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

• Bernoulli
• Bernstein
• Cardan
• Charlier
• Chebyshev
• Euler
• Gegenbauer
• Hermite
• Jacobi
• Krawtchouk
• Laguerre
• Legendre
• Meixner
• Zernike

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.

CHEBYSHEV_POLYNOMIAL, a C++ library which evaluates the Chebyshev polynomial and associated functions.

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 evaluates elementary and special functions, by Wayne Fullerton.

GSL a C++ library which evaluates many special functions.

HERMITE_POLYNOMIAL, a C++ library which evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial, the Hermite function, and related functions.

JACOBI_POLYNOMIAL, a C++ library which evaluates the Jacobi polynomial and associated functions.

LAGUERRE_POLYNOMIAL, a C++ library which evaluates the Laguerre polynomial, the generalized Laguerre polynomial, and the Laguerre function.

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

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:

1. Milton Abramowitz, Irene Stegun,
   Handbook of Mathematical Functions,
   National Bureau of Standards, 1964,
   ISBN: 0-486-61272-4,
   LC: QA47.A34.
2. Robert Banks,
   Slicing Pizzas, Racing Turtles, and Further Adventures in Applied Mathematics,
   Princeton, 1999,
   ISBN13: 9780691059471,
   LC: QA93.B358.
3. Frank Benford,
   The Law of Anomalous Numbers,
   Proceedings of the American Philosophical Society,
   Volume 78, 1938, pages 551-572.
4. Paul Bratley, Bennett Fox, Linus Schrage,
   A Guide to Simulation,
   Second Edition,
   Springer, 1987,
   ISBN: 0387964673,
   LC: QA76.9.C65.B73.
5. Chad Brewbaker,
   Lonesum (0,1)-matrices and poly-Bernoulli numbers of negative index,
   Master of Science Thesis,
   Computer Science Department,
   Iowa State University, 2005.
6. 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.
7. Theodore Chihara,
   An Introduction to Orthogonal Polynomials,
   Gordon and Breach, 1978,
   ISBN: 0677041500,
   LC: QA404.5 C44.
8. William Cody,
   Rational Chebyshev Approximations for the Error Function,
   Mathematics of Computation,
   Volume 23, Number 107, July 1969, pages 631-638.
9. 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.
10. 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.
11. Walter Gautschi,
    Orthogonal Polynomials: Computation and Approximation,
    Oxford, 2004,
    ISBN: 0-19-850672-4,
    LC: QA404.5 G3555.
12. Ralph Hartley,
    A More Symmetrical Fourier Analysis Applied to Transmission Problems,
    Proceedings of the Institute of Radio Engineers,
    Volume 30, 1942, pages 144-150.
13. Brian Hayes,
    The Vibonacci Numbers,
    American Scientist,
    Volume 87, Number 4, July-August 1999, pages 296-301.
14. Brian Hayes,
    Why W?,
    American Scientist,
    Volume 93, Number 2, March-April 2005, pages 104-108.
15. Ted Hill,
    The First Digit Phenomenon,
    American Scientist,
    Volume 86, Number 4, July/August 1998, pages 358-363.
16. Douglas Hofstadter,
    Goedel, Escher, Bach,
    Basic Books, 1979,
    ISBN: 0465026567,
    LC: QA9.8H63.
17. Masanobu Kaneko,
    Poly-Bernoulli Numbers,
    Journal Theorie des Nombres Bordeaux,
    Volume 9, Number 1, 1997, pages 221-228.
18. Cleve Moler,
    Trigonometry is a Complex Subject,
    MATLAB News and Notes, Summer 1998.
19. Thomas Osler,
    Cardan Polynomials and the Reduction of Radicals,
    Mathematics Magazine,
    Volume 74, Number 1, February 2001, pages 26-32.
20. J Simoes Pereira,
    Algorithm 234: Poisson-Charliers Polynomials,
    Communications of the ACM,
    Volume 7, Number 7, July 1964, page 420.
21. Charles Pinter,
    A Book of Abstract Algebra,
    Second Edition,
    McGraw Hill, 2003,
    ISBN: 0072943505,
    LC: QA162.P56.
22. Ralph Raimi,
    The Peculiar Distribution of First Digits,
    Scientific American,
    December 1969, pages 109-119.
23. Dennis Stanton, Dennis White,
    Constructive Combinatorics,
    Springer, 1986,
    ISBN: 0387963472,
    LC: QA164.S79.
24. Gabor Szego,
    Orthogonal Polynomials,
    American Mathematical Society, 1992,
    ISBN: 0821810235,
    LC: QA3.A5.v23.
25. Daniel Velleman, Gregory Call,
    Permutations and Combination Locks,
    Mathematics Magazine,
    Volume 68, Number 4, October 1995, pages 243-253.
26. Divakar Viswanath,
    Random Fibonacci sequences and the number 1.13198824,
    Mathematics of Computation,
    Volume 69, Number 231, July 2000, pages 1131-1155.
27. Michael Waterman,
    Introduction to Computational Biology,
    Chapman and Hall, 1995,
    ISBN: 0412993910,
    LC: QH438.4.M33.W38.
28. Eric Weisstein,
    CRC Concise Encyclopedia of Mathematics,
    CRC Press, 2002,
    Second edition,
    ISBN: 1584883472,
    LC: QA5.W45
29. Stephen Wolfram,
    The Mathematica Book,
    Fourth Edition,
    Cambridge University Press, 1999,
    ISBN: 0-521-64314-7,
    LC: QA76.95.W65.
30. 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.
31. Shanjie Zhang, Jianming Jin,
    Computation of Special Functions,
    Wiley, 1996,
    ISBN: 0-471-11963-6,
    LC: QA351.C45.
32. Daniel Zwillinger, editor,
    CRC Standard Mathematical Tables and Formulae,
    30th Edition,
    CRC Press, 1996,
    ISBN: 0-8493-2479-3,
    LC: QA47.M315.

Source Code:

• polpak.cpp, the source code;
• polpak.hpp, the include file for POLPAK;

Examples and Tests:

• polpak_test.cpp, the calling program;
• polpak_test.txt, the output file.

List of Routines:

• AGM computes the arithmetic-geometric mean of A and B.
• AGM_VALUES returns some values of the AGM.
• AGUD evaluates the inverse Gudermannian function.
• ALIGN_ENUM counts the alignments of two sequences of M and N elements.
• ARC_COSINE computes the arc cosine function, with argument truncation.
• ARC_SINE computes the arc sine function, with argument truncation.
• ASINH2 returns the inverse hyperbolic sine of a number.
• ATAN4 computes the inverse tangent of the ratio Y / X.
• ATANH2 returns the inverse hyperbolic tangent of a number.
• BELL returns the Bell numbers from 0 to N.
• BELL_VALUES returns some values of the Bell numbers.
• BENFORD returns the Benford probability of one or more significant digits.
• BERNOULLI_NUMBER computes the value of the Bernoulli numbers B(0) through B(N).
• BERNOULLI_NUMBER2 evaluates the Bernoulli numbers.
• BERNOULLI_NUMBER3 computes the value of the Bernoulli number B(N).
• BERNOULLI_NUMBER_VALUES returns some values of the Bernoulli numbers.
• BERNOULLI_POLY evaluates the Bernoulli polynomial of order N at X.
• BERNOULLI_POLY2 evaluates the N-th Bernoulli polynomial at X.
• BERNSTEIN_POLY evaluates the Bernstein polynomials at a point X.
• BERNSTEIN_POLY_VALUES returns some values of the Bernstein polynomials.
• BETA returns the value of the Beta function.
• BETA_VALUES returns some values of the Beta function.
• BPAB evaluates at X the Bernstein polynomials based in [A,B].
• CARDAN evaluates the Cardan polynomials.
• CARDAN_POLY_COEF computes the coefficients of the N-th Cardan polynomial.
• CARDINAL_COS evaluates the J-th cardinal cosine basis function.
• CARDINAL_SIN evaluates the J-th cardinal sine basis function.
• CATALAN computes the Catalan numbers, from C(0) to C(N).
• CATALAN_ROW_NEXT computes row N of Catalan's triangle.
• CATALAN_VALUES returns some values of the Catalan numbers.
• CHARLIER evaluates Charlier polynomials at a point.
• CHEBY_T_POLY evaluates Chebyshev polynomials T(n,x).
• CHEBY_T_POLY_COEF evaluates coefficients of Chebyshev polynomials T(n,x).
• CHEBY_T_POLY_VALUES returns values of Chebyshev polynomials T(n,x).
• CHEBY_T_POLY_ZERO returns zeroes of Chebyshev polynomials T(n,x).
• CHEBY_U_POLY evaluates Chebyshev polynomials U(n,x).
• CHEBY_U_POLY_COEF evaluates coefficients of Chebyshev polynomials U(n,x).
• CHEBY_U_POLY_VALUES returns values of Chebyshev polynomials U(n,x).
• CHEBY_U_POLY_ZERO returns zeroes of Chebyshev polynomials U(n,x).
• CHEBYSHEV_DISCRETE evaluates discrete Chebyshev polynomials at a point.
• COLLATZ_COUNT counts the number of terms in a Collatz sequence.
• COLLATZ_COUNT_MAX seeks the maximum Collatz count for 1 through N.
• COLLATZ_COUNT_VALUES returns some values of the Collatz count function.
• COMB_ROW computes row N of Pascal's triangle.
• COMMUL computes a multinomial combinatorial coefficient.
• COMPLETE_SYMMETRIC_POLY evaluates a complete symmetric polynomial.
• COS_DEG returns the cosine of an angle given in degrees.
• COS_POWER_INT evaluates the cosine power integral.
• COS_POWER_INT_VALUES returns some values of the sine power integral.
• E_CONSTANT returns the value of the base of the natural logarithm system.
• ERF_VALUES returns some values of the ERF or "error" function.
• ERROR_F evaluates the error function ERF(X).
• ERROR_F_INVERSE inverts the error function ERF.
• EULER_CONSTANT returns the value of the Euler-Mascheroni constant.
• EULER_NUMBER computes the Euler numbers.
• EULER_NUMBER2 computes the Euler numbers.
• EULER_NUMBER_VALUES returns some values of the Euler numbers.
• EULER_POLY evaluates the N-th Euler polynomial at X.
• EULERIAN computes the Eulerian number E(N,K).
• F_HOFSTADTER computes the Hofstadter F sequence.
• FIBONACCI_DIRECT computes the N-th Fibonacci number directly.
• FIBONACCI_FLOOR returns the largest Fibonacci number less or equal to N.
• FIBONACCI_RECURSIVE computes the first N Fibonacci numbers.
• G_HOFSTADTER computes the Hofstadter G sequence.
• GAMMA_LOG_VALUES returns some values of the Log Gamma function.
• GAMMA_VALUES returns some values of the Gamma function.
• GEGENBAUER_POLY computes the Gegenbauer polynomials C(0:N,ALPHA,X).
• GEGENBAUER_POLY_VALUES returns some values of the Gegenbauer polynomials.
• GEN_HERMITE_POLY evaluates the generalized Hermite polynomials at X.
• GEN_LAGUERRE_POLY evaluates generalized Laguerre polynomials.
• GUD evaluates the Gudermannian function.
• GUD_VALUES returns some values of the Gudermannian function.
• H_HOFSTADTER computes the Hofstadter H sequence.
• HAIL computes the hail function.
• HERMITE_POLY evaluates the Hermite polynomials at X.
• HERMITE_POLY_COEF evaluates the coefficients of the physicist's Hermite polynomial H(n,x).
• HERMITE_POLY_VALUES returns some values of the Hermite polynomial.
• HYPERGEOMETRIC_CDF_VALUES returns some values of the hypergeometric function 2F1.
• I4_CHOOSE computes the binomial coefficient C(N,K).
• I4_FACTOR factors an integer into prime factors.
• I4_FACTORIAL computes the factorial of N.
• I4_FACTORIAL_VALUES returns values of the factorial function.
• I4_FACTORIAL2 computes the double factorial function.
• I4_FACTORIAL2_VALUES returns values of the double factorial function.
• I4_GCD finds the greatest common divisor of two I4's.
• I4_IS_PRIME reports whether an I4 is prime.
• I4_IS_TRIANGULAR determines whether an I4 is triangular.
• I4_MAX returns the maximum of two I4's.
• I4_MIN returns the smaller of two I4's.
• I4_MODP returns the nonnegative remainder of I4 division.
• I4_PARTITION_DISTINCT_COUNT returns any value of Q(N).
• I4_POCHHAMMER returns the value of ( I * (I+1) * ... * (J-1) * J ).
• I4_SIGN returns the sign of an integer.
• I4_SWAP switches two integer values.
• I4_TO_TRIANGLE converts an integer to triangular coordinates.
• I4_UNIFORM_AB returns a scaled pseudorandom I4.
• I4MAT_PRINT prints an I4MAT.
• I4MAT_PRINT_SOME prints some of an I4MAT.
• JACOBI_POLY evaluates the Jacobi polynomials at X.
• JACOBI_POLY_VALUES returns some values of the Jacobi polynomial.
• JACOBI_SYMBOL evaluates the Jacobi symbol (Q/P).
• KRAWTCHOUK evaluates the Krawtchouk polynomials at X.
• LAGUERRE_ASSOCIATED evaluates the associated Laguerre polynomials L(N,M,X) at X.
• LAGUERRE_POLY evaluates the Laguerre polynomials at X.
• LAGUERRE_POLY_COEF evaluates the Laguerre polynomial coefficients.
• LAGUERRE_POLYNOMIAL_VALUES returns some values of the Laguerre polynomial.
• LEGENDRE_ASSOCIATED evaluates the associated Legendre functions.
• LEGENDRE_ASSOCIATED_NORMALIZED: normalized associated Legendre functions.
• LEGENDRE_ASSOCIATED_NORMALIZED_VALUES: normalied associated Legendre.
• LEGENDRE_ASSOCIATED_VALUES returns values of associated Legendre functions.
• LEGENDRE_FUNCTION_Q evaluates the Legendre Q functions.
• LEGENDRE_FUNCTION_Q_VALUES returns values of the Legendre Q function.
• LEGENDRE_POLY evaluates the Legendre polynomials.
• LEGENDRE_POLY_COEF evaluates the Legendre polynomial coefficients.
• LEGENDRE_POLY_VALUES returns values of the Legendre polynomials.
• LEGENDRE_SYMBOL evaluates the Legendre symbol (Q/P).
• LERCH estimates the Lerch transcendent function.
• LERCH_VALUES returns some values of the Lerch transcendent function.
• LOCK returns the number of codes for a lock with N buttons.
• MEIXNER evaluates Meixner polynomials at a point.
• MERTENS evaluates the Mertens function.
• MERTENS_VALUES returns some values of the Mertens function.
• MOEBIUS returns the value of MU(N), the Moebius function of N.
• MOEBIUS_VALUES returns some values of the Moebius function.
• MOTZKIN returns the Motzkin numbers up to order N.
• NORMAL_01_CDF_INV inverts the standard normal CDF.
• OMEGA returns OMEGA(N), the number of distinct prime divisors of N.
• OMEGA_VALUES returns some values of the OMEGA function.
• PARTITION_COUNT_VALUES returns values of the integer partition count.
• PARTITION_DISTINCT_COUNT_VALUES returns some values of Q(N).
• PENTAGON_NUM computes the N-th pentagonal number.
• PHI computes the number of relatively prime predecessors of an integer.
• PHI_VALUES returns some values of the PHI function.
• PLANE_PARTITION_NUM returns the number of plane partitions of the integer N.
• POLY_BERNOULLI evaluates the poly-Bernolli numbers with negative index.
• POLY_COEF_COUNT: polynomial coefficient count given dimension and degree.
• PRIME returns any of the first PRIME_MAX prime numbers.
• PSI_VALUES returns some values of the Psi or Digamma function.
• PYRAMID_NUM returns the N-th pyramidal number.
• R8_ABS returns the absolute value of an R8.
• R8_ACOSH returns the inverse hyperbolic cosine of a number.
• R8_ASINH returns the inverse hyperbolic sine of a number.
• R8_ATANH returns the inverse hyperbolic tangent of a number.
• R8_CHOOSE computes the combinatorial coefficient C(N,K).
• R8_COT returns the cotangent of an angle.
• R8_COT_DEG returns the cotangent of an angle given in degrees.
• R8_CSC returns the cosecant of X.
• R8_CSC_DEG returns the cosecant of an angle given in degrees.
• R8_EPSILON returns the R8 roundoff unit.
• R8_FACTORIAL returns the factorial function as an R8.
• R8_FACTORIAL_LOG computes the natural logarithm of the factorial function.
• R8_FACTORIAL_LOG_VALUES returns values of log(factorial(n)).
• R8_FACTORIAL_VALUES returns values of the real factorial function.
• R8_FACTORIAL2 computes the double factorial function.
• R8_GAMMA evaluates Gamma(X) for a real argument.
• R8_GAMMA_LOG calculates the natural logarithm of GAMMA ( X ) for positive X.
• R8_HUGE returns a "huge" R8.
• R8_HYPER_2F1 evaluates the hypergeometric function F(A,B,C,X).
• R8_MAX returns the maximum of two R8's.
• R8_MIN returns the minimum of two R8's.
• R8_MOP returns the I-th power of -1 as an R8 value.
• R8_NINT returns the integer that is nearest to a real value.
• R8_PI returns the value of PI.
• R8_PSI evaluates the function Psi(X).
• R8_UNIFORM_01 is a portable pseudorandom number generator.
• R8POLY_DEGREE returns the degree of a polynomial.
• R8POLY_PRINT prints out a polynomial.
• R8POLY_VALUE evaluates a double precision polynomial.
• R8VEC_LINSPACE_NEW creates a vector of linearly spaced values.
• R8VEC_PRINT prints an R8VEC.
• R8VEC_ZERO zeroes a real vector.
• S_LEN_TRIM returns the length of a string to the last nonblank.
• SEC_DEG returns the secant of an angle given in degrees.
• SIGMA returns the value of SIGMA(N), the divisor sum.
• SIGMA_VALUES returns some values of the Sigma function.
• SIN_DEG returns the sine of an angle given in degrees.
• SIN_POWER_INT evaluates the sine power integral.
• SIN_POWER_INT_VALUES returns some values of the sine power integral.
• SLICE: maximum number of pieces created by a given number of slices.
• SPHERICAL_HARMONIC evaluates spherical harmonic functions.
• SPHERICAL_HARMONIC_VALUES returns values of spherical harmonic functions.
• STIRLING1 computes the Stirling numbers of the first kind.
• STIRLING2 computes the Stirling numbers of the second kind.
• TAN_DEG returns the tangent of an angle given in degrees.
• TAU returns the value of TAU(N), the number of distinct divisors of N.
• TAU_VALUES returns some values of the Tau function.
• TETRAHEDRON_NUM returns the N-th tetrahedral number.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TRIANGLE_NUM returns the N-th triangular number.
• TRIANGLE_TO_I4 converts a triangular coordinate to an integer.
• V_HOFSTADTER computes the Hofstadter V sequence.
• VIBONACCI computes the first N Vibonacci numbers.
• ZECKENDORF produces the Zeckendorf decomposition of a positive integer.
• ZERNIKE_POLY evaluates a Zernike polynomial at RHO.
• ZERNIKE_POLY_COEF: coefficients of a Zernike polynomial.
• ZETA estimates the Riemann Zeta function.
• ZETA_VALUES returns some values of the Riemann Zeta function.

You can go up one level to the C++ source codes.