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<html>
<head>
<title>
POLPAK - Recursive Polynomials
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
POLPAK <br> Recursive Polynomials
</h1>
<hr>
<p>
<b>POLPAK</b>
is a C++ library which
evaluates a variety of mathematical functions.
</p>
<p>
It includes routines to evaluate the
recursively-defined polynomial families of
<ul>
<li>
Bernoulli
</li>
<li>
Bernstein
</li>
<li>
Cardan
</li>
<li>
Charlier
</li>
<li>
Chebyshev
</li>
<li>
Euler
</li>
<li>
Gegenbauer
</li>
<li>
Hermite
</li>
<li>
Jacobi
</li>
<li>
Krawtchouk
</li>
<li>
Laguerre
</li>
<li>
Legendre
</li>
<li>
Meixner
</li>
<li>
Zernike
</li>
</ul>
A variety of other polynomials and functions have been added.
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>POLPAK</b> is available in
<a href = "../../c_src/polpak/polpak.html">a C version</a> and
<a href = "../../cpp_src/polpak/polpak.html">a C++ version</a> and
<a href = "../../f77_src/polpak/polpak.html">a FORTRAN77 version</a> and
<a href = "../../f_src/polpak/polpak.html">a FORTRAN90 version</a> and
<a href = "../../m_src/polpak/polpak.html">a MATLAB version</a> and
<a href = "../../py_src/polpak/polpak.html">a Python version</a>
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/bernstein_polynomial/bernstein_polynomial.html">
BERNSTEIN_POLYNOMIAL</a>,
a C++ library which
evaluates the Bernstein polynomials,
useful for uniform approximation of functions;
</p>
<p>
<a href = "../../cpp_src/chebyshev_polynomial/chebyshev_polynomial.html">
CHEBYSHEV_POLYNOMIAL</a>,
a C++ library which
evaluates the Chebyshev polynomial and associated functions.
</p>
<p>
<a href = "../../cpp_src/cordic/cordic.html">
CORDIC</a>,
a C++ library which
uses the CORDIC method to compute certain elementary functions.
</p>
<p>
<a href = "../../cpp_src/fn/fn.html">
FN</a>,
a C++ library which
evaluates elementary and special functions,
by Wayne Fullerton.
</p>
<p>
<a href = "../../cpp_src/gsl/gsl.html">
GSL</a>
a C++ library which
evaluates many special functions.
</p>
<p>
<a href = "../../cpp_src/hermite_polynomial/hermite_polynomial.html">
HERMITE_POLYNOMIAL</a>,
a C++ library which
evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial,
the Hermite function, and related functions.
</p>
<p>
<a href = "../../cpp_src/jacobi_polynomial/jacobi_polynomial.html">
JACOBI_POLYNOMIAL</a>,
a C++ library which
evaluates the Jacobi polynomial and associated functions.
</p>
<p>
<a href = "../../cpp_src/laguerre_polynomial/laguerre_polynomial.html">
LAGUERRE_POLYNOMIAL</a>,
a C++ library which
evaluates the Laguerre polynomial, the generalized Laguerre polynomial,
and the Laguerre function.
</p>
<p>
<a href = "../../cpp_src/legendre_polynomial/legendre_polynomial.html">
LEGENDRE_POLYNOMIAL</a>,
a C++ library which
evaluates the Legendre polynomial and associated functions.
</p>
<p>
<a href = "../../cpp_src/legendre_product_polynomial/legendre_product_polynomial.html">
LEGENDRE_PRODUCT_POLYNOMIAL</a>,
a C++ library which
defines Legendre product polynomials, creating a multivariate
polynomial as the product of univariate Legendre polynomials.
</p>
<p>
<a href = "../../cpp_src/lobatto_polynomial/lobatto_polynomial.html">
LOBATTO_POLYNOMIAL</a>,
a C++ library which
evaluates Lobatto polynomials, similar to Legendre polynomials
except that they are zero at both endpoints.
</p>
<p>
<a href = " ../../cpp_src/test_values/test_values.html">
TEST_VALUES</a>,
a C++ library which
stores values of many mathematical functions, and can be used for
testing.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Milton Abramowitz, Irene Stegun,<br>
Handbook of Mathematical Functions,<br>
National Bureau of Standards, 1964,<br>
ISBN: 0-486-61272-4,<br>
LC: QA47.A34.
</li>
<li>
Robert Banks,<br>
Slicing Pizzas, Racing Turtles, and Further Adventures in Applied Mathematics,<br>
Princeton, 1999,<br>
ISBN13: 9780691059471,<br>
LC: QA93.B358.
</li>
<li>
Frank Benford,<br>
The Law of Anomalous Numbers,<br>
Proceedings of the American Philosophical Society,<br>
Volume 78, 1938, pages 551-572.
</li>
<li>
Paul Bratley, Bennett Fox, Linus Schrage,<br>
A Guide to Simulation,<br>
Second Edition,<br>
Springer, 1987,<br>
ISBN: 0387964673,<br>
LC: QA76.9.C65.B73.
</li>
<li>
Chad Brewbaker,<br>
Lonesum (0,1)-matrices and poly-Bernoulli numbers of negative
index,<br>
Master of Science Thesis,<br>
Computer Science Department,<br>
Iowa State University, 2005.
</li>
<li>
William Briggs, Van Emden Henson,<br>
The DFT: An Owner's Manual for the Discrete Fourier Transform,<br>
SIAM, 1995,<br>
ISBN13: 978-0-898713-42-8,<br>
LC: QA403.5.B75.
</li>
<li>
Theodore Chihara,<br>
An Introduction to Orthogonal Polynomials,<br>
Gordon and Breach, 1978,<br>
ISBN: 0677041500,<br>
LC: QA404.5 C44.
</li>
<li>
William Cody,<br>
Rational Chebyshev Approximations for the Error Function,<br>
Mathematics of Computation,<br>
Volume 23, Number 107, July 1969, pages 631-638.
</li>
<li>
Robert Corless, Gaston Gonnet, David Hare, David Jeffrey,
Donald Knuth,<br>
On the Lambert W Function,<br>
Advances in Computational Mathematics,<br>
Volume 5, Number 1, December 1996, pages 329-359.
</li>
<li>
Bennett Fox,<br>
Algorithm 647:
Implementation and Relative Efficiency of Quasirandom
Sequence Generators,<br>
ACM Transactions on Mathematical Software,<br>
Volume 12, Number 4, December 1986, pages 362-376.
</li>
<li>
Walter Gautschi,<br>
Orthogonal Polynomials: Computation and Approximation,<br>
Oxford, 2004,<br>
ISBN: 0-19-850672-4,<br>
LC: QA404.5 G3555.
</li>
<li>
Ralph Hartley,<br>
A More Symmetrical Fourier Analysis Applied to Transmission
Problems,<br>
Proceedings of the Institute of Radio Engineers,<br>
Volume 30, 1942, pages 144-150.
</li>
<li>
Brian Hayes,<br>
The Vibonacci Numbers,<br>
American Scientist,<br>
Volume 87, Number 4, July-August 1999, pages 296-301.
</li>
<li>
Brian Hayes,<br>
Why W?,<br>
American Scientist,<br>
Volume 93, Number 2, March-April 2005, pages 104-108.
</li>
<li>
Ted Hill,<br>
The First Digit Phenomenon,<br>
American Scientist,<br>
Volume 86, Number 4, July/August 1998, pages 358-363.
</li>
<li>
Douglas Hofstadter,<br>
Goedel, Escher, Bach,<br>
Basic Books, 1979,<br>
ISBN: 0465026567,<br>
LC: QA9.8H63.
</li>
<li>
Masanobu Kaneko,<br>
Poly-Bernoulli Numbers,<br>
Journal Theorie des Nombres Bordeaux,<br>
Volume 9, Number 1, 1997, pages 221-228.
</li>
<li>
Cleve Moler,<br>
Trigonometry is a Complex Subject,<br>
MATLAB News and Notes, Summer 1998.
</li>
<li>
Thomas Osler,<br>
Cardan Polynomials and the Reduction of Radicals,<br>
Mathematics Magazine, <br>
Volume 74, Number 1, February 2001, pages 26-32.
</li>
<li>
J Simoes Pereira,<br>
Algorithm 234: Poisson-Charliers Polynomials,<br>
Communications of the ACM,<br>
Volume 7, Number 7, July 1964, page 420.
</li>
<li>
Charles Pinter,<br>
A Book of Abstract Algebra,<br>
Second Edition,<br>
McGraw Hill, 2003,<br>
ISBN: 0072943505,<br>
LC: QA162.P56.
</li>
<li>
Ralph Raimi,<br>
The Peculiar Distribution of First Digits,<br>
Scientific American,<br>
December 1969, pages 109-119.
</li>
<li>
Dennis Stanton, Dennis White,<br>
Constructive Combinatorics,<br>
Springer, 1986,<br>
ISBN: 0387963472,<br>
LC: QA164.S79.
</li>
<li>
Gabor Szego,<br>
Orthogonal Polynomials,<br>
American Mathematical Society, 1992,<br>
ISBN: 0821810235,<br>
LC: QA3.A5.v23.
</li>
<li>
Daniel Velleman, Gregory Call,<br>
Permutations and Combination Locks,<br>
Mathematics Magazine,<br>
Volume 68, Number 4, October 1995, pages 243-253.
</li>
<li>
Divakar Viswanath,<br>
Random Fibonacci sequences and the number 1.13198824,<br>
Mathematics of Computation,<br>
Volume 69, Number 231, July 2000, pages 1131-1155.
</li>
<li>
Michael Waterman,<br>
Introduction to Computational Biology,<br>
Chapman and Hall, 1995,<br>
ISBN: 0412993910,<br>
LC: QH438.4.M33.W38.
</li>
<li>
Eric Weisstein,<br>
CRC Concise Encyclopedia of Mathematics,<br>
CRC Press, 2002,<br>
Second edition,<br>
ISBN: 1584883472,<br>
LC: QA5.W45
</li>
<li>
Stephen Wolfram,<br>
The Mathematica Book,<br>
Fourth Edition,<br>
Cambridge University Press, 1999,<br>
ISBN: 0-521-64314-7,<br>
LC: QA76.95.W65.
</li>
<li>
ML Wolfson, HV Wright,<br>
ACM Algorithm 160: Combinatorial of M Things Taken N at a Time,<br>
Communications of the ACM,<br>
Volume 6, Number 4, April 1963, page 161.
</li>
<li>
Shanjie Zhang, Jianming Jin,<br>
Computation of Special Functions,<br>
Wiley, 1996,<br>
ISBN: 0-471-11963-6,<br>
LC: QA351.C45.
</li>
<li>
Daniel Zwillinger, editor,<br>
CRC Standard Mathematical Tables and Formulae,<br>
30th Edition,<br>
CRC Press, 1996,<br>
ISBN: 0-8493-2479-3,<br>
LC: QA47.M315.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "polpak.cpp">polpak.cpp</a>, the source code;
</li>
<li>
<a href = "polpak.hpp">polpak.hpp</a>, the include file for POLPAK;
</li>
<li>
<a href = "polpak.sh">polpak.sh</a>, commands to compile the code;
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "polpak_prb.cpp">polpak_prb.cpp</a>, the calling program;
</li>
<li>
<a href = "polpak_prb.sh">polpak_prb.sh</a>, commands to
compile, link and run the sample problem;
</li>
<li>
<a href = "polpak_prb_output.txt">polpak_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>AGM</b> computes the arithmetic-geometric mean of A and B.
</li>
<li>
<b>AGM_VALUES</b> returns some values of the AGM.
</li>
<li>
<b>AGUD</b> evaluates the inverse Gudermannian function.
</li>
<li>
<b>ALIGN_ENUM</b> counts the alignments of two sequences of M and N elements.
</li>
<li>
<b>ARC_COSINE</b> computes the arc cosine function, with argument truncation.
</li>
<li>
<b>ARC_SINE</b> computes the arc sine function, with argument truncation.
</li>
<li>
<b>ASINH2</b> returns the inverse hyperbolic sine of a number.
</li>
<li>
<b>ATAN4</b> computes the inverse tangent of the ratio Y / X.
</li>
<li>
<b>ATANH2</b> returns the inverse hyperbolic tangent of a number.
</li>
<li>
<b>BELL</b> returns the Bell numbers from 0 to N.
</li>
<li>
<b>BELL_VALUES</b> returns some values of the Bell numbers.
</li>
<li>
<b>BENFORD</b> returns the Benford probability of one or more significant digits.
</li>
<li>
<b>BERNOULLI_NUMBER</b> computes the value of the Bernoulli numbers B(0) through B(N).
</li>
<li>
<b>BERNOULLI_NUMBER2</b> evaluates the Bernoulli numbers.
</li>
<li>
<b>BERNOULLI_NUMBER3</b> computes the value of the Bernoulli number B(N).
</li>
<li>
<b>BERNOULLI_NUMBER_VALUES</b> returns some values of the Bernoulli numbers.
</li>
<li>
<b>BERNOULLI_POLY</b> evaluates the Bernoulli polynomial of order N at X.
</li>
<li>
<b>BERNOULLI_POLY2</b> evaluates the N-th Bernoulli polynomial at X.
</li>
<li>
<b>BERNSTEIN_POLY</b> evaluates the Bernstein polynomials at a point X.
</li>
<li>
<b>BERNSTEIN_POLY_VALUES</b> returns some values of the Bernstein polynomials.
</li>
<li>
<b>BETA</b> returns the value of the Beta function.
</li>
<li>
<b>BETA_VALUES</b> returns some values of the Beta function.
</li>
<li>
<b>BPAB</b> evaluates at X the Bernstein polynomials based in [A,B].
</li>
<li>
<b>CARDAN</b> evaluates the Cardan polynomials.
</li>
<li>
<b>CARDAN_POLY_COEF</b> computes the coefficients of the N-th Cardan polynomial.
</li>
<li>
<b>CARDINAL_COS</b> evaluates the J-th cardinal cosine basis function.
</li>
<li>
<b>CARDINAL_SIN</b> evaluates the J-th cardinal sine basis function.
</li>
<li>
<b>CATALAN</b> computes the Catalan numbers, from C(0) to C(N).
</li>
<li>
<b>CATALAN_ROW_NEXT</b> computes row N of Catalan's triangle.
</li>
<li>
<b>CATALAN_VALUES</b> returns some values of the Catalan numbers.
</li>
<li>
<b>CHARLIER</b> evaluates Charlier polynomials at a point.
</li>
<li>
<b>CHEBY_T_POLY</b> evaluates Chebyshev polynomials T(n,x).
</li>
<li>
<b>CHEBY_T_POLY_COEF</b> evaluates coefficients of Chebyshev polynomials T(n,x).
</li>
<li>
<b>CHEBY_T_POLY_VALUES</b> returns values of Chebyshev polynomials T(n,x).
</li>
<li>
<b>CHEBY_T_POLY_ZERO</b> returns zeroes of Chebyshev polynomials T(n,x).
</li>
<li>
<b>CHEBY_U_POLY</b> evaluates Chebyshev polynomials U(n,x).
</li>
<li>
<b>CHEBY_U_POLY_COEF</b> evaluates coefficients of Chebyshev polynomials U(n,x).
</li>
<li>
<b>CHEBY_U_POLY_VALUES</b> returns values of Chebyshev polynomials U(n,x).
</li>
<li>
<b>CHEBY_U_POLY_ZERO</b> returns zeroes of Chebyshev polynomials U(n,x).
</li>
<li>
<b>CHEBYSHEV_DISCRETE</b> evaluates discrete Chebyshev polynomials at a point.
</li>
<li>
<b>COLLATZ_COUNT</b> counts the number of terms in a Collatz sequence.
</li>
<li>
<b>COLLATZ_COUNT_MAX</b> seeks the maximum Collatz count for 1 through N.
</li>
<li>
<b>COLLATZ_COUNT_VALUES</b> returns some values of the Collatz count function.
</li>
<li>
<b>COMB_ROW</b> computes row N of Pascal's triangle.
</li>
<li>
<b>COMMUL</b> computes a multinomial combinatorial coefficient.
</li>
<li>
<b>COMPLETE_SYMMETRIC_POLY</b> evaluates a complete symmetric polynomial.
</li>
<li>
<b>COS_DEG</b> returns the cosine of an angle given in degrees.
</li>
<li>
<b>COS_POWER_INT</b> evaluates the cosine power integral.
</li>
<li>
<b>COS_POWER_INT_VALUES</b> returns some values of the sine power integral.
</li>
<li>
<b>E_CONSTANT</b> returns the value of the base of the natural logarithm system.
</li>
<li>
<b>ERF_VALUES</b> returns some values of the ERF or "error" function.
</li>
<li>
<b>ERROR_F</b> evaluates the error function ERF(X).
</li>
<li>
<b>ERROR_F_INVERSE</b> inverts the error function ERF.
</li>
<li>
<b>EULER_CONSTANT</b> returns the value of the Euler-Mascheroni constant.
</li>
<li>
<b>EULER_NUMBER</b> computes the Euler numbers.
</li>
<li>
<b>EULER_NUMBER2</b> computes the Euler numbers.
</li>
<li>
<b>EULER_NUMBER_VALUES</b> returns some values of the Euler numbers.
</li>
<li>
<b>EULER_POLY</b> evaluates the N-th Euler polynomial at X.
</li>
<li>
<b>EULERIAN</b> computes the Eulerian number E(N,K).
</li>
<li>
<b>F_HOFSTADTER</b> computes the Hofstadter F sequence.
</li>
<li>
<b>FIBONACCI_DIRECT</b> computes the N-th Fibonacci number directly.
</li>
<li>
<b>FIBONACCI_FLOOR</b> returns the largest Fibonacci number less or equal to N.
</li>
<li>
<b>FIBONACCI_RECURSIVE</b> computes the first N Fibonacci numbers.
</li>
<li>
<b>G_HOFSTADTER</b> computes the Hofstadter G sequence.
</li>
<li>
<b>GAMMA_LOG_VALUES</b> returns some values of the Log Gamma function.
</li>
<li>
<b>GAMMA_VALUES</b> returns some values of the Gamma function.
</li>
<li>
<b>GEGENBAUER_POLY</b> computes the Gegenbauer polynomials C(0:N,ALPHA,X).
</li>
<li>
<b>GEGENBAUER_POLY_VALUES</b> returns some values of the Gegenbauer polynomials.
</li>
<li>
<b>GEN_HERMITE_POLY</b> evaluates the generalized Hermite polynomials at X.
</li>
<li>
<b>GEN_LAGUERRE_POLY</b> evaluates generalized Laguerre polynomials.
</li>
<li>
<b>GUD</b> evaluates the Gudermannian function.
</li>
<li>
<b>GUD_VALUES</b> returns some values of the Gudermannian function.
</li>
<li>
<b>H_HOFSTADTER</b> computes the Hofstadter H sequence.
</li>
<li>
<b>HAIL</b> computes the hail function.
</li>
<li>
<b>HERMITE_POLY</b> evaluates the Hermite polynomials at X.
</li>
<li>
<b>HERMITE_POLY_COEF</b> evaluates the coefficients of the physicist's Hermite polynomial H(n,x).
</li>
<li>
<b>HERMITE_POLY_VALUES</b> returns some values of the Hermite polynomial.
</li>
<li>
<b>HYPERGEOMETRIC_CDF_VALUES</b> returns some values of the hypergeometric function 2F1.
</li>
<li>
<b>I4_CHOOSE</b> computes the binomial coefficient C(N,K).
</li>
<li>
<b>I4_FACTOR</b> factors an integer into prime factors.
</li>
<li>
<b>I4_FACTORIAL</b> computes the factorial of N.
</li>
<li>
<b>I4_FACTORIAL_VALUES</b> returns values of the factorial function.
</li>
<li>
<b>I4_FACTORIAL2</b> computes the double factorial function.
</li>
<li>
<b>I4_FACTORIAL2_VALUES</b> returns values of the double factorial function.
</li>
<li>
<b>I4_GCD</b> finds the greatest common divisor of two I4's.
</li>
<li>
<b>I4_IS_PRIME</b> reports whether an I4 is prime.
</li>
<li>
<b>I4_IS_TRIANGULAR</b> determines whether an I4 is triangular.
</li>
<li>
<b>I4_MAX</b> returns the maximum of two I4's.
</li>
<li>
<b>I4_MIN</b> returns the smaller of two I4's.
</li>
<li>
<b>I4_MODP</b> returns the nonnegative remainder of I4 division.
</li>
<li>
<b>I4_PARTITION_DISTINCT_COUNT</b> returns any value of Q(N).
</li>
<li>
<b>I4_POCHHAMMER</b> returns the value of ( I * (I+1) * ... * (J-1) * J ).
</li>
<li>
<b>I4_SIGN</b> returns the sign of an integer.
</li>
<li>
<b>I4_SWAP</b> switches two integer values.
</li>
<li>
<b>I4_TO_TRIANGLE</b> converts an integer to triangular coordinates.
</li>
<li>
<b>I4_UNIFORM_AB</b> returns a scaled pseudorandom I4.
</li>
<li>
<b>I4MAT_PRINT</b> prints an I4MAT.
</li>
<li>
<b>I4MAT_PRINT_SOME</b> prints some of an I4MAT.
</li>
<li>
<b>JACOBI_POLY</b> evaluates the Jacobi polynomials at X.
</li>
<li>
<b>JACOBI_POLY_VALUES</b> returns some values of the Jacobi polynomial.
</li>
<li>
<b>JACOBI_SYMBOL</b> evaluates the Jacobi symbol (Q/P).
</li>
<li>
<b>KRAWTCHOUK</b> evaluates the Krawtchouk polynomials at X.
</li>
<li>
<b>LAGUERRE_ASSOCIATED</b> evaluates the associated Laguerre polynomials L(N,M,X) at X.
</li>
<li>
<b>LAGUERRE_POLY</b> evaluates the Laguerre polynomials at X.
</li>
<li>
<b>LAGUERRE_POLY_COEF</b> evaluates the Laguerre polynomial coefficients.
</li>
<li>
<b>LAGUERRE_POLYNOMIAL_VALUES</b> returns some values of the Laguerre polynomial.
</li>
<li>
<b>LEGENDRE_ASSOCIATED</b> evaluates the associated Legendre functions.
</li>
<li>
<b>LEGENDRE_ASSOCIATED_NORMALIZED:</b> normalized associated Legendre functions.
</li>
<li>
<b>LEGENDRE_ASSOCIATED_NORMALIZED_VALUES:</b> normalied associated Legendre.
</li>
<li>
<b>LEGENDRE_ASSOCIATED_VALUES</b> returns values of associated Legendre functions.
</li>
<li>
<b>LEGENDRE_FUNCTION_Q</b> evaluates the Legendre Q functions.
</li>
<li>
<b>LEGENDRE_FUNCTION_Q_VALUES</b> returns values of the Legendre Q function.
</li>
<li>
<b>LEGENDRE_POLY</b> evaluates the Legendre polynomials.
</li>
<li>
<b>LEGENDRE_POLY_COEF</b> evaluates the Legendre polynomial coefficients.
</li>
<li>
<b>LEGENDRE_POLY_VALUES</b> returns values of the Legendre polynomials.
</li>
<li>
<b>LEGENDRE_SYMBOL</b> evaluates the Legendre symbol (Q/P).
</li>
<li>
<b>LERCH</b> estimates the Lerch transcendent function.
</li>
<li>
<b>LERCH_VALUES</b> returns some values of the Lerch transcendent function.
</li>
<li>
<b>LOCK</b> returns the number of codes for a lock with N buttons.
</li>
<li>
<b>MEIXNER</b> evaluates Meixner polynomials at a point.
</li>
<li>
<b>MERTENS</b> evaluates the Mertens function.
</li>
<li>
<b>MERTENS_VALUES</b> returns some values of the Mertens function.
</li>
<li>
<b>MOEBIUS</b> returns the value of MU(N), the Moebius function of N.
</li>
<li>
<b>MOEBIUS_VALUES</b> returns some values of the Moebius function.
</li>
<li>
<b>MOTZKIN</b> returns the Motzkin numbers up to order N.
</li>
<li>
<b>NORMAL_01_CDF_INV</b> inverts the standard normal CDF.
</li>
<li>
<b>OMEGA</b> returns OMEGA(N), the number of distinct prime divisors of N.
</li>
<li>
<b>OMEGA_VALUES</b> returns some values of the OMEGA function.
</li>
<li>
<b>PARTITION_COUNT_VALUES</b> returns values of the integer partition count.
</li>
<li>
<b>PARTITION_DISTINCT_COUNT_VALUES</b> returns some values of Q(N).
</li>
<li>
<b>PENTAGON_NUM</b> computes the N-th pentagonal number.
</li>
<li>
<b>PHI</b> computes the number of relatively prime predecessors of an integer.
</li>
<li>
<b>PHI_VALUES</b> returns some values of the PHI function.
</li>
<li>
<b>PLANE_PARTITION_NUM</b> returns the number of plane partitions of the integer N.
</li>
<li>
<b>POLY_BERNOULLI</b> evaluates the poly-Bernolli numbers with negative index.
</li>
<li>
<b>POLY_COEF_COUNT:</b> polynomial coefficient count given dimension and degree.
</li>
<li>
<b>PRIME</b> returns any of the first PRIME_MAX prime numbers.
</li>
<li>
<b>PSI_VALUES</b> returns some values of the Psi or Digamma function.
</li>
<li>
<b>PYRAMID_NUM</b> returns the N-th pyramidal number.
</li>
<li>
<b>R8_ABS</b> returns the absolute value of an R8.
</li>
<li>
<b>R8_ACOSH</b> returns the inverse hyperbolic cosine of a number.
</li>
<li>
<b>R8_ASINH</b> returns the inverse hyperbolic sine of a number.
</li>
<li>
<b>R8_ATANH</b> returns the inverse hyperbolic tangent of a number.
</li>
<li>
<b>R8_CHOOSE</b> computes the combinatorial coefficient C(N,K).
</li>
<li>
<b>R8_COT</b> returns the cotangent of an angle.
</li>
<li>
<b>R8_COT_DEG</b> returns the cotangent of an angle given in degrees.
</li>
<li>
<b>R8_CSC</b> returns the cosecant of X.
</li>
<li>
<b>R8_CSC_DEG</b> returns the cosecant of an angle given in degrees.
</li>
<li>
<b>R8_EPSILON</b> returns the R8 roundoff unit.
</li>
<li>
<b>R8_FACTORIAL</b> returns the factorial function as an R8.
</li>
<li>
<b>R8_FACTORIAL_LOG</b> computes the natural logarithm of the factorial function.
</li>
<li>
<b>R8_FACTORIAL_LOG_VALUES</b> returns values of log(factorial(n)).
</li>
<li>
<b>R8_FACTORIAL_VALUES</b> returns values of the real factorial function.
</li>
<li>
<b>R8_FACTORIAL2</b> computes the double factorial function.
</li>
<li>
<b>R8_GAMMA</b> evaluates Gamma(X) for a real argument.
</li>
<li>
<b>R8_GAMMA_LOG</b> calculates the natural logarithm of GAMMA ( X ) for positive X.
</li>
<li>
<b>R8_HUGE</b> returns a "huge" R8.
</li>
<li>
<b>R8_HYPER_2F1</b> evaluates the hypergeometric function F(A,B,C,X).
</li>
<li>
<b>R8_MAX</b> returns the maximum of two R8's.
</li>
<li>
<b>R8_MIN</b> returns the minimum of two R8's.
</li>
<li>
<b>R8_MOP</b> returns the I-th power of -1 as an R8 value.
</li>
<li>
<b>R8_NINT</b> returns the integer that is nearest to a real value.
</li>
<li>
<b>R8_PI</b> returns the value of PI.
</li>
<li>
<b>R8_PSI</b> evaluates the function Psi(X).
</li>
<li>
<b>R8_UNIFORM_01</b> is a portable pseudorandom number generator.
</li>
<li>
<b>R8POLY_DEGREE</b> returns the degree of a polynomial.
</li>
<li>
<b>R8POLY_PRINT</b> prints out a polynomial.
</li>
<li>
<b>R8POLY_VALUE</b> evaluates a double precision polynomial.
</li>
<li>
<b>R8VEC_LINSPACE_NEW</b> creates a vector of linearly spaced values.
</li>
<li>
<b>R8VEC_PRINT</b> prints an R8VEC.
</li>
<li>
<b>R8VEC_ZERO</b> zeroes a real vector.
</li>
<li>
<b>S_LEN_TRIM</b> returns the length of a string to the last nonblank.
</li>
<li>
<b>SEC_DEG</b> returns the secant of an angle given in degrees.
</li>
<li>
<b>SIGMA</b> returns the value of SIGMA(N), the divisor sum.
</li>
<li>
<b>SIGMA_VALUES</b> returns some values of the Sigma function.
</li>
<li>
<b>SIN_DEG</b> returns the sine of an angle given in degrees.
</li>
<li>
<b>SIN_POWER_INT</b> evaluates the sine power integral.
</li>
<li>
<b>SIN_POWER_INT_VALUES</b> returns some values of the sine power integral.
</li>
<li>
<b>SLICE:</b> maximum number of pieces created by a given number of slices.
</li>
<li>
<b>SPHERICAL_HARMONIC</b> evaluates spherical harmonic functions.