POLPAK
Recursive Polynomials

POLPAK, a MATLAB 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 MATLAB library which evaluates the Bernstein polynomials, useful for uniform approximation of functions;

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

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

CORDIC, a MATLAB library which use the CORDIC method to compute certain elementary functions.

FN, a MATLAB library which approximates elementary and special functions using Chebyshev polynomials, by Wayne Fullerton.

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

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

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

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

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

LEGENDRE_PRODUCT_POLYNOMIAL, a MATLAB library which defines Legendre product polynomials, creating a multivariate polynomial as the product of univariate Legendre polynomials.

LOBATTO_POLYNOMIAL, a MATLAB library which evaluates Lobatto polynomials, similar to Legendre polynomials except that they are zero at both endpoints.

polpak_test

SPHERICAL_HARMONIC, a MATLAB library which evaluates spherical harmonic functions.

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

Source Code:

• agm_values.m, returns selected values of the arithmetic geometric mean function.
• agud.m, evaluates the inverse Gudermannian function;
• align_enum.m, counts the alignments of two sequences of M and N elements;
• bell.m, evaluates the Bell numbers;
• bell_poly_coef.m, returns the coefficients of a Bell polynomial;
• bell_values.m, evaluates some values of the Bell numbers;
• benford.m, returns the Benford probability of one or more significant digits;
• bernoulli_number.m, returns the Bernoulli numbers;
• bernoulli_number_values.m, returns some values of the Bernoulli numbers;
• bernoulli_number2.m, returns the Bernoulli numbers;
• bernoulli_number3.m, computes the Bernoulli numbers;
• bernoulli_poly.m, evaluates a Bernoulli polynomial;
• bernoulli_poly2.m, evaluates the Nth Bernoulli polynomial at X;
• bernstein_poly.m, evaluates the Bernstein polynomials at a point;
• bernstein_poly_01_values.m, returns some values of the Bernstein polynomials;
• beta_values.m, returns some values of the Beta function;
• bpab.m, evaluates the Bernstein polynomials defined on the interval [A,B];
• cardan_poly.m, evaluates the Cardan polynomials;
• cardan_poly_coef.m, returns the coefficients of a Cardan polynomial;
• cardinal_cos.m, evaluates the cardinal cosine basis functions.
• cardinal_sin.m, evaluates the cardinal sine basis functions.
• catalan.m, evaluates the Catalan numbers;
• catalan_row_next.m, computes the next row of Catalan numbers;
• catalan_values.m, evaluates some values of the Catalan numbers;
• charlier.m, evaluates the Charlier polynomials;
• cheby_t_poly.m, evaluates the Chebyshev T polynomials;
• cheby_t_poly_coef.m, returns the coefficients of the Chebyshev T polynomials;
• cheby_t_values.m, returns some values of the Chebyshev T polynomials;
• cheby_t_poly_zero.m, returns the zeros of the Chebyshev T polynomials;
• cheby_u_poly.m, evaluates the Chebyshev U polynomials;
• cheby_u_poly_coef.m, returns the coefficients of the Chebyshev U polynomials;
• cheby_u_values.m, returns some values of the Chebyshev U polynomials;
• cheby_u_poly_zero.m, returns the zeros of the Chebyshev U polynomials;
• chebyshev_discrete.m, evaluates the discrete Chebyshev polynomials;
• collatz_count.m, counts the length of the Collatz sequence for a seed of N;
• collatz_count_max.m, seeks the maximum Collatz count from 1 to N.
• collatz_count_values.m, stores some values of the Collatz count function;
• comb_row_next.m, computes the next row of Pascal's triangle;
• commul.m, computes a multinomial coefficient;
• complete_symmetric_poly.m, evaluates the complete symmetric polynomial tau(n,r).
• cos_power_int.m, returns the value of the integral of the N-th power of the cosine function;
• cos_power_int_values.m, returns some values of the integral of the N-th power of the cosine function;
• delannoy.m, computes the Delannoy numbers;
• domino_tiling_num.m, computes the the number of ways of tiling an MxN chessboard with dominoes.
• erf_values.m, returns some values of the error function;
• euler_constant.m, returns the value of the Euler-Mascheroni constant;
• euler_number.m, evaluates the Euler numbers;
• euler_number2.m, computes the Euler numbers;
• euler_number_values.m, returns some values of the Euler numbers;
• euler_poly.m, evaluates the N-th Euler polynomial at X;
• eulerian.m, computes the eulerian number E(N,K);
• f_hofstadter.m, computes the Hofstadter F function;
• fibonacci_direct.m, computes the N-th Fibonacci number directly;
• fibonacci_floor.m, computes the largest Fibonacci number less than or equal to N;
• fibonacci_recursive.m, computes the first N Fibonacci numbers recursively;
• g_hofstadter.m, computes the Hofstadter G function;
• gamma_values.m, returns some values of the Gamma function;
• gamma_log_values.m, returns some values of the logarithm of the Gamma function;
• gegenbauer_poly.m, evaluates the Gegenbauer polynomials at a point;
• gegenbauer_poly_values.m, returns some values of the Gegenbauer polynomials;
• gen_hermite_poly.m, evaluates the generalized Hermite polynomials;
• gen_laguerre_poly.m, evaluates the generalized Laguerre polynomials;
• gud.m, evaluates the Gudermannian function;
• gud_values.m, returns some values of the Gudermannian function;
• h_hofstadter.m, computes the Hofstadter H function;
• hail.m, computes the hail function;
• hermite_poly_phys.m, evaluates the Hermite physicist polynomials;
• hermite_poly_coef.m, evaluates the Hermite polynomial coefficients;
• hermite_poly_phys_values.m, returns some values of the Hermite physicist polynomials;
• hyper_2f1_values.m, returns some values of the 2F1 hypergeometric function.
• i4_choose.m, computes the combinatorial coefficient C(N,K);
• i4_factor.m, factors an integer into prime factors;
• i4_factorial.m, computes N! for small values of N;
• i4_factorial_values.m, returns some values of the (integer) factorial function;
• i4_factorial2.m, computes the double factorial function;
• i4_factorial2_values.m, returns some values of the double factorial function;
• i4_is_fibonacci.m, determines whether an integer is a Fibonacci number;
• i4_is_prime.m, determines whether an integer is prime;
• i4_is_triangular.m, determines whether an integer is triangular;
• i4_partition_distinct_count_values.m, computes the value of Q(N);
• i4_to_triangle_lower.m, converts an integer to lower triangular coordinates;
• i4_to_triangle_upper.m, converts an integer to upper triangular coordinates;
• i4_uniform_ab.m, returns a random integer in a given range;
• i4mat_print.m, prints an I4MAT;
• i4mat_print_some.m, prints some of an I4MAT;
• jacobi_poly.m, evaluates the Jacobi polynomials at a point;
• jacobi_poly_values.m, returns some values of the Jacobi polynomials;
• jacobi_symbol.m, evaluates the Jacobi symbol (Q/P);
• krawtchouk.m, evaluates the Krawtchouk polynomials;
• laguerre_associated.m, evaluates the associated Laguerre polynomials L(N,M)(X);
• laguerre_poly.m, evaluates the Laguerre polynomials;
• laguerre_poly_coef.m, evaluates the Laguerre polynomial coefficients;
• laguerre_polynomial_values.m, returns some values of the Laguerre polynomials;
• legendre_associated.m, evaluates the associated Legendre functions P(N,M)(X);
• legendre_associated_values.m, returns some values of the associated Legendre function;
• legendre_associated_normalized.m, evaluates the associated Legendre functions P(N,M)(X), normalized for use in spherical harmonic calculations;
• legendre_associated_normalized_sphere_values.m, returns some values of the associated Legendre function normalized for use in spherical harmonic calculations;
• legendre_function_q.m, evaluates the Legendre functions Q(N)(X);
• legendre_function_q_values.m, returns some values of the Legendre Q(N) functions;
• legendre_poly.m, evaluates the Legendre polynomials P(N)(X);
• legendre_poly_coef.m, evaluates the coefficients of the Legendre polynomials P(N)(X);
• legendre_poly_values.m, returns some values of the Legendre polynomials;
• legendre_symbol.m, evaluates the Legendre symbol (Q/P);
• lerch.m, evaluates the Lerch transcendent function;
• lerch_values.m, returns some values of the Lerch transcendent function;
• lock.m, returns the number of codes for a lock with N buttons;
• meixner.m, evaluates the Meixner polynomials;
• mertens.m, returns the value of the Mertens function;
• mertens_values.m, returns some tabulated values of the Mertens function;
• moebius.m, returns the value of MU(N), the Moebius function of N;
• moebius_values.m, returns some tabulated values of the Moebius function;
• motzkin.m, returns the Motzkin numbers up to order N;
• normal_01_cdf_inverse.m inverts the Normal 01 CDF.
• normal_01_cdf_values.m returns some values of the Normal 01 CDF.
• omega.m, returns the number of distinct prime divisors of an integer;
• omega_values.m, returns some values of the Omega function;
• partition_distinct_count_values.m, returns some values of Q(N);
• pentagon_num.m, returns the N-th pentagonal number;
• phi.m, returns the number of relatively prime predecessors of an integer;
• phi_values.m, returns some values of the Phi function;
• plane_partition_num.m, returns the number of plane partitions of an integer.
• poly_bernoulli.m, evaluates the poly-Bernoulli numbers of negative index;
• poly_coef_count.m, counts the coefficients in a polynomial of given degree and dimension.
• prime.m, returns any of the first MAXPRIME primes;
• psi_values.m, returns some values of the Psi function;
• pyramid_num.m, returns the N-th pyramidal number;
• pyramid_square_num.m, returns the N-th pyramidal square number;
• r8_agm.m, computes the arithmetic geometric mean of two numbers;
• r8_beta.m, evaluates the Beta function;
• r8_choose.m, computes the combinatorial coefficient C(N,K);
• r8_erf.m, evaluates the error function;
• r8_erf_inverse.m, inverts the error function;
• r8_factorial.m, computes the (real) factorial function;
• r8_factorial_log.m, computes the logarithm of the (real) factorial function;
• r8_factorial_log_values.m, returns some values of the logarithm of the (real) factorial function;
• r8_factorial_values.m, returns some values of the (real) factorial function;
• r8_gamma.m, returns the Gamma function;
• r8_huge.m, returns a "huge" R8;
• r8_hyper_2f1.m, evaluates the 2F1 hypergeometric function.
• r8_nint.m, rounds an R8 to the nearest integer;
• r8_pi.m, returns the value of Pi;
• r8_psi.m, returns the Psi function;
• r8_uniform_01.m, returns a unit pseudorandom R8;
• r8_uniform_ab.m, returns a pseudorandom R8 in [A,B];
• r8poly_degree.m, returns the degree of a polynomial;
• r8poly_print.m, prints a polynomial;
• r8poly_value_horner.m, evaluates a polynomial using Horner's method;
• r8vec_linspace.m, returns evenly spaced values from A to B as an R8VEC.
• r8vec_print.m, prints an R8VEC.
• s_len_trim.m, returns the length of a string to the last nonblank;
• sigma.m, returns the value of SIGMA(N), the divisor sum;
• sigma_values.m, returns some values of the Sigma function;
• simplex_num.m
• sin_power_int.m, returns the value of the integral of the N-th power of the sine function;
• sin_power_int_values.m, returns some values of the integral of the N-th power of the sine function;
• slices.m, maximum number of pieces created by a given number of slices.
• spherical_harmonic.m, evaluates the spherical harmonic function;
• spherical_harmonic_values.m, returns some values of the spherical harmonic function;
• stirling1.m, returns the Stirling numbers of the first kind;
• stirling2.m, returns the Stirling numbers of the second kind;
• tau.m, evaluates the Tau function, the number of distinct divisors;
• tau_values.m, returns some values of the Tau function;
• tetrahedron_num.m, returns the N-th tetrahedral number;
• timestamp.m, prints the current YMDHMS date as a timestamp;
• triangle_num.m, returns the N-th triangle number;
• triangle_lower_to_i4.m, converts a lower triangular coordinate to an integer;
• triangle_upper_to_i4.m, converts an upper triangular coordinate to an integer;
• tribonacci_recursive.m, computes the first N Tribonacci numbers recursively;
• trinomial.m, computes a trinomial coefficient;
• v_hofstadter.m, computes the Hofstadter V function;
• vibonacci.m, computes the Vibonacci numbers;
• zeckendorf.m, produces the Zeckendorf decomposition of a positive integer;
• zernike_poly.m, evaluates a Zernike polynomial;
• zernike_poly_coef.m, returns the coefficients of a Zernike polynomial;
• zeta_m1.m, approximates the Riemann Zeta Minus One function;
• zeta_m1_values.m, returns some stored values of the Riemann Zeta Minus One function;
• zeta_naive.m, approximates the Riemann Zeta function;
• zeta_values.m, returns some stored values of the Riemann Zeta function;