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 FORTRAN77 version and a FORTRAN90 version and a MATLAB version and a Python version

Related Data and Programs:

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_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;
• 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_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.py, converts an integer to 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_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;
• slice.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_to_i4.py, converts a triangular coordinate to an integer;
• v_hofstadter.py, computes the Hofstadter V function;
• vibonacci.py, computes the Vibonacci numbers;
• zernike_poly.py, evaluates a Zernike polynomial;
• zeta.py, approximates the Riemann Zeta function;
• zeta_values.py, returns some stored values of the Riemann Zeta function;

NOT CONVERTED YET

• r8_hyper_2f1.py, evaluates the 2F1 hypergeometric function.
• zeckendorf.py, produces the Zeckendorf decomposition of a positive integer;
• zernike_poly_coef.py, returns the coefficients of a Zernike polynomial;

Examples and Tests:

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

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