POLPAK - Recursive Polynomials

POLPAK evaluates a variety of mathematical functions. It includes routines to evaluate the classic recursively defined polynomials of Chebyshev, Gegenbauer, Hermite, Jacobi, Laguerre and Legendre. A variety of other polynomials and functions have been added.

Files you may copy include:

• polpak.f90, the source code;
• polpak_prb.f90, the calling program;
• polpak_prb.out, the sample output.

The list of routines includes:

• ALIGN_ENUM counts the number of alignments of two sequences of M and N elements.
• BELL returns the Bell numbers from 0 to N.
• BENFORD returns the Benford probability of one or more significant digits.
• BERN computes the value of the Bernoulli numbers B(0) through B(N).
• BERN2 evaluates the Bernoulli numbers.
• BERN_POLY evaluates the Bernoulli polynomial of order N at X.
• BERN_POLY2 evaluates the N-th Bernoulli polynomial at X.
• BETA returns the value of the Beta function.
• BETA_VALUES returns some values of the Beta function for testing.
• BP01 computes the values of the Bernstein polynomials at a point X.
• BPAB evaluates at X the Bernstein polynomials based in [A,B].
• CARDAN evaluates the Cardan polynomials.
• CARDAN_COEFF computes the coefficients of the N-th Cardan polynomial.
• CATALAN computes the Catalan numbers, from C(0) to C(N).
• CATALAN_ROW computes row N of Catalan's triangle.
• CHEBY1 evaluates the Chebyshev polynomials of the first kind.
• CHEBY2 evaluates the Chebyshev polynomials of the second kind.
• COMBIN computes the combinatorial coefficient C(N,K).
• COMBIN2 computes the binomial coefficient C(N,K).
• COMB_ROW computes row N of Pascal's triangle.
• COMMUL computes a multinomial combinatorial coefficient.
• D_BERN3 computes the value of the Bernoulli number B(N).
• D_EULER2 computes the Euler numbers.
• D_FACTORIAL computes the factorial of N, also denoted "N!".
• D_PI returns the double precision value of Pi.
• E_CONSTANT returns the value of the base of the natural logarithm system.
• EULER computes the Euler numbers.
• EULER_CONSTANT returns the value of the Euler-Mascheroni constant.
• EULERIAN computes the Eulerian number E(N,K).
• EULER_POLY evaluates the N-th Euler polynomial at X.
• F_HOFSTADTER computes the Hofstadter F sequence.
• FIBONACCI_DIRECT computes the N-th Fibonacci number directly.
• FIBONACCI_FLOOR returns the largest Fibonacci number less than or equal to N.
• FIBONACCI_RECURSIVE computes the first N Fibonacci numbers.
• G_HOFSTADTER computes the Hofstadter G sequence.
• GAMMA returns the value of the Gamma function at X.
• GAMMA_LOG calculates the natural logarithm of GAMMA ( X ) for positive X.
• GAMMA_VALUES returns some values of the Gamma function for testing.
• GEGENBAUER computes the Gegenbauer polynomials C(I,ALFA)(X) for I = 1 to N.
• HAIL computes the hail function.
• H_HOFSTADTER computes the Hofstadter H sequence.
• HERMITE evaluates the Hermite polynomials at X.
• I_FACTORIAL computes the factorial N!
• I_FACTORIAL2 computes the factorial N!!
• I_RANDOM returns a random integer in a given range.
• JACOBI evaluates the Jacobi polynomials at X.
• LAGUERRE evaluates the Laguerre polynomials at X.
• LAGUERRE_GEN evaluates the generalized Laguerre polynomials at X.
• LAGUERRE_LNM evaluates the associated Laguerre polynomials Lnm at X.
• LEGENDRE_FUNCTION_VALUES returns values of associated Legendre functions.
• LEGENDRE_PN evaluates the Legendre polynomials P(N)(X) at X.
• LEGENDRE_PNM evaluates the associated Legendre polynomial Pnm(x) of the first kind.
• LEGENDRE_QN evaluates the Legendre polynomials Qn(X).
• LOCK returns the number of codes for a lock with N buttons.
• LOG_FACTORIAL computes the natural logarithm of the factorial N!
• PENTAGON_NUM computes the N-th pentagonal number.
• PYRAMID_NUM returns the N-th pyramidal number.
• R_FACTORIAL computes the factorial N!
• R_FACTORIAL_VALUES returns values of the real factorial function for testing.
• R_PI returns the value of pi.
• RPOLY_VAL_HORNER evaluates a polynomial using Horner's method.
• STIRLING1 computes the Stirling numbers of the first kind.
• STIRLING2 computes the Stirling numbers of the second kind.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TRIANGLE_NUM returns the N-th triangular number.
• V_HOFSTADTER computes the Hofstadter V sequence.
• VIBONACCI computes the first N Vibonacci numbers.
• ZECKENDORF produces the Zeckendorf decomposition of a positive integer.

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