POLPAK
Recursive Polynomials

POLPAK is a Python 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:

ASA103, a Python library which evaluates the digamma or psi function, by Jose Bernardo. This is a version of Applied Statistics Algorithm 103.

CHEBYSHEV_POLYNOMIAL, a Python library which considers the Chebyshev polynomials T(i,x), U(i,x), V(i,x) and W(i,x). Functions are provided to evaluate the polynomials, determine their zeros, produce their polynomial coefficients, produce related quadrature rules, project other functions onto these polynomial bases, and integrate double and triple products of the polynomials.

CLAUSEN, a Python library which evaluates a Chebyshev interpolant to the Clausen function Cl2(x).

FN, a Python library which approximates elementary and special functions using Chebyshev polynomials; functions include Airy, Bessel I, Bessel J, Bessel K, Bessel Y, beta, confluent hypergeometric, cosine integral, Dawson's integral, digamma (psi), error, exponential integral, gamma, hyperbolic cosine integral, hyperbolic sine integral, incomplete gamma, log gamma, logarithmic integral, Pochhammer, psi, sine integral, Spence; by Wayne Fullerton.

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

TEST_VALUES, a Python library which contains some sample values of many mathematical functions.

Source Code:

• agm_values.py, returns selected values of the arithmetic geometric mean function.
• agud.py, computes the inverse Gudermannian.
• align_enum.py, counts the alignments of two sequences of M and N elements;
• bell.py, evaluates the Bell numbers;
• bell_poly_coef.py, returns the coefficients of a Bell polynomial;
• bell_values.py, returns selected values of the Bell numbers.
• benford.py, returns the Benford probability of one or more significant digits;
• bernoulli_number.py, returns the Bernoulli numbers;
• bernoulli_number2.py, returns the Bernoulli numbers;
• bernoulli_number3.py, computes the Bernoulli numbers;
• bernoulli_number_values.py, returns selected values of the Bernoulli numbers.
• bernoulli_poly.py, evaluates a Bernoulli polynomial;
• bernoulli_poly2.py, evaluates the Nth Bernoulli polynomial at X;
• bernstein_poly.py, evaluates the Bernstein polynomials at a point;
• bernstein_poly_01_values.py, returns some values of the Bernstein polynomials;
• beta_values.py returns some values of the Beta function.
• bpab.py, evaluates the Bernstein polynomials defined on the interval [A,B];
• cardan_poly.py, evaluates the Cardan polynomials;
• cardan_poly_coef.py, returns the coefficients of a Cardan polynomial;
• cardinal_cos.py, evaluates the cardinal cosine basis functions.
• cardinal_sin.py, evaluates the cardinal sine basis functions.
• catalan.py, evaluates the Catalan numbers;
• catalan_row_next.py, computes the next row of Catalan numbers;
• catalan_values.py returns some values of the Catalan numbers.
• charlier.py, evaluates the Charlier polynomials;
• cheby_t_poly.py, evaluates the Chebyshev T polynomials;
• cheby_t_poly_coef.py, returns the coefficients of the Chebyshev T polynomials;
• cheby_t_values.py, returns some values of the Chebyshev T polynomials;
• cheby_t_poly_zero.py, returns the zeros of the Chebyshev T polynomials;
• cheby_u_poly.py, evaluates the Chebyshev U polynomials;
• cheby_u_poly_coef.py, returns the coefficients of the Chebyshev U polynomials;
• cheby_u_values.py, returns some values of the Chebyshev U polynomials;
• cheby_u_poly_zero.py, returns the zeros of the Chebyshev U polynomials;
• chebyshev_discrete.py, evaluates the discrete Chebyshev polynomials;
• collatz_count.py, counts the length of the Collatz sequence for a seed of N;
• collatz_count_max.py, seeks the maximum Collatz count from 1 to N.
• collatz_count_values.py, stores some values of the Collatz count function;
• comb_row_next.py, computes the next row of Pascal's triangle;
• commul.py, computes a multinomial coefficient;
• complete_symmetric_poly.py, evaluates the complete symmetric polynomial tau(n,r).
• cos_power_int.py, returns the value of the integral of the N-th power of the cosine function;
• cos_power_int_values.py, returns some values of the integral of the N-th power of the cosine function;
• delannoy.py, computes the Delannoy numbers;
• domino_tiling_num.py, computes the number of distinct ways to tile a rectangle with dominoes;
• erf_values.py, returns selected values of the error function.
• euler_number.py, evaluates the Euler numbers;
• euler_number2.py, computes the Euler numbers;
• euler_number_values.py, returns some values of the Euler numbers;
• euler_poly.py, evaluates the N-th Euler polynomial at X;
• eulerian.py, computes the eulerian number E(N,K);
• f_hofstadter.py, computes the Hofstadter F function;
• fibonacci_direct.py, computes the N-th Fibonacci number directly;
• fibonacci_floor.py, computes the largest Fibonacci number less than or equal to N;
• fibonacci_recursive.py, computes the first N Fibonacci numbers recursively;
• g_hofstadter.py, computes the Hofstadter G function;
• gamma_values.py, returns selected values of the Gamma function.
• gamma_log_values.py, returns selected values of the Log(Gamma) function.
• gegenbauer_poly.py, evaluates the Gegenbauer polynomials at a point;
• gegenbauer_poly_values.py, returns some values of the Gegenbauer polynomials;
• gen_hermite_poly.py, evaluates the generalized Hermite polynomials;
• gen_laguerre_poly.py, evaluates the generalized Laguerre polynomials;
• gud.py, computes the Gudermannian function.
• gud_values.py, returns selected values of the Gudermannian function.
• hail.py, computes the hail function;
• h_hofstadter.py, computes the Hofstadter H function;
• hermite_poly_phys.py, evaluates the Hermite physicist polynomials;
• hermite_poly_phys_coef.py, evaluates the Hermite physicist polynomial coefficients;
• hermite_poly_phys_values.py, returns some values of the Hermite physicist polynomials;
• hyper_2f1_values.py, returns some values of the 2F1 hypergeometric function.
• i4_choose.py, computes the binomial coefficient C(N,K) as an I4.
• i4_factor.py, factors an integer into prime factors;
• i4_factorial.py, evaluates the factorial function.
• i4_factorial_values.py, returns selected values of the factorial function.
• i4_factorial2.py, computes the double factorial function;
• i4_factorial2_values.py, returns some values of the double factorial function;
• i4_is_fibonacci.py, determines whether an integer is a Fibonacci number;
• i4_is_prime.py, determines whether an integer is prime;
• i4_is_triangular.py, determines whether an integer is triangular;
• i4_partition_distinct_count_values.py, computes the value of Q(N);
• i4_to_triangle_lower.py, converts an integer to lower triangular coordinates;
• i4_to_triangle_upper.py, converts an integer to upper triangular coordinates;
• i4_uniform_ab.py, returns a scaled uniform I4 between A and B.
• i4mat_print.py, prints an I4MAT.
• i4mat_print_some.py, prints some of an I4MAT.
• i4vec_print.py, prints an I4VEC.
• jacobi_poly.py, evaluates the Jacobi polynomials at a point;
• jacobi_poly_values.py, returns some values of the Jacobi polynomials;
• jacobi_symbol.py, evaluates the Jacobi symbol (Q/P);
• krawtchouk.py, evaluates the Krawtchouk polynomials;
• laguerre_associated.py, evaluates the associated Laguerre polynomials L(N,M)(X);
• laguerre_poly.py, evaluates the Laguerre polynomials;
• laguerre_poly_coef.py, evaluates the Laguerre polynomial coefficients;
• laguerre_polynomial_values.py, returns some values of the Laguerre polynomials;
• legendre_associated.py, evaluates the associated Legendre functions P(N,M)(X);
• legendre_associated_values.py, returns some values of the associated Legendre function;
• legendre_associated_normalized.py, evaluates the associated Legendre functions P(N,M)(X), normalized for use in spherical harmonic calculations;
• legendre_associated_normalized_sphere_values.py, returns some values of the normalized associated Legendre function;
• legendre_function_q.py, evaluates the Legendre functions Q(N)(X);
• legendre_function_q_values.py, returns some values of the Legendre Q(N) functions;
• legendre_poly_values.py, returns some values of the Legendre polynomials;
• legendre_poly.py, evaluates the Legendre polynomials P(N)(X);
• legendre_poly_coef.py, evaluates the coefficients of the Legendre polynomials P(N)(X);
• legendre_symbol.py, evaluates the Legendre symbol (Q/P);
• lerch.py, evaluates the Lerch transcendent function;
• lerch_values.py, returns some values of the Lerch transcendent function;
• lock.py, returns the number of codes for a lock with N buttons;
• meixner.py, evaluates the Meixner polynomials;
• mertens.py, returns the value of the Mertens function;
• mertens_values.py, returns some tabulated values of the Mertens function;
• moebius.py, returns the value of MU(N), the Moebius function of N;
• moebius_values.py, returns some tabulated values of the Moebius function;
• motzkin.py, returns the Motzkin numbers up to order N;
• normal_01_cdf_inverse.py inverts the Normal 01 CDF.
• normal_01_cdf_values.py returns values of the Normal01 CDF.
• omega.py, returns the number of distinct prime divisors of an integer;
• omega_values.py, returns some values of the Omega function;
• partition_distinct_count_values.py, returns some values of Q(N);
• pentagon_num.py, returns the N-th pentagonal number;
• phi.py, returns the number of relatively prime predecessors of an integer;
• phi_values.py, returns some values of the Phi function;
• plane_partition_num.py, returns the number of plane partitions of an integer.
• poly_bernoulli.py, evaluates the poly-Bernoulli numbers of negative index;
• poly_coef_count.py, counts the coefficients in a polynomial of given degree and dimension.
• prime.py, returns a prime number from a stored table.
• psi_values.py, returns some values of the Psi function;
• pyramid_num.py, returns the N-th pyramidal number;
• pyramid_square_num.py, returns the N-th pyramidal square number;
• r8_agm.py, computes the arithmetic geometric mean of two numbers;
• r8_beta.py, evaluates the Beta function;
• r8_choose.py, computes the combinatorial coefficient C(N,K);
• r8_erf.py, evaluates the error function;
• r8_erf_inverse.py, inverts the error function;
• r8_euler_constant.py, returns the value of the Euler-Mascheroni constant;
• r8_factorial.py, evaluates the factorial function.
• r8_factorial_values.py returns values of the real factorial function.
• r8_factorial_log.py, computes the logarithm of the (real) factorial function;
• r8_factorial_log_values.py, returns some values of the logarithm of the (real) factorial function;
• r8_gamma.py returns values of the gamma function (only here temporarily, because our local Python installation is not making gamma available.)
• r8_gamma_log.py returns values of the gamma function (only here temporarily, because our local Python installation is not making lgamma available.)
• r8_huge.py, returns a "huge" R8;
• r8_hyper_2f1.py, evaluates the hypergeometric function 2F1.
• r8_mop.py, returns a power of -1 as an R8;
• r8_nint.py, rounds an R8 to the nearest integer;
• r8_pi.py, returns the value of Pi;
• r8_psi.py, returns the Psi function;
• r8_uniform_01.py, returns a unit pseudorandom R8.
• r8_uniform_ab.py, returns a scaled pseudorandom R8.
• r8poly_degree.py returns the degree of an R8POLY.
• r8poly_print.py, prints a polynomial;
• r8poly_value_horner.py, evaluates a polynomial using Horner's method;
• r8vec_linspace.py, returns an R8VEC of values evenly spaced between given limits.
• r8vec_print.py, prints an R8VEC.
• sigma.py, returns the value of SIGMA(N), the divisor sum;
• sigma_values.py, returns some values of the Sigma function;
• simplex_num.py, returns the N-th simplex number in M dimensions;
• sin_power_int.py, returns the value of the integral of the N-th power of the sine function;
• sin_power_int_values.py, returns some values of the integral of the N-th power of the sine function;
• slices.py, maximum number of pieces created by a given number of slices.
• spherical_harmonic.py, evaluates the spherical harmonic function.
• spherical_harmonic_values.py, returns some values of the spherical harmonic function;
• stirling1.py, returns the Stirling numbers of the first kind;
• stirling2.py, returns the Stirling numbers of the second kind;
• tau.py, evaluates the Tau function, the number of distinct divisors;
• tau_values.py, returns some values of the Tau function;
• tetrahedron_num.py, returns the N-th tetrahedral number;
• timestamp.py, prints the current YMDHMS date as a timestamp;
• triangle_num.py, returns the N-th triangle number;
• triangle_lower_to_i4.py, converts a lower triangular coordinate to an integer;
• triangle_upper_to_i4.py, converts a upper triangular coordinate to an integer;
• tribonacci_recursive.py, computes the first N Tribonacci numbers recursively;
• trinomial.py, computes trinomial coefficients.
• v_hofstadter.py, computes the Hofstadter V function;
• vibonacci.py, computes the Vibonacci numbers;
• zeckendorf.py, produces the Zeckendorf decomposition of a positive integer;
• zernike_poly.py, evaluates a Zernike polynomial;
• zernike_poly_coef.py, returns the coefficients of a Zernike polynomial;
• zeta.py, approximates the Riemann Zeta function;

Examples and Tests:

• polpak_test.py, calls all the tests;
• polpak_test.sh, runs all the tests;
• polpak_test.txt, is the output from the tests;

You can go up one level to the Python source codes.