUARE_CDF_VALUES returns some values of the Chi-Square CDF.
• CUMBET evaluates the cumulative incomplete beta distribution.
• CUMBIN evaluates the cumulative binomial distribution.
• CUMCHI evaluates the cumulative chi-square distribution.
• CUMCHN evaluates the cumulative noncentral chi-square distribution.
• CUMF evaluates the cumulative F distribution.
• CUMFNC evaluates the cumulative noncentral F distribution.
• CUMGAM evaluates the cumulative incomplete gamma distribution.
• CUMNBN evaluates the cumulative negative binomial distribution.
• CUMNOR computes the cumulative normal distribution.
• CUMPOI evaluates the cumulative Poisson distribution.
• CUMT evaluates the cumulative T distribution.
• DBETRM computes the Sterling remainder for the complete beta function.
• DEXPM1 evaluates the function EXP(X) - 1.
• DINVNR computes the inverse of the normal distribution.
• DINVR bounds the zero of the function and invokes DZROR.
• DSTINV SeT INverse finder - Reverse Communication
• DLANOR evaluates the logarithm of the asymptotic Normal CDF.
• DSTREM computes the Sterling remainder ln ( Gamma ( Z ) ) - Sterling ( Z ).
• DT1 computes an approximate inverse of the cumulative T distribution.
• DZROR seeks a zero of a function, using reverse communication.
• DSTZR - SeT ZeRo finder - Reverse communication version
• ERF_VALUES returns some values of the ERF or "error" function.
• ERROR_F evaluates the error function ERF.
• ERROR_FC evaluates the complementary error function ERFC.
• ESUM evaluates exp ( MU + X ).
• EVAL_POL evaluates a polynomial at X.
• EXPARG returns the largest or smallest legal argument for EXP.
• F_CDF_VALUES returns some values of the F CDF test function.
• F_NONCENTRAL_CDF_VALUES returns some values of the F CDF test function.
• FPSER evaluates IX(A,B)(X) for very small B.
• GAM1 computes 1 / GAMMA(A+1) - 1 for -0.5 <= A <= 1.5
• GAMMA evaluates the gamma function.
• GAMMA_INC evaluates the incomplete gamma ratio functions P(A,X) and Q(A,X).
• GAMMA_INC_INV computes the inverse incomplete gamma ratio function.
• GAMMA_INC_VALUES returns some values of the incomplete Gamma function.
• GAMMA_LN1 evaluates ln ( Gamma ( 1 + A ) ), for -0.2 <= A <= 1.25.
• GAMMA_LOG evaluates ln ( Gamma ( A ) ) for positive A.
• GAMMA_RAT1 evaluates the incomplete gamma ratio functions P(A,X) and Q(A,X).
• GAMMA_VALUES returns some values of the Gamma function.
• GSUMLN evaluates the function ln(Gamma(A + B)).
• IPMPAR returns integer machine constants.
• NEGATIVE_BINOMIAL_CDF_VALUES returns values of the negative binomial CDF.
• NORMAL_01_CDF_VALUES returns some values of the Normal 01 CDF.
• NORMAL_CDF_VALUES returns some values of the Normal CDF.
• POISSON_CDF_VALUES returns some values of the Poisson CDF.
• PSI evaluates the psi or digamma function, d/dx ln(gamma(x)).
• PSI_VALUES returns some values of the Psi or Digamma function.
• R8_SWAP swaps two R8 values.
• RCOMP evaluates exp(-X) * X**A / Gamma(A).
• REXP evaluates the function EXP(X) - 1.
• RLOG computes X - 1 - LN(X).
• RLOG1 evaluates the function X - ln ( 1 + X ).
• STUDENT_CDF_VALUES returns some values of the Student CDF.
• STVALN provides starting values for the inverse of the normal distribution.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

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