poly

A Python package for native object polynomials and vectorized numpy.polynomial.polynomial functions.

>>> from poly import polyval
>>> from fractions import Fraction
>>> p = (Fraction(1, 2), Fraction(3, 4))
>>> x = Fraction(5, 6)
>>> polyval(p, x)
Fraction(9, 8)

Installation

pip install git+https://github.com/goessl/poly.git

Contents

This package consists of two modules:

• poly: native polynomial operations &
• nppoly: row wise vectorized versions of numpy.polynomial.polynomials functions to operate on many polynomials in parallel.

poly

This module provides functions, similar to the ones found in numpy.polynomial.polynomial, but in pure Python. This means it can handle Pythons native unlimited ints, Fractions and every object that implements the needed scalar arithmetic without numpys lossy conversion.

To represent polynomials sequences like lists and tuples are used (coefficients in ascending order). The results are always returned as tuples. The arguments are consumed but not altered. The functions signatures try to mimic numpy.polynomial.polynomial, most often with less parameters because of less functionality. Operations that are associative, like polyadd and polymul, allow arbitrary many arguments, even none.

If actual polyzeros ((), not some untrimmed polynomial like (0,)) are passed as arguments, the functions also correctly return actual polyzeros (()) or polyones ((1,)). The user is responsible to polytrim arguments and results.

Provided are fundamental polynomials:

• polyzero: $0$,
• polyone: $1$,
• polyx: $x$,
• polymono(n, c=1): $c\cdot x^n$;

evaluation:

• polyval(p, x): $p(x)$,
• polycom(p, q): $p(q)$;

arithmetic operations:

• polyadd(*ps): $p_0 + p_1 + \dots$,
• polysub(p, q): $p - q$,
• polymul(*ps): $p_0 \cdot p_1 \cdot \dots$,
• polydiv(n, d): (q, r) such that $n=qd+r$,
• polypow(p, n): $p^n$;

calculus:

• polyder(p): $p'$,
• polyint(p, c=0): $\int p(x)dx$

nppoly

This module provides vectorised versions of functions found in numpy.polynomial.polynomial. Polynomials are represented as rows in numpy.ndarrays. The functions then work row by row (but are implemented faster than simply using numpy.apply_along_axis).

Difference to poly:

• polyzero is represented by [0] like numpy.polynomial.polynomial.polyzero and not by () like poly does.
• Associative operations (polyadd & polymul) don't handle no arguments, as there is no way to determine the required number of rows to be returned.
• numpy.polynomial.polynomial.polyval is already vectorized and therefore not implemented again.

Provided are fundamental polynomials:

• polyzero(d): [[0], [0], ..., [0]],
• polyone(d): [[1], [1], ..., [1]],
• polymono(d, n): [[0, 0, ..., 0, 1], [0, 0, ..., 0, 1], ..., [0, 0, ..., 0, 1]];

evaluation:

• polycom(p, q): $p(q)$;

arithmetic operations:

• polyadd(*ps): $p_0 + p_1 + \dots$,
• polysub(p, q): $p - q$,
• polymul(*ps): $p_0 \cdot p_1 \cdot \dots$,
• polydiv(n, d): (q, r) such that $n=qd+r$,
• polypow(p, n): $p^n$;

calculus:

• polyder(p): $p'$,
• polyint(p, c=0): $\int p(x)dx$

License (MIT)

Copyright (c) 2024 Sebastian Gössl

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.