TEST_VALUES
Sample Function Values

TEST_VALUES is a C++ library which stores a few selected values of various mathematical functions.

The intent of TEST_VALUES is to provide a means of making very simple tests for correctness of software designed to compute a variety of functions. The testing can be done automatically. The data provided is generally skimpy, and might not test the algorithm over a suitably wide range. It does, however, provide a small amount of reassurance that a given computation is (or is not) computing the appropriate quantity, and doing so reasonably accurately.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

TEST_VALUES is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATHEMATICA version and a MATLAB version and a Python version.

Related Programs:

CORDIC, a C++ library which uses the CORDIC method to compute certain elementary functions.

FN, a C++ library which contains routines by Wayne Fullerton for evaluating elementary and special functions.

GSL, a C++ library which includes routines that evaluate many special functions.

LEGENDRE_POLYNOMIAL, a C++ library which evaluates the Legendre polynomial and associated functions.

LOBATTO_POLYNOMIAL, a C++ library which evaluates Lobatto polynomials, similar to Legendre polynomials except that they are zero at both endpoints.

POLPAK, a C++ library which computes various mathematical functions; test values for many of these functions are available in TEST_VALUES.

PROB, a C++ library which computes various statistical functions; test values for many of these functions are available in TEST_VALUES.

Reference:

1. Milton Abramowitz, Irene Stegun,
   Handbook of Mathematical Functions,
   National Bureau of Standards, 1964,
   ISBN: 0-486-61272-4,
   LC: QA47.A34.
2. Robert Corless, Gaston Gonnet, David Hare, David Jeffrey, Donald Knuth,
   On the Lambert W Function,
   Advances in Computational Mathematics,
   Volume 5, 1996, pages 329-359.
3. Gerard Cornuejols, Regina Urbaniak, Robert Weismantel, Laurence Wolsey,
   Decomposition of Integer Programs and of Generating Sets,
   in Algorithms - ESA '97,
   Lecture Notes in Computer Science 1284,
   edited by R Burkard, G Woeginger,
   Springer, 1997, pages 92-103,
   LC: QA76.9.A43.E83.
4. Marc Deleglise, Joel Rivat,
   Computing the Summation of the Moebius Function,
   Experimental Mathematics,
   Volume 5, 1996, pages 291-295.
5. Lester Haar, John Gallagher, George Kell,
   NBS/NRC Steam Tables:
   Thermodynamic and Transport Properties and Computer Programs for Vapor and Liquid States of Water in SI Units,
   Hemisphere Publishing Corporation, Washington, 1984,
   ISBN: 0-89116-353-0,
   LC: TJ270.H3.
6. Brian Hayes,
   "Why W?",
   The American Scientist,
   Volume 93, March-April 2005, pages 104-108.
7. Kanti Mardia, Peter Jupp,
   Directional Statistics,
   Wiley, 2000,
   ISBN: 0471953334,
   LC: QA276.M335
8. Allan McLeod,
   Algorithm 757: MISCFUN: A software package to compute uncommon special functions,
   ACM Transactions on Mathematical Software,
   Volume 22, Number 3, September 1996, pages 288-301.
9. National Bureau of Standards,
   Tables of the Bivariate Normal Distribution and Related Functions,
   Applied Mathematics Series, Number 50, 1959.
10. Robert Owens,
    An Algorithm to Solve the Frobenius Problem,
    Mathematics Magazine,
    Volume 76, Number 4, October 2003, 264-275.
11. Karl Pearson,
    Tables of the Incomplete Beta Function,
    Cambridge University Press, 1968,
    ISBN: 0521059224,
    LC: QA351.P38.
12. Frank Powell,
    Statistical Tables for Sociology, Biology and Physical Sciences,
    Cambridge University Press, 1982,
    ISBN: 0521284732,
    LC: QA276.25.S73.
13. Edward Reingold, Nachum Dershowitz,
    Calendrical Calculations: The Millennium Edition,
    Cambridge University Press, 2001,
    ISBN: 0-521-77752-6,
    LC: CE12.R45.
14. Johannes van der Corput,
    Verteilungsfunktionen,
    Proc Akad Amsterdam,
    Volume 38, 1935,
    Volume 39, 1936.
15. Eric Weisstein,
    CRC Concise Encyclopedia of Mathematics,
    CRC Press, 2002,
    Second edition,
    ISBN: 1584883472,
    LC: QA5.W45
16. Stephen Wolfram,
    The Mathematica Book,
    Fourth Edition,
    Cambridge University Press, 1999,
    ISBN: 0-521-64314-7,
    LC: QA76.95.W65.
17. Shanjie Zhang, Jianming Jin,
    Computation of Special Functions,
    Wiley, 1996,
    ISBN: 0-471-11963-6,
    LC: QA351.C45.
18. Daniel Zwillinger, editor,
    CRC Standard Mathematical Tables and Formulae,
    30th Edition,
    CRC Press, 1996,
    ISBN: 0-8493-2479-3,
    LC: QA47.M315.
19. Daniel Zwillinger, Steven Kokoska,
    Standard Probability and Statistical Tables,
    CRC Press, 2000,
    ISBN: 1-58488-059-7,
    LC: QA273.3.Z95.

Source Code:

• test_values.cpp, the source code.
• test_values.hpp, the include file.
• test_values.sh, commands to compile the source code.

Examples and Tests:

• test_values_prb.cpp, a sample calling program.
• test_values_prb.sh, commands to compile and run the sample program.
• test_values_prb_output.txt, the output file.

List of Routines:

• ABRAM0_VALUES returns some values of the Abramowitz0 function.
• ABRAM1_VALUES returns some values of the Abramowitz1 function.
• ABRAM2_VALUES returns some values of the Abramowitz2 function.
• AGM_VALUES returns some values of the AGM.
• AIRY_AI_VALUES returns some values of the Airy Ai(x) function.
• AIRY_AI_INT_VALUES returns some values of the integral of the Airy function.
• AIRY_AI_PRIME_VALUES returns some values of the Airy function Ai'(x).
• AIRY_BI_VALUES returns some values of the Airy Bi(x) function.
• AIRY_BI_INT_VALUES returns some values of the integral of the Airy function.
• AIRY_BI_PRIME_VALUES returns some values of the Airy function Bi'(x).
• AIRY_CAI_VALUES returns some values of the Airy Ai(x) for complex argument.
• AIRY_CBI_VALUES returns some values of the Airy Bi(x) for complex argument.
• AIRY_GI_VALUES returns some values of the Airy Gi function.
• AIRY_HI_VALUES returns some values of the Airy Hi function.
• ARCCOS_VALUES returns some values of the arc cosine function.
• ARCCOSH_VALUES returns some values of the hyperbolic arc cosine function.
• ARCSIN_VALUES returns some values of the arc sine function.
• ARCSINH_VALUES returns some values of the hyperbolic arc sine function.
• ARCTAN_VALUES returns some values of the arc tangent function.
• ARCTAN_INT_VALUES returns some values of the inverse tangent integral.
• ARCTAN2_VALUES: arc tangent function of two arguments.
• ARCTANH_VALUES returns some values of the hyperbolic arc tangent function.
• BEI0_VALUES returns some values of the Kelvin BEI function of order NU = 0.
• BEI1_VALUES returns some values of the Kelvin BEI function of order NU = 1.
• BELL_VALUES returns some values of the Bell numbers.
• BER0_VALUES returns some values of the Kelvin BER function of order NU = 0.
• BER1_VALUES returns some values of the Kelvin BER function of order NU = 1.
• BERNOULLI_NUMBER_VALUES returns some values of the Bernoulli numbers.
• BERNOULLI_POLY_VALUES returns some values of the Bernoulli polynomials.
• BERNSTEIN_POLY_01_VALUES returns some values of the Bernstein polynomials.
• BESSEL_I0_INT_VALUES returns some values of the Bessel I0 integral.
• BESSEL_I0_SPHERICAL_VALUES returns some values of the Spherical Bessel function i0.
• BESSEL_I0_VALUES returns some values of the I0 Bessel function.
• BESSEL_I1_SPHERICAL_VALUES returns some values of the Spherical Bessel function i1.
• BESSEL_I1_VALUES returns some values of the I1 Bessel function.
• BESSEL_IN_VALUES returns some values of the In Bessel function.
• BESSEL_IX_VALUES returns some values of the Ix Bessel function.
• BESSEL_J0_INT_VALUES returns some values of the Bessel J0 integral.
• BESSEL_J0_SPHERICAL_VALUES returns some values of the Spherical Bessel function j0.
• BESSEL_J0_VALUES returns some values of the J0 Bessel function.
• BESSEL_J1_SPHERICAL_VALUES returns some values of the Spherical Bessel function j1.
• BESSEL_J1_VALUES returns some values of the J1 Bessel function.
• BESSEL_JN_VALUES returns some values of the Jn Bessel function.
• BESSEL_JX_VALUES returns some values of the Jx Bessel function.
• BESSEL_K0_VALUES returns some values of the K0 Bessel function.
• BESSEL_K0_INT_VALUES returns some values of the Bessel K0 integral.
• BESSEL_K1_VALUES returns some values of the K1 Bessel function.
• BESSEL_KN_VALUES returns some values of the Kn Bessel function.
• BESSEL_KX_VALUES returns some values of the Kx Bessel function.
• BESSEL_Y0_VALUES returns some values of the Y0 Bessel function.
• BESSEL_Y0_INT_VALUES returns some values of the Bessel Y0 integral.
• BESSEL_Y0_SPHERICAL_VALUES returns some values of the Spherical Bessel function y0.
• BESSEL_Y1_VALUES returns some values of the Y1 Bessel function.
• BESSEL_Y1_SPHERICAL_VALUES returns some values of the Spherical Bessel function y1.
• BESSEL_YN_VALUES returns some values of the Yn Bessel function.
• BESSEL_YX_VALUES returns some values of the Yx Bessel function.
• BETA_CDF_VALUES returns some values of the Beta CDF.
• BETA_INC_VALUES returns some values of the incomplete Beta function.
• BETA_LOG_VALUES returns some values of the logarithm of the Beta function.
• BETA_NONCENTRAL_CDF_VALUES returns some values of the noncentral Beta CDF.
• BETA_VALUES returns some values of the Beta function.
• BINOMIAL_VALUES returns some values of the binomial coefficients.
• BINOMIAL_CDF_VALUES returns some values of the binomial CDF.
• BIVARIATE_NORMAL_CDF_VALUES returns some values of the bivariate normal CDF.
• CATALAN_VALUES returns some values of the Catalan numbers.
• CAUCHY_CDF_VALUES returns some values of the Cauchy CDF.
• CBRT_VALUES returns some values of the cube root function.
• CHEBY_T_POLY_VALUES returns values of Chebyshev polynomials T(n,x).
• CHEBY_U_POLY_VALUES returns values of Chebyshev polynomials U(n,x).
• CHEBY_V_POLY_VALUES returns values of Chebyshev polynomials V(n,x).
• CHEBY_W_POLY_VALUES returns values of Chebyshev polynomials W(n,x).
• CHI_VALUES returns some values of the hyperbolic cosine integral function.
• CHI_SQUARE_CDF_VALUES returns some values of the Chi-Square CDF.
• CHI_SQUARE_NONCENTRAL_CDF_VALUES returns values of the noncentral chi CDF.
• CI_VALUES returns some values of the cosine integral function.
• CIN_VALUES returns some values of the alternate cosine integral function.
• CINH_VALUES returns some values of the alternate hyperbolic cosine integral.
• CLAUSEN_VALUES returns some values of the Clausen's integral.
• CLEBSCH_GORDAN_VALUES returns some values of the Clebsch-Gordan function.
• COLLATZ_COUNT_VALUES returns some values of the Collatz count function.
• COS_VALUES returns some values of the cosine function.
• COS_DEGREE_VALUES: values of the cosine function for degree arguments.
• COS_POWER_INT_VALUES returns some values of the sine power integral.
• COSH_VALUES returns some values of the hyperbolic cosine function.
• COT_VALUES returns some values of the cotangent function.
• CP_VALUES returns some values of the specific heat at constant pressure.
• DAWSON_VALUES returns some values of Dawson's integral.
• DEBYE1_VALUES returns some values of Debye's function of order 1.
• DEBYE2_VALUES returns some values of Debye's function of order 2.
• DEBYE3_VALUES returns some values of Debye's function of order 3.
• DEBYE4_VALUES returns some values of Debye's function of order 4.
• DIELECTRIC_VALUES returns some values of the static dielectric constant.
• DEDEKIND_SUM_VALUES returns some values of the Dedekind sum.
• DILOGARITHM_VALUES returns some values of the dilogarithm function.
• E1_VALUES returns some values of the exponential integral function E1(X).
• EI_VALUES returns some values of the exponential integral function EI(X).
• ELLIPTIC_EA_VALUES returns values of the complete elliptic integral E(ALPHA).
• ELLIPTIC_EM_VALUES returns values of the complete elliptic integral E(M).
• ELLIPTIC_KA_VALUES returns values of the complete elliptic integral K(ALPHA).
• ELLIPTIC_KM_VALUES returns values of the complete elliptic integral K(M).
• ERF_VALUES returns some values of the ERF or "error" function.
• ERFC_VALUES returns some values of the ERFC or "complementary error" function.
• EULER_NUMBER_VALUES returns some values of the Euler numbers.
• EULER_POLY_VALUES returns some values of the Euler polynomials.
• EXP_VALUES returns some values of the exponential function.
• EXP3_INT_VALUES returns some values of the EXP3 integral function.
• EXPONENTIAL_CDF_VALUES returns some values of the Exponential CDF.
• EXTREME_VALUES_CDF_VALUES returns some values of the Extreme Values CDF.
• 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.
• FACTORIAL_VALUES returns values of the factorial function.
• FACTORIAL2_VALUES returns values of the double factorial function.
• FACTORIAL_RISING_VALUES returns values of the int *Pochhammer function.
• FRESNEL_COS_VALUES returns values of the Fresnel cosine integral function.
• FRESNEL_SIN_VALUES returns some values of the Fresnel sine integral function.
• FROBENIUS_NUMBER_DATA_VALUES returns data for the Frobenius problem.
• FROBENIUS_NUMBER_ORDER_VALUES returns orders of the Frobenius problem.
• FROBENIUS_NUMBER_ORDER2_VALUES returns values of the order 2 Frobenius number.
• GAMMA_VALUES returns some values of the Gamma function.
• GAMMA_CDF_VALUES returns some values of the Gamma CDF.
• GAMMA_INC_P_VALUES: values of normalized incomplete Gamma function P(A,X).
• GAMMA_INC_Q_VALUES: values of normalized incomplete Gamma function Q(A,X).
• GAMMA_INC_TRICOMI_VALUES: values of Tricomi's incomplete Gamma function.
• GAMMA_INC_VALUES returns some values of the incomplete Gamma function.
• GAMMA_LOG_VALUES returns some values of the Log Gamma function.
• GEGENBAUER_POLY_VALUES returns some values of the Gegenbauer polynomials.
• GEOMETRIC_CDF_VALUES returns values of the geometric CDF.
• GOODWIN_VALUES returns some values of the Goodwin and Staton function.
• GUD_VALUES returns some values of the Gudermannian function.
• HERMITE_FUNCTION_VALUES returns some values of the Hermite function.
• HERMITE_POLY_PHYS_VALUES returns some values of the physicist's Hermite polynomial.
• HERMITE_POLY_PROB_VALUES: values of the probabilist's Hermite polynomial.
• HYPER_1F1_VALUES returns some values of the hypergeometric function 1F1.
• HYPER_2F1_VALUES returns some values of the hypergeometric function 2F1.
• HYPERGEOMETRIC_CDF_VALUES returns some values of the hypergeometric CDF.
• HYPERGEOMETRIC_PDF_VALUES returns some values of the hypergeometric PDF.
• HYPERGEOMETRIC_U_VALUES: some values of the hypergeometric function U(a,b,x).
• I0ML0_VALUES returns some values of the I0ML0 function.
• I1ML1_VALUES returns some values of the I1ML1 function.
• INT_VALUES returns some values of the "integer part" function.
• JACOBI_CN_VALUES returns some values of the Jacobi elliptic function CN(A,X).
• JACOBI_DN_VALUES returns some values of the Jacobi elliptic function DN(A,X).
• JACOBI_POLY_VALUES returns some values of the Jacobi polynomial.
• JACOBI_SN_VALUES returns some values of the Jacobi elliptic function SN(A,X).
• JED_CE_VALUES returns the Common Era dates for Julian Ephemeris Dates.
• JED_MJD_VALUES returns the MJD for Julian Ephemeris Dates.
• JED_RD_VALUES returns the RD for Julian Ephemeris Dates.
• JED_WEEKDAY_VALUES returns the day of the week for Julian Ephemeris Dates.
• KEI0_VALUES returns some values of the Kelvin KEI function of order NU = 0.
• KEI1_VALUES returns some values of the Kelvin KEI function of order NU = 1.
• KER0_VALUES returns some values of the Kelvin KER function of order NU = 0.
• KER1_VALUES returns some values of the Kelvin KER function of order NU = 1.
• LAGUERRE_ASSOCIATED_VALUES returns some values of the associated Laguerre polynomials.
• LAGUERRE_GENERAL_VALUES returns some values of the generalized Laguerre function.
• LAGUERRE_POLYNOMIAL_VALUES returns some values of the Laguerre polynomial.
• LAMBERT_W_VALUES returns some values of the Lambert W function.
• LAPLACE_CDF_VALUES returns some values of the Laplace CDF.
• LEGENDRE_ASSOCIATED_VALUES returns values of associated Legendre functions.
• LEGENDRE_ASSOCIATED_NORMALIZED_SPHERE_VALUES: normalized associated Legendre.
• LEGENDRE_ASSOCIATED_NORMALIZED_VALUES: normalized associated Legendre.
• LEGENDRE_FUNCTION_Q_VALUES returns values of the Legendre Q function.
• LEGENDRE_POLY_VALUES returns values of the Legendre polynomials.
• LOBATTO_POLYNOMIAL_VALUES returns values of the completed Lobatto polynomials.
• LOBATTO_POLYNOMIAL_DERIVATIVES: derivatives of the completed Lobatto polynomials.
• LERCH_VALUES returns some values of the Lerch transcendent function.
• LINEAR_SYSTEM_VALUES returns some linear systems.
• LOBACHEVSKY_VALUES returns some values of the Lobachevsky function.
• LOG_VALUES returns some values of the natural logarithm function.
• LOG_NORMAL_CDF_VALUES returns some values of the Log Normal CDF.
• LOG_SERIES_CDF_VALUES returns some values of the log series CDF.
• LOG10_VALUES returns some values of the logarithm base 10 function.
• LOGARITHMIC_INTEGRAL_VALUES returns values of the logarithmic integral LI(X).
• LOGISTIC_CDF_VALUES returns some values of the Logistic CDF.
• MERTENS_VALUES returns some values of the Mertens function.
• MOEBIUS_VALUES returns some values of the Moebius function.
• NEGATIVE_BINOMIAL_CDF_VALUES returns values of the negative binomial CDF.
• NINE_J_VALUES returns some values of the Wigner 9J function.
• NORMAL_CDF_VALUES returns some values of the Normal CDF.
• NORMAL_01_CDF_VALUES returns some values of the Normal 01 CDF.
• OMEGA_VALUES returns some values of the OMEGA function.
• OWEN_VALUES returns some values of Owen's T function.
• PARTITION_COUNT_VALUES returns some values of the int *partition count.
• PARTITION_DISTINCT_COUNT_VALUES returns some values of Q(N).
• PHI_VALUES returns some values of the PHI function.
• PI_VALUES returns values of the Pi function.
• POCHHAMMER_VALUES returns some values of the Pochhammer function.
• POISSON_CDF_VALUES returns some values of the Poisson CDF.
• POLYLOGARITHM_VALUES returns some values of the polylogarithm.
• PRANDTL_VALUES returns some values of the Prandtl number.
• PRIME_VALUES returns values of the prime function.
• PSAT_VALUES returns some values of the saturation pressure.
• PSI_VALUES returns some values of the Psi or Digamma function.
• R8_FACTORIAL_VALUES returns values of the real factorial function.
• R8_FACTORIAL_LOG_VALUES returns values of log(n!).
• RAYLEIGH_CDF_VALUES returns some values of the Rayleigh CDF.
• SECVIR_VALUES returns some values of the second virial coefficient.
• SHI_VALUES returns some values of the hyperbolic sine integral function.
• SI_VALUES returns some values of the sine integral function.
• SIGMA_VALUES returns some values of the Sigma function.
• SIN_VALUES returns some values of the sine function.
• SIN_DEGREE_VALUES: the sine function with argument in degrees.
• SIN_POWER_INT_VALUES returns some values of the sine power integral.
• SINH_VALUES returns some values of the hyperbolic sine function.
• SIX_J_VALUES returns some values of the Wigner 6J function.
• SOUND_VALUES returns some values of the speed of sound.
• SPHERE_UNIT_AREA_VALUES returns some areas of the unit sphere in ND.
• SPHERE_UNIT_VOLUME_VALUES returns some volumes of the unit sphere in ND.
• SPHERICAL_HARMONIC_VALUES returns values of spherical harmonic functions.
• SQRT_VALUES returns some values of the square root function.
• STIRLING1_VALUES returns some values of the Stirling numbers, kind 1.
• STIRLING2_VALUES returns some values of the Stirling numbers, kind 2.
• STROMGEN_VALUES returns some values of the Stromgen function.
• STRUVE_H0_VALUES returns some values of the Struve H0 function.
• STRUVE_H1_VALUES returns some values of the Struve H1 function.
• STRUVE_L0_VALUES returns some values of the Struve L0 function.
• STRUVE_L1_VALUES returns some values of the Struve L1 function.
• STUDENT_CDF_VALUES returns some values of the Student CDF.
• STUDENT_NONCENTRAL_CDF_VALUES returns values of the noncentral Student CDF.
• SUBFACTORIAL_VALUES returns values of the subfactorial function.
• SURTEN_VALUES returns some values of the surface tension.
• SYNCH1_VALUES returns some values of the synchrotron radiation function.
• SYNCH2_VALUES returns some values of the synchrotron radiation function.
• TAN_VALUES returns some values of the tangent function.
• TANH_VALUES returns some values of the hyperbolic tangent function.
• TAU_VALUES returns some values of the Tau function.
• THERCON_VALUES returns some values of the thermal conductivity.
• THREE_J_VALUES returns some values of the Wigner 3J function.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TRAN02_VALUES returns some values of the order 2 transportation function.
• TRAN03_VALUES returns some values of the order 3 transportation function.
• TRAN04_VALUES returns some values of the order 4 transportation function.
• TRAN05_VALUES returns some values of the order 5 transportation function.
• TRAN06_VALUES returns some values of the order 6 transportation function.
• TRAN07_VALUES returns some values of the order 7 transportation function.
• TRAN08_VALUES returns some values of the order 8 transportation function.
• TRAN09_VALUES returns some values of the order 9 transportation function.
• TRIGAMMA_VALUES returns some values of the TriGamma function.
• TRUNCATED_NORMAL_AB_CDF_VALUES: values of the Truncated Normal CDF.
• TRUNCATED_NORMAL_AB_PDF_VALUES: values of the Truncated Normal PDF.
• TRUNCATED_NORMAL_A_CDF_VALUES: values of the Lower Truncated Normal CDF.
• TRUNCATED_NORMAL_A_PDF_VALUES: values of the Lower Truncated Normal PDF.
• TRUNCATED_NORMAL_B_CDF_VALUES: values of the upper Truncated Normal CDF.
• TRUNCATED_NORMAL_B_PDF_VALUES: values of the Upper Truncated Normal PDF.
• TSAT_VALUES returns some values of the saturation temperature.
• VAN_DER_CORPUT_VALUES returns some values of the van der Corput sequence.
• VISCOSITY_VALUES returns some values of the viscosity function.
• VON_MISES_CDF_VALUES returns some values of the von Mises CDF.
• WEEKDAY_VALUES returns the day of the week for various dates.
• WEIBULL_CDF_VALUES returns some values of the Weibull CDF.
• ZETA_VALUES returns some values of the Riemann Zeta function.

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