POLPAK
Recursive Polynomials
POLPAK
is a C library which
evaluates a variety of mathematical functions.
It includes routines to evaluate the
recursively-defined polynomial families of
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Bernoulli
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Bernstein
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Cardan
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Charlier
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Chebyshev
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Euler
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Gegenbauer
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Hermite
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Jacobi
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Krawtchouk
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Laguerre
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Legendre
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Meixner
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Zernike
A variety of other polynomials and functions have been added.
Licensing:
The computer code and data files described and made available on this web page
are distributed under
the GNU LGPL license.
Languages:
POLPAK is available in
a C version and
a C++ version and
a FORTRAN90 version and
a MATLAB version and
a Python version
Related Data and Programs:
BERNSTEIN_POLYNOMIAL,
a C library which
evaluates the Bernstein polynomials,
useful for uniform approximation of functions;
BESSELJ,
a C library which
evaluates Bessel J functions of noninteger order.
CLAUSEN,
a C library which
evaluates a Chebyshev interpolant to the Clausen function Cl2(x).
CORDIC,
a C library which
uses the CORDIC method to compute certain elementary functions.
FN,
a C library which
approximates elementary and special functions using Chebyshev polynomials,
by Wayne Fullerton.
LEGENDRE_PRODUCT_POLYNOMIAL,
a C library which
defines Legendre product polynomials, creating a multivariate
polynomial as the product of univariate Legendre polynomials.
LOBATTO_POLYNOMIAL,
a C library which
evaluates Lobatto polynomials, similar to Legendre polynomials
except that they are zero at both endpoints.
TEST_VALUES,
a C library which
stores values of many mathematical functions, and can be used for
testing.
Reference:
-
Milton Abramowitz, Irene Stegun,
Handbook of Mathematical Functions,
National Bureau of Standards, 1964,
ISBN: 0-486-61272-4,
LC: QA47.A34.
-
Robert Banks,
Slicing Pizzas, Racing Turtles, and Further Adventures in Applied
Mathematics,
Princeton, 1999,
ISBN13: 9780691059471,
LC: QA93.B358.
-
Frank Benford,
The Law of Anomalous Numbers,
Proceedings of the American Philosophical Society,
Volume 78, 1938, pages 551-572.
-
Paul Bratley, Bennett Fox, Linus Schrage,
A Guide to Simulation,
Second Edition,
Springer, 1987,
ISBN: 0387964673,
LC: QA76.9.C65.B73.
-
Chad Brewbaker,
Lonesum (0,1)-matrices and poly-Bernoulli numbers of negative
index,
Master of Science Thesis,
Computer Science Department,
Iowa State University, 2005.
-
William Briggs, Van Emden Henson,
The DFT: An Owner's Manual for the Discrete Fourier Transform,
SIAM, 1995,
ISBN13: 978-0-898713-42-8,
LC: QA403.5.B75.
-
Theodore Chihara,
An Introduction to Orthogonal Polynomials,
Gordon and Breach, 1978,
ISBN: 0677041500,
LC: QA404.5 C44.
-
William Cody,
Rational Chebyshev Approximations for the Error Function,
Mathematics of Computation,
Volume 23, Number 107, July 1969, pages 631-638.
-
Robert Corless, Gaston Gonnet, David Hare, David Jeffrey,
Donald Knuth,
On the Lambert W Function,
Advances in Computational Mathematics,
Volume 5, Number 1, December 1996, pages 329-359.
-
Bennett Fox,
Algorithm 647:
Implementation and Relative Efficiency of Quasirandom
Sequence Generators,
ACM Transactions on Mathematical Software,
Volume 12, Number 4, December 1986, pages 362-376.
-
Walter Gautschi,
Orthogonal Polynomials: Computation and Approximation,
Oxford, 2004,
ISBN: 0-19-850672-4,
LC: QA404.5 G3555.
-
Ralph Hartley,
A More Symmetrical Fourier Analysis Applied to Transmission
Problems,
Proceedings of the Institute of Radio Engineers,
Volume 30, 1942, pages 144-150.
-
Brian Hayes,
The Vibonacci Numbers,
American Scientist,
Volume 87, Number 4, July-August 1999, pages 296-301.
-
Brian Hayes,
Why W?,
American Scientist,
Volume 93, Number 2, March-April 2005, pages 104-108.
-
Ted Hill,
The First Digit Phenomenon,
American Scientist,
Volume 86, Number 4, July/August 1998, pages 358-363.
-
Douglas Hofstadter,
Goedel, Escher, Bach,
Basic Books, 1979,
ISBN: 0465026567,
LC: QA9.8H63.
-
Masanobu Kaneko,
Poly-Bernoulli Numbers,
Journal Theorie des Nombres Bordeaux,
Volume 9, Number 1, 1997, pages 221-228.
-
Cleve Moler,
Trigonometry is a Complex Subject,
MATLAB News and Notes, Summer 1998.
-
Thomas Osler,
Cardan Polynomials and the Reduction of Radicals,
Mathematics Magazine,
Volume 74, Number 1, February 2001, pages 26-32.
-
J Simoes Pereira,
Algorithm 234: Poisson-Charliers Polynomials,
Communications of the ACM,
Volume 7, Number 7, July 1964, page 420.
-
Charles Pinter,
A Book of Abstract Algebra,
Second Edition,
McGraw Hill, 2003,
ISBN: 0072943505,
LC: QA162.P56.
-
Ralph Raimi,
The Peculiar Distribution of First Digits,
Scientific American,
December 1969, pages 109-119.
-
Dennis Stanton, Dennis White,
Constructive Combinatorics,
Springer, 1986,
ISBN: 0387963472,
LC: QA164.S79.
-
Gabor Szego,
Orthogonal Polynomials,
American Mathematical Society, 1992,
ISBN: 0821810235,
LC: QA3.A5.v23.
-
Daniel Velleman, Gregory Call,
Permutations and Combination Locks,
Mathematics Magazine,
Volume 68, Number 4, October 1995, pages 243-253.
-
Divakar Viswanath,
Random Fibonacci sequences and the number 1.13198824,
Mathematics of Computation,
Volume 69, Number 231, July 2000, pages 1131-1155.
-
Michael Waterman,
Introduction to Computational Biology,
Chapman and Hall, 1995,
ISBN: 0412993910,
LC: QH438.4.M33.W38.
-
Eric Weisstein,
CRC Concise Encyclopedia of Mathematics,
CRC Press, 2002,
Second edition,
ISBN: 1584883472,
LC: QA5.W45
-
Stephen Wolfram,
The Mathematica Book,
Fourth Edition,
Cambridge University Press, 1999,
ISBN: 0-521-64314-7,
LC: QA76.95.W65.
-
ML Wolfson, HV Wright,
ACM Algorithm 160: Combinatorial of M Things Taken N at a Time,
Communications of the ACM,
Volume 6, Number 4, April 1963, page 161.
-
Shanjie Zhang, Jianming Jin,
Computation of Special Functions,
Wiley, 1996,
ISBN: 0-471-11963-6,
LC: QA351.C45.
-
Daniel Zwillinger, editor,
CRC Standard Mathematical Tables and Formulae,
30th Edition,
CRC Press, 1996,
ISBN: 0-8493-2479-3,
LC: QA47.M315.
Source Code:
Examples and Tests:
List of Routines:
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AGM computes the arithmetic-geometric mean of A and B.
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AGM_VALUES returns some values of the AGM.
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AGUD evaluates the inverse Gudermannian function.
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ALIGN_ENUM counts the alignments of two sequences of M and N elements.
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ARC_COSINE computes the arc cosine function, with argument truncation.
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ARC_SINE computes the arc sine function, with argument truncation.
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ATAN4 computes the inverse tangent of the ratio Y / X.
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ATANH2 returns the inverse hyperbolic tangent of a number.
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BELL returns the Bell numbers from 0 to N.
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BELL_VALUES returns some values of the Bell numbers.
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BENFORD returns the Benford probability of one or more significant digits.
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BERNOULLI_NUMBER computes the value of the Bernoulli numbers B(0) through B(N).
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BERNOULLI_NUMBER2 evaluates the Bernoulli numbers.
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BERNOULLI_NUMBER3 computes the value of the Bernoulli number B(N).
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BERNOULLI_NUMBER_VALUES returns some values of the Bernoulli numbers.
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BERNOULLI_POLY evaluates the Bernoulli polynomial of order N at X.
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BERNOULLI_POLY2 evaluates the N-th Bernoulli polynomial at X.
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BERNSTEIN_POLY evaluates the Bernstein polynomials at a point X.
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BERNSTEIN_POLY_VALUES returns some values of the Bernstein polynomials.
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BETA returns the value of the Beta function.
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BETA_VALUES returns some values of the Beta function.
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BPAB evaluates at X the Bernstein polynomials based in [A,B].
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CARDAN evaluates the Cardan polynomials.
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CARDAN_POLY_COEF computes the coefficients of the N-th Cardan polynomial.
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CARDINAL_COS evaluates the J-th cardinal cosine basis function.
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CARDINAL_SIN evaluates the J-th cardinal sine basis function.
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CATALAN computes the Catalan numbers, from C(0) to C(N).
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CATALAN_ROW_NEXT computes row N of Catalan's triangle.
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CATALAN_VALUES returns some values of the Catalan numbers.
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CHARLIER evaluates Charlier polynomials at a point.
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CHEBY_T_POLY evaluates Chebyshev polynomials T(n,x).
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CHEBY_T_POLY_COEF evaluates coefficients of Chebyshev polynomials T(n,x).
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CHEBY_T_POLY_VALUES returns values of Chebyshev polynomials T(n,x).
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CHEBY_T_POLY_ZERO returns zeroes of Chebyshev polynomials T(n,x).
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CHEBY_U_POLY evaluates Chebyshev polynomials U(n,x).
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CHEBY_U_POLY_COEF evaluates coefficients of Chebyshev polynomials U(n,x).
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CHEBY_U_POLY_VALUES returns values of Chebyshev polynomials U(n,x).
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CHEBY_U_POLY_ZERO returns zeroes of Chebyshev polynomials U(n,x).
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CHEBYSHEV_DISCRETE evaluates discrete Chebyshev polynomials at a point.
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COLLATZ_COUNT counts the number of terms in a Collatz sequence.
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COLLATZ_COUNT_MAX seeks the maximum Collatz count for 1 through N.
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COLLATZ_COUNT_VALUES returns some values of the Collatz count function.
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COMB_ROW computes row N of Pascal's triangle.
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COMMUL computes a multinomial combinatorial coefficient.
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COMPLETE_SYMMETRIC_POLY evaluates a complete symmetric polynomial.
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COS_DEG returns the cosine of an angle given in degrees.
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COS_POWER_INT evaluates the cosine power integral.
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COS_POWER_INT_VALUES returns some values of the cosine power integral.
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E_CONSTANT returns the value of the base of the natural logarithm system.
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ERF_VALUES returns some values of the ERF or "error" function.
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ERROR_F evaluates the error function ERF(X).
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ERROR_F_INVERSE inverts the error function ERF.
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EULER_CONSTANT returns the value of the Euler-Mascheroni constant.
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EULER_NUMBER computes the Euler numbers.
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EULER_NUMBER2 computes the Euler numbers.
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EULER_NUMBER_VALUES returns some values of the Euler numbers.
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EULER_POLY evaluates the N-th Euler polynomial at X.
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EULERIAN computes the Eulerian number E(N,K).
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F_HOFSTADTER computes the Hofstadter F sequence.
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FIBONACCI_DIRECT computes the N-th Fibonacci number directly.
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FIBONACCI_FLOOR returns the largest Fibonacci number less or equal to N.
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FIBONACCI_RECURSIVE computes the first N Fibonacci numbers.
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G_HOFSTADTER computes the Hofstadter G sequence.
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GAMMA_LOG_VALUES returns some values of the Log Gamma function.
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GAMMA_VALUES returns some values of the Gamma function.
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GEGENBAUER_POLY computes the Gegenbauer polynomials C(0:N,ALPHA,X).
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GEGENBAUER_POLY_VALUES returns some values of the Gegenbauer polynomials.
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GEN_HERMITE_POLY evaluates the generalized Hermite polynomials at X.
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GEN_LAGUERRE_POLY evaluates generalized Laguerre polynomials.
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GUD evaluates the Gudermannian function.
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GUD_VALUES returns some values of the Gudermannian function.
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H_HOFSTADTER computes the Hofstadter H sequence.
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HAIL computes the hail function.
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HERMITE_POLY evaluates the Hermite polynomials at X.
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HERMITE_POLY_COEF evaluates the Hermite polynomial coefficients.
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HERMITE_POLY_VALUES returns some values of the Hermite polynomial.
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HYPERGEOMETRIC_CDF_VALUES returns some values of the hypergeometric function 2F1.
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I4_CHOOSE computes the binomial coefficient C(N,K).
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I4_FACTOR factors an integer into prime factors.
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I4_FACTORIAL computes the factorial of N.
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I4_FACTORIAL_VALUES returns values of the factorial function.
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I4_FACTORIAL2 computes the double factorial function.
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I4_FACTORIAL2_VALUES returns values of the double factorial function.
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I4_GCD finds the greatest common divisor of two I4's.
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I4_IS_PRIME reports whether an I4 is prime.
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I4_IS_TRIANGULAR determines whether an I4 is triangular.
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I4_MAX returns the maximum of two I4's.
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I4_MIN returns the smaller of two I4's.
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I4_MODP returns the nonnegative remainder of I4 division.
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I4_PARTITION_DISTINCT_COUNT returns any value of Q(N).
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I4_POCHHAMMER returns the value of ( I * (I+1) * ... * (J-1) * J ).
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I4_SIGN returns the sign of an I4.
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I4_SWAP switches two I4's.
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I4_TO_TRIANGLE converts an integer to triangular coordinates.
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I4_UNIFORM_AB returns a scaled pseudorandom I4.
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I4MAT_PRINT prints an I4MAT.
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I4MAT_PRINT_SOME prints some of an I4MAT.
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JACOBI_POLY evaluates the Jacobi polynomials at X.
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JACOBI_POLY_VALUES returns some values of the Jacobi polynomial.
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JACOBI_SYMBOL evaluates the Jacobi symbol (Q/P).
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LAGUERRE_ASSOCIATED evaluates the associated Laguerre polynomials L(N,M,X) at X.
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LAGUERRE_POLY evaluates the Laguerre polynomials at X.
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LAGUERRE_POLY_COEF evaluates the Laguerre polynomial coefficients.
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LAGUERRE_POLYNOMIAL_VALUES returns some values of the Laguerre polynomial.
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LEGENDRE_ASSOCIATED evaluates the associated Legendre functions.
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LEGENDRE_ASSOCIATED_NORMALIZED: normalized associated Legendre functions.
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LEGENDRE_ASSOCIATED_NORMALIZED_VALUES: normalied associated Legendre.
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LEGENDRE_ASSOCIATED_VALUES returns values of associated Legendre functions.
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LEGENDRE_FUNCTION_Q evaluates the Legendre Q functions.
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LEGENDRE_FUNCTION_Q_VALUES returns values of the Legendre Q function.
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LEGENDRE_POLY evaluates the Legendre polynomials.
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LEGENDRE_POLY_COEF evaluates the Legendre polynomial coefficients.
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LEGENDRE_POLY_VALUES returns values of the Legendre polynomials.
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LEGENDRE_SYMBOL evaluates the Legendre symbol (Q/P).
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LERCH estimates the Lerch transcendent function.
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LERCH_VALUES returns some values of the Lerch transcendent function.
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LOCK returns the number of codes for a lock with N buttons.
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MEIXNER evaluates Meixner polynomials at a point.
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MERTENS evaluates the Mertens function.
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MERTENS_VALUES returns some values of the Mertens function.
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MOEBIUS returns the value of MU(N), the Moebius function of N.
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MOEBIUS_VALUES returns some values of the Moebius function.
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MOTZKIN returns the Motzkin numbers up to order N.
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NORMAL_01_CDF_INV inverts the standard normal CDF.
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OMEGA returns OMEGA(N), the number of distinct prime divisors of N.
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OMEGA_VALUES returns some values of the OMEGA function.
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PARTITION_COUNT_VALUES returns some values of the int *partition count.
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PARTITION_DISTINCT_COUNT_VALUES returns some values of Q(N).
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PENTAGON_NUM computes the N-th pentagonal number.
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PHI computes the number of relatively prime predecessors of an integer.
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PHI_VALUES returns some values of the PHI function.
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PLANE_PARTITION_NUM returns the number of plane partitions of the integer N.
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POLY_BERNOULLI evaluates the poly-Bernolli numbers with negative index.
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POLY_COEF_COUNT: polynomial coefficient count given dimension and degree.
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PRIME returns any of the first PRIME_MAX prime numbers.
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PSI_VALUES returns some values of the Psi or Digamma function.
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PYRAMID_NUM returns the N-th pyramidal number.
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R8_ABS returns the absolute value of an R8.
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R8_ACOSH returns the inverse hyperbolic cosine of a number.
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R8_ASINH returns the inverse hyperbolic sine of a number.
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R8_ATANH returns the inverse hyperbolic tangent of a number.
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R8_CHOOSE computes the binomial coefficient C(N,K) as an R8.
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R8_COT returns the cotangent of an angle.
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R8_COT_DEG returns the cotangent of an angle given in degrees.
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R8_CSC returns the cosecant of X.
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R8_CSC_DEG returns the cosecant of an angle given in degrees.
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R8_EPSILON returns the R8 roundoff unit.
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R8_FACTORIAL computes the factorial of N.
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R8_FACTORIAL_LOG computes the natural logarithm of the factorial function.
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R8_FACTORIAL_LOG_VALUES returns values of log(n!).
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R8_FACTORIAL_VALUES returns values of the real factorial function.
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R8_FACTORIAL2 computes the double factorial function.
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R8_GAMMA evaluates Gamma(X) for a real argument.
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R8_GAMMA_LOG calculates the natural logarithm of GAMMA ( X ) for positive X.
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R8_HUGE returns a "huge" R8.
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R8_HYPER_2F1 evaluates the hypergeometric function 2F1(A,B,C,X).
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R8_MAX returns the maximum of two R8's.
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R8_MIN returns the minimum of two R8's.
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R8_MOP returns the I-th power of -1 as an R8 value.
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R8_NINT returns the nearest integer to an R8.
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R8_PI returns the value of PI to 16 digits.
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R8_PSI evaluates the function Psi(X).
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R8_UNIFORM_01 returns a unit pseudorandom R8.
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R8POLY_DEGREE returns the degree of an R8POLY.
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R8POLY_PRINT prints out a polynomial.
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R8POLY_VALUE evaluates a double precision polynomial.
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R8VEC_LINSPACE_NEW creates a vector of linearly spaced values.
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R8VEC_PRINT prints an R8VEC.
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R8VEC_ZERO zeroes an R8VEC.
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S_LEN_TRIM returns the length of a string to the last nonblank.
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SEC_DEG returns the secant of an angle given in degrees.
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SIGMA returns the value of SIGMA(N), the divisor sum.
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SIGMA_VALUES returns some values of the Sigma function.
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SIN_DEG returns the sine of an angle given in degrees.
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SIN_POWER_INT evaluates the sine power integral.
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SIN_POWER_INT_VALUES returns some values of the sine power integral.
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SLICE: maximum number of pieces created by a given number of slices.
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SPHERICAL_HARMONIC evaluates spherical harmonic functions.
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SPHERICAL_HARMONIC_VALUES returns values of spherical harmonic functions.
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STIRLING1 computes the Stirling numbers of the first kind.
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STIRLING2 computes the Stirling numbers of the second kind.
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TAN_DEG returns the tangent of an angle given in degrees.
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TAU returns the value of TAU(N), the number of distinct divisors of N.
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TAU_VALUES returns some values of the Tau function.
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TETRAHEDRON_NUM returns the N-th tetrahedral number.
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TIMESTAMP prints the current YMDHMS date as a time stamp.
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TRIANGLE_NUM returns the N-th triangular number.
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TRIANGLE_TO_I4 converts a triangular coordinate to an integer.
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V_HOFSTADTER computes the Hofstadter V sequence.
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VIBONACCI computes the first N Vibonacci numbers.
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ZECKENDORF produces the Zeckendorf decomposition of a positive integer.
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ZERNIKE_POLY evaluates a Zernike polynomial at RHO.
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ZERNIKE_POLY_COEF: coefficients of a Zernike polynomial.
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ZETA estimates the Riemann Zeta function.
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ZETA_VALUES returns some values of the Riemann Zeta function.
You can go up one level to
the C source codes.
Last revised on 13 May 2014.