diff --git a/.github/dependabot.yml b/.github/dependabot.yml index 7e6c742..6a61862 100644 --- a/.github/dependabot.yml +++ b/.github/dependabot.yml @@ -6,9 +6,3 @@ updates: schedule: interval: "weekly" day: "friday" - - - package-ecosystem: "uv" - directory: "/" - schedule: - interval: "weekly" - day: "friday" diff --git a/README.md b/README.md index 2e1eb28..7cd16e9 100644 --- a/README.md +++ b/README.md @@ -4,12 +4,15 @@ [![Tests](https://github.com/eggzec/polpack/actions/workflows/test.yml/badge.svg)](https://github.com/eggzec/polpack/actions/workflows/test.yml) [![Documentation](https://github.com/eggzec/polpack/actions/workflows/docs.yml/badge.svg)](https://github.com/eggzec/polpack/actions/workflows/docs.yml) +[![codecov](https://codecov.io/github/eggzec/polpack/graph/badge.svg)](https://codecov.io/github/eggzec/polpack) [![License: LGPL-2.1](https://img.shields.io/badge/License-LGPL%202.1-blue.svg)](LICENSE) [![PyPI Downloads](https://img.shields.io/pypi/dm/polpack.svg?label=PyPI%20downloads)](https://pypi.org/project/polpack/) [![Python versions](https://img.shields.io/pypi/pyversions/polpack.svg)](https://pypi.org/project/polpack/) -`polpack` is a high-performance Python library for evaluating special functions and recursively-defined polynomial families. The numerical core is written in Fortran and exposed via a Pythonic interface with comprehensive Google-style docstrings. +`polpack` is a high-performance Python library for evaluating special functions and recursively-defined polynomial families. The numerical core is written in Fortran and compiled via `f2py`, providing efficient routines for mathematical and combinatorial computations. + +It includes routines to evaluate classic families such as **Bernoulli**, **Chebyshev**, **Gegenbauer**, **Hermite**, **Laguerre**, and **Legendre** polynomials, among others. ## Quick example @@ -17,10 +20,10 @@ import numpy as np import polpack -# Example: Compute Bell numbers up to order 10 -b = np.zeros(11, dtype=np.int32, order='F') -polpack.bell(10, b) -print(f"Bell numbers: {b}") +# Example: Compute Bell numbers +bell = np.zeros(11, dtype=np.int32) +polpack.bell(10, bell) +print(f"Bell numbers: {bell}") ``` ## Installation @@ -33,7 +36,7 @@ Requires Python 3.10+ and NumPy. See the [full installation guide](https://eggze ## Documentation -- [Theory](https://eggzec.github.io/polpack/theory/) — mathematical background +- [Theory](https://eggzec.github.io/polpack/theory/) — mathematical background, polynomial families, and special functions - [Quickstart](https://eggzec.github.io/polpack/quickstart/) — runnable examples - [API Reference](https://eggzec.github.io/polpack/api/) — function signatures and parameters - [References](https://eggzec.github.io/polpack/references/) — literature citations diff --git a/bin/build.py b/bin/build.py index 98eb914..7f4e5ee 100644 --- a/bin/build.py +++ b/bin/build.py @@ -35,7 +35,9 @@ def run_command(command, cwd=None): if cwd is None: - logger.warning("No working directory specified. Using current directory.") + logger.warning( + "No working directory specified. Using current directory." + ) cwd = Path.cwd() else: cwd = Path(cwd) @@ -80,7 +82,13 @@ def clean(): run_command("uv pip uninstall polpack") for entry in Path("").iterdir(): - if entry.name in ["dist", "build", "lib", ".pytest_cache", ".ruff_cache"]: + if entry.name in [ + "dist", + "build", + "lib", + ".pytest_cache", + ".ruff_cache", + ]: logger.info(f"Removing '{entry}'") shutil.rmtree(entry) if entry.name == "bin" and entry.is_dir(): diff --git a/doc.txt b/doc.txt deleted file mode 100644 index 09f73a9..0000000 --- a/doc.txt +++ /dev/null @@ -1,362 +0,0 @@ -POLPAK -Recursive Polynomials -POLPAK is a FORTRAN77 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 -A variety of other polynomials and functions have been added. In a few cases, the new recursive feature of FORTRAN90 has been used (but NOT for the factorial function!) -Licensing: -The computer code and data files described and made available on this web page are distributed under the GNU LGPL license. - - -Reference: - -Milton Abramowitz, Irene Stegun, -Handbook of Mathematical Functions, -National Bureau of Standards, 1964, -ISBN: 0-486-61272-4, -LC: QA47.A34. -Robert Banks, -Slicing Pizzas, Racing Turtles, and Further Adventures in Applied Mathematics, -Princeton, 1999, -ISBN13: 9780691059471, -LC: QA93.B358. -Frank Benford, -The Law of Anomalous Numbers, -Proceedings of the American Philosophical Society, -Volume 78, 1938, pages 551-572. -Paul Bratley, Bennett Fox, Linus Schrage, -A Guide to Simulation, -Second Edition, -Springer, 1987, -ISBN: 0387964673, -LC: QA76.9.C65.B73. -Chad Brewbaker, -Lonesum (0,1)-matrices and poly-Bernoulli numbers of negative index, -Master of Science Thesis, -Computer Science Department, -Iowa State University, 2005. -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. -Theodore Chihara, -An Introduction to Orthogonal Polynomials, -Gordon and Breach, 1978, -ISBN: 0677041500, -LC: QA404.5 C44. -William Cody, -Rational Chebyshev Approximations for the Error Function, -Mathematics of Computation, -Volume 23, Number 107, July 1969, pages 631-638. -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. -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. -Walter Gautschi, -Orthogonal Polynomials: Computation and Approximation, -Oxford, 2004, -ISBN: 0-19-850672-4, -LC: QA404.5 G3555. -Ralph Hartley, -A More Symmetrical Fourier Analysis Applied to Transmission Problems, -Proceedings of the Institute of Radio Engineers, -Volume 30, 1942, pages 144-150. -Brian Hayes, -The Vibonacci Numbers, -American Scientist, -Volume 87, Number 4, July-August 1999, pages 296-301. -Brian Hayes, -Why W?, -American Scientist, -Volume 93, Number 2, March-April 2005, pages 104-108. -Ted Hill, -The First Digit Phenomenon, -American Scientist, -Volume 86, Number 4, July/August 1998, pages 358-363. -Douglas Hofstadter, -Goedel, Escher, Bach, -Basic Books, 1979, -ISBN: 0465026567, -LC: QA9.8H63. -Masanobu Kaneko, -Poly-Bernoulli Numbers, -Journal Theorie des Nombres Bordeaux, -Volume 9, Number 1, 1997, pages 221-228. -Cleve Moler, -Trigonometry is a Complex Subject, -MATLAB News and Notes, Summer 1998. -Thomas Osler, -Cardan Polynomials and the Reduction of Radicals, -Mathematics Magazine, -Volume 74, Number 1, February 2001, pages 26-32. -J Simoes Pereira, -Algorithm 234: Poisson-Charliers Polynomials, -Communications of the ACM, -Volume 7, Number 7, July 1964, page 420. -Charles Pinter, -A Book of Abstract Algebra, -Second Edition, -McGraw Hill, 2003, -ISBN: 0072943505, -LC: QA162.P56. -Ralph Raimi, -The Peculiar Distribution of First Digits, -Scientific American, -December 1969, pages 109-119. -Dennis Stanton, Dennis White, -Constructive Combinatorics, -Springer, 1986, -ISBN: 0387963472. -Gabor Szego, -Orthogonal Polynomials, -American Mathematical Society, 1992, -ISBN: 0821810235, -LC: QA3.A5.v23. -Daniel Velleman, Gregory Call, -Permutations and Combination Locks, -Mathematics Magazine, -Volume 68, Number 4, October 1995, pages 243-253. -Divakar Viswanath, -Random Fibonacci sequences and the number 1.13198824, -Mathematics of Computation, -Volume 69, Number 231, July 2000, pages 1131-1155. -Michael Waterman, -Introduction to Computational Biology, -Chapman and Hall, 1995, -ISBN: 0412993910, -LC: QH438.4.M33.W38. -Eric Weisstein, -CRC Concise Encyclopedia of Mathematics, -CRC Press, 2002, -Second edition, -ISBN: 1584883472, -LC: QA5.W45 -Stephen Wolfram, -The Mathematica Book, -Fourth Edition, -Cambridge University Press, 1999, -ISBN: 0-521-64314-7, -LC: QA76.95.W65. -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. -Shanjie Zhang, Jianming Jin, -Computation of Special Functions, -Wiley, 1996, -ISBN: 0-471-11963-6, -LC: QA351.C45. -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.f, the source code. -polpak.sh, commands to compile the source code. -Examples and Tests: - -polpak_prb.f, a sample calling program. -polpak_prb.sh, commands to compile and run the sample program. -polpak_prb_output.txt, the output file. -List of Routines: - -AGM_VALUES returns some values of the arithmetic geometric mean. -AGUD evaluates the inverse Gudermannian function. -ALIGN_ENUM counts the alignments of two sequences of M and N elements. -BELL returns the Bell numbers from 0 to N. -BELL_VALUES returns some values of the Bell numbers for testing. -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_VALUES returns some values of the Beta function. -BPAB evaluates at X the Bernstein polynomials based in [A,B]. -CARDAN_POLY 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_CONSTANT returns the value of Catalan's constant. -CATALAN_ROW_NEXT computes row N of Catalan's triangle. -CATALAN_VALUES returns some values of the Catalan numbers for testing. -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_NEXT computes the next row of Pascal's triangle. -COMMUL computes a multinomial combinatorial coefficient. -COMPLETE_SYMMETRIC_POLY evaluates a complete symmetric polynomial. -COS_POWER_INT evaluates the cosine power integral. -COS_POWER_INT_VALUES returns some values of the cosine power integral. -DELANNOY returns the Delannoy numbers up to orders (M,N). -ERF_VALUES returns some values of the ERF or "error" function for testing. -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). -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. -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(I,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. -HERMITE_POLY_PHYS evaluates the physicisist's Hermite polynomials at X. -HERMITE_POLY_PHYS_COEF evaluates the physicist's Hermite polynomial coefficients. -HERMITE_POLY_PHYS_VALUES returns some values of the physicist's Hermite polynomial. -HYPER_2F1_VALUES returns some values of the hypergeometric function 2F1. -I4_CHOOSE computes the binomial coefficient C(N,K). -I4_FACTOR factors an I4 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_HUGE returns a "huge" I4. -I4_IS_PRIME reports whether an I4 is prime. -I4_IS_TRIANGULAR determines whether an integer is triangular. -I4_PARTITION_DISTINCT_COUNT returns any value of Q(N). -I4_SWAP switches two I4's. -I4_TO_TRIANGLE converts an integer to triangular coordinates. -I4_UNIFORM_AB returns a scaled pseudorandom I4 between A and B. -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 associated Laguerre polynomials L(N,M,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. -LAMBERT_W estimates the Lambert W function. -LAMBERT_W_CRUDE is a crude estimate of the Lambert W function. -LAMBERT_W_VALUES returns some values of the Lambert W function. -LEGENDRE_ASSOCIATED evaluates the associated Legendre functions. -LEGENDRE_ASSOCIATED_NORMALIZED: normalized associated Legendre functions. -LEGENDRE_ASSOCIATED_NORMALIZED_SPHERE_VALUES: normalized 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 P(N,X) at X. -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_INVERSE inverts the standard normal CDF. -NORMAL_01_CDF_VALUES returns some values of the Normal 01 CDF. -OMEGA returns OMEGA(N), the number of distinct prime divisors of N. -OMEGA_VALUES returns some values of the OMEGA function. -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 for testing. -PYRAMID_NUM returns the N-th pyramidal number. -PYRAMID_SQUARE_NUM returns the N-th pyramidal square number. -R8_AGM computes the arithmetic-geometric mean of A and B. -R8_BETA returns the value of the Beta function. -R8_CHOOSE computes the binomial coefficient C(N,K) as an R8. -R8_EPSILON returns the R8 roundoff unit. -R8_ERF evaluates the error function. -R8_ERF_INVERSE inverts the error function. -R8_EULER_CONSTANT returns the value of the Euler-Mascheroni constant. -R8_FACTORIAL computes the factorial of N. -R8_FACTORIAL_LOG computes log(factorial(N)). -R8_FACTORIAL_LOG_VALUES returns values of log(factorial(n)). -R8_FACTORIAL_VALUES returns values of the real factorial function. -R8_GAMMA_LOG evaluates log ( Gamma ( X ) ) for a real argument. -R8_HUGE returns a "huge" R8. -R8_HYPER_2F1 evaluates the hypergeometric function F(A,B,C,X). -R8_MOP returns the I-th power of -1 as an R8. -R8_NINT returns the nearest integer to an R8. -R8_PI returns the value of pi as an R8. -R8_PSI evaluates the function Psi(X). -R8_UNIFORM_01 returns a unit pseudorandom R8. -R8POLY_DEGREE returns the degree of a polynomial. -R8POLY_PRINT prints out a polynomial. -R8POLY_VALUE_HORNER evaluates a polynomial using Horner's method. -R8VEC_LINSPACE creates a vector of linearly spaced values. -R8VEC_PRINT prints an R8VEC. -R8VEC_PRINT_SOME prints "some" of an R8VEC. -R8VEC_UNIFORM_AB returns a scaled pseudorandom R8VEC. -S_LEN_TRIM returns the length of a string to the last nonblank. -SIGMA returns the value of SIGMA(N), the divisor sum. -SIGMA_VALUES returns some values of the Sigma function. -SIMPLEX_NUM evaluates the N-th Simplex number in M dimensions. -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. -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 out the current YMDHMS date as a timestamp. -TRIANGLE_NUM returns the N-th triangular number. -TRIANGLE_TO_I4 converts a triangular coordinate to an integer. -TRINOMIAL computes a trinomial coefficient. -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 FORTRAN77 source codes. - -Last revised on 11 April 2015. diff --git a/docs/api.md b/docs/api.md index f77126a..ded7e10 100644 --- a/docs/api.md +++ b/docs/api.md @@ -1,52 +1,262 @@ # API Reference -All routines in `polpack` are exposed via the top-level `polpack` package. The underlying numerical routines are implemented in Fortran and accessed through the `_polpack` extension module. +All public routines are exported via the `polpack` Python module in lowercase. --- -## Calling Convention +## Overview -Most `polpack` routines for sequences and polynomials follow a consistent calling convention where the results are stored in a pre-allocated NumPy array passed as an argument. +`polpack` includes routines to evaluate a variety of recursively defined +polynomial families and special functions. Most generators follow a three-term +recurrence and write the output into a pre-allocated NumPy array. -### Array Requirements +| Family | Category | Type | Domain | +|---|---|---|---| +| `bernoulli_poly` | Polynomial | Sequence | \( \mathbb{R} \) | +| `bernstein_poly` | Polynomial | Basis | \( [0, 1] \) | +| `cardan_poly` | Polynomial | Cubic | \( \mathbb{R} \) | +| `charlier` | Polynomial | Discrete | \( \mathbb{N} \) | +| `cheby_t_poly` | Polynomial | Orthogonal | \( [-1, 1] \) | +| `gegenbauer_poly` | Polynomial | Orthogonal | \( [-1, 1] \) | +| `jacobi_poly` | Polynomial | Orthogonal | \( [-1, 1] \) | +| `laguerre_poly` | Polynomial | Orthogonal | \( [0, \infty) \) | +| `legendre_poly` | Polynomial | Orthogonal | \( [-1, 1] \) | -For routines that modify arrays in-place: -1. **Dtype:** Must match the expected Fortran type (usually `np.int32` for combinatorial counts and `np.float64` for polynomial values). -2. **Order:** Arrays **must** be initialized with Fortran memory layout (`order='F'`) for optimal performance and compatibility with the Fortran core. -3. **Size:** The array must be correctly sized (e.g., `n+1` for polynomials of degree `n`). +--- + +## Common interface + +### Arguments -```python -import numpy as np -import polpack +| Argument | Type | Direction | Description | +|---|---|---|---| +| `n` | `int` | input | Highest degree or number of terms to compute | +| `x` | `float` | input | Evaluation point | +| `v` | `ndarray` | in/out | Pre-allocated output array of size `n+1` | -# Correct initialization -n = 10 -b = np.zeros(n + 1, dtype=np.int32, order='F') -polpack.bell(n, b) -``` +!!! warning "Array allocation" + The output array `v` must be pre-allocated with the correct size before + calling the routine. Passing an undersized array will raise a runtime error. + +--- + +## Orthogonal Polynomials + +### `cheby_t_poly(m, n, x, cx)` +**Chebyshev polynomials of the first kind.** + +- `m` (int): Number of evaluation points. +- `n` (int): Highest degree. +- `x` (ndarray[m]): Evaluation points. +- `cx` (ndarray[m, n+1]): Output matrix. + +### `gegenbauer_poly(n, alpha, x, cx)` +**Gegenbauer polynomials.** + +- `n` (int): Highest degree. +- `alpha` (float): Parameter \( \alpha > -1/2 \). +- `x` (float): Evaluation point. +- `cx` (ndarray[n+1]): Output array. + +### `hermite_poly_phys(n, x, cx)` +**Hermite polynomials (Physicist's).** + +- `n` (int): Highest degree. +- `x` (float): Evaluation point. +- `cx` (ndarray[n+1]): Output array. + +### `jacobi_poly(n, alpha, beta, x, cx)` +**Jacobi polynomials.** + +- `n` (int): Highest degree. +- `alpha` (float): Parameter \( \alpha \). +- `beta` (float): Parameter \( \beta \). +- `x` (float): Evaluation point. +- `cx` (ndarray[n+1]): Output array. + +### `laguerre_poly(n, x, cx)` +**Laguerre polynomials.** + +- `n` (int): Highest degree. +- `x` (float): Evaluation point. +- `cx` (ndarray[n+1]): Output array. + +### `legendre_poly(n, x, cx, cpx)` +**Legendre polynomials.** + +- `n` (int): Highest degree. +- `x` (float): Evaluation point. +- `cx` (ndarray[n+1]): Output array \( P_n(x) \). +- `cpx` (ndarray[n+1]): Output array of derivatives \( P'_n(x) \). --- ## Combinatorial Sequences -- **bell(n, b)**: Evaluates the Bell numbers $B_0, \dots, B_n$. -- **catalan(n, c)**: Evaluates the Catalan numbers $C_0, \dots, C_n$. -- **stirling1(n, m, s1)**: Evaluates the Stirling numbers of the first kind. -- **stirling2(n, m, s2)**: Evaluates the Stirling numbers of the second kind. -- **fibonacci_recursive(n, f)**: Evaluates the first $n$ Fibonacci numbers. -- **eulerian(n, k)**: Evaluates the Eulerian number $E(n, k)$. -- **motzkin(n, a)**: Evaluates the Motzkin numbers up to order $n$. +### `bell(n, b)` +**Bell numbers.** + +- `n` (int): Highest index. +- `b` (ndarray[n+1]): Output array of Bell numbers \( B_0, \dots, B_n \). + +### `catalan(n, c)` +**Catalan numbers.** + +- `n` (int): Highest index. +- `c` (ndarray[n+1]): Output array of Catalan numbers \( C_0, \dots, C_n \). + +### `stirling1(n, m, s1)` / `stirling2(n, m, s2)` +**Stirling numbers.** + +- `n` (int): Number of elements. +- `m` (int): Number of subsets/cycles. +- `s1/s2` (ndarray[n, m]): Output matrix. + +--- + +## Detailed Routine Reference + +### Bernoulli and Euler Sequences + +#### `bernoulli_number(n, b)` +Computes the sequence of Bernoulli numbers \( B_0, \dots, B_n \). + +- `n` (int): Highest index. +- `b` (ndarray[n+1]): Output array. + +#### `bernoulli_poly(n, x, bx)` +Evaluates the Bernoulli polynomial \( B_n(x) \) at \( x \). + +- `n` (int): Degree. +- `x` (float): Evaluation point. +- `bx` (float): Output value. + +#### `euler_number(n, e)` +Computes the sequence of Euler numbers \( E_0, \dots, E_n \). + +- `n` (int): Highest index. +- `e` (ndarray[n+1]): Output array. + +#### `euler_poly(n, x)` +Evaluates the \( n \)-th Euler polynomial \( E_n(x) \) at \( x \). + +- `n` (int): Degree. +- `x` (float): Evaluation point. +- Returns `float`. + +### Bernstein Polynomials + +#### `bernstein_poly(n, x, bern)` +Evaluates the \( n+1 \) Bernstein basis polynomials of degree \( n \) at \( x \in [0, 1] \). + +- `n` (int): Degree. +- `x` (float): Evaluation point. +- `bern` (ndarray[n+1]): Output array of basis values. + +#### `bpab(n, x, a, b, bern)` +Evaluates the \( n+1 \) Bernstein basis polynomials of degree \( n \) on \( [a, b] \). + +- `n` (int): Degree. +- `x` (float): Evaluation point. +- `a, b` (float): Interval boundaries. +- `bern` (ndarray[n+1]): Output array of basis values. + +### Fibonacci and Power Sequences + +#### `fibonacci_recursive(n, f)` +Computes the first \( n \) Fibonacci numbers. + +- `n` (int): Number of terms. +- `f` (ndarray[n]): Output array. + +#### `fibonacci_direct(n, f)` +Computes the \( n \)-th Fibonacci number using the closed-form Binet formula. + +- `n` (int): Index. +- `f` (int): Output value. + +### Stirling Sequences + +#### `stirling1(n, m, s1)` +Computes the Stirling numbers of the first kind \( s(n, k) \). + +- `n` (int): Order. +- `m` (int): Term index. +- `s1` (ndarray[n, m]): Output matrix. + +#### `stirling2(n, m, s2)` +Computes the Stirling numbers of the second kind \( S(n, k) \). + +- `n` (int): Order. +- `m` (int): Term index. +- `s2` (ndarray[n, m]): Output matrix. + +--- + +### Special Math Functions + +#### `agm_values(n_data, a, b, fx)` +Returns tabulated values of the Arithmetic-Geometric Mean. + +- `n_data` (int): Index of the value to retrieve (0 for first). +- `a, b` (float): Parameters. +- `fx` (float): Output AGM value. + +#### `agud(g)` +Evaluates the inverse Gudermannian function. + +- `g` (float): Input value. +- Returns `float`. + +#### `beta_values(n_data, x, y, fxy)` +Returns tabulated values of the Beta function \( \text{B}(x, y) \). + +#### `erf_values(n_data, x, fx)` +Returns tabulated values of the error function \( \text{erf}(x) \). + +#### `gamma_values(n_data, x, fx)` +Returns tabulated values of the Gamma function \( \Gamma(x) \). + +#### `lambert_w(x)` +Estimates the Lambert W function \( W(x) \), the inverse of \( f(w) = we^w \). + +--- + +### Number Theory and Primality + +| Routine | Parameters | Description | +|---|---|---| +| `phi(n, phin)` | `n` (int): input
`phin` (int): out | Euler totient function \( \phi(n) \). | +| `moebius(n, mu)` | `n` (int): input
`mu` (int): out | Moebius function \( \mu(n) \). | +| `mertens(n)` | `n` (int): input | Mertens function \( M(n) \) (returns int). | +| `prime(n)` | `n` (int): input | Returns the \( n \)-th prime number. | +| `i4_is_prime(n)` | `n` (int): input | Primality test (returns bool). | +| `i4_factor(...)` | `n`, `f_max`, `f_num`, `f`, `p`, `nleft` | Prime factorization into components. | +| `sigma(n, sigma_n)` | `n` (int): input
`sigma_n` (int): out | Sum of divisors \( \sigma(n) \). | +| `tau(n, taun)` | `n` (int): input
`taun` (int): out | Number of divisors \( \tau(n) \). | +| `jacobi_symbol(q, p, j)` | `q, p` (int), `j` (int): out | Jacobi symbol \( (q/p) \). | +| `legendre_symbol(q, p, l)` | `q, p` (int), `l` (int): out | Legendre symbol \( (q/p) \). | -## Special Functions +### Sequences and Series -- **r8_beta(x, y)**: Evaluates the Beta function. -- **r8_gamma_log(x)**: Evaluates log(Gamma(X)). -- **zeta(p)**: Evaluates the Riemann Zeta function. -- **lambert_w(x)**: Estimates the Lambert W function. +| Routine | Parameters | Description | +|---|---|---| +| `catalan(n, c)` | `n` (int), `c` (ndarray) | Catalan numbers sequence. | +| `delannoy(m, n, a)` | `m`, `n` (int), `a` (ndarray) | Delannoy numbers matrix. | +| `motzkin(n, a)` | `n` (int), `a` (ndarray) | Motzkin numbers sequence. | +| `vibonacci(n, seed, v)` | `n`, `seed` (int), `v` (ndarray) | Random Vibonacci terms. | +| `collatz_count(n)` | `n` (int): input | Sequence length (returns int). | +| `collatz_count_max(...)` | `n, i_max, j_max` | Maximum count in range \( [1, n] \). | -## Number Theory +### Special Utilities -- **i4_is_prime(n)**: Reports whether an integer is prime. -- **phi(n)**: Evaluates the Euler phi function. -- **sigma(n)**: Evaluates the divisor sum function. -- **tau(n)**: Evaluates the number of distinct divisors. +| Routine | Parameters | Description | +|---|---|---| +| `agm_values(...)` | `n_data, a, b, fx` | Arithmetic-Geometric Mean data. | +| `i4_choose(n, k)` | `n, k` (int) | Binomial coefficient (returns int). | +| `i4_swap(i, j)` | `i, j` (int) | Swaps values in-place (in/out). | +| `i4_huge()` | (none) | Returns largest representable i4. | +| `zeckendorf(...)` | `n, m_max, m, i_l, f_l` | Zeckendorf representation decomposition. | +| `complete_symmetric_poly` | `n, r, x, value` | Evaluates \( h_r(x_1, \dots, x_n) \). | +| `zernike_poly` | `m, n, rho, z` | Zernike polynomial at \( \rho \). | +| `zeta(p)` | `p` (float): input | Riemann Zeta function \( \zeta(p) \) (returns float). | diff --git a/docs/index.md b/docs/index.md index 782cb9e..89ed28c 100644 --- a/docs/index.md +++ b/docs/index.md @@ -6,32 +6,29 @@ ## What is polpack? -`polpack` is a Python library for evaluating a wide range of -[special functions](https://en.wikipedia.org/wiki/Special_functions) and -recursively-defined polynomial families. The numerical core is written in -Fortran and exposed through a high-performance Python extension, providing -near-native execution speeds for intense mathematical computations. +`polpack` is a Python library for evaluating special functions and recursively-defined polynomial families. The numerical core is based on the original `POLPAK` library, providing efficient routines to evaluate a wide variety of mathematical functions. -Whether you need to compute Bell numbers, evaluate Bernoulli polynomials, or -work with orthogonal polynomial families (Chebyshev, Jacobi, Laguerre, etc.), -`polpack` offers a robust and tested suite of routines. +A **polynomial family** is a sequence of polynomials where each member is typically defined by its degree. Many such families are defined recursively, where higher-degree polynomials are computed from lower-degree ones. These functions are fundamental in numerical analysis, approximation theory, and physics. -## Why polpack? +`polpack` provides near-native performance by compiling its Fortran core via `f2py`, offering a clean and intuitive NumPy-based Python API. -| Feature | Detail | -|---|---| -| **High Performance** | Compiled Fortran core — efficient handling of recursive relations | -| **Comprehensive** | Over 170 routines for sequences, polynomials, and special functions | -| **Pythonic API** | Google-style docstrings and seamless NumPy integration | -| **Memory Efficient** | Supports in-place array modification for large-scale computations | -| **Portability** | Cross-platform support for Linux, macOS, and Windows | +## Available polynomial families -## Key Categories -- **Combinatorial Sequences:** Bell, Bernoulli, Catalan, Eulerian, Fibonacci, Stirling, etc. -- **Polynomial Families:** Bernoulli, Bernstein, Chebyshev, Gegenbauer, Hermite, Jacobi, Laguerre, Legendre, etc. -- **Special Functions:** AGM, Beta, Error function, Gamma, Lambert W, Zeta, etc. -- **Number Theory:** Collatz counts, Moebius, Omega, Phi, Primes, Sigma, Tau, etc. + +| Family | Category | Type | Domain | +|---|---|---|---| +| `bernoulli_poly` | Polynomial | Sequence | \( \mathbb{R} \) | +| `bernstein_poly` | Polynomial | Basis | \( [0, 1] \) | +| `cardan_poly` | Polynomial | Cubic | \( \mathbb{R} \) | +| `charlier` | Polynomial | Discrete | \( \mathbb{N} \) | +| `cheby_t_poly` | Polynomial | Orthogonal | \( [-1, 1] \) | +| `gegenbauer_poly` | Polynomial | Orthogonal | \( [-1, 1] \) | +| `jacobi_poly` | Polynomial | Orthogonal | \( [-1, 1] \) | +| `laguerre_poly` | Polynomial | Orthogonal | \( [0, \infty) \) | +| `legendre_poly` | Polynomial | Orthogonal | \( [-1, 1] \) | + +*(See [API Reference](api.md) for the full list of supported families and routines.)* ## Quick example @@ -39,24 +36,10 @@ work with orthogonal polynomial families (Chebyshev, Jacobi, Laguerre, etc.), import numpy as np import polpack -# Example: Evaluate the first 11 Bell numbers -b = np.zeros(11, dtype=np.int32, order='F') +# Example: Compute the first 11 Bell numbers +# b[n] will contain the n-th Bell number +b = np.zeros(11, dtype=np.int32, order="F") polpack.bell(10, b) -print(f"Bell numbers B(0..10): {b}") -# Example: Evaluate Legendre polynomials at a point -cx = np.zeros(6, dtype=np.float64, order='F') -cpx = np.zeros(6, dtype=np.float64, order='F') -polpack.legendre_poly(5, 0.5, cx, cpx) -print(f"Legendre P_5(0.5): {cx[5]}") +print(f"Bell numbers B_0 to B_10: {b}") ``` - -## Licensing - -The computer code and data files are distributed under the [GNU LGPL license](LICENSE). - -## References - -1. Milton Abramowitz, Irene Stegun, **Handbook of Mathematical Functions**, National Bureau of Standards, 1964. -2. Frank Benford, **The Law of Anomalous Numbers**, Proceedings of the American Philosophical Society, Vol. 78, 1938. -3. Robert Corless, et al., **On the Lambert W Function**, Advances in Computational Mathematics, Vol. 5, 1996. diff --git a/docs/installation.md b/docs/installation.md index 58f8283..8b1cf96 100644 --- a/docs/installation.md +++ b/docs/installation.md @@ -1,34 +1,137 @@ # Installation -`polpack` can be installed via `pip`, `conda`, or built from source using `meson-python`. +`polpack` is distributed as a compiled wheel on PyPI and can also be installed +from source via GitHub. + +--- ## Prerequisites -- **Python:** 3.10 or later. -- **NumPy:** 2.0 or later recommended. -- **Fortran Compiler:** `gfortran` (required for source builds). +- **Python 3.10+** +- **NumPy** (installed automatically as a dependency) + +For source builds you additionally need: + +- A Fortran compiler (`gfortran` recommended) +- `meson` and `meson-python` build system +- `numpy` (for `f2py` compilation) + +## PyPI (recommended) + +### pip + +```bash +pip install --upgrade polpack +``` + +### pyproject.toml dependency + +```toml +[project] +dependencies = [ + "polpack" +] +``` + +### requirements.txt + +```text +polpack +``` + +## Package managers + +### uv + +```bash +# Add to a uv project +uv add polpack + +# Or install into the current environment +uv pip install polpack +``` + +### pipenv + +```bash +pipenv install polpack +``` + +### poetry + +```bash +poetry add polpack +``` + +### pdm + +```bash +pdm add polpack +``` + +### hatch + +```bash +hatch add polpack +``` + +## Installing from source (GitHub) + +Install the latest development version directly from the repository: + +```bash +pip install --upgrade "git+https://github.com/eggzec/polpack.git#egg=polpack" +``` -## Using pip +### Building locally -The simplest way to install the latest release is from PyPI: +Clone and build from source if you want to modify the Fortran code or test +local changes: ```bash -pip install polpack +git clone https://github.com/eggzec/polpack.git +cd polpack +pip install -e . ``` -## Building from source +This invokes the `meson` build system to compile the Fortran sources via +`f2py` and install the resulting extension module in development mode. -To build `polpack` from the source repository, you will need `uv` and a Fortran compiler. +!!! warning "Fortran compiler required" + Source builds require a working Fortran compiler. On most Linux + distributions install `gfortran`: + +```bash +# Debian/Ubuntu +sudo apt install gfortran + +# Fedora +sudo dnf install gcc-gfortran + +# macOS (Homebrew) +brew install gcc +``` + +On Windows, install MinGW-w64 with gfortran or use MSYS2. + +## Verifying the installation + +After installation, verify that the package loads correctly: + +```python +import polpack +import numpy as np + +n = 10 +b = np.zeros(n + 1, dtype=np.int32, order="F") +polpack.bell(n, b) +print("polpack is working! Bell numbers:", b) +``` -1. **Clone the repository:** - ```bash - git clone https://github.com/eggzec/polpack - cd polpack - ``` +## Dependencies -2. **Install using the build script:** - ```bash - uv run bin/build.py install - ``` +| Package | Purpose | +|---|---| +| `numpy` | Array handling, `f2py` integration | -This will automatically configure Meson, compile the Fortran source, and install the package into your active environment. +No other runtime dependencies are required. diff --git a/docs/quickstart.md b/docs/quickstart.md index 33c0514..9d858c9 100644 --- a/docs/quickstart.md +++ b/docs/quickstart.md @@ -2,6 +2,30 @@ This guide will help you get started with polpack in Python. +## Working with Sequences + +Beyond polynomial families, `polpack` provides routines for various integer +sequences. These typically follow a similar pattern but return integers +(or floats for very large values). + +### Bell Numbers +The Bell numbers $B_n$ count the number of partitions of a set. + +```python +import numpy as np +import polpack + +# Compute Bell numbers up to n=10 +n = 10 +b = np.zeros(n + 1) +polpack.bell(n, b) + +print(f"Bell numbers B_0 to B_{n}:") +print(b) +``` + +--- + ## Basic Usage Evaluate the first 10 Bernoulli numbers: @@ -14,38 +38,82 @@ n = 10 # Initialize an array of size n+1 in Fortran order b = np.zeros(n + 1, dtype=np.float64, order="F") polpack.bernoulli_number(n, b) -This guide shows you how to use `polpack` for common mathematical tasks. +``` + +This guide provides walk-through examples of how to use `polpack` to evaluate +common polynomial families and combinatorial sequences in Python. -## Combinatorial counts +--- -To compute sequences like Bell numbers or Catalan numbers, pass a pre-allocated NumPy array to the routine. +## Recursive Polynomials: Chebyshev First Kind + +Chebyshev polynomials are fundamental in approximation theory and spectral +methods. In `polpack`, you can evaluate the entire sequence $T_0(x), \dots, T_n(x)$ +efficiently in a single call. ```python import numpy as np import polpack -# Compute first 11 Bell numbers -n = 10 -b = np.zeros(n + 1, dtype=np.int32, order='F') -polpack.bell(n, b) -print(f"Bell numbers: {b}") +# 1. Configuration +m = 1 # Number of points +n = 5 # Highest degree +x = 0.5 # Evaluation point +v = np.zeros(n + 1, dtype=np.float64, order="F") + +# 2. Evaluation +polpack.cheby_t_poly(m, n, x, v) + +# 3. Results +print(f"Chebyshev T_0({x}) to T_{n}({x}):") +for i, val in enumerate(v): + print(f" T_{i}({x}) = {val:.4f}") ``` -## Polynomial evaluation +## Orthogonal Polynomials: Legendre Polynomials + +Legendre polynomials are orthogonal on $[-1, 1]$ and arise naturally in +potential theory and spherical harmonics. + +```python +import numpy as np +import polpack + +# 1. Evaluate Legendre polynomials P_0 to P_10 at x=0.0 +n = 10 +x = 0.0 +p = np.zeros(n + 1, dtype=np.float64, order="F") +# Output array for first derivatives (not used here) +dp = np.zeros(n + 1, dtype=np.float64, order="F") + +polpack.legendre_poly(n, x, p, dp) + +# 2. Inspect results +# P_n(0.0) = 0 for odd n +print(f"Legendre P_n(0.0) for n=0..10:\n{p}") +``` -`polpack` provides routines for evaluating polynomial families at specific points or computing their coefficients. +## Combinatorial Sequences: Bell Numbers -### Point evaluation +Bell numbers $B_n$ count the number of partitions of a set of size $n$. These +increase rapidly, so `polpack` typically uses `int32` or `float64` depending +on the routine variant. ```python import numpy as np import polpack +# 1. Compute the first 15 Bell numbers +n = 14 +b = np.zeros(n + 1, dtype=np.int32, order="F") + +polpack.bell(n, b) + # Evaluate the 5th Legendre polynomial at x = 0.5 n = 5 x = 0.5 -cx = np.zeros(n + 1, dtype=np.float64, order='F') -cpx = np.zeros(n + 1, dtype=np.float64, order='F') # for derivatives +cx = np.zeros(n + 1, dtype=np.float64, order="F") +cpx = np.zeros(n + 1, dtype=np.float64, order="F") # for derivatives polpack.legendre_poly(n, x, cx, cpx) @@ -62,7 +130,7 @@ import polpack # Get coefficients of the 4th-order Chebyshev polynomial T_4(x) n = 4 # Space for (n+1) x (n+1) coefficient matrix -c = np.zeros((n + 1, n + 1), dtype=np.float64, order='F') +c = np.zeros((n + 1, n + 1), dtype=np.float64, order="F") polpack.cheby_t_poly_coef(n, c) print("Chebyshev T_4 coefficients (highest power first):") diff --git a/docs/references.md b/docs/references.md index 4e84525..2821d5f 100644 --- a/docs/references.md +++ b/docs/references.md @@ -1,34 +1,133 @@ # References -The algorithms implemented in `polpack` are based on established mathematical literature and numerical libraries. - -## Key Sources - -1. **Abramowitz, M. and Stegun, I. A. (Eds.).** *Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables*. National Bureau of Standards Applied Mathematics Series, 55. U.S. Government Printing Office, 1972. -2. **Burkardt, John.** *polpack: Mathematical functions, polynomial families, and sequences*. [Original Fortran versions](https://people.sc.fsu.edu/~jburkardt/f_src/polpak/polpak.html). -3. **Knuth, Donald E.** *The Art of Computer Programming, Volume 1: Fundamental Algorithms*. Addison-Wesley, 1997. (For combinatorial sequences). -4. **Press, W. H., et al.** *Numerical Recipes: The Art of Scientific Computing*. Cambridge University Press, 2007. - -## Specific Algorithms - -- **Chebyshev Polynomials:** Based on standard recurrences and zeroes from approximation theory. -- **Lambert W Function:** Implementation follows iterative methods for real-valued branches. -- **RNG:** Standard Lehmer (Park-Miller) linear congruential generator for seeded random numbers. - -3. Frank Benford, **The Law of Anomalous Numbers**, Proceedings of the American Philosophical Society, Vol. 78, 1938, pages 551-572. -4. Paul Bratley, Bennett Fox, Linus Schrage, **A Guide to Simulation**, Second Edition, Springer, 1987, ISBN: 0387964673. -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. -7. Theodore Chihara, **An Introduction to Orthogonal Polynomials**, Gordon and Breach, 1978, ISBN: 0677041500. -8. William Cody, **Rational Chebyshev Approximations for the Error Function**, Mathematics of Computation, Vol. 23, No. 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, Vol. 5, No. 1, Dec 1996, pages 329-359. -10. Bennett Fox, **Algorithm 647: Implementation and Relative Efficiency of Quasirandom Sequence Generators**, ACM Transactions on Mathematical Software, Vol. 12, No. 4, Dec 1986, pages 362-376. -11. Walter Gautschi, **Orthogonal Polynomials: Computation and Approximation**, Oxford, 2004, ISBN: 0-19-850672-4. -12. Brian Hayes, **The Vibonacci Numbers**, American Scientist, Vol. 87, No. 4, July-August 1999, pages 296-301. -13. Douglas Hofstadter, **Goedel, Escher, Bach**, Basic Books, 1979, ISBN: 0465026567. -14. Masanobu Kaneko, **Poly-Bernoulli Numbers**, Journal Theorie des Nombres Bordeaux, Vol. 9, No. 1, 1997, pages 221-228. -15. Thomas Osler, **Cardan Polynomials and the Reduction of Radicals**, Mathematics Magazine, Vol. 74, No. 1, Feb 2001, pages 26-32. -16. Gabor Szego, **Orthogonal Polynomials**, American Mathematical Society, 1992, ISBN: 0821810235. -17. Divakar Viswanath, **Random Fibonacci sequences and the number 1.13198824**, Mathematics of Computation, Vol. 69, No. 231, July 2000, pages 1131-1155. -18. Eric Weisstein, **CRC Concise Encyclopedia of Mathematics**, CRC Press, 2002, ISBN: 1584883472. -19. Shanjie Zhang, Jianming Jin, **Computation of Special Functions**, Wiley, 1996, ISBN: 0-471-11963-6. +## Core Mathematical References + +1. **Abramowitz, M., Stegun, I.** + *Handbook of Mathematical Functions*. + National Bureau of Standards, 1964. + +2. **Chihara, T.** + *An Introduction to Orthogonal Polynomials*. + Gordon and Breach, 1978. + +3. **Gautschi, W.** + *Orthogonal Polynomials: Computation and Approximation*. + Oxford, 2004. + +4. **Szego, G.** + *Orthogonal Polynomials*. + American Mathematical Society, 1992. + +5. **Zhang, S., Jin, J.** + *Computation of Special Functions*. + Wiley, 1996. + +6. **Knuth, D.** + *The Art of Computer Programming, Volume 2: Seminumerical Algorithms*. + Addison-Wesley, 1997. + +## Specialized Literature + +7. **Banks, R.** + *Slicing Pizzas, Racing Turtles, and Further Adventures in Applied Mathematics*. + Princeton, 1999. + +8. **Benford, F.** + “The Law of Anomalous Numbers.” + *Proceedings of the American Philosophical Society*, 78, 1938. + +9. **Bratley, P., Fox, B., Schrage, L.** + *A Guide to Simulation*. + Second Edition, Springer, 1987. + +10. **Brewbaker, C.** + “Lonesum (0,1)-matrices and poly-Bernoulli numbers of negative index.” + *Master of Science Thesis*, Iowa State University, 2005. + +11. **Briggs, W., Henson, V.** + *The DFT: An Owner's Manual for the Discrete Fourier Transform*. + SIAM, 1995. + +12. **Cody, W.** + “Rational Chebyshev Approximations for the Error Function.” + *Mathematics of Computation*, 23(107), 1969. + +13. **Corless, R., Gonnet, G., Hare, D., Jeffrey, D., Knuth, D.** + “On the Lambert W Function.” + *Advances in Computational Mathematics*, 5(1), 1996. + +14. **Fox, B.** + “Algorithm 647: Implementation and Relative Efficiency of Quasirandom Sequence Generators.” + *ACM Transactions on Mathematical Software*, 12(4), 1986. + +15. **Hartley, R.** + “A More Symmetrical Fourier Analysis Applied to Transmission Problems.” + *Proceedings of the Institute of Radio Engineers*, 30, 1942. + +16. **Hayes, B.** + “The Vibonacci Numbers.” + *American Scientist*, 87(4), 1999. + +17. **Hill, T.** + “The First Digit Phenomenon.” + *American Scientist*, 86(4), 1998. + +18. **Hofstadter, D.** + *Goedel, Escher, Bach*. + Basic Books, 1979. + +19. **Kaneko, M.** + “Poly-Bernoulli Numbers.” + *Journal Theorie des Nombres Bordeaux*, 9(1), 1997. + +20. **Osler, T.** + “Cardan Polynomials and the Reduction of Radicals.” + *Mathematics Magazine*, 74(1), 2001. + +21. **Pereira, J.** + “Algorithm 234: Poisson-Charliers Polynomials.” + *Communications of the ACM*, 7(7), 1964. + +22. **Pinter, C.** + *A Book of Abstract Algebra*. + Second Edition, McGraw Hill, 2003. + +23. **Raimi, R.** + “The Peculiar Distribution of First Digits.” + *Scientific American*, 1969. + +24. **Stanton, D., White, D.** + *Constructive Combinatorics*. + Springer, 1986. + +25. **Velleman, D., Call, G.** + “Permutations and Combination Locks.” + *Mathematics Magazine*, 68(4), 1995. + +26. **Viswanath, D.** + “Random Fibonacci sequences and the number 1.13198824.” + *Mathematics of Computation*, 69(231), 2000. + +27. **Waterman, M.** + *Introduction to Computational Biology*. + Chapman and Hall, 1995. + +28. **Weisstein, E.** + *CRC Concise Encyclopedia of Mathematics*. + Second Edition, CRC Press, 2002. + +29. **Wolfram, S.** + *The Mathematica Book*. + Fourth Edition, Cambridge University Press, 1999. + +30. **Wolfson, M., Wright, H.** + “ACM Algorithm 160: Combinatorial of M Things Taken N at a Time.” + *Communications of the ACM*, 6(4), 1963. + +31. **Zwillinger, D. (ed.)** + *CRC Standard Mathematical Tables and Formulae*. + 30th Edition, CRC Press, 1996. + +--- + +*Last revised on 11 April 2015.* diff --git a/docs/theory.md b/docs/theory.md index a156af2..3afb1ee 100644 --- a/docs/theory.md +++ b/docs/theory.md @@ -1,38 +1,347 @@ # Theory -`polpack` implements numerical algorithms for a broad set of mathematical objects, primarily focused on recursive relations and orthogonal properties. +This page covers the mathematical foundations behind `polpack`: the definition +of polynomial families, the stable recurrence relations used to evaluate them, +and the combinatorial sequences supported by the library. -## Orthogonal Polynomials +--- + +## 1) Background: Special Functions and Polynomials + +A **polynomial family** is a sequence of polynomials $\{P_n(x)\}_{n=0}^\infty$ +where $n$ is the degree of the polynomial. Many of these families are +**orthogonal polynomials**, meaning they satisfy a specific inner product +relationship over an interval $[a, b]$ with respect to a weight function $w(x)$. + +Special functions and polynomials arise naturally across science and engineering: + +- **Physics** — quantum mechanics (Hermite, Laguerre), optics (Zernike), Laplace's equation (Legendre) +- **Numerical Analysis** — function approximation (Chebyshev), Gaussian quadrature, spectral methods +- **Combinatorics** — counting set partitions (Bell numbers), mountain ranges (Catalan numbers) +- **Probability** — orthogonal basis expansion, stochastic processes + +### Orthogonality condition + +The structure in polynomial families is often characterized by: + +1. **Interval of support** — the range $[a, b]$ where the polynomials are defined +2. **Weight function** $w(x)$ — a non-negative function defining the inner product +3. **Orthogonality** — $\int_a^b P_m(x) P_n(x) w(x) \, dx = 0$ for $m \neq n$ +4. **Normalization** — the value of the integral for $m = n$ (often $C_n$) + +While sdepack focuses on sample paths of Wiener processes, `polpack` focuses on +the stable evaluation of these deterministic bases. + +### Explicit vs. Recursive interpretation + +There are two principal ways to define and evaluate polynomial families: + +| | **Explicit Formulas** | **Recurrence Relations** | +|---|---|---| +| Evaluation point | Direct calculation from $x$ | Computed step-wise from $P_0, P_1, \dots$ | +| Numerical Stability | Often poor for large $n$ (cancellation errors) | Typically excellent and stable | +| Implementation | Combinatorial sums, binomial coefficients | Three-term recurrence loops | +| Typical use | Symbolic manipulation | Numerical computing, library implementations | + +`polpack` uses **recurrence relations** exclusively for its numerical core. + +!!! info "Christoffel–Darboux formula" + For orthogonal polynomials, the sum of products follows a specific + closed form: + + $$ \sum_{k=0}^n \frac{P_k(x)P_k(y)}{h_k} = \frac{k_n}{k_{n+1}h_n} \frac{P_{n+1}(x)P_n(y) - P_n(x)P_{n+1}(y)}{x - y} $$ + + The additional terms arising in this identity (analogous to Itô's lemma) + are fundamental to understanding the convergence of spectral expansions. + +## 2) Orthogonal Polynomial Families -Many of the polynomial families in `polpack` share the property of **orthogonality** with respect to a weight function $w(x)$ over an interval $[a, b]$: +All routines in `polpack` target specific polynomial forms: $$ -\int_a^b w(x) P_n(x) P_m(x) \, dx = \delta_{nm} h_n, +\int_a^b P_m(x) P_n(x) w(x) \, dx = \delta_{mn} C_n. $$ -where $\delta_{nm}$ is the Kronecker delta and $h_n$ is a normalization constant. +Within the library, this is implemented as: -### Three-term recurrence +$$ +P_{n+1}(x) = (A_n x + B_n) P_n(x) - C_n P_{n-1}(x), +$$ + +where: + +- $x$ is the evaluation point +- $A_n, B_n, C_n$ are method-specific recurrence coefficients +- $k$ denotes the degree or index of the recurrence -Most orthogonal polynomials are computed using a stable three-term recurrence relation of the form: +For **degree-invariant** routines, the coefficients $A, B, C$ are pre-defined +constants for a given degree $n$. + +## Polynomial families + +`polpack` evaluates a variety of recursively defined polynomial families, +including those orthogonal with respect to specific weights and intervals. + +### Bernoulli Polynomials +The Bernoulli polynomials $B_n(x)$ are defined by the generating function: +$\frac{t e^{xt}}{e^t - 1} = \sum_{n=0}^{\infty} B_n(x) \frac{t^n}{n!}$ +They satisfy the recurrence: $$ -P_{n+1}(x) = (A_n x + B_n) P_n(x) - C_n P_{n-1}(x). +B_n(x) = \sum_{k=0}^n \binom{n}{k} B_k x^{n-k} $$ -`polpack` evaluates these recurrences efficiently in Fortran, ensuring high precision and performance. +### Bernstein Polynomials +The Bernstein basis polynomials $b_{\nu, n}(x)$ are defined on $[0,1]$ as: +$b_{\nu, n}(x) = \binom{n}{\nu} x^{\nu} (1-x)^{n-\nu}$ +They form a basis for the space of polynomials of degree at most $n$. + +### Cardan Polynomials +Cardan polynomials $C_n(x, s)$ are associated with the solution of cubic +equations. They satisfy the three-term recurrence: + +$$ +C_{n+1}(x, s) = x C_n(x, s) - s C_{n-1}(x, s) +$$ + +### Charlier Polynomials +The Charlier polynomials $C_n(x; a)$ are orthogonal with respect to the +Poisson distribution. They satisfy: + +$$ +a C_{n+1}(x; a) = (n + a - x) C_n(x; a) - n C_{n-1}(x; a) +$$ + +### Chebyshev Polynomials +**First Kind ($T_n$):** Orthogonal on $[-1, 1]$ with weight $(1-x^2)^{-1/2}$. + +$$ +T_{n+1}(x) = 2x T_n(x) - T_{n-1}(x) +$$ + +**Second Kind ($U_n$):** Orthogonal on $[-1, 1]$ with weight $(1-x^2)^{1/2}$. + +$$ +U_{n+1}(x) = 2x U_n(x) - U_{n-1}(x) +$$ + +### Gegenbauer Polynomials +The Gegenbauer polynomials $C_n^{(\alpha)}(x)$ are orthogonal on $[-1, 1]$ with +weight $(1-x^2)^{\alpha - 1/2}$. + +$$ +(n+1)C_{n+1}^{(\alpha)}(x) = 2(n+\alpha)x C_n^{(\alpha)}(x) - (n+2\alpha-1)C_{n-1}^{(\alpha)}(x) +$$ + +### Hermite Polynomials (Physicist's) +Orthogonal on $(-\infty, \infty)$ with weight $e^{-x^2}$. + +$$ +H_{n+1}(x) = 2x H_n(x) - 2n H_{n-1}(x) +$$ + +### Jacobi Polynomials +The Jacobi polynomials $P_n^{(\alpha, \beta)}(x)$ are orthogonal on $[-1, 1]$ +with weight $(1-x)^{\alpha}(1+x)^{\beta}$. They generalize both Chebyshev and +Legendre polynomials. + +### Krawtchouk Polynomials +Discrete orthogonal polynomials $K_n(x; p, N)$ related to the binomial +distribution. + +### Laguerre Polynomials +**Associated Laguerre ($L_n^{(\alpha)}$):** Orthogonal on $[0, \infty)$ with +weight $x^{\alpha}e^{-x}$. + +$$ +(n+1)L_{n+1}^{(\alpha)}(x) = (2n + \alpha + 1 - x) L_n^{(\alpha)}(x) - (n+\alpha)L_{n-1}^{(\alpha)}(x) +$$ + +### Legendre Polynomials +Orthogonal on $[-1, 1]$ with weight 1. + +$$ +(n+1)P_{n+1}(x) = (2n+1)x P_n(x) - n P_{n-1}(x) +$$ + +### Meixner Polynomials +Discrete orthogonal polynomials $M_n(x; \beta, c)$ related to the negative +binomial distribution. + +### Zernike Polynomials +Polynomials defined on the unit disk, widely used in optics to describe +wavefront aberrations. + +--- ## Combinatorial Sequences +`polpack` provides routines for several important integer sequences. + ### Bell Numbers -The Bell number $B_n$ counts the number of partitions of a set of $n$ elements. +The Bell number $B_n$ counts the number of ways a set of $n$ elements can be +partitioned into non-empty subsets. + +$$ +B_{n+1} = \sum_{k=0}^n \binom{n}{k} B_k +$$ ### Catalan Numbers -Catalan numbers $C_n$ count various objects in combinatorial geometry and computer science, such as the number of ways to triangulate a convex polygon with $n+2$ vertices. +The Catalan number $C_n$ occurs in various counting problems (e.g., number +of binary trees with $n$ nodes). + +$$ +C_n = \frac{1}{n+1} \binom{2n}{n} +$$ + +### Stirling Numbers +**First Kind ($s(n,k)$):** Number of permutations of $n$ elements with $k$ cycles. +**Second Kind ($S(n,k)$):** Number of ways to partition a set of $n$ elements +into $k$ non-empty subsets. + +--- + +## Number Theory and Probability + +### Benford's Law +Benford's law, also called the Newcomb–Benford law or the law of anomalous +numbers, defines the probability distribution of the first digit $d$ in many +real-life sets of numerical data: + +$$ +P(d) = \log_{10}(1 + 1/d) +$$ + +### Collatz Conjecture +The Collatz sequence for a starting integer $n$ is defined by: + +- If $n$ is even, $n_{i+1} = n_i / 2$ +- If $n$ is odd, $n_{i+1} = 3n_i + 1$ +`polpack` provides routines to count the number of steps to reach 1. + +### Arithmetical Functions + +- **Euler Totient ($\phi(n)$):** Number of positive integers less than $n$ + that are relatively prime to $n$. +- **Moebius ($\mu(n)$):** $\mu(n) = (-1)^k$ if $n$ is square-free with $k$ prime + factors, else 0. +- **Mertens ($M(n)$):** The sum of the Moebius function $M(n) = \sum_{k=1}^n \mu(k)$. +- **Divisor Sum ($\sigma(n)$):** Sum of all positive divisors of $n$. + +## 3) Three-term recurrence and evaluation scaling + +Given an interval $[a, b]$ and degree $N$: +$$ +P_0(x) = \alpha, \qquad P_1(x) = \beta x + \gamma. +$$ + +At each stage $n$, the polynomial value is generated: +$$ +P_{n+1}(x) = (A_n x + B_n)P_n(x) - C_n P_{n-1}(x), +$$ + +where $A_n, B_n, C_n$ are coefficients that ensure the correct scaling at each +stage. When multiplied in the core loops, this produces the high-order +polynomial values required for scientific applications. + +## 4) Chebyshev Polynomials — RK1 analog + +The simplest and most widely used family in numerical analysis. Named after +Pafnuty Chebyshev, they are the deterministic analog of many foundational +integrators. + +For both first and second kind variants: + +$$ +T_{n+1}(x) = 2x T_n(x) - T_{n-1}(x), +$$ + +$$ +T_0(x) = 1,\quad T_1(x)=x. +$$ +This recurrence is notoriously stable for $x \in [-1, 1]$. + +!!! info "Minimax property" + The Chebyshev polynomials minimize the maximum error in function + approximation, analogously to how Euler-Maruyama minimizes one-step + error in SDEs. + +## 5) Legendre Polynomials — Two-stage recurrence + +A classic family providing improved accuracy in spherical harmonic expansions: + +$$ +\begin{aligned} +A_{n} &= \frac{2n+1}{n+1},\\ +B_{n} &= 0,\\ +C_{n} &= \frac{n}{n+1},\\ +(n+1)P_{n+1}(x) &= (2n+1)x P_n(x) - n P_{n-1}(x). +\end{aligned} +$$ -### Fibonacci Numbers -The Fibonacci sequence is defined by $F_0=0, F_1=1$, and $F_{n} = F_{n-1} + F_{n-2}$. +Coefficients: +$$ +P_0=1,\quad P_1=x. +$$ --- -*Refer to [References](references.md) for detailed literature on these mathematical objects.* +## 6) Hermite Polynomials — physicist form (time-invariant only) + +A three-term recurrence providing higher accuracy for systems involving +Gaussian weights: +$$ +H_{n+1}(x) = 2x H_n(x) - 2n H_{n-1}(x), +$$ + +with coefficients: +$$ +\begin{aligned} +H_0&=1,\\ +H_1&=2x. +\end{aligned} +$$ + +!!! note + The probabilist's form $He_n(x)$ uses different scaling parameters. Both + are available in `polpack`. + +## 7) Jacobi and Gegenbauer Polynomials + +The highest-order methods in `polpack`, using four terms or generalized +parameters for maximum flexibility. + +The general update formula is: +$$ +P_{n+1}(x) = (A_n x + B_n) P_n(x) - C_n P_{n-1}(x). +$$ + +The Jacobi and Gegenbauer variants use **different** coefficient sets +optimized for their respective weighting intervals. See the [API Reference](api.md) +for the complete recurrence tables. + +## 8) Convergence and stability considerations + +### Strong vs. weak stability + +When assessing the quality of a numerical polynomial evaluation, two notions of +stability are distinguished: + +- **Forward stability** — accuracy of individual evaluations. +- **Backward stability** — whether the computed $P_n(x)$ is the exact solution + to a slightly perturbed recurrence. + +Forward stability is often sufficient for function approximation, while +backward stability matters in eigenvalue problems. + +### Relationship between degree and order + +The three-term recurrence schemes in `polpack` are designed to achieve higher +numerical precision. The Euler-like first-order recursions (RK1 analog) are +stable but may accumulate error at very high degrees ($n > 1000$). Higher-degree +polynomials generally benefit from the stable recurrence architecture. + +!!! tip "Evaluation guidance" + Unlike simple power series, the recurrence $P_{n+1}(x)$ only reduces the + error by a factor of 2 per degree. For very large $n$, use specialized + asymptotic routines if available. diff --git a/main.py b/main.py deleted file mode 100644 index 33a93d1..0000000 --- a/main.py +++ /dev/null @@ -1,6 +0,0 @@ -def main(): - print("polpack v0.0.1") - - -if __name__ == "__main__": - main() diff --git a/mkdocs.yml b/mkdocs.yml index 85563fe..a32db6c 100644 --- a/mkdocs.yml +++ b/mkdocs.yml @@ -1,44 +1,39 @@ site_name: polpack -site_url: https://eggzec.github.io/polpack/ +site_description: Multidimensional Quadrature Using Sparse Grids for Python +site_url: https://eggzec.github.io/polpack repo_url: https://github.com/eggzec/polpack -edit_uri: edit/main/docs/ +repo_name: eggzec/polpack + +extra: + social: + - icon: fontawesome/brands/github + link: https://github.com/eggzec/polpack + - icon: fontawesome/brands/python + link: https://pypi.org/project/polpack/ theme: name: material - palette: - - media: "(prefers-color-scheme: light)" - scheme: default - primary: indigo - accent: indigo - toggle: - icon: material/brightness-7 - name: Switch to dark mode - - media: "(prefers-color-scheme: dark)" - scheme: slate - primary: indigo - accent: indigo - toggle: - icon: material/brightness-4 - name: Switch to light mode features: - - navigation.tabs - - navigation.sections - - navigation.top - - search.suggest - - search.highlight - content.code.copy +markdown_extensions: + - admonition + - tables + - fenced_code + - pymdownx.superfences + - pymdownx.highlight + - pymdownx.arithmatex: + generic: true + - toc: + permalink: true + +extra_javascript: + - https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js + nav: - - Welcome: index.md + - Home: index.md + - Theory: theory.md - Installation: installation.md - Quickstart: quickstart.md - API Reference: api.md - References: references.md - -markdown_extensions: - - admonition - - pymdownx.highlight: - anchor_linenums: true - - pymdownx.inlinehilite - - pymdownx.snippets - - pymdownx.superfences diff --git a/pyproject.toml b/pyproject.toml index 6e60b90..eec5117 100644 --- a/pyproject.toml +++ b/pyproject.toml @@ -50,14 +50,22 @@ issues = "https://github.com/eggzec/polpack/issues" [dependency-groups] dev = [ + {include-group = "build"}, {include-group = "docs"}, - {include-group = "test"}, - "meson-python", - "ninja", + {include-group = "lint"}, + {include-group = "test"} +] +build = [ + "meson-python", + "ninja", + "setuptools_scm>=8", ] docs = [ "zensical>=0.0.23" ] +lint = [ + "ruff==0.15.*", +] test = [ "pytest>=8.3.5", "pytest-cov>=6.0", @@ -67,7 +75,7 @@ test = [ [build-system] requires = [ "meson-python", - "numpy>=2.0.0", + "numpy", "packaging", "setuptools_scm>=8" ] @@ -79,3 +87,56 @@ local_scheme = "no-local-version" [tool.cibuildwheel] enable = ["cpython-prerelease"] + +[tool.ruff] +line-length = 80 +indent-width = 4 +preview = true + +# Output serialization format for violations. The default serialization +# format is "full" [env: RUFF_OUTPUT_FORMAT=] [possible values: +# concise, full, json, json-lines, junit, grouped, github, gitlab, +# pylint, rdjson, azure, sarif] +output-format = "grouped" + +[tool.ruff.lint] +isort.lines-after-imports = 2 +isort.split-on-trailing-comma = false + +select = [ + "ANN", # flake8-annotations (required strict type annotations for public functions) + "S", # flake8-bandit (checks basic security issues in code) + "BLE", # flake8-blind-except (checks the except blocks that do not specify exception) + "FBT", # flake8-boolean-trap (ensure that boolean args can be used with kw only) + "E", # pycodestyle errors (PEP 8 style guide violations) + "W", # pycodestyle warnings (e.g., extra spaces, indentation issues) + "DOC", # pydoclint issues (e.g., extra or missing return, yield, warnings) + "A", # flake8-buitins (check variable and function names to not shadow builtins) + "N", # Naming convention checks (e.g., PEP 8 variable and function names) + "F", # Pyflakes errors (e.g., unused imports, undefined variables) + "I", # isort (Ensures imports are sorted properly) + "B", # flake8-bugbear (Detects likely bugs and bad practices) + "TID", # flake8-tidy-imports (Checks for banned or misplaced imports) + "UP", # pyupgrade (Automatically updates old Python syntax) + "YTT", # flake8-2020 (Detects outdated Python 2/3 compatibility issues) + "FLY", # flynt (Converts old-style string formatting to f-strings) + "PIE", # flake8-pie + "PL", # pylint + "RUF", # Ruff-specific rules (Additional optimizations and best practices) +] + +ignore = [] + +[tool.ruff.lint.per-file-ignores] +"**/__init__.py" = ["RUF067"] +"**/tests/*.py" = [ + "ANN001", # Missing type annotation for function argument + "ANN202", # Missing return type annotation for private function + "DOC201", # `return` is not documented in docstring + "PLR6301", # Method could be a function, class method, or static method + "S101", # Use of `assert` detected +] + +[tool.ruff.format] +docstring-code-format = true +skip-magic-trailing-comma = true diff --git a/src/polpack.pyf b/src/polpack.pyf index 9ce17e4..4afca10 100644 --- a/src/polpack.pyf +++ b/src/polpack.pyf @@ -46,9 +46,9 @@ python module _polpack ! in double precision :: c end subroutine bernoulli_number_values subroutine bernoulli_poly(n,x,bx) - integer :: n - double precision :: x - double precision :: bx + integer intent(in) :: n + double precision intent(in) :: x + double precision intent(out) :: bx end subroutine bernoulli_poly subroutine bernoulli_poly2(n,x,bx) integer :: n @@ -220,7 +220,7 @@ python module _polpack ! in subroutine delannoy(m,n,a) integer intent(in) :: m integer intent(in) :: n - integer dimension(1 + m,1 + n), intent(in, out), depend(m) :: a + integer dimension(1 + m,1 + n), intent(in, out), depend(m,n) :: a end subroutine delannoy subroutine erf_values(n_data,x,fx) integer :: n_data @@ -250,8 +250,8 @@ python module _polpack ! in integer dimension(n,n), intent(in, out), depend(n) :: e end subroutine eulerian subroutine fibonacci_direct(n,f) - integer :: n - integer :: f + integer intent(in) :: n + integer intent(out) :: f end subroutine fibonacci_direct subroutine fibonacci_floor(n,f,i) integer :: n diff --git a/src/polpack/__init__.py b/src/polpack/__init__.py index 9744472..543d3e9 100644 --- a/src/polpack/__init__.py +++ b/src/polpack/__init__.py @@ -5,31 +5,39 @@ of mathematical functions, polynomials, and combinatorial sequences. """ -from . import _polpack import numpy as np -def agm_values(n_data, a, b, fx): - """returns some values of the arithmetic geometric mean. +from . import _polpack + + +def agm_values(n_data: int, a: float, b: float, fx: float) -> None: + """Returns some values of the arithmetic geometric mean. Args: - n_data (int): Description for n_data. - a (float): Description for a. - b (float): Description for b. - fx (float): Description for fx. + n_data (int): Number of data points. + a (float): First value. + b (float): Second value. + fx (float): Function value. + Returns: + None """ return _polpack.agm_values(n_data, a, b, fx) -def bell(n, b): - """returns the Bell numbers from 0 to N. + +def bell(n: int, b: np.ndarray) -> None: + """Returns the Bell numbers from 0 to N. Args: - n (int): Description for n. - b (int): Description for b. + n (int): order of recursion. + b (np.ndarray): array to store Bell numbers. + Returns: + None """ return _polpack.bell(n, b) -def bell_values(n_data, n, c): - """returns some values of the Bell numbers for testing. + +def bell_values(n_data: int, n: int, c: np.ndarray) -> None: + """Returns some values of the Bell numbers for testing. Args: n_data (int): Description for n_data. @@ -38,8 +46,9 @@ def bell_values(n_data, n, c): """ return _polpack.bell_values(n_data, n, c) -def bernoulli_number(n, b): - """computes the value of the Bernoulli numbers B(0) through B(N). + +def bernoulli_number(n: int, b: np.ndarray) -> None: + """Computes the value of the Bernoulli numbers B(0) through B(N). Args: n (int): Description for n. @@ -47,8 +56,9 @@ def bernoulli_number(n, b): """ return _polpack.bernoulli_number(n, b) -def bernoulli_number2(n, b): - """evaluates the Bernoulli numbers. + +def bernoulli_number2(n: int, b: np.ndarray) -> None: + """Evaluates the Bernoulli numbers. Args: n (int): Description for n. @@ -56,8 +66,9 @@ def bernoulli_number2(n, b): """ return _polpack.bernoulli_number2(n, b) -def bernoulli_number3(n, b): - """computes the value of the Bernoulli number B(N). + +def bernoulli_number3(n: int, b: np.ndarray) -> None: + """Computes the value of the Bernoulli number B(N). Args: n (int): Description for n. @@ -65,248 +76,319 @@ def bernoulli_number3(n, b): """ return _polpack.bernoulli_number3(n, b) -def bernoulli_number_values(n_data, n, c): - """returns some values of the Bernoulli numbers. + +def bernoulli_number_values(n_data: int, n: int, c: float) -> None: + """Returns some values of the Bernoulli numbers. Args: - n_data (int): Description for n_data. - n (int): Description for n. - c (float): Description for c. + n_data (int): Number of data points. + n (int): value of n. + c (float): value of Bernoulli number. + Returns: + None """ return _polpack.bernoulli_number_values(n_data, n, c) -def bernoulli_poly(n, x, bx): - """evaluates the Bernoulli polynomial of order N at X. + +def bernoulli_poly(n: int, x: float) -> float: + """Evaluates the Bernoulli polynomial of order N at X. Args: - n (int): Description for n. - x (float): Description for x. - bx (float): Description for bx. + n (int): order of the polynomial. + x (float): evaluation point. + Returns: + float: value of the Bernoulli polynomial. """ - return _polpack.bernoulli_poly(n, x, bx) + return _polpack.bernoulli_poly(n, x) + -def bernoulli_poly2(n, x, bx): - """evaluates the N-th Bernoulli polynomial at X. +def bernoulli_poly2(n: int, x: float, bx: float) -> None: + """Evaluates the N-th Bernoulli polynomial at X. Args: - n (int): Description for n. - x (float): Description for x. - bx (float): Description for bx. + n (int): order of polynomial. + x (float): evaluation point. + bx (float): value of Bernoulli polynomial. + Returns: + None """ return _polpack.bernoulli_poly2(n, x, bx) -def bernstein_poly(n, x, bern): - """evaluates the Bernstein polynomials at a point X. + +def bernstein_poly(n: int, x: float, bern: np.ndarray) -> None: + """Evaluates the Bernstein polynomials at a point X. Args: - n (int): Description for n. - x (float): Description for x. - bern (float): Description for bern. + n (int): degree of polynomial. + x (float): evaluation point. + bern (np.ndarray): values of Bernstein polynomials. + Returns: + None """ return _polpack.bernstein_poly(n, x, bern) -def bernstein_poly_values(n_data, n, k, x, b): - """returns some values of the Bernstein polynomials. + +def bernstein_poly_values( + n_data: int, n: int, k: int, x: float, b: float +) -> None: + """Returns some values of the Bernstein polynomials. Args: - n_data (int): Description for n_data. - n (int): Description for n. - k (int): Description for k. - x (float): Description for x. - b (float): Description for b. + n_data (int): Number of data points. + n (int): degree of polynomial. + k (int): index of polynomial. + x (float): evaluation point. + b (float): value of Bernstein polynomial. + Returns: + None """ return _polpack.bernstein_poly_values(n_data, n, k, x, b) -def beta_values(n_data, x, y, fxy): - """returns some values of the Beta function. + +def beta_values(n_data: int, x: float, y: float, fxy: float) -> None: + """Returns some values of the Beta function. Args: - n_data (int): Description for n_data. - x (float): Description for x. - y (float): Description for y. - fxy (float): Description for fxy. + n_data (int): Number of data points. + x (float): First argument. + y (float): Second argument. + fxy (float): Function value. + Returns: + None """ return _polpack.beta_values(n_data, x, y, fxy) -def bpab(n, x, a, b, bern): - """evaluates at X the Bernstein polynomials based in [A,B]. + +def bpab(n: int, x: float, a: float, b: float, bern: np.ndarray) -> None: + """Evaluates at X the Bernstein polynomials based in [A,B]. Args: - n (int): Description for n. - x (float): Description for x. - a (float): Description for a. - b (float): Description for b. - bern (float): Description for bern. + n (int): degree of polynomial. + x (float): evaluation point. + a (float): left endpoint. + b (float): right endpoint. + bern (np.ndarray): values of Bernstein polynomials. + Returns: + None """ return _polpack.bpab(n, x, a, b, bern) -def cardan_poly(n, x, s, cx): - """evaluates the Cardan polynomials. + +def cardan_poly(n: int, x: float, s: float, cx: np.ndarray) -> None: + """Evaluates the Cardan polynomials. Args: - n (int): Description for n. - x (float): Description for x. - s (float): Description for s. - cx (float): Description for cx. + n (int): degree of polynomial. + x (float): evaluation point. + s (float): parameter s. + cx (np.ndarray): value of Cardan polynomial. + Returns: + None """ return _polpack.cardan_poly(n, x, s, cx) -def cardan_poly_coef(n, s, c): - """computes the coefficients of the N-th Cardan polynomial. + +def cardan_poly_coef(n: int, s: float, c: np.ndarray) -> None: + """Computes the coefficients of the N-th Cardan polynomial. Args: - n (int): Description for n. - s (float): Description for s. - c (float): Description for c. + n (int): degree of polynomial. + s (float): parameter s. + c (np.ndarray): coefficients of Cardan polynomial. + Returns: + None """ return _polpack.cardan_poly_coef(n, s, c) -def cardinal_cos(j, m, n, t, c): - """evaluates the J-th cardinal cosine basis function. + +def cardinal_cos(j: int, m: int, n: int, t: float, c: np.ndarray) -> None: + """Evaluates the J-th cardinal cosine basis function. Args: - j (int): Description for j. - m (int): Description for m. - n (int): Description for n. - t (float): Description for t. - c (float): Description for c. + j (int): index of basis function. + m (int): parameter m. + n (int): parameter n. + t (float): evaluation point. + c (np.ndarray): value of basis function. + Returns: + None """ return _polpack.cardinal_cos(j, m, n, t, c) -def cardinal_sin(j, m, n, t, s): - """evaluates the J-th cardinal sine basis function. + +def cardinal_sin(j: int, m: int, n: int, t: float, s: np.ndarray) -> None: + """Evaluates the J-th cardinal sine basis function. Args: - j (int): Description for j. - m (int): Description for m. - n (int): Description for n. - t (float): Description for t. - s (float): Description for s. + j (int): index of basis function. + m (int): parameter m. + n (int): parameter n. + t (float): evaluation point. + s (np.ndarray): value of basis function. + Returns: + None """ return _polpack.cardinal_sin(j, m, n, t, s) -def catalan(n, c): - """computes the Catalan numbers, from C(0) to C(N). + +def catalan(n: int, c: np.ndarray) -> None: + """Computes the Catalan numbers, from C(0) to C(N). Args: - n (int): Description for n. - c (int): Description for c. + n (int): degree. + c (np.ndarray): Catalan numbers. + Returns: + None """ return _polpack.catalan(n, c) -def catalan_row_next(ido, n, irow): - """computes row N of Catalan's triangle. + +def catalan_row_next(ido: int, n: int, irow: np.ndarray) -> None: + """Computes row N of Catalan's triangle. Args: - ido (int): Description for ido. - n (int): Description for n. - irow (int): Description for irow. + ido (int): initialization flag. + n (int): row index. + irow (np.ndarray): row values. + Returns: + None """ return _polpack.catalan_row_next(ido, n, irow) -def catalan_values(n_data, n, c): - """returns some values of the Catalan numbers for testing. + +def catalan_values(n_data: int, n: int, c: int) -> None: + """Returns some values of the Catalan numbers for testing. Args: - n_data (int): Description for n_data. - n (int): Description for n. - c (int): Description for c. + n_data (int): Number of points. + n (int): index. + c (int): value. + Returns: + None """ return _polpack.catalan_values(n_data, n, c) -def charlier(n, a, x, value): - """evaluates Charlier polynomials at a point. + +def charlier(n: int, a: float, x: float, value: np.ndarray) -> None: + """Evaluates Charlier polynomials at a point. Args: - n (int): Description for n. - a (float): Description for a. - x (float): Description for x. - value (float): Description for value. + n (int): degree. + a (float): parameter. + x (float): point. + value (np.ndarray): values. + Returns: + None """ return _polpack.charlier(n, a, x, value) -def cheby_t_poly(m, n, x, cx): - """evaluates Chebyshev polynomials T(n,x). + +def cheby_t_poly(m: int, n: int, x: float, cx: np.ndarray) -> None: + """Evaluates Chebyshev polynomials T(n,x). Args: - m (int): Description for m. - n (int): Description for n. - x (float): Description for x. - cx (float): Description for cx. + m (int): number of points. + n (int): max degree. + x (float): point. + cx (np.ndarray): values. + Returns: + None """ return _polpack.cheby_t_poly(m, n, x, cx) -def cheby_t_poly_coef(n, c): - """evaluates coefficients of Chebyshev polynomials T(n,x). + +def cheby_t_poly_coef(n: int, c: np.ndarray) -> None: + """Evaluates coefficients of Chebyshev polynomials T(n,x). Args: - n (int): Description for n. - c (float): Description for c. + n (int): degree. + c (np.ndarray): coefficients. + Returns: + None """ return _polpack.cheby_t_poly_coef(n, c) -def cheby_t_poly_values(n_data, n, x, fx): - """returns values of Chebyshev polynomials T(n,x). + +def cheby_t_poly_values(n_data: int, n: int, x: float, fx: float) -> None: + """Returns values of Chebyshev polynomials T(n,x). Args: - n_data (int): Description for n_data. - n (int): Description for n. - x (float): Description for x. - fx (float): Description for fx. + n_data (int): points. + n (int): degree. + x (float): point. + fx (float): value. + Returns: + None """ return _polpack.cheby_t_poly_values(n_data, n, x, fx) -def cheby_t_poly_zero(n, z): - """returns zeroes of Chebyshev polynomials T(n,x). + +def cheby_t_poly_zero(n: int, z: np.ndarray) -> None: + """Returns zeroes of Chebyshev polynomials T(n,x). Args: - n (int): Description for n. - z (float): Description for z. + n (int): degree. + z (np.ndarray): zeroes. + Returns: + None """ return _polpack.cheby_t_poly_zero(n, z) -def cheby_u_poly(m, n, x, cx): - """evaluates Chebyshev polynomials U(n,x). + +def cheby_u_poly(m: int, n: int, x: float, cx: np.ndarray) -> None: + """Evaluates Chebyshev polynomials U(n,x). Args: - m (int): Description for m. - n (int): Description for n. - x (float): Description for x. - cx (float): Description for cx. + m (int): number of points. + n (int): max degree. + x (float): point. + cx (np.ndarray): values. + Returns: + None """ return _polpack.cheby_u_poly(m, n, x, cx) -def cheby_u_poly_coef(n, c): - """evaluates coefficients of Chebyshev polynomials U(n,x). + +def cheby_u_poly_coef(n: int, c: np.ndarray) -> None: + """Evaluates coefficients of Chebyshev polynomials U(n,x). Args: - n (int): Description for n. - c (float): Description for c. + n (int): degree. + c (np.ndarray): coefficients. + Returns: + None """ return _polpack.cheby_u_poly_coef(n, c) -def cheby_u_poly_values(n_data, n, x, fx): - """returns values of Chebyshev polynomials U(n,x). + +def cheby_u_poly_values(n_data: int, n: int, x: float, fx: float) -> None: + """Returns values of Chebyshev polynomials U(n,x). Args: - n_data (int): Description for n_data. - n (int): Description for n. - x (float): Description for x. - fx (float): Description for fx. + n_data (int): points. + n (int): degree. + x (float): point. + fx (float): value. + Returns: + None """ return _polpack.cheby_u_poly_values(n_data, n, x, fx) -def cheby_u_poly_zero(n, z): - """returns zeroes of Chebyshev polynomials U(n,x). + +def cheby_u_poly_zero(n: int, z: np.ndarray) -> None: + """Returns zeroes of Chebyshev polynomials U(n,x). Args: - n (int): Description for n. - z (float): Description for z. + n (int): degree. + z (np.ndarray): zeroes. + Returns: + None """ return _polpack.cheby_u_poly_zero(n, z) + def chebyshev_discrete(n, m, x, v): - """evaluates discrete Chebyshev polynomials at a point. + """Evaluates discrete Chebyshev polynomials at a point. Args: n (int): Description for n. @@ -316,866 +398,1162 @@ def chebyshev_discrete(n, m, x, v): """ return _polpack.chebyshev_discrete(n, m, x, v) -def collatz_count_max(n, i_max, j_max): - """seeks the maximum Collatz count for 1 through N. + +def collatz_count_max(n: int, i_max: np.ndarray, j_max: np.ndarray) -> None: + """Seeks the maximum Collatz count for 1 through N. Args: - n (int): Description for n. - i_max (int): Description for i_max. - j_max (int): Description for j_max. + n (int): maximum value to check. + i_max (np.ndarray): value achieving the maximum count. + j_max (np.ndarray): the maximum count. + Returns: + None """ return _polpack.collatz_count_max(n, i_max, j_max) -def collatz_count_values(n_data, n, count): - """returns some values of the Collatz count function. + +def collatz_count_values(n_data: int, n: int, count: int) -> None: + """Returns some values of the Collatz count function. Args: - n_data (int): Description for n_data. - n (int): Description for n. - count (int): Description for count. + n_data (int): Number of points. + n (int): value of n. + count (int): Collatz count. + Returns: + None """ return _polpack.collatz_count_values(n_data, n, count) -def comb_row_next(n, row): - """computes the next row of Pascal's triangle. + +def comb_row_next(n: int, row: np.ndarray) -> None: + """Computes the next row of Pascal's triangle. Args: - n (int): Description for n. - row (int): Description for row. + n (int): current row index. + row (np.ndarray): current row / next row. + Returns: + None """ return _polpack.comb_row_next(n, row) -def commul(n, nfactor, factor, ncomb): - """computes a multinomial combinatorial coefficient. + +def commul(n: int, nfactor: int, factor: np.ndarray, ncomb: int) -> None: + """Computes a multinomial combinatorial coefficient. Args: - n (int): Description for n. - nfactor (int): Description for nfactor. - factor (int): Description for factor. - ncomb (int): Description for ncomb. + n (int): total number of items. + nfactor (int): number of factors. + factor (np.ndarray): counts of each factor. + ncomb (int): result coefficient. + Returns: + None """ return _polpack.commul(n, nfactor, factor, ncomb) -def complete_symmetric_poly(n, r, x, value): - """evaluates a complete symmetric polynomial. + +def complete_symmetric_poly( + n: int, r: int, x: np.ndarray, value: float +) -> None: + """Evaluates a complete symmetric polynomial. Args: - n (int): Description for n. - r (int): Description for r. - x (float): Description for x. - value (float): Description for value. + n (int): number of variables. + r (int): degree of polynomial. + x (np.ndarray): variables. + value (float): result value. + Returns: + None """ return _polpack.complete_symmetric_poly(n, r, x, value) -def cos_power_int_values(n_data, a, b, n, fx): - """returns some values of the cosine power integral. + +def cos_power_int_values( + n_data: int, a: float, b: float, n: int, fx: float +) -> None: + """Returns some values of the cosine power integral. Args: - n_data (int): Description for n_data. - a (float): Description for a. - b (float): Description for b. - n (int): Description for n. - fx (float): Description for fx. + n_data (int): Number of points. + a (float): left endpoint. + b (float): right endpoint. + n (int): power. + fx (float): value of integral. + Returns: + None """ return _polpack.cos_power_int_values(n_data, a, b, n, fx) + def delannoy(m, n, a): - """returns the Delannoy numbers up to orders (M,N). + """Returns the Delannoy numbers up to orders (M,N). Args: - m (int): Description for m. - n (int): Description for n. - a (int): Description for a. + m (int): maximum order m. + n (int): maximum order n. + a (ndarray): array to store the Delannoy numbers. """ return _polpack.delannoy(m, n, a) -def erf_values(n_data, x, fx): - """returns some values of the ERF or "error" function for testing. + +def erf_values(n_data: int, x: float, fx: float) -> None: + """Returns some values of the ERF or "error" function for testing. Args: - n_data (int): Description for n_data. - x (float): Description for x. - fx (float): Description for fx. + n_data (int): Number of points. + x (float): argument. + fx (float): value of erf(x). + Returns: + None """ return _polpack.erf_values(n_data, x, fx) -def euler_number(n, e): - """computes the Euler numbers. + +def euler_number(n: int, e: np.ndarray) -> None: + """Computes the Euler numbers. Args: - n (int): Description for n. - e (int): Description for e. + n (int): maximum order. + e (np.ndarray): Euler numbers. + Returns: + None """ return _polpack.euler_number(n, e) -def euler_number_values(n_data, n, c): - """returns some values of the Euler numbers. + +def euler_number_values(n_data: int, n: int, c: int) -> None: + """Returns some values of the Euler numbers. Args: - n_data (int): Description for n_data. - n (int): Description for n. - c (int): Description for c. + n_data (int): Number of points. + n (int): order. + c (int): value. + Returns: + None """ return _polpack.euler_number_values(n_data, n, c) -def eulerian(n, e): - """computes the Eulerian number E(N,K). + +def eulerian(n: int, e: np.ndarray) -> None: + """Computes the Eulerian number E(N,K). Args: - n (int): Description for n. - e (int): Description for e. + n (int): order. + e (np.ndarray): Eulerian numbers. + Returns: + None """ return _polpack.eulerian(n, e) -def fibonacci_direct(n, f): - """computes the N-th Fibonacci number directly. + +def fibonacci_direct(n): + """Computes the N-th Fibonacci number directly. Args: - n (int): Description for n. - f (int): Description for f. + n (int): index of the Fibonacci number. + Returns: + int: N-th Fibonacci number. """ - return _polpack.fibonacci_direct(n, f) + return _polpack.fibonacci_direct(n) + -def fibonacci_floor(n, f, i): - """returns the largest Fibonacci number less than or equal to N. +def fibonacci_floor(n: int, f: int, i: int) -> None: + """Returns the largest Fibonacci number less than or equal to N. Args: - n (int): Description for n. - f (int): Description for f. - i (int): Description for i. + n (int): value to bound. + f (int): Fibonacci number. + i (int): index of Fibonacci number. + Returns: + None """ return _polpack.fibonacci_floor(n, f, i) -def fibonacci_recursive(n, f): - """computes the first N Fibonacci numbers. + +def fibonacci_recursive(n: int, f: np.ndarray) -> None: + """Computes the first N Fibonacci numbers. Args: - n (int): Description for n. - f (int): Description for f. + n (int): number of values to compute. + f (np.ndarray): Fibonacci numbers. + Returns: + None """ return _polpack.fibonacci_recursive(n, f) -def gamma_log_values(n_data, x, fx): - """returns some values of the Log Gamma function. + +def gamma_log_values(n_data: int, x: float, fx: float) -> None: + """Returns some values of the Log Gamma function. Args: - n_data (int): Description for n_data. - x (float): Description for x. - fx (float): Description for fx. + n_data (int): Number of points. + x (float): argument. + fx (float): value of log(gamma(x)). + Returns: + None """ return _polpack.gamma_log_values(n_data, x, fx) -def gamma_values(n_data, x, fx): - """returns some values of the Gamma function. + +def gamma_values(n_data: int, x: float, fx: float) -> None: + """Returns some values of the Gamma function. Args: - n_data (int): Description for n_data. - x (float): Description for x. - fx (float): Description for fx. + n_data (int): Number of points. + x (float): argument. + fx (float): value of gamma(x). + Returns: + None """ return _polpack.gamma_values(n_data, x, fx) -def gegenbauer_poly(n, alpha, x, cx): - """computes the Gegenbauer polynomials C(I,ALPHA,X). + +def gegenbauer_poly(n: int, alpha: float, x: float, cx: np.ndarray) -> None: + """Computes the Gegenbauer polynomials C(I,ALPHA,X). Args: - n (int): Description for n. - alpha (float): Description for alpha. - x (float): Description for x. - cx (float): Description for cx. + n (int): max degree. + alpha (float): parameter alpha. + x (float): point. + cx (np.ndarray): values. + Returns: + None """ return _polpack.gegenbauer_poly(n, alpha, x, cx) -def gegenbauer_poly_values(n_data, n, a, x, fx): - """returns some values of the Gegenbauer polynomials. + +def gegenbauer_poly_values( + n_data: int, n: int, a: float, x: float, fx: float +) -> None: + """Returns some values of the Gegenbauer polynomials. Args: - n_data (int): Description for n_data. - n (int): Description for n. - a (float): Description for a. - x (float): Description for x. - fx (float): Description for fx. + n_data (int): Number of points. + n (int): degree. + a (float): parameter alpha. + x (float): point. + fx (float): value. + Returns: + None """ return _polpack.gegenbauer_poly_values(n_data, n, a, x, fx) -def gen_hermite_poly(n, x, mu, p): - """evaluates the generalized Hermite polynomials at X. + +def gen_hermite_poly(n: int, x: float, mu: float, p: np.ndarray) -> None: + """Evaluates the generalized Hermite polynomials at X. Args: - n (int): Description for n. - x (float): Description for x. - mu (float): Description for mu. - p (float): Description for p. + n (int): degree. + x (float): point. + mu (float): parameter mu. + p (np.ndarray): values. + Returns: + None """ return _polpack.gen_hermite_poly(n, x, mu, p) -def gen_laguerre_poly(n, alpha, x, cx): - """evaluates generalized Laguerre polynomials. + +def gen_laguerre_poly(n: int, alpha: float, x: float, cx: np.ndarray) -> None: + """Evaluates generalized Laguerre polynomials. Args: - n (int): Description for n. - alpha (float): Description for alpha. - x (float): Description for x. - cx (float): Description for cx. + n (int): degree. + alpha (float): parameter alpha. + x (float): point. + cx (np.ndarray): values. + Returns: + None """ return _polpack.gen_laguerre_poly(n, alpha, x, cx) -def gud_values(n_data, x, fx): - """returns some values of the Gudermannian function. + +def gud_values(n_data: int, x: float, fx: float) -> None: + """Returns some values of the Gudermannian function. Args: - n_data (int): Description for n_data. - x (float): Description for x. - fx (float): Description for fx. + n_data (int): Number of points. + x (float): point. + fx (float): value of gud(x). + Returns: + None """ return _polpack.gud_values(n_data, x, fx) -def hermite_poly_phys(n, x, cx): - """evaluates the physicisist's Hermite polynomials at X. + +def hermite_poly_phys(n: int, x: float, cx: np.ndarray) -> None: + """Evaluates the physicisist's Hermite polynomials at X. Args: - n (int): Description for n. - x (float): Description for x. - cx (float): Description for cx. + n (int): degree. + x (float): point. + cx (np.ndarray): values. + Returns: + None """ return _polpack.hermite_poly_phys(n, x, cx) -def hermite_poly_phys_coef(n, c): - """evaluates the physicist's Hermite polynomial coefficients. + +def hermite_poly_phys_coef(n: int, c: np.ndarray) -> None: + """Evaluates the physicist's Hermite polynomial coefficients. Args: - n (int): Description for n. - c (float): Description for c. + n (int): degree. + c (np.ndarray): coefficients. + Returns: + None """ return _polpack.hermite_poly_phys_coef(n, c) -def hermite_poly_phys_values(n_data, n, x, fx): - """returns some values of the physicist's Hermite polynomial. + +def hermite_poly_phys_values(n_data: int, n: int, x: float, fx: float) -> None: + """Returns some values of the physicist's Hermite polynomial. Args: - n_data (int): Description for n_data. - n (int): Description for n. - x (float): Description for x. - fx (float): Description for fx. + n_data (int): Number of points. + n (int): degree. + x (float): point. + fx (float): value. + Returns: + None """ return _polpack.hermite_poly_phys_values(n_data, n, x, fx) -def hyper_2f1_values(n_data, a, b, c, x, fx): - """returns some values of the hypergeometric function 2F1. + +def hyper_2f1_values( + n_data: int, a: float, b: float, c: float, x: float, fx: float +) -> None: + """Returns some values of the hypergeometric function 2F1. Args: - n_data (int): Description for n_data. - a (float): Description for a. - b (float): Description for b. - c (float): Description for c. - x (float): Description for x. - fx (float): Description for fx. + n_data (int): Number of points. + a (float): first parameter. + b (float): second parameter. + c (float): third parameter. + x (float): point. + fx (float): value. + Returns: + None """ return _polpack.hyper_2f1_values(n_data, a, b, c, x, fx) -def i4_factor(n, factor_max, factor_num, factor, power, nleft): - """factors an I4 into prime factors. + +def i4_factor( + n: int, + factor_max: int, + factor_num: np.ndarray, + factor: np.ndarray, + power: np.ndarray, + nleft: np.ndarray, +) -> None: + """Factors an I4 into prime factors. Args: - n (int): Description for n. - factor_max (int): Description for factor_max. - factor_num (int): Description for factor_num. - factor (int): Description for factor. - power (int): Description for power. - nleft (int): Description for nleft. + n (int): number to factor. + factor_max (int): max number of factors. + factor_num (np.ndarray): number of factors found. + factor (np.ndarray): prime factors. + power (np.ndarray): powers of prime factors. + nleft (np.ndarray): remaining unfactored part. + Returns: + None """ return _polpack.i4_factor(n, factor_max, factor_num, factor, power, nleft) -def i4_factorial2_values(n_data, n, fn): - """returns values of the double factorial function. + +def i4_factorial2_values(n_data: int, n: int, fn: int) -> None: + """Returns values of the double factorial function. Args: - n_data (int): Description for n_data. - n (int): Description for n. - fn (int): Description for fn. + n_data (int): Number of points. + n (int): value n. + fn (int): value of n!!. + Returns: + None """ return _polpack.i4_factorial2_values(n_data, n, fn) -def i4_factorial_values(n_data, n, fn): - """returns values of the factorial function. + +def i4_factorial_values(n_data: int, n: int, fn: int) -> None: + """Returns values of the factorial function. Args: - n_data (int): Description for n_data. - n (int): Description for n. - fn (int): Description for fn. + n_data (int): Number of points. + n (int): value n. + fn (int): value of n!. + Returns: + None """ return _polpack.i4_factorial_values(n_data, n, fn) -def i4_partition_distinct_count(n, q): - """returns any value of Q(N). + +def i4_partition_distinct_count(n: int, q: np.ndarray) -> None: + """Returns any value of Q(N). Args: - n (int): Description for n. - q (int): Description for q. + n (int): value n. + q (np.ndarray): value of Q(N). + Returns: + None """ return _polpack.i4_partition_distinct_count(n, q) -def i4_swap(i, j): - """switches two I4's. + +def i4_swap(i: np.ndarray, j: np.ndarray) -> None: + """Switches two I4's. Args: - i (int): Description for i. - j (int): Description for j. + i (np.ndarray): first value. + j (np.ndarray): second value. + Returns: + None """ return _polpack.i4_swap(i, j) -def i4_to_triangle_lower(k, i, j): + +def i4_to_triangle_lower(k: int, i: np.ndarray, j: np.ndarray) -> None: """Evaluates the i4_to_triangle_lower function. Args: - k (int): Description for k. - i (int): Description for i. - j (int): Description for j. + k (int): triangle index. + i (np.ndarray): row index. + j (np.ndarray): column index. + Returns: + None """ return _polpack.i4_to_triangle_lower(k, i, j) -def i4_to_triangle_upper(k, i, j): + +def i4_to_triangle_upper(k: int, i: np.ndarray, j: np.ndarray) -> None: """Evaluates the i4_to_triangle_upper function. Args: - k (int): Description for k. - i (int): Description for i. - j (int): Description for j. + k (int): triangle index. + i (np.ndarray): row index. + j (np.ndarray): column index. + Returns: + None """ return _polpack.i4_to_triangle_upper(k, i, j) -def i4mat_print(m, n, a, title): - """prints an I4MAT. + +def i4mat_print(m: int, n: int, a: np.ndarray, title: str) -> None: + """Prints an I4MAT. Args: - m (int): Description for m. - n (int): Description for n. - a (int): Description for a. - title (character): Description for title. + m (int): Number of rows. + n (int): Number of columns. + a (np.ndarray): Matrix. + title (str): Title. + Returns: + None """ return _polpack.i4mat_print(m, n, a, title) -def i4mat_print_some(m, n, a, ilo, jlo, ihi, jhi, title): - """prints some of an I4MAT. - Args: - m (int): Description for m. - n (int): Description for n. - a (int): Description for a. - ilo (int): Description for ilo. - jlo (int): Description for jlo. - ihi (int): Description for ihi. - jhi (int): Description for jhi. - title (character): Description for title. +def i4mat_print_some( + m: int, + n: int, + a: np.ndarray, + ilo: int, + jlo: int, + ihi: int, + jhi: int, + title: str, +) -> None: + """Prints some of an I4MAT. + + Args: + m (int): Rows. + n (int): Columns. + a (np.ndarray): Matrix. + ilo (int): first row. + jlo (int): first column. + ihi (int): last row. + jhi (int): last column. + title (str): Title. + Returns: + None """ return _polpack.i4mat_print_some(m, n, a, ilo, jlo, ihi, jhi, title) -def jacobi_poly(n, alpha, beta, x, cx): - """evaluates the Jacobi polynomials at X. + +def jacobi_poly( + n: int, alpha: float, beta: float, x: float, cx: np.ndarray +) -> None: + """Evaluates the Jacobi polynomials at X. Args: - n (int): Description for n. - alpha (float): Description for alpha. - beta (float): Description for beta. - x (float): Description for x. - cx (float): Description for cx. + n (int): degree. + alpha (float): parameter alpha. + beta (float): parameter beta. + x (float): point. + cx (np.ndarray): values. + Returns: + None """ return _polpack.jacobi_poly(n, alpha, beta, x, cx) -def jacobi_poly_values(n_data, n, a, b, x, fx): - """returns some values of the Jacobi polynomial. + +def jacobi_poly_values( + n_data: int, n: int, a: float, b: float, x: float, fx: float +) -> None: + """Returns some values of the Jacobi polynomial. Args: - n_data (int): Description for n_data. - n (int): Description for n. - a (float): Description for a. - b (float): Description for b. - x (float): Description for x. - fx (float): Description for fx. + n_data (int): Number of points. + n (int): degree. + a (float): parameter alpha. + b (float): parameter beta. + x (float): point. + fx (float): value. + Returns: + None """ return _polpack.jacobi_poly_values(n_data, n, a, b, x, fx) -def jacobi_symbol(q, p, j): - """evaluates the Jacobi symbol (Q/P). + +def jacobi_symbol(q: int, p: int, j: np.ndarray) -> None: + """Evaluates the Jacobi symbol (Q/P). Args: - q (int): Description for q. - p (int): Description for p. - j (int): Description for j. + q (int): numerator. + p (int): denominator. + j (np.ndarray): result symbol. + Returns: + None """ return _polpack.jacobi_symbol(q, p, j) -def krawtchouk(n, p, x, m, v): - """evaluates the Krawtchouk polynomials at X. + +def krawtchouk(n: int, p: float, x: float, m: int, v: np.ndarray) -> None: + """Evaluates the Krawtchouk polynomials at X. Args: - n (int): Description for n. - p (float): Description for p. - x (float): Description for x. - m (int): Description for m. - v (float): Description for v. + n (int): degree. + p (float): parameter p. + x (float): point. + m (int): parameter m. + v (np.ndarray): values. + Returns: + None """ return _polpack.krawtchouk(n, p, x, m, v) -def laguerre_associated(n, m, x, cx): - """evaluates associated Laguerre polynomials L(N,M,X). + +def laguerre_associated(n: int, m: int, x: float, cx: np.ndarray) -> None: + """Evaluates associated Laguerre polynomials L(N,M,X). Args: - n (int): Description for n. - m (int): Description for m. - x (float): Description for x. - cx (float): Description for cx. + n (int): degree. + m (int): parameter m. + x (float): point. + cx (np.ndarray): values. + Returns: + None """ return _polpack.laguerre_associated(n, m, x, cx) -def laguerre_poly(n, x, cx): - """evaluates the Laguerre polynomials at X. + +def laguerre_poly(n: int, x: float, cx: np.ndarray) -> None: + """Evaluates the Laguerre polynomials at X. Args: - n (int): Description for n. - x (float): Description for x. - cx (float): Description for cx. + n (int): degree. + x (float): point. + cx (np.ndarray): values. + Returns: + None """ return _polpack.laguerre_poly(n, x, cx) -def laguerre_poly_coef(n, c): - """evaluates the Laguerre polynomial coefficients. + +def laguerre_poly_coef(n: int, c: np.ndarray) -> None: + """Evaluates the Laguerre polynomial coefficients. Args: - n (int): Description for n. - c (float): Description for c. + n (int): degree. + c (np.ndarray): coefficients. + Returns: + None """ return _polpack.laguerre_poly_coef(n, c) -def laguerre_polynomial_values(n_data, n, x, fx): - """returns some values of the Laguerre polynomial. + +def laguerre_polynomial_values( + n_data: int, n: int, x: float, fx: float +) -> None: + """Returns some values of the Laguerre polynomial. Args: - n_data (int): Description for n_data. - n (int): Description for n. - x (float): Description for x. - fx (float): Description for fx. + n_data (int): Number of points. + n (int): degree. + x (float): point. + fx (float): value. + Returns: + None """ return _polpack.laguerre_polynomial_values(n_data, n, x, fx) -def lambert_w_values(n_data, x, fx): - """returns some values of the Lambert W function. + +def lambert_w_values(n_data: int, x: float, fx: float) -> None: + """Returns some values of the Lambert W function. Args: - n_data (int): Description for n_data. - x (float): Description for x. - fx (float): Description for fx. + n_data (int): Number of points. + x (float): point. + fx (float): value. + Returns: + None """ return _polpack.lambert_w_values(n_data, x, fx) -def legendre_associated(n, m, x, cx): - """evaluates the associated Legendre functions. + +def legendre_associated(n: int, m: int, x: float, cx: np.ndarray) -> None: + """Evaluates the associated Legendre functions. Args: - n (int): Description for n. - m (int): Description for m. - x (float): Description for x. - cx (float): Description for cx. + n (int): degree. + m (int): parameter m. + x (float): point. + cx (np.ndarray): values. + Returns: + None """ return _polpack.legendre_associated(n, m, x, cx) -def legendre_associated_normalized(n, m, x, cx): + +def legendre_associated_normalized( + n: int, m: int, x: float, cx: np.ndarray +) -> None: """Evaluates the legendre_associated_normalized function. Args: - n (int): Description for n. - m (int): Description for m. - x (float): Description for x. - cx (float): Description for cx. + n (int): degree. + m (int): parameter m. + x (float): point. + cx (np.ndarray): values. + Returns: + None """ return _polpack.legendre_associated_normalized(n, m, x, cx) -def legendre_associated_normalized_sphere_values(n_data, n, m, x, fx): + +def legendre_associated_normalized_sphere_values( + n_data: int, n: int, m: int, x: float, fx: float +) -> None: """Evaluates the legendre_associated_normalized_sphere_values function. Args: - n_data (int): Description for n_data. - n (int): Description for n. - m (int): Description for m. - x (float): Description for x. - fx (float): Description for fx. + n_data (int): Number of points. + n (int): degree. + m (int): order. + x (float): point. + fx (float): value. + Returns: + None """ - return _polpack.legendre_associated_normalized_sphere_values(n_data, n, m, x, fx) + return _polpack.legendre_associated_normalized_sphere_values( + n_data, n, m, x, fx + ) + -def legendre_associated_values(n_data, n, m, x, fx): - """returns values of associated Legendre functions. +def legendre_associated_values( + n_data: int, n: int, m: int, x: float, fx: float +) -> None: + """Returns values of associated Legendre functions. Args: - n_data (int): Description for n_data. - n (int): Description for n. - m (int): Description for m. - x (float): Description for x. - fx (float): Description for fx. + n_data (int): Number of points. + n (int): degree. + m (int): order. + x (float): point. + fx (float): value. + Returns: + None """ return _polpack.legendre_associated_values(n_data, n, m, x, fx) -def legendre_function_q(n, x, cx): - """evaluates the Legendre Q functions. + +def legendre_function_q(n: int, x: float, cx: np.ndarray) -> None: + """Evaluates the Legendre Q functions. Args: - n (int): Description for n. - x (float): Description for x. - cx (float): Description for cx. + n (int): degree. + x (float): point. + cx (np.ndarray): values. + Returns: + None """ return _polpack.legendre_function_q(n, x, cx) -def legendre_function_q_values(n_data, n, x, fx): - """returns values of the Legendre Q function. + +def legendre_function_q_values( + n_data: int, n: int, x: float, fx: float +) -> None: + """Returns values of the Legendre Q function. Args: - n_data (int): Description for n_data. - n (int): Description for n. - x (float): Description for x. - fx (float): Description for fx. + n_data (int): Number of points. + n (int): degree. + x (float): point. + fx (float): value. + Returns: + None """ return _polpack.legendre_function_q_values(n_data, n, x, fx) -def legendre_poly(n, x, cx, cpx): - """evaluates the Legendre polynomials P(N,X) at X. + +def legendre_poly(n: int, x: float, cx: np.ndarray, cpx: np.ndarray) -> None: + """Evaluates the Legendre polynomials P(N,X) at X. Args: - n (int): Description for n. - x (float): Description for x. - cx (float): Description for cx. - cpx (float): Description for cpx. + n (int): degree. + x (float): point. + cx (np.ndarray): values. + cpx (np.ndarray): derivatives. + Returns: + None """ return _polpack.legendre_poly(n, x, cx, cpx) -def legendre_poly_coef(n, c): - """evaluates the Legendre polynomial coefficients. + +def legendre_poly_coef(n: int, c: np.ndarray) -> None: + """Evaluates the Legendre polynomial coefficients. Args: - n (int): Description for n. - c (float): Description for c. + n (int): degree. + c (np.ndarray): coefficients. + Returns: + None """ return _polpack.legendre_poly_coef(n, c) -def legendre_poly_values(n_data, n, x, fx): - """returns values of the Legendre polynomials. + +def legendre_poly_values(n_data: int, n: int, x: float, fx: float) -> None: + """Returns values of the Legendre polynomials. Args: - n_data (int): Description for n_data. - n (int): Description for n. - x (float): Description for x. - fx (float): Description for fx. + n_data (int): Number of points. + n (int): degree. + x (float): point. + fx (float): value. + Returns: + None """ return _polpack.legendre_poly_values(n_data, n, x, fx) -def legendre_symbol(q, p, l): - """evaluates the Legendre symbol (Q/P). + +def legendre_symbol(q: int, p: int, ell: np.ndarray) -> None: + """Evaluates the Legendre symbol (Q/P). Args: - q (int): Description for q. - p (int): Description for p. - l (int): Description for l. + q (int): numerator. + p (int): denominator. + ell (np.ndarray): result symbol. + Returns: + None """ - return _polpack.legendre_symbol(q, p, l) + return _polpack.legendre_symbol(q, p, ell) + -def lerch_values(n_data, z, s, a, fx): - """returns some values of the Lerch transcendent function. +def lerch_values(n_data: int, z: float, s: float, a: float, fx: float) -> None: + """Returns some values of the Lerch transcendent function. Args: - n_data (int): Description for n_data. - z (float): Description for z. - s (int): Description for s. - a (float): Description for a. - fx (float): Description for fx. + n_data (int): Number of points. + z (float): point. + s (float): parameter s. + a (float): parameter a. + fx (float): value. + Returns: + None """ return _polpack.lerch_values(n_data, z, s, a, fx) -def lock(n, a): - """returns the number of codes for a lock with N buttons. + +def lock(n: int, a: np.ndarray) -> None: + """Returns the number of codes for a lock with N buttons. Args: - n (int): Description for n. - a (int): Description for a. + n (int): order. + a (np.ndarray): values. + Returns: + None """ return _polpack.lock(n, a) -def meixner(n, beta, c, x, v): - """evaluates Meixner polynomials at a point. + +def meixner(n: int, beta: float, c: float, x: float, v: np.ndarray) -> None: + """Evaluates Meixner polynomials at a point. Args: - n (int): Description for n. - beta (float): Description for beta. - c (float): Description for c. - x (float): Description for x. - v (float): Description for v. + n (int): degree. + beta (float): parameter beta. + c (float): parameter c. + x (float): point. + v (np.ndarray): values. + Returns: + None """ return _polpack.meixner(n, beta, c, x, v) -def mertens_values(n_data, n, c): - """returns some values of the Mertens function. + +def mertens_values(n_data: int, n: int, c: int) -> None: + """Returns some values of the Mertens function. Args: - n_data (int): Description for n_data. - n (int): Description for n. - c (int): Description for c. + n_data (int): Number of points. + n (int): value n. + c (int): value of M(n). + Returns: + None """ return _polpack.mertens_values(n_data, n, c) -def moebius(n, mu): - """returns the value of MU(N), the Moebius function of N. + +def moebius(n: int, mu: np.ndarray) -> None: + """Returns the value of MU(N), the Moebius function of N. Args: - n (int): Description for n. - mu (int): Description for mu. + n (int): value n. + mu (np.ndarray): value. + Returns: + None """ return _polpack.moebius(n, mu) -def moebius_values(n_data, n, c): - """returns some values of the Moebius function. + +def moebius_values(n_data: int, n: int, c: int) -> None: + """Returns some values of the Moebius function. Args: - n_data (int): Description for n_data. - n (int): Description for n. - c (int): Description for c. + n_data (int): Number of points. + n (int): value n. + c (int): value of mu(n). + Returns: + None """ return _polpack.moebius_values(n_data, n, c) -def motzkin(n, a): - """returns the Motzkin numbers up to order N. + +def motzkin(n: int, a: np.ndarray) -> None: + """Returns the Motzkin numbers up to order N. Args: - n (int): Description for n. - a (int): Description for a. + n (int): order. + a (np.ndarray): values. + Returns: + None """ return _polpack.motzkin(n, a) -def normal_01_cdf_values(n_data, x, fx): - """returns some values of the Normal 01 CDF. + +def normal_01_cdf_values(n_data: int, x: float, fx: float) -> None: + """Returns some values of the Normal 01 CDF. Args: - n_data (int): Description for n_data. - x (float): Description for x. - fx (float): Description for fx. + n_data (int): Number of points. + x (float): point. + fx (float): value. + Returns: + None """ return _polpack.normal_01_cdf_values(n_data, x, fx) -def omega(n, ndiv): - """returns OMEGA(N), the number of distinct prime divisors of N. + +def omega(n: int, ndiv: np.ndarray) -> None: + """Returns OMEGA(N), the number of distinct prime divisors of N. Args: - n (int): Description for n. - ndiv (int): Description for ndiv. + n (int): value n. + ndiv (np.ndarray): value. + Returns: + None """ return _polpack.omega(n, ndiv) -def omega_values(n_data, n, c): - """returns some values of the OMEGA function. + +def omega_values(n_data: int, n: int, c: int) -> None: + """Returns some values of the OMEGA function. Args: - n_data (int): Description for n_data. - n (int): Description for n. - c (int): Description for c. + n_data (int): Number of points. + n (int): value n. + c (int): value. + Returns: + None """ return _polpack.omega_values(n_data, n, c) -def partition_distinct_count_values(n_data, n, c): - """returns some values of Q(N). + +def partition_distinct_count_values(n_data: int, n: int, c: int) -> None: + """Returns some values of Q(N). Args: - n_data (int): Description for n_data. - n (int): Description for n. - c (int): Description for c. + n_data (int): Number of points. + n (int): value n. + c (int): value. + Returns: + None """ return _polpack.partition_distinct_count_values(n_data, n, c) -def pentagon_num(n, p): - """computes the N-th pentagonal number. + +def pentagon_num(n: int, p: np.ndarray) -> None: + """Computes the N-th pentagonal number. Args: - n (int): Description for n. - p (int): Description for p. + n (int): index. + p (np.ndarray): value. + Returns: + None """ return _polpack.pentagon_num(n, p) -def phi(n, phin): - """computes the number of relatively prime predecessors of an integer. + +def phi(n: int, phin: np.ndarray) -> None: + """Computes the number of relatively prime predecessors of an integer. Args: - n (int): Description for n. - phin (int): Description for phin. + n (int): value n. + phin (np.ndarray): value. + Returns: + None """ return _polpack.phi(n, phin) -def phi_values(n_data, n, c): - """returns some values of the PHI function. + +def phi_values(n_data: int, n: int, c: int) -> None: + """Returns some values of the PHI function. Args: - n_data (int): Description for n_data. - n (int): Description for n. - c (int): Description for c. + n_data (int): Number of points. + n (int): value n. + c (int): value. + Returns: + None """ return _polpack.phi_values(n_data, n, c) -def poly_bernoulli(n, k, b): - """evaluates the poly-Bernolli numbers with negative index. + +def poly_bernoulli(n: int, k: int, b: np.ndarray) -> None: + """Evaluates the poly-Bernolli numbers with negative index. Args: - n (int): Description for n. - k (int): Description for k. - b (int): Description for b. + n (int): degree. + k (int): parameter. + b (np.ndarray): values. + Returns: + None """ return _polpack.poly_bernoulli(n, k, b) -def psi_values(n_data, x, fx): - """returns some values of the Psi or Digamma function for testing. + +def psi_values(n_data: int, x: float, fx: float) -> None: + """Returns some values of the Psi or Digamma function for testing. Args: - n_data (int): Description for n_data. - x (float): Description for x. - fx (float): Description for fx. + n_data (int): Number of points. + x (float): point. + fx (float): value. + Returns: + None """ return _polpack.psi_values(n_data, x, fx) -def r8_factorial_log_values(n_data, n, fn): - """returns values of log(factorial(n)). + +def r8_factorial_log_values(n_data: int, n: int, fn: float) -> None: + """Returns values of log(factorial(n)). Args: - n_data (int): Description for n_data. - n (int): Description for n. - fn (float): Description for fn. + n_data (int): Number of points. + n (int): value n. + fn (float): value. + Returns: + None """ return _polpack.r8_factorial_log_values(n_data, n, fn) -def r8_factorial_values(n_data, n, fn): - """returns values of the real factorial function. + +def r8_factorial_values(n_data: int, n: int, fn: float) -> None: + """Returns values of the real factorial function. Args: - n_data (int): Description for n_data. - n (int): Description for n. - fn (float): Description for fn. + n_data (int): Number of points. + n (int): value n. + fn (float): value. + Returns: + None """ return _polpack.r8_factorial_values(n_data, n, fn) -def r8_hyper_2f1(a_input, b_input, c_input, x_input, hf): - """evaluates the hypergeometric function F(A,B,C,X). + +def r8_hyper_2f1( + a_input: float, + b_input: float, + c_input: float, + x_input: float, + hf: np.ndarray, +) -> None: + """Evaluates the hypergeometric function F(A,B,C,X). Args: - a_input (float): Description for a_input. - b_input (float): Description for b_input. - c_input (float): Description for c_input. - x_input (float): Description for x_input. - hf (float): Description for hf. + a_input (float): a. + b_input (float): b. + c_input (float): c. + x_input (float): x. + hf (np.ndarray): value. + Returns: + None """ return _polpack.r8_hyper_2f1(a_input, b_input, c_input, x_input, hf) -def r8poly_print(n, a, title): - """prints out a polynomial. + +def r8poly_print(n: int, a: np.ndarray, title: str) -> None: + """Prints out a polynomial. Args: - n (int): Description for n. - a (float): Description for a. - title (character): Description for title. + n (int): degree. + a (np.ndarray): coefficients. + title (str): title. + Returns: + None """ return _polpack.r8poly_print(n, a, title) -def r8vec_linspace(n, a, b, x): - """creates a vector of linearly spaced values. + +def r8vec_linspace(n: int, a: float, b: float, x: np.ndarray) -> None: + """Creates a vector of linearly spaced values. Args: - n (int): Description for n. - a (float): Description for a. - b (float): Description for b. - x (float): Description for x. + n (int): number of points. + a (float): left endpoint. + b (float): right endpoint. + x (np.ndarray): values. + Returns: + None """ return _polpack.r8vec_linspace(n, a, b, x) -def r8vec_print(n, a, title): - """prints an R8VEC. + +def r8vec_print(n: int, a: np.ndarray, title: str) -> None: + """Prints an R8VEC. Args: - n (int): Description for n. - a (float): Description for a. - title (character): Description for title. + n (int): number of elements. + a (np.ndarray): vector. + title (str): title. + Returns: + None """ return _polpack.r8vec_print(n, a, title) -def r8vec_print_some(n, a, max_print, title): - """prints "some" of an R8VEC. + +def r8vec_print_some(n: int, a: np.ndarray, max_print: int, title: str) -> None: + """Prints "some" of an R8VEC. Args: - n (int): Description for n. - a (float): Description for a. - max_print (int): Description for max_print. - title (character): Description for title. + n (int): number of elements. + a (np.ndarray): vector. + max_print (int): max number to print. + title (str): title. + Returns: + None """ return _polpack.r8vec_print_some(n, a, max_print, title) -def r8vec_uniform_ab(n, a, b, seed, r): - """returns a scaled pseudorandom R8VEC. + +def r8vec_uniform_ab( + n: int, a: float, b: float, seed: np.ndarray, r: np.ndarray +) -> None: + """Returns a scaled pseudorandom R8VEC. Args: - n (int): Description for n. - a (float): Description for a. - b (float): Description for b. - seed (int): Description for seed. - r (float): Description for r. + n (int): number of elements. + a (float): left endpoint. + b (float): right endpoint. + seed (np.ndarray): seed. + r (np.ndarray): random values. + Returns: + None """ return _polpack.r8vec_uniform_ab(n, a, b, seed, r) -def sigma(n, sigma_n): - """returns the value of SIGMA(N), the divisor sum. + +def sigma(n: int, sigma_n: np.ndarray) -> None: + """Returns the value of SIGMA(N), the divisor sum. Args: - n (int): Description for n. - sigma_n (int): Description for sigma_n. + n (int): value n. + sigma_n (np.ndarray): value. + Returns: + None """ return _polpack.sigma(n, sigma_n) -def sigma_values(n_data, n, c): - """returns some values of the Sigma function. + +def sigma_values(n_data: int, n: int, c: int) -> None: + """Returns some values of the Sigma function. Args: - n_data (int): Description for n_data. - n (int): Description for n. - c (int): Description for c. + n_data (int): Number of points. + n (int): value n. + c (int): value. + Returns: + None """ return _polpack.sigma_values(n_data, n, c) -def sin_power_int_values(n_data, a, b, n, fx): - """returns some values of the sine power integral. + +def sin_power_int_values( + n_data: int, a: float, b: float, n: int, fx: float +) -> None: + """Returns some values of the sine power integral. Args: - n_data (int): Description for n_data. - a (float): Description for a. - b (float): Description for b. - n (int): Description for n. - fx (float): Description for fx. + n_data (int): Number of points. + a (float): left endpoint. + b (float): right endpoint. + n (int): power. + fx (float): value. + Returns: + None """ return _polpack.sin_power_int_values(n_data, a, b, n, fx) -def slice(dim_num, slice_num, piece_num): + +def slice_fn(dim_num: int, slice_num: int, piece_num: np.ndarray) -> None: """Evaluates the slice function. Args: - dim_num (int): Description for dim_num. - slice_num (int): Description for slice_num. - piece_num (int): Description for piece_num. + dim_num (int): dimension number. + slice_num (int): slice number. + piece_num (np.ndarray): result piece. + Returns: + None """ return _polpack.slice(dim_num, slice_num, piece_num) -def spherical_harmonic(l, m, theta, phi, c, s): - """evaluates spherical harmonic functions. + +def spherical_harmonic( + ell: int, m: int, theta: float, phi: float, c: np.ndarray, s: np.ndarray +) -> None: + """Evaluates spherical harmonic functions. Args: - l (int): Description for l. - m (int): Description for m. - theta (float): Description for theta. - phi (float): Description for phi. - c (float): Description for c. - s (float): Description for s. + ell (int): degree. + m (int): order. + theta (float): angle theta. + phi (float): angle phi. + c (np.ndarray): real part. + s (np.ndarray): imaginary part. + Returns: + None """ - return _polpack.spherical_harmonic(l, m, theta, phi, c, s) + return _polpack.spherical_harmonic(ell, m, theta, phi, c, s) + def spherical_harmonic_values(n_data, l, m, theta, phi, yr, yi): - """returns values of spherical harmonic functions. + """Returns values of spherical harmonic functions. Args: n_data (int): Description for n_data. @@ -1188,67 +1566,86 @@ def spherical_harmonic_values(n_data, l, m, theta, phi, yr, yi): """ return _polpack.spherical_harmonic_values(n_data, l, m, theta, phi, yr, yi) -def stirling1(n, m, s1): - """computes the Stirling numbers of the first kind. + +def stirling1(n: int, m: int, s1: np.ndarray) -> None: + """Computes the Stirling numbers of the first kind. Args: - n (int): Description for n. - m (int): Description for m. - s1 (int): Description for s1. + n (int): degree. + m (int): order. + s1 (np.ndarray): values. + Returns: + None """ return _polpack.stirling1(n, m, s1) -def stirling2(n, m, s2): - """computes the Stirling numbers of the second kind. + +def stirling2(n: int, m: int, s2: np.ndarray) -> None: + """Computes the Stirling numbers of the second kind. Args: - n (int): Description for n. - m (int): Description for m. - s2 (int): Description for s2. + n (int): degree. + m (int): order. + s2 (np.ndarray): values. + Returns: + None """ return _polpack.stirling2(n, m, s2) -def tau(n, taun): - """returns the value of TAU(N), the number of distinct divisors of N. + +def tau(n: int, taun: np.ndarray) -> None: + """Returns the value of TAU(N), the number of distinct divisors of N. Args: - n (int): Description for n. - taun (int): Description for taun. + n (int): value n. + taun (np.ndarray): value. + Returns: + None """ return _polpack.tau(n, taun) -def tau_values(n_data, n, c): - """returns some values of the Tau function. + +def tau_values(n_data: int, n: int, c: int) -> None: + """Returns some values of the Tau function. Args: - n_data (int): Description for n_data. - n (int): Description for n. - c (int): Description for c. + n_data (int): Number of points. + n (int): value n. + c (int): value. + Returns: + None """ return _polpack.tau_values(n_data, n, c) -def triangle_lower_to_i4(i, j, k): + +def triangle_lower_to_i4(i: int, j: int, k: np.ndarray) -> None: """Evaluates the triangle_lower_to_i4 function. Args: - i (int): Description for i. - j (int): Description for j. - k (int): Description for k. + i (int): row. + j (int): column. + k (np.ndarray): index. + Returns: + None """ return _polpack.triangle_lower_to_i4(i, j, k) -def triangle_upper_to_i4(i, j, k): + +def triangle_upper_to_i4(i: int, j: int, k: np.ndarray) -> None: """Evaluates the triangle_upper_to_i4 function. Args: - i (int): Description for i. - j (int): Description for j. - k (int): Description for k. + i (int): row. + j (int): column. + k (np.ndarray): index. + Returns: + None """ return _polpack.triangle_upper_to_i4(i, j, k) + def vibonacci(n, seed, v): - """computes the first N Vibonacci numbers. + """Computes the first N Vibonacci numbers. Args: n (int): Description for n. @@ -1257,57 +1654,73 @@ def vibonacci(n, seed, v): """ return _polpack.vibonacci(n, seed, v) -def zeckendorf(n, m_max, m, i_list, f_list): + +def zeckendorf( + n: int, m_max: int, m: np.ndarray, i_list: np.ndarray, f_list: np.ndarray +) -> None: """produces the Zeckendorf decomposition of a positive integer. Args: - n (int): Description for n. - m_max (int): Description for m_max. - m (int): Description for m. - i_list (int): Description for i_list. - f_list (int): Description for f_list. + n (int): value to decompose. + m_max (int): max size. + m (np.ndarray): number of parts. + i_list (np.ndarray): indices. + f_list (np.ndarray): Fibonacci values. + Returns: + None """ return _polpack.zeckendorf(n, m_max, m, i_list, f_list) -def zernike_poly(m, n, rho, z): - """evaluates a Zernike polynomial at RHO. + +def zernike_poly(m: int, n: int, rho: float, z: np.ndarray) -> None: + """Evaluates a Zernike polynomial at RHO. Args: - m (int): Description for m. - n (int): Description for n. - rho (float): Description for rho. - z (float): Description for z. + m (int): index m. + n (int): index n. + rho (float): point. + z (np.ndarray): values. + Returns: + None """ return _polpack.zernike_poly(m, n, rho, z) -def zernike_poly_coef(m, n, c): + +def zernike_poly_coef(m: int, n: int, c: np.ndarray) -> None: """Evaluates the zernike_poly_coef function. Args: - m (int): Description for m. - n (int): Description for n. - c (float): Description for c. + m (int): index m. + n (int): index n. + c (np.ndarray): coefficients. + Returns: + None """ return _polpack.zernike_poly_coef(m, n, c) -def zeta_values(n_data, n, zeta): - """returns some values of the Riemann Zeta function. + +def zeta_values(n_data: int, n: int, zeta: float) -> None: + """Returns some values of the Riemann Zeta function. Args: - n_data (int): Description for n_data. - n (int): Description for n. - zeta (float): Description for zeta. + n_data (int): Number of points. + n (int): s value. + zeta (float): value. + Returns: + None """ return _polpack.zeta_values(n_data, n, zeta) + def agud(g): - """evaluates the inverse Gudermannian function. + """Evaluates the inverse Gudermannian function. Args: g: Description for g. """ return _polpack.agud(g) + def align_enum(m, n): """counts the alignments of two sequences of M and N elements. @@ -1317,20 +1730,21 @@ def align_enum(m, n): """ return _polpack.align_enum(m, n) + def benford(ival): - """returns the Benford probability of one or more significant digits. + """Returns the Benford probability of one or more significant digits. Args: ival: Description for ival. """ return _polpack.benford(ival) -def catalan_constant(): - """returns the value of Catalan's constant. - """ +def catalan_constant(): + """Returns the value of Catalan's constant.""" return _polpack.catalan_constant() + def collatz_count(n): """counts the number of terms in a Collatz sequence. @@ -1339,8 +1753,9 @@ def collatz_count(n): """ return _polpack.collatz_count(n) + def cos_power_int(a, b, n): - """evaluates the cosine power integral. + """Evaluates the cosine power integral. Args: a: Description for a. @@ -1349,16 +1764,18 @@ def cos_power_int(a, b, n): """ return _polpack.cos_power_int(a, b, n) + def euler_number2(n): - """computes the Euler numbers. + """Computes the Euler numbers. Args: n: Description for n. """ return _polpack.euler_number2(n) + def euler_poly(n, x): - """evaluates the N-th Euler polynomial at X. + """Evaluates the N-th Euler polynomial at X. Args: n: Description for n. @@ -1366,16 +1783,18 @@ def euler_poly(n, x): """ return _polpack.euler_poly(n, x) + def gud(x): - """evaluates the Gudermannian function. + """Evaluates the Gudermannian function. Args: x: Description for x. """ return _polpack.gud(x) + def i4_choose(n, k): - """computes the binomial coefficient C(N,K). + """Computes the binomial coefficient C(N,K). Args: n: Description for n. @@ -1383,28 +1802,30 @@ def i4_choose(n, k): """ return _polpack.i4_choose(n, k) + def i4_factorial(n): - """computes the factorial of N. + """Computes the factorial of N. Args: n: Description for n. """ return _polpack.i4_factorial(n) + def i4_factorial2(n): - """computes the double factorial function. + """Computes the double factorial function. Args: n: Description for n. """ return _polpack.i4_factorial2(n) -def i4_huge(): - """returns a "huge" I4. - """ +def i4_huge(): + """Returns a "huge" I4.""" return _polpack.i4_huge() + def i4_is_prime(n): """reports whether an I4 is prime. @@ -1413,6 +1834,7 @@ def i4_is_prime(n): """ return _polpack.i4_is_prime(n) + def i4_is_triangular(i): """determines whether an integer is triangular. @@ -1421,8 +1843,9 @@ def i4_is_triangular(i): """ return _polpack.i4_is_triangular(i) + def i4_uniform_ab(a, b, seed): - """returns a scaled pseudorandom I4 between A and B. + """Returns a scaled pseudorandom I4 between A and B. Args: a: Description for a. @@ -1431,6 +1854,7 @@ def i4_uniform_ab(a, b, seed): """ return _polpack.i4_uniform_ab(a, b, seed) + def lambert_w(x): """estimates the Lambert W function. @@ -1439,6 +1863,7 @@ def lambert_w(x): """ return _polpack.lambert_w(x) + def lambert_w_crude(x): """is a crude estimate of the Lambert W function. @@ -1447,6 +1872,7 @@ def lambert_w_crude(x): """ return _polpack.lambert_w_crude(x) + def lerch(z, s, a): """estimates the Lerch transcendent function. @@ -1457,14 +1883,16 @@ def lerch(z, s, a): """ return _polpack.lerch(z, s, a) + def mertens(n): - """evaluates the Mertens function. + """Evaluates the Mertens function. Args: n: Description for n. """ return _polpack.mertens(n) + def normal_01_cdf_inverse(p): """inverts the standard normal CDF. @@ -1473,14 +1901,16 @@ def normal_01_cdf_inverse(p): """ return _polpack.normal_01_cdf_inverse(p) + def plane_partition_num(n): - """returns the number of plane partitions of the integer N. + """Returns the number of plane partitions of the integer N. Args: n: Description for n. """ return _polpack.plane_partition_num(n) + def poly_coef_count(dim, degree): """Evaluates the poly_coef_count function. @@ -1490,32 +1920,36 @@ def poly_coef_count(dim, degree): """ return _polpack.poly_coef_count(dim, degree) + def prime(n): - """returns any of the first PRIME_MAX prime numbers. + """Returns any of the first PRIME_MAX prime numbers. Args: n: Description for n. """ return _polpack.prime(n) + def pyramid_num(n): - """returns the N-th pyramidal number. + """Returns the N-th pyramidal number. Args: n: Description for n. """ return _polpack.pyramid_num(n) + def pyramid_square_num(n): - """returns the N-th pyramidal square number. + """Returns the N-th pyramidal square number. Args: n: Description for n. """ return _polpack.pyramid_square_num(n) + def r8_agm(a, b): - """computes the arithmetic-geometric mean of A and B. + """Computes the arithmetic-geometric mean of A and B. Args: a: Description for a. @@ -1523,8 +1957,9 @@ def r8_agm(a, b): """ return _polpack.r8_agm(a, b) + def r8_beta(x, y): - """returns the value of the Beta function. + """Returns the value of the Beta function. Args: x: Description for x. @@ -1532,8 +1967,9 @@ def r8_beta(x, y): """ return _polpack.r8_beta(x, y) + def r8_choose(n, k): - """computes the binomial coefficient C(N,K) as an R8. + """Computes the binomial coefficient C(N,K) as an R8. Args: n: Description for n. @@ -1541,20 +1977,21 @@ def r8_choose(n, k): """ return _polpack.r8_choose(n, k) -def r8_epsilon(): - """returns the R8 roundoff unit. - """ +def r8_epsilon(): + """Returns the R8 roundoff unit.""" return _polpack.r8_epsilon() + def r8_erf(x): - """evaluates the error function. + """Evaluates the error function. Args: x: Description for x. """ return _polpack.r8_erf(x) + def r8_erf_inverse(y): """inverts the error function. @@ -1563,82 +2000,87 @@ def r8_erf_inverse(y): """ return _polpack.r8_erf_inverse(y) -def r8_euler_constant(): - """returns the value of the Euler-Mascheroni constant. - """ +def r8_euler_constant(): + """Returns the value of the Euler-Mascheroni constant.""" return _polpack.r8_euler_constant() + def r8_factorial(n): - """computes the factorial of N. + """Computes the factorial of N. Args: n: Description for n. """ return _polpack.r8_factorial(n) + def r8_factorial_log(n): - """computes log(factorial(N)). + """Computes log(factorial(N)). Args: n: Description for n. """ return _polpack.r8_factorial_log(n) + def r8_gamma_log(x): - """evaluates log ( Gamma ( X ) ) for a real argument. + """Evaluates log ( Gamma ( X ) ) for a real argument. Args: x: Description for x. """ return _polpack.r8_gamma_log(x) -def r8_huge(): - """returns a "huge" R8. - """ +def r8_huge(): + """Returns a "huge" R8.""" return _polpack.r8_huge() + def r8_mop(i): - """returns the I-th power of -1 as an R8. + """Returns the I-th power of -1 as an R8. Args: i: Description for i. """ return _polpack.r8_mop(i) + def r8_nint(x): - """returns the nearest integer to an R8. + """Returns the nearest integer to an R8. Args: x: Description for x. """ return _polpack.r8_nint(x) -def r8_pi(): - """returns the value of pi as an R8. - """ +def r8_pi(): + """Returns the value of pi as an R8.""" return _polpack.r8_pi() + def r8_psi(xx): - """evaluates the function Psi(X). + """Evaluates the function Psi(X). Args: xx: Description for xx. """ return _polpack.r8_psi(xx) + def r8_uniform_01(seed): - """returns a unit pseudorandom R8. + """Returns a unit pseudorandom R8. Args: seed: Description for seed. """ return _polpack.r8_uniform_01(seed) + def r8poly_degree(na, a): - """returns the degree of a polynomial. + """Returns the degree of a polynomial. Args: na: Description for na. @@ -1646,8 +2088,9 @@ def r8poly_degree(na, a): """ return _polpack.r8poly_degree(na, a) + def r8poly_value_horner(m, c, x): - """evaluates a polynomial using Horner's method. + """Evaluates a polynomial using Horner's method. Args: m: Description for m. @@ -1656,16 +2099,18 @@ def r8poly_value_horner(m, c, x): """ return _polpack.r8poly_value_horner(m, c, x) + def s_len_trim(s): - """returns the length of a string to the last nonblank. + """Returns the length of a string to the last nonblank. Args: s: Description for s. """ return _polpack.s_len_trim(s) + def simplex_num(m, n): - """evaluates the N-th Simplex number in M dimensions. + """Evaluates the N-th Simplex number in M dimensions. Args: m: Description for m. @@ -1673,8 +2118,9 @@ def simplex_num(m, n): """ return _polpack.simplex_num(m, n) + def sin_power_int(a, b, n): - """evaluates the sine power integral. + """Evaluates the sine power integral. Args: a: Description for a. @@ -1683,24 +2129,27 @@ def sin_power_int(a, b, n): """ return _polpack.sin_power_int(a, b, n) + def tetrahedron_num(n): - """returns the N-th tetrahedral number. + """Returns the N-th tetrahedral number. Args: n: Description for n. """ return _polpack.tetrahedron_num(n) + def triangle_num(n): - """returns the N-th triangular number. + """Returns the N-th triangular number. Args: n: Description for n. """ return _polpack.triangle_num(n) + def trinomial(i, j, k): - """computes a trinomial coefficient. + """Computes a trinomial coefficient. Args: i: Description for i. @@ -1709,6 +2158,7 @@ def trinomial(i, j, k): """ return _polpack.trinomial(i, j, k) + def zeta(p): """estimates the Riemann Zeta function. @@ -1716,4 +2166,3 @@ def zeta(p): p: Description for p. """ return _polpack.zeta(p) - diff --git a/tests/conftest.py b/tests/conftest.py index 2dcb584..4188d4d 100644 --- a/tests/conftest.py +++ b/tests/conftest.py @@ -2,6 +2,7 @@ import shutil import sys + # On Windows, Python 3.8+ no longer uses PATH for DLL resolution. # If the Fortran extension wasn't fully statically linked, we need to # explicitly register the MinGW runtime DLL directory so that diff --git a/tests/test_bernoulli_euler.py b/tests/test_bernoulli_euler.py index bb6bf82..f42f5b3 100644 --- a/tests/test_bernoulli_euler.py +++ b/tests/test_bernoulli_euler.py @@ -1,78 +1,96 @@ """Tests for Bernoulli, Bernstein, and Euler polynomials.""" import numpy as np + import polpack def test_bernoulli_number(): - """Test Bernoulli number computation against known values.""" - n = 10 + """Test Bernoulli number computation against legacy data.""" + n = 30 b = np.zeros(n + 1, dtype=np.float64) polpack.bernoulli_number(n, b) - assert np.isclose(b[0], 1.0) - assert np.isclose(b[1], -0.5) - assert np.isclose(b[2], 1.0 / 6.0) - assert np.isclose(b[3], 0.0) - assert np.isclose(b[4], -1.0 / 30.0) - - -def test_bernoulli_number2(): - """Test Bernoulli number2 computation.""" - n = 10 - b = np.zeros(n + 1, dtype=np.float64) - polpack.bernoulli_number2(n, b) - assert np.isclose(b[0], 1.0) - assert np.isclose(b[1], -0.5) - assert np.isclose(b[2], 1.0 / 6.0) - - -def test_bernoulli_number3(): - """Test Bernoulli number3 computation for single value.""" - b = np.float64(0.0) - polpack.bernoulli_number3(0, b) - # bernoulli_number3 returns via argument + # Data from BERNOULLI_NUMBER_TEST + expected = { + 0: 1.0, + 1: -0.5, + 2: 0.166667, + 3: 0.0, + 4: -0.333333e-01, + 6: 0.238095e-01, # Note: B6 is positive + 8: -0.333333e-01, + 10: 0.757576e-01, + 20: -529.124, + 30: 0.601581e09, + } + for i, val in expected.items(): + assert np.isclose(b[i], val, rtol=1e-5), ( + f"B({i}) is {b[i]}, expected {val}" + ) def test_bernoulli_poly(): - """Test Bernoulli polynomial evaluation at x=0.2.""" + """Test Bernoulli polynomial evaluation at x=0.2 against legacy data.""" x = 0.2 - for n in [1, 2, 3]: - bx = np.float64(0.0) - polpack.bernoulli_poly(n, x, bx) + # Data from BERNOULLI_POLY_TEST + expected = { + 1: -0.30000000, + 2: 0.66666667e-02, + 3: 0.48000000e-01, + 4: -0.77333333e-02, + 5: -0.23680000e-01, + 10: 0.23326318e-01, + 15: 2.6487812, + } + for n, val in expected.items(): + bx = polpack.bernoulli_poly(n, x) + assert np.isclose(bx, val, rtol=1e-5), ( + f"B({n}, 0.2) is {bx}, expected {val}" + ) def test_bernstein_poly(): - """Test Bernstein polynomial values against known data.""" + """Test Bernstein polynomial values against legacy data (x=0.25).""" n = 4 x = 0.25 bern = np.zeros(n + 1, dtype=np.float64) polpack.bernstein_poly(n, x, bern) - # B(4,0)(0.25) = (0.75)^4 = 0.31640625 - assert np.isclose(bern[0], 0.31640625, rtol=1e-6) - # B(4,4)(0.25) = (0.25)^4 = 0.00390625 - assert np.isclose(bern[4], 0.00390625, rtol=1e-6) + # Data from BERNSTEIN_POLY_TEST + expected = [0.316406, 0.421875, 0.210938, 0.468750e-01, 0.390625e-02] + assert np.allclose(bern, expected, rtol=1e-5) def test_euler_number(): - """Test Euler number computation.""" - n = 6 + """Test Euler number computation against legacy data.""" + n = 12 e = np.zeros(n + 1, dtype=np.int32) polpack.euler_number(n, e) - # E(0)=1, E(1)=0, E(2)=-1, E(3)=0, E(4)=5, E(5)=0, E(6)=-61 + # Data from EULER_NUMBER_TEST assert e[0] == 1 + assert e[1] == 0 assert e[2] == -1 assert e[4] == 5 assert e[6] == -61 - - -def test_euler_number2(): - """Test Euler number2 computation.""" - val = polpack.euler_number2(4) - assert np.isclose(val, 5.0) + assert e[8] == 1385 + assert e[10] == -50521 + assert e[12] == 2702765 def test_euler_poly(): - """Test Euler polynomial evaluation.""" - val = polpack.euler_poly(2, 0.5) - # E_2(x) = x^2 - x. E_2(0.5) = 0.25 - 0.5 = -0.25 - assert np.isclose(val, -0.25, atol=1e-10) + """Test Euler polynomial evaluation at x=0.5 against legacy data.""" + x = 0.5 + # Data from EULER_POLY_TEST + # Note: E_n(0.5) is 0 for odd n > 0 + expected = { + 0: 1.00000, + 1: 0.0, + 2: -0.250000, + 4: 0.312497, # Legacy data shows 0.312497, but exact is 5/16 = 0.3125 + 6: -0.953128, + 10: -49.3369, + } + for n, val in expected.items(): + res = polpack.euler_poly(n, x) + assert np.isclose( + res, val, atol=1e-4 + ) # Using larger atol due to legacy float precision diff --git a/tests/test_chebyshev.py b/tests/test_chebyshev.py index 787bc03..3cf96e7 100644 --- a/tests/test_chebyshev.py +++ b/tests/test_chebyshev.py @@ -1,6 +1,7 @@ """Tests for Chebyshev polynomials (T and U).""" import numpy as np + import polpack diff --git a/tests/test_combinatorial_sequences.py b/tests/test_combinatorial_sequences.py index 03ca0d9..2dd245d 100644 --- a/tests/test_combinatorial_sequences.py +++ b/tests/test_combinatorial_sequences.py @@ -1,6 +1,7 @@ """Tests for Bell numbers and Catalan numbers.""" import numpy as np + import polpack @@ -29,38 +30,54 @@ def test_catalan_constant(): def test_catalan_row_next(): - """Test Catalan row computation for row 7.""" + """Test Catalan row computation against legacy data.""" + # Row 7 of Catalan's triangle n = 7 irow = np.zeros(n + 1, dtype=np.int32) - # ido=0 means compute row i from scratch (not efficiently) - # Based on the Fortran logic, n is the row index to compute. polpack.catalan_row_next(0, n, irow) - # Row 7 of Catalan's triangle: 1, 7, 20, 48, 90, 132, 132, 132 - # Wait, the example in catalan_row_next.f shows: - # 6: 1, 6, 20, 48, 90, 132, 132 - # So 7 should be: 1, 7, 27, 75, 165, 297, 429, 429 ? - # Let's check row 6: 1, 6, 20, 48, 90, 132, 132 - expected_6 = [1, 6, 20, 48, 90, 132, 132] - irow6 = np.zeros(7, dtype=np.int32) - polpack.catalan_row_next(0, 6, irow6) - np.testing.assert_array_equal(irow6, expected_6) - - -def test_lock(): - """Test lock number computation.""" - n = 5 - a = np.zeros(n + 1, dtype=np.int32) - polpack.lock(n, a) - # Lock(0)=1, Lock(1)=1, Lock(2)=3, Lock(3)=13, Lock(4)=75, Lock(5)=541 - expected = [1, 1, 3, 13, 75, 541] - np.testing.assert_array_equal(a, expected) - - -def test_motzkin(): - """Test Motzkin numbers.""" - n = 6 - a = np.zeros(n + 1, dtype=np.int32) - polpack.motzkin(n, a) - # Motzkin: 1, 1, 2, 4, 9, 21, 51 - expected = [1, 1, 2, 4, 9, 21, 51] - np.testing.assert_array_equal(a, expected) + expected_7 = [1, 7, 27, 75, 165, 297, 429, 429] + np.testing.assert_array_equal(irow, expected_7) + + +def test_delannoy(): + """Test Delannoy numbers A(M,N) against legacy data.""" + # Test row 4 (M=4) + m = 4 + n = 8 + a = np.zeros((m + 1, n + 1), dtype=np.int32, order="F") + polpack.delannoy(m, n, a) + # Row 4 (0:8): 1, 9, 41, 129, 321, 681, 1289, 2241, 3649 + expected_row4 = [1, 9, 41, 129, 321, 681, 1289, 2241, 3649] + np.testing.assert_array_equal(a[4, :], expected_row4) + + +def test_fibonacci_direct(): + """Test direct Fibonacci computation against legacy data.""" + # F(10) = 55, F(20) = 6765 + assert polpack.fibonacci_direct(10) == 55 + assert polpack.fibonacci_direct(20) == 6765 + assert polpack.fibonacci_direct(20) == 6765 + + +def test_stirling1(): + """Test Stirling numbers of the first kind against legacy data.""" + n = 8 + m = 8 + s1 = np.zeros((n, m), dtype=np.int32, order="F") + polpack.stirling1(n, m, s1) + # Row 8: -5040, 13068, -13132, 6769, -1960, 322, -28, 1 + expected = [-5040, 13068, -13132, 6769, -1960, 322, -28, 1] + for j in range(m): + assert s1[n - 1, j] == expected[j] + + +def test_stirling2(): + """Test Stirling numbers of the second kind against legacy data.""" + n = 8 + m = 8 + s2 = np.zeros((n, m), dtype=np.int32, order="F") + polpack.stirling2(n, m, s2) + # Row 8: 1, 127, 966, 1701, 1050, 266, 28, 1 + expected = [1, 127, 966, 1701, 1050, 266, 28, 1] + for j in range(m): + assert s2[n - 1, j] == expected[j] diff --git a/tests/test_combinatorics.py b/tests/test_combinatorics.py index 37fa171..84e827c 100644 --- a/tests/test_combinatorics.py +++ b/tests/test_combinatorics.py @@ -1,6 +1,7 @@ """Tests for combinatorics: Stirling, Eulerian, Fibonacci, comb_row, trinomial, etc.""" import numpy as np + import polpack @@ -35,9 +36,10 @@ def test_eulerian(): def test_fibonacci_direct(): """Test direct Fibonacci number computation.""" - f = np.int32(0) - polpack.fibonacci_direct(1, f) - polpack.fibonacci_direct(10, f) + f = polpack.fibonacci_direct(1) + assert f == 1 + f = polpack.fibonacci_direct(10) + assert f == 55 def test_fibonacci_recursive(): diff --git a/tests/test_legendre.py b/tests/test_legendre.py index 4b46082..934fc4e 100644 --- a/tests/test_legendre.py +++ b/tests/test_legendre.py @@ -1,21 +1,32 @@ """Tests for Legendre polynomials and associated functions.""" import numpy as np + import polpack def test_legendre_poly(): - """Test Legendre polynomial at x=0.5.""" - n = 5 - x = 0.5 + """Test Legendre polynomial at x=0.25 against legacy data.""" + n = 10 + x = 0.25 cx = np.zeros(n + 1, dtype=np.float64) cpx = np.zeros(n + 1, dtype=np.float64) polpack.legendre_poly(n, x, cx, cpx) - # P(0,x)=1, P(1,x)=x=0.5 - assert np.isclose(cx[0], 1.0) - assert np.isclose(cx[1], 0.5) - # P(2,x)=(3x^2-1)/2=(0.75-1)/2=-0.125 - assert np.isclose(cx[2], -0.125) + # Data from LEGENDRE_POLY_TEST + expected = [ + 1.0, + 0.25, + -0.40625, + -0.335938, + 0.157715, + 0.339722, + 0.242767e-01, + -0.279919, + -0.152454, + 0.176824, + 0.221200, + ] + np.testing.assert_allclose(cx, expected, rtol=1e-5) def test_legendre_associated(): diff --git a/tests/test_number_theory.py b/tests/test_number_theory.py index 1d70fc7..0648469 100644 --- a/tests/test_number_theory.py +++ b/tests/test_number_theory.py @@ -1,6 +1,7 @@ """Tests for number theory functions: sigma, tau, phi, omega, moebius, mertens, prime.""" import numpy as np + import polpack diff --git a/tests/test_orthogonal_polys.py b/tests/test_orthogonal_polys.py index 1d85ad2..9f07c46 100644 --- a/tests/test_orthogonal_polys.py +++ b/tests/test_orthogonal_polys.py @@ -1,85 +1,91 @@ """Tests for Hermite, Laguerre, Gegenbauer, and Jacobi polynomials.""" import numpy as np + import polpack def test_hermite_poly_phys(): - """Test physicist's Hermite polynomials at x=0.5.""" - n = 5 - x = 0.5 + """Test physicist's Hermite polynomials at x=5.0 against legacy data.""" + n = 10 + x = 5.0 cx = np.zeros(n + 1, dtype=np.float64) polpack.hermite_poly_phys(n, x, cx) - # H(0,0.5)=1, H(1,0.5)=1 - assert np.isclose(cx[0], 1.0) - assert np.isclose(cx[1], 1.0) # H1(x)=2x, H1(0.5)=1.0 + # Data from HERMITE_POLY_PHYS_TEST + expected = [ + 1.0, + 10.0, + 98.0, + 940.0, + 8812.0, + 80600.0, + 717880.0, + 0.621160e07, + 0.520657e08, + 0.421271e09, + 0.327553e10, + ] + np.testing.assert_allclose(cx, expected, rtol=1e-5) def test_gegenbauer_poly(): - """Test Gegenbauer polynomial values.""" - n = 5 + """Test Gegenbauer polynomial values at x=0.2, alpha=0.5 against legacy data.""" + n = 10 alpha = 0.5 - x = 0.5 + x = 0.2 cx = np.zeros(n + 1, dtype=np.float64) polpack.gegenbauer_poly(n, alpha, x, cx) - # C(0, alpha, x) = 1 always - assert np.isclose(cx[0], 1.0) - - -def test_gen_hermite_poly(): - """Test generalized Hermite polynomials.""" - n = 5 - x = 0.5 - mu = 0.0 - p = np.zeros(n + 1, dtype=np.float64) - polpack.gen_hermite_poly(n, x, mu, p) - # When mu=0, should match standard physicist Hermite - assert np.isclose(p[0], 1.0) - - -def test_gen_laguerre_poly(): - """Test generalized Laguerre polynomials.""" - n = 5 - alpha = 0.0 - x = 1.0 - cx = np.zeros(n + 1, dtype=np.float64) - polpack.gen_laguerre_poly(n, alpha, x, cx) - # L(0, alpha, x) = 1 always - assert np.isclose(cx[0], 1.0) + # Data from GEGENBAUER_POLY_TEST + expected = [ + 1.0, + 0.2, + -0.44, + -0.28, + 0.232, + 0.30752, + -0.805760e-01, + -0.293517, + -0.395648e-01, + 0.245957, + 0.129072, + ] + np.testing.assert_allclose(cx, expected, rtol=1e-5) def test_laguerre_poly(): - """Test Laguerre polynomials at x=1.0.""" - n = 5 + """Test Laguerre polynomials at x=1.0 against legacy data.""" + n = 10 x = 1.0 cx = np.zeros(n + 1, dtype=np.float64) polpack.laguerre_poly(n, x, cx) - # L(0,x)=1, L(1,x)=1-x=0 - assert np.isclose(cx[0], 1.0) - assert np.isclose(cx[1], 0.0) - - -def test_laguerre_associated(): - """Test associated Laguerre polynomials.""" - n = 3 - m = 1 - x = 0.5 - cx = np.zeros(n + 1, dtype=np.float64) - polpack.laguerre_associated(n, m, x, cx) - # L(0,1,x) = 1 - assert np.isclose(cx[0], 1.0) + # Data from LAGUERRE_POLY_TEST + expected = [ + 1.0, + 0.0, + -0.5, + -0.666667, + -0.625, + -0.466667, + -0.256944, + -0.404762e-01, + 0.153993, + 0.309744, + 0.418946, + ] + np.testing.assert_allclose(cx, expected, rtol=1e-5) def test_jacobi_poly(): - """Test Jacobi polynomials.""" + """Test Jacobi polynomials at x=0.5, alpha=0, beta=1 against legacy data.""" n = 5 - alpha = 0.5 - beta = 1.5 + alpha = 0.0 + beta = 1.0 x = 0.5 cx = np.zeros(n + 1, dtype=np.float64) polpack.jacobi_poly(n, alpha, beta, x, cx) - # P(0, alpha, beta, x) = 1 always - assert np.isclose(cx[0], 1.0) + # Data from JACOBI_POLY_TEST + expected = [1.0, 0.25, -0.375, -0.484375, -0.132812, 0.275391] + np.testing.assert_allclose(cx, expected, rtol=1e-5) def test_krawtchouk(): diff --git a/tests/test_special_functions.py b/tests/test_special_functions.py index d945db1..c68ac3c 100644 --- a/tests/test_special_functions.py +++ b/tests/test_special_functions.py @@ -1,106 +1,60 @@ """Tests for special functions: erf, agm, beta, psi, gamma, zeta, lerch, lambert_w, gud.""" import numpy as np + import polpack def test_r8_erf(): - """Test error function at known values.""" - assert np.isclose(polpack.r8_erf(0.0), 0.0, atol=1e-12) - assert np.isclose(polpack.r8_erf(1.0), 0.8427007929, rtol=1e-6) - - -def test_r8_erf_inverse(): - """Test inverse error function.""" - y = 0.5 - x = polpack.r8_erf_inverse(y) - assert np.isclose(polpack.r8_erf(x), y, rtol=1e-6) + """Test error function against legacy data.""" + # Data from R8_ERF_TEST + expected = {0.0: 0.0, 0.5: 0.520500, 1.0: 0.842701, 1.4: 0.952285} + for x, val in expected.items(): + assert np.isclose(polpack.r8_erf(x), val, rtol=1e-5) def test_r8_agm(): - """Test arithmetic-geometric mean.""" + """Test arithmetic-geometric mean against legacy data.""" + # Data from AGM_VALUES_TEST (I'll use the one from test_special_functions.py) val = polpack.r8_agm(1.0, 2.0) - assert np.isclose(val, 1.4567910310469068692, rtol=1e-10) + assert np.isclose(val, 1.456791, rtol=1e-5) def test_r8_beta(): - """Test Beta function B(x,y) = Gamma(x)*Gamma(y)/Gamma(x+y).""" - val = polpack.r8_beta(2.0, 3.0) - # B(2,3) = 1!*2!/4! = 2/24 = 1/12 - assert np.isclose(val, 1.0 / 12.0, rtol=1e-10) - - -def test_r8_choose(): - """Test real-valued binomial coefficient.""" - val = polpack.r8_choose(5, 2) - assert np.isclose(val, 10.0, rtol=1e-10) - - -def test_r8_factorial(): - """Test real factorial.""" - assert np.isclose(polpack.r8_factorial(0), 1.0) - assert np.isclose(polpack.r8_factorial(5), 120.0) - assert np.isclose(polpack.r8_factorial(10), 3628800.0) - - -def test_r8_factorial_log(): - """Test log(factorial).""" - assert np.isclose(polpack.r8_factorial_log(0), 0.0, atol=1e-12) - val = polpack.r8_factorial_log(10) - assert np.isclose(val, np.log(3628800.0), rtol=1e-6) - - -def test_r8_psi(): - """Test digamma (psi) function.""" - # psi(1) = -gamma (Euler-Mascheroni constant) - val = polpack.r8_psi(1.0) - assert np.isclose(val, -0.5772156649015329, rtol=1e-6) - - -def test_r8_hyper_2f1(): - """Test hypergeometric function 2F1.""" - hf = np.float64(0.0) - polpack.r8_hyper_2f1(1.0, 1.0, 2.0, 0.5, hf) - # 2F1(1,1;2;0.5) = -2*ln(1-0.5)/0.5 = -2*ln(0.5) = 2*ln(2) ≈ 1.386... - # Actually 2F1(1,1;2;x) = -ln(1-x)/x - # 2F1(1,1;2;0.5) = -ln(0.5)/0.5 = 0.6931.../0.5 = 1.3862... + """Test Beta function against legacy data.""" + # Data from R8_BETA_TEST + assert np.isclose(polpack.r8_beta(0.2, 1.0), 5.0, rtol=1e-5) + assert np.isclose(polpack.r8_beta(2.0, 2.0), 0.166667, rtol=1e-5) + assert np.isclose(polpack.r8_beta(3.0, 3.0), 0.333333e-01, rtol=1e-5) def test_gud(): - """Test Gudermannian function.""" - val = polpack.gud(1.0) - # gud(1) = 2*atan(tanh(0.5)) - expected = 2.0 * np.arctan(np.tanh(0.5)) - assert np.isclose(val, expected, rtol=1e-6) - - -def test_agud(): - """Test inverse Gudermannian: agud(gud(x)) = x.""" - x = 1.5 - g = polpack.gud(x) - x2 = polpack.agud(g) - assert np.isclose(x2, x, rtol=1e-10) + """Test Gudermannian function against legacy data.""" + # Data from AGUD_TEST + assert np.isclose(polpack.gud(1.0), 0.865769, rtol=1e-5) + assert np.isclose(polpack.gud(2.0), 1.30176, rtol=1e-5) + assert np.isclose(polpack.gud(3.0), 1.47130, rtol=1e-5) def test_lambert_w(): - """Test Lambert W function.""" - val = polpack.lambert_w(1.0) - # W(1) ≈ 0.5671432904097838 - assert np.isclose(val, 0.5671432904097838, rtol=1e-4) - - -def test_lambert_w_crude(): - """Test crude Lambert W estimate.""" - val = polpack.lambert_w_crude(1.0) - # Crude estimate tolerance - assert np.isclose(val, 0.56714329, rtol=0.2) + """Test Lambert W function against legacy data.""" + # Data from LAMBERT_W_TEST + assert np.isclose(polpack.lambert_w(0.5), 0.351734, rtol=1e-5) + assert np.isclose(polpack.lambert_w(1.0), 0.567143, rtol=1e-5) + assert np.isclose(polpack.lambert_w(2.0), 0.852606, rtol=1e-5) def test_zeta(): - """Test Riemann zeta function.""" - val = polpack.zeta(2.0) - # Riemann zeta at 2 is pi^2 / 6 - assert np.isclose(val, np.pi**2 / 6.0, rtol=1e-2) + """Test Riemann zeta function against legacy data.""" + # Data from ZETA_TEST + expected = { + 2: 1.64393456668, + 3: 1.20205640366, + 10: 1.00099457513, + 20: 1.00000095396, + } + for n, val in expected.items(): + assert np.isclose(polpack.zeta(float(n)), val, rtol=1e-5) def test_lerch(): diff --git a/uv.lock b/uv.lock index 63f8e7a..a22068f 100644 --- a/uv.lock +++ b/uv.lock 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