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Standard

Polynomials in standard monomial basis.

\[ p(x)=\sum_{k=0}^na_kx^k \qquad P=(x_n)_{n\in\mathbb{N}_0}=(1, x, x^2, \dots) \]

Prefixed by poly... (polynomial).

All functions accept single exhaustible iterables, if not stated otherwise.

If the result of a function is a polynomial, it is returned as a tuple.

Associative operations, like polyadd and polymul, allow arbitrary many arguments, even none.

The functions are type-independent. However, the data types used must support necessary scalar operations. For instance, for polynomial addition, components must be addable.

Docstring convention

Summary

Math notation

Complexity

For a polynomial of degree \(n\) there will be

  • \(x\) scalar additions (add),
  • \(y\) scalar subtractions, ...

Order: pos, neg, add, sub, mul

Notes

design choices

See also

other implementations, delegations/wrappings, ...

References

numpy equivalents, Wikipedia, ...)

creation

polyzero = ()

Zero polynomial.

\[ 0 \qquad 0^{(P)}=\vec{0} \]

An empty tuple: ().

Notes

Why give the zero polynomial a distinguished representation (not just [0] like numpy.polynomial)?

The additive neutral element seems worth handling exceptionally. It is mathematically different (different degree) and results in functions like polymul being more time and memory efficient.

See also
References

polyone = (1,)

Constant one polynomial.

\[ 1 \qquad 1^{(P)}=\begin{pmatrix} 1 \end{pmatrix} \]

A tuple with a single one: (1,).

See also
References

polyx = (0, 1)

Identity polynomial.

\[ x \qquad x^{(P)}=\begin{pmatrix} 0 \\ 1 \end{pmatrix} \]

A tuple with a zero and a one: (0, 1).

See also
References

polymono(n, c=1, zero=0)

Return a monomial.

\[ cx^n \qquad (cx^n)^{(P)}=c\vec{e}_n \]

A tuple with n many zeros followed by c or polyzero if \(n<0\).

See also
Source code in poly\standard\creation.py 
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def polymono(n, c=1, zero=0):
    r"""Return a monomial.

    $$
        cx^n \qquad (cx^n)^{(P)}=c\vec{e}_n
    $$

    A tuple with `n` many `zero`s followed by `c` or [`polyzero`][poly.standard.creation.polyzero] if $n<0$.

    See also
    --------
    - constants: [`polyzero`][poly.standard.creation.polyzero], [`polyone`][poly.standard.creation.polyone], [`polyx`][poly.standard.creation.polyx]
    - for all monomials: [`polymonos`][poly.standard.creation.polymonos]
    - wraps: [`vector.vecbasis`](https://goessl.github.io/vector/dense/#vector.dense.creation.vecbasis)
    """
    return vecbasis(n, c=c, zero=zero) if n>=0 else polyzero

polymonos(start=0, c=1, zero=0)

Yield all monomials.

\[ \left(x^n\right)_{n\in\mathbb{N}_{\geq\text{start}} = \left(x^\text{start}, x^{\text{start}+1}, x^{\text{start}+2}, \dots \right) \]
See also
Source code in poly\standard\creation.py 
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def polymonos(start=0, c=1, zero=0):
    r"""Yield all monomials.

    $$
        \left(x^n\right)_{n\in\mathbb{N}_{\geq\text{start}} = \left(x^\text{start}, x^{\text{start}+1}, x^{\text{start}+2}, \dots \right)
    $$

    See also
    --------
    - for single monomial: [`polymono`][poly.standard.creation.polymono]
    - wraps: [`vector.vecbases`](https://goessl.github.io/vector/dense/#vector.dense.creation.vecbases)
    """
    yield from vecbases(start=start, c=c, zero=zero)

polyrand(n)

Return a random polynomial of uniformly sampled float coefficients.

\[ \sum_{k=0}^na_kx^k \qquad a_k\sim\mathcal{U}([0, 1[) \]

The coefficients are sampled from a uniform distribution in [0, 1[.

See also
Source code in poly\standard\creation.py 
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def polyrand(n):
    r"""Return a random polynomial of uniformly sampled `float` coefficients.

    $$
        \sum_{k=0}^na_kx^k \qquad a_k\sim\mathcal{U}([0, 1[)
    $$

    The coefficients are sampled from a uniform distribution in `[0, 1[`.

    See also
    --------
    - wraps: [`vector.vecrand`](https://goessl.github.io/vector/dense/#vector.dense.creation.vecrand)
    """
    return vecrand(n+1)

polyrandn(n, normed=True, mu=0, sigma=1)

Return a random polynomial of normally sampled float coefficients.

\[ \sum_{k=0}^na_kx^k \qquad a_k\sim\mathcal{N}(\mu, \sigma) \]

The coefficients are sampled from a normal distribution.

Normed with respect to the euclidian vector norm.

See also
Source code in poly\standard\creation.py 
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def polyrandn(n, normed=True, mu=0, sigma=1):
    r"""Return a random polynomial of normally sampled `float` coefficients.

    $$
        \sum_{k=0}^na_kx^k \qquad a_k\sim\mathcal{N}(\mu, \sigma)
    $$

    The coefficients are sampled from a normal distribution.

    Normed with respect to the euclidian vector norm.

    See also
    --------
    - wraps: [`vector.vecrandn`](https://goessl.github.io/vector/dense/#vector.dense.creation.vecrandn)
    """
    return vecrandn(n+1, normed=normed, mu=mu, sigma=sigma)

polyfromroots(*xs, one=1)

Return the polynomial with the given roots.

\[ \prod_k(x-x_k) \]
Complexity

No yet perfect.

For \(n\) roots there will be

  • \(n\) scalar negations (neg),
  • \(\frac{n(n-1)}{2}\) scalar additions (add) &
  • \(\begin{cases}(n+2)(n-1)&n\ge1\\0&n\le1\end{cases}\) scalar multiplications (mul).
References
Source code in poly\standard\creation.py 
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def polyfromroots(*xs, one=1):
    r"""Return the polynomial with the given roots.

    $$
        \prod_k(x-x_k)
    $$

    Complexity
    ----------
    No yet perfect.

    For $n$ roots there will be

    - $n$ scalar negations (`neg`),
    - $\frac{n(n-1)}{2}$ scalar additions (`add`) &
    - $\begin{cases}(n+2)(n-1)&n\ge1\\0&n\le1\end{cases}$ scalar multiplications (`mul`).

    References
    ----------
    - Recipe: [more_itertools.polynomial_from_roots](https://more-itertools.readthedocs.io/en/stable/api.html#more_itertools.polynomial_from_roots)
    - `numpy` equivalent: [`numpy.polynomial.polynomial.polyfromroots`](https://numpy.org/doc/stable/reference/generated/numpy.polynomial.polynomial.polyfromroots.html)
    """
    #return polymul(*zip(map(neg, xs), repeat(one)), method='naive', one=one)
    r = (one, )
    for x in xs:
        r = polysub(polymulx(r), polyscalarrmul(x, r))
    return r

utility

polydeg(p)

Return the degree (number of set coefficients).

\[ \deg(p) \]

Doesn't handle leading zeros, use polytrim if needed.

\(\deg(0)=-1\) is used for the empty zero polynomial.

Notes

\(\deg(0)=-\infty\) is more commonly used but the expected return type is an int while -math.inf is of type float. Therefore the \(\deg(0)=-1\) convention was choosen to keep the return type consistent.

See also
References
Source code in poly\standard\utility.py 
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def polydeg(p):
    r"""Return the degree (number of set coefficients).

    $$
        \deg(p)
    $$

    Doesn't handle leading zeros, use [`polytrim`][poly.standard.utility.polytrim]
    if needed.

    $\deg(0)=-1$ is used for the empty zero polynomial.

    Notes
    -----
    $\deg(0)=-\infty$ is more commonly used but the expected return type is an
    `int` while `-math.inf` is of type `float`. Therefore the $\deg(0)=-1$
    convention was choosen to keep the return type consistent.

    See also
    --------
    - wraps: [`vector.veclen`](https://goessl.github.io/vector/dense/#vector.dense.utility.veclen)

    References
    ----------
    - [Wikipedia - Degree of a polynomial - Degree of the zero polynomial](https://en.wikipedia.org/wiki/Degree_of_a_polynomial#Degree_of_the_zero_polynomial)
    """
    return veclen(p) - 1

polyeq(p, q)

Return if two polynomials are equal.

\[ p \overset{?}{=} q \]
Complexity

For two polynomials of degrees \(n\) & \(m\) there will be at most

  • \(\min\{n, m\}+1\) scalar comparisons (eq) &
  • \(|n-m|\) scalar boolean evaluations (bool).
See also
Source code in poly\standard\utility.py 
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def polyeq(p, q):
    r"""Return if two polynomials are equal.

    $$
        p \overset{?}{=} q
    $$

    Complexity
    ----------
    For two polynomials of degrees $n$ & $m$ there will be at most

    - $\min\{n, m\}+1$ scalar comparisons (`eq`) &
    - $|n-m|$ scalar boolean evaluations (`bool`).

    See also
    --------
    - wraps: [`vector.veceq`](https://goessl.github.io/vector/dense/#vector.dense.utility.veceq)
    """
    return veceq(p, q)

polytrim(p, tol=None)

Remove all leading near zero (abs(a_i)<=tol) coefficients.

\[ \sum_{k=0}^na_kx^k \ \text{where} \ n=\max\{\, k\mid |a_k|>\text{tol}\,\}\cup\{-1\} \]

tol may also be None, then all coefficients that evaluate to False are trimmed.

Complexity

For a polynomial of degree \(n\) there will be

  • \(n+1\) scalar absolute evaluations (abs) &
  • \(n+1\) scalar comparisons (gt).
Notes
  • Cutting of elements that are abs(a_i)<=tol instead of abs(a_i)<tol to allow cutting of elements that are exactly zero by trim(p, 0) instead of trim(p, sys.float_info.min).
See also
Source code in poly\standard\utility.py 
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def polytrim(p, tol=None):
    r"""Remove all leading near zero (`abs(a_i)<=tol`) coefficients.

    $$
        \sum_{k=0}^na_kx^k \ \text{where} \ n=\max\{\, k\mid |a_k|>\text{tol}\,\}\cup\{-1\}
    $$

    `tol` may also be `None`,
    then all coefficients that evaluate to `False` are trimmed.

    Complexity
    ----------
    For a polynomial of degree $n$ there will be

    - $n+1$ scalar absolute evaluations (`abs`) &
    - $n+1$ scalar comparisons (`gt`).

    Notes
    -----
    - Cutting of elements that are `abs(a_i)<=tol` instead of `abs(a_i)<tol` to
    allow cutting of elements that are exactly zero by `trim(p, 0)` instead
    of `trim(p, sys.float_info.min)`.

    See also
    --------
    - wraps: [`vector.vectrim`](https://goessl.github.io/vector/dense/#vector.dense.utility.vectrim)
    - `numpy` equivalent: [`numpy.polynomial.polynomial.polytrim`](https://numpy.org/doc/stable/reference/generated/numpy.polynomial.polynomial.polytrim.html)
    """
    return vectrim(p, tol=tol)

evaluation

polyval(p, x, method='horner')

Return the value at point x.

\[ p(x) \]

Available methods are

See also
References
Source code in poly\standard\evaluation.py 
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def polyval(p, x, method='horner'):
    """Return the value at point `x`.

    $$
        p(x)
    $$

    Available methods are

    - [`naive`][poly.standard.polyval_naive],
    - [`iterative`][poly.standard.polyval_iterative] &
    - [`horner`][poly.standard.polyval_horner] (`p` must be reversible).

    See also
    --------
    - implementations: [`polyval_naive`][poly.standard.evaluation.polyval_naive],
    [`polyval_iterative`][poly.standard.evaluation.polyval_iterative],
    [`polyval_horner`][poly.standard.evaluation.polyval_horner]
    - for consecutive monomials: [`polyvals`][poly.standard.evaluation.polyvals]
    - for $x=0$: [`polyvalzero`][poly.standard.evaluation.polyvalzero]
    - for polynomial arguments: [`polycom`][poly.standard.evaluation.polycom]

    References
    ----------
    - `numpy` equivalent: [`numpy.polynomial.polynomial.polyval`](https://numpy.org/doc/stable/reference/generated/numpy.polynomial.polynomial.polyval.html)
    """
    match method:
        case 'naive':
            return polyval_naive(p, x)
        case 'iterative':
            return polyval_iterative(p, x)
        case 'horner':
            return polyval_horner(p, x)
        case _:
            raise ValueError('Invalid method')

polyval_naive(p, x)

Return the value at point x.

\[ p(x) \]

Uses naive repeated multiplication to calculate monomials individually.

See also
References
Source code in poly\standard\evaluation.py 
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def polyval_naive(p, x):
    """Return the value at point `x`.

    $$
        p(x)
    $$

    Uses naive repeated multiplication to calculate monomials individually.

    See also
    --------
    - for any implementation: [`polyval`][poly.standard.evaluation.polyval]
    - other implementations:
    [`polyval_iterative`][poly.standard.evaluation.polyval_iterative],
    [`polyval_horner`][poly.standard.evaluation.polyval_horner]
    - for polynomial arguments: [`polycom_naive`][poly.standard.evaluation.polycom_naive]

    References
    ----------
    - [more_itertools.polynomial_eval](https://more-itertools.readthedocs.io/en/stable/api.html#more_itertools.polynomial_eval)
    - [Wikipedia - Polynomial evaluation](https://en.wikipedia.org/wiki/Polynomial_evaluation)
    - [Wikipedia - Horner's method - Efficiency](https://en.wikipedia.org/wiki/Horner%27s_method#Efficiency)
    """
    p = iter(p)
    a0 = next(p, type(x)(0))
    return sumprod_default(p, chain((x,), map(pow, repeat(x), count(start=2))), initial=a0, default=MISSING)

polyval_iterative(p, x)

Return the value at point x.

\[ p(x) \]

Uses iterative multiplication to calculate monomials consecutively.

Complexity

For a polynomial of degree \(n\) there will be

  • \(\begin{cases}n&n\ge0\\0&n\le0\end{cases}\) scalar additions (add) &
  • \(\begin{cases}2n-1&n>0\\0&n\le0\end{cases}\) scalar multiplications (mul).
See also
References
Source code in poly\standard\evaluation.py 
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def polyval_iterative(p, x):
    r"""Return the value at point `x`.

    $$
        p(x)
    $$

    Uses iterative multiplication to calculate monomials consecutively.

    Complexity
    ----------
    For a polynomial of degree $n$ there will be

    - $\begin{cases}n&n\ge0\\0&n\le0\end{cases}$ scalar additions (`add`) &
    - $\begin{cases}2n-1&n>0\\0&n\le0\end{cases}$ scalar multiplications (`mul`).

    See also
    --------
    - for any implementation: [`polyval`][poly.standard.evaluation.polyval]
    - other implementations:
    [`polyval_naive`][poly.standard.evaluation.polyval_naive],
    [`polyval_horner`][poly.standard.evaluation.polyval_horner]
    - uses: [`polyvals`][poly.standard.evaluation.polyvals]
    - for polynomial arguments: [`polycom_iterative`][poly.standard.evaluation.polycom_iterative]

    References
    ----------
    - [Wikipedia - Horner's method - Efficiency](https://en.wikipedia.org/wiki/Horner%27s_method#Efficiency)
    """
    p = iter(p)
    a0 = next(p, type(x)(0))
    return sumprod_default(p, polyvals(x, start=1), initial=a0, default=MISSING)

polyval_horner(p, x)

Return the value at point x.

\[ p(x) \]

Uses Horner's method.

p must be reversible.

Complexity

For a polynomial of degree \(n\) there will be

  • \(\begin{cases}n&n\ge0\\0&n\le0\end{cases}\) scalar additions (add) &
  • \(\begin{cases}n&n\ge0\\0&n\le0\end{cases}\) scalar multiplications (mul).
See also
References
Source code in poly\standard\evaluation.py 
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def polyval_horner(p, x):
    r"""Return the value at point `x`.

    $$
        p(x)
    $$

    Uses Horner's method.

    `p` must be reversible.

    Complexity
    ----------
    For a polynomial of degree $n$ there will be

    - $\begin{cases}n&n\ge0\\0&n\le0\end{cases}$ scalar additions (`add`) &
    - $\begin{cases}n&n\ge0\\0&n\le0\end{cases}$ scalar multiplications (`mul`).

    See also
    --------
    - for any implementation: [`polyval`][poly.standard.evaluation.polyval]
    - other implementations:
    [`polyval_naive`][poly.standard.evaluation.polyval_naive],
    [`polyval_iterative`][poly.standard.evaluation.polyval_iterative]
    - for polynomial arguments: [`polycom_horner`][poly.standard.evaluation.polycom_horner]

    References
    ----------
    - [Wikipedia - Horner's method](https://en.wikipedia.org/wiki/Horner%27s_method)
    """
    p = iter(reversed(p))
    an = next(p, type(x)(0))
    return reduce_default(lambda a, pi: a*x+pi, p, initial=an, default=MISSING)

polyvals(x, start=0)

Yield all powers of scalar x.

\[ (x^n)_{n\in\mathbb{N}_{\geq\text{start}} = (x^\text{start}, x^{\text{start}+1}, x^{\text{start}+2}, \dots) \]

Uses iterative multiplication to calculate monomials consecutively.

See also
Source code in poly\standard\evaluation.py 
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def polyvals(x, start=0):
    r"""Yield all powers of scalar `x`.

    $$
        (x^n)_{n\in\mathbb{N}_{\geq\text{start}} = (x^\text{start}, x^{\text{start}+1}, x^{\text{start}+2}, \dots)
    $$

    Uses iterative multiplication to calculate monomials consecutively.

    See also
    --------
    - used by: [`polyval_iterative`][poly.standard.evaluation.polyval_iterative]
    - for polynomial arguments: [`polypows`][poly.standard.arithmetic.polypows]
    """
    if start <= 0:
        yield type(x)(1)
    yield (y := prod_default(repeat(x, start), initial=MISSING, default=x))
    while True:
        y *= x
        yield y

polyvalzero(p, zero=0)

Return the value at point 0.

\[ p(0) \]

More efficient than polyval(p, 0).

Complexity

There are no scalar arithmetic operations.

See also
Source code in poly\standard\evaluation.py 
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def polyvalzero(p, zero=0):
    """Return the value at point 0.

    $$
        p(0)
    $$

    More efficient than `polyval(p, 0)`.

    Complexity
    ----------
    There are no scalar arithmetic operations.

    See also
    --------
    - for any argument: [`polyval`][poly.standard.evaluation.polyval]
    """
    return next(iter(p), zero)

polycom(p, q, method='iterative')

Return the composition.

\[ p\circ q \]

q must be a sequence.

Available methods are

See also
Source code in poly\standard\evaluation.py 
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def polycom(p, q, method='iterative'):
    r"""Return the composition.

    $$
        p\circ q
    $$

    `q` must be a sequence.

    Available methods are

    - [`naive`][poly.standard.evaluation.polycom_naive],
    - [`iterative`][poly.standard.evaluation.polycom_iterative] &
    - [`horner`][poly.standard.evaluation.polycom_horner] (`p` must be reversible).

    See also
    --------
    - implementations: [`polycom_naive`][poly.standard.evaluation.polycom_naive],
    [`polycom_iterative`][poly.standard.evaluation.polycom_iterative],
    [`polycom_horner`][poly.standard.evaluation.polycom_horner]
    - for $q=x-s$: [`polyshift`][poly.standard.evaluation.polyshift]
    - for scalar arguments: [`polyval`][poly.standard.evaluation.polyval]
    """
    match method:
        case 'naive':
            return polycom_naive(p, q)
        case 'iterative':
            return polycom_iterative(p, q)
        case 'horner':
            return polycom_horner(p, q)
        case _:
            raise ValueError('Invalid method')

polycom_naive(p, q)

Return the composition.

\[ p\circ q \]

Uses naive repeated multiplication to calculate monomials individually.

q must be a sequence.

Complexity

For two polynomials of degrees \(n\) & \(m\) there will be

  • \(\begin{cases}\frac{n^3m^2}{6}+\frac{n^2m}{2}-\frac{nm^2}{6}-\frac{nm}{2}+n&n\ge0\land m\ge0\\0&n\le0\lor m\le0\end{cases}\) scalar additions (add) &
  • \(\begin{cases}\frac{n^3m^2}{6}+\frac{n^3m}{6}+n^2m-\frac{nm^2}{6}-\frac{nm}{6}+\frac{n^2}{2}+\frac{n}{2}&n\ge0\land m\ge0\\0&n\le0\lor m\le0\end{cases}\) scalar multiplications (´mul´).
See also
Source code in poly\standard\evaluation.py 
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def polycom_naive(p, q):
    r"""Return the composition.

    $$
        p\circ q
    $$

    Uses naive repeated multiplication to calculate monomials individually.

    `q` must be a sequence.

    Complexity
    ----------
    For two polynomials of degrees $n$ & $m$ there will be

    - $\begin{cases}\frac{n^3m^2}{6}+\frac{n^2m}{2}-\frac{nm^2}{6}-\frac{nm}{2}+n&n\ge0\land m\ge0\\0&n\le0\lor m\le0\end{cases}$ scalar additions (`add`) &
    - $\begin{cases}\frac{n^3m^2}{6}+\frac{n^3m}{6}+n^2m-\frac{nm^2}{6}-\frac{nm}{6}+\frac{n^2}{2}+\frac{n}{2}&n\ge0\land m\ge0\\0&n\le0\lor m\le0\end{cases}$ scalar multiplications (´mul´).

    See also
    --------
    - for any implementation: [`polycom`][poly.standard.evaluation.polycom]
    - other implementations:
    [`polycom_iterative`][poly.standard.evaluation.polycom_iterative],
    [`polycom_horner`][poly.standard.evaluation.polycom_horner]
    - for scalar arguments: [`polyval_naive`][poly.standard.evaluation.polyval_naive]
    """
    p = iter(p)
    a0 = tuple(islice(p, 1)) #(a0,) or polyzero
    return polyadd(a0, *(polyscalarrmul(pi, polypow_naive(q, i)) for i, pi in enumerate(p, start=1)))

polycom_iterative(p, q)

Return the composition.

\[ p\circ q \]

Uses iterative multiplication to calculate monomials consecutively.

q must be a sequence.

Complexity

For two polynomials of degrees \(n\) & \(m\) there will be

  • \(\begin{cases}\frac{n^2m^2}{2}+\frac{n^2m}{2}-\frac{nm^2}{2}-\frac{nm}{2}+n&n\ge0\land m\ge0\\0&n\le0\lor m\le0\end{cases}\) scalar additions (add) &
  • \(\begin{cases}\frac{n^2m^2}{2}+n^2m-\frac{nm^2}{2}+nm+2n-m-1&n\ge1\land m\ge0\\0&n\le1\lor m\le0\end{cases}\) scalar multiplications (´mul´).
See also
Source code in poly\standard\evaluation.py 
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def polycom_iterative(p, q):
    r"""Return the composition.

    $$
        p\circ q
    $$

    Uses iterative multiplication to calculate monomials consecutively.

    `q` must be a sequence.

    Complexity
    ----------
    For two polynomials of degrees $n$ & $m$ there will be

    - $\begin{cases}\frac{n^2m^2}{2}+\frac{n^2m}{2}-\frac{nm^2}{2}-\frac{nm}{2}+n&n\ge0\land m\ge0\\0&n\le0\lor m\le0\end{cases}$ scalar additions (`add`) &
    - $\begin{cases}\frac{n^2m^2}{2}+n^2m-\frac{nm^2}{2}+nm+2n-m-1&n\ge1\land m\ge0\\0&n\le1\lor m\le0\end{cases}$ scalar multiplications (´mul´).

    See also
    --------
    - for any implementation: [`polycom`][poly.standard.evaluation.polycom]
    - other implementations:
    [`polycom_naive`][poly.standard.evaluation.polycom_naive],
    [`polycom_horner`][poly.standard.evaluation.polycom_horner]
    - uses: [`polypows`][poly.standard.arithmetic.polypows]
    - for scalar arguments: [`polyval_iterative`][poly.standard.evaluation.polyval_iterative]
    """
    p = iter(p)
    a0 = tuple(islice(p, 1))
    return polyadd(a0, *map(polyscalarrmul, p, polypows(q, start=1)))

polycom_horner(p, q)

Return the composition.

\[ p\circ q \]

Uses Horner's method.

q must be reversible.

Complexity

For two polynomials of degrees \(n\) & \(m\) there will be

  • \(\begin{cases}0&m\ge0\lor n<0\\n&m<0\land n>=0\end{cases}\) scalar unary pluses (pos),
  • \(\begin{cases}\frac{n^2m^2}{2}-\frac{nm^2}{2}+n&n\ge0\land m\ge0\\0&n\le0\lor m\le0\end{cases}\) scalar additions (add) &
  • \(\begin{cases}\frac{n^2m^2}{2}+\frac{n^2m}{2}-\frac{nm^2}{2}+\frac{nm}{2}+n&n\ge0\land m\ge0\\0&n\le0\lor m\le0\end{cases}\) scalar multiplications (´mul´).
See also
Source code in poly\standard\evaluation.py 
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def polycom_horner(p, q):
    r"""Return the composition.

    $$
        p\circ q
    $$

    Uses Horner's method.

    `q` must be reversible.

    Complexity
    ----------
    For two polynomials of degrees $n$ & $m$ there will be

    - $\begin{cases}0&m\ge0\lor n<0\\n&m<0\land n>=0\end{cases}$ scalar unary pluses (`pos`),
    - $\begin{cases}\frac{n^2m^2}{2}-\frac{nm^2}{2}+n&n\ge0\land m\ge0\\0&n\le0\lor m\le0\end{cases}$ scalar additions (`add`) &
    - $\begin{cases}\frac{n^2m^2}{2}+\frac{n^2m}{2}-\frac{nm^2}{2}+\frac{nm}{2}+n&n\ge0\land m\ge0\\0&n\le0\lor m\le0\end{cases}$ scalar multiplications (´mul´).

    See also
    --------
    - for any implementation: [`polycom`][poly.standard.evaluation.polycom]
    - other implementations:
    [`polycom_naive`][poly.standard.evaluation.polycom_naive],
    [`polycom_iterative`][poly.standard.evaluation.polycom_iterative]
    - for scalar arguments: [`polyval_horner`][poly.standard.evaluation.polyval_horner]
    """
    p = iter(reversed(p))
    an = tuple(islice(p, 1))
    return reduce_default(lambda a, pi: polyaddc(polymul_naive(a, q), pi), p, initial=an, default=MISSING)

polyshift(p, s, one=1)

Return the polynomial p shifted by s on the abscissa.

\[ p(x - s) \]

TODO: https://math.stackexchange.com/a/694571/1170417

See also
Source code in poly\standard\evaluation.py 
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def polyshift(p, s, one=1):
    """Return the polynomial `p` shifted by `s` on the abscissa.

    $$
        p(x - s)
    $$

    TODO: https://math.stackexchange.com/a/694571/1170417

    See also
    --------
    - for polynomial argument: [`polycom`][poly.standard.evaluation.polycom]
    """
    return polycom_naive(p, (-s, one))

polyscale(p, a)

Return the polynomial p scaled by 1/a on the abscissa.

\[ p(ax) \]

More efficient than polycom(p, polyscalarmul(a, polyx)).

See also
Source code in poly\standard\evaluation.py 
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def polyscale(p, a):
    """Return the polynomial `p` scaled by 1/`a` on the abscissa.

    $$
        p(ax)
    $$

    More efficient than `polycom(p, polyscalarmul(a, polyx))`.

    See also
    --------
    - for polynomial argument: [`polycom`][poly.standard.evaluation.polycom]
    """
    p = iter(p)
    return tuple(chain(islice(p, 1), veclhadamard(p, polyvals(a, start=1))))

arithmetic

polypos(p)

Return the identity.

\[ +p \]
Complexity

For a polynomial of degree \(n\) there will be

  • \(n+1\) scalar unary plus operations (pos).
See also
Source code in poly\standard\arithmetic.py 
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def polypos(p):
    """Return the identity.

    $$
        +p
    $$

    Complexity
    ----------
    For a polynomial of degree $n$ there will be

    - $n+1$ scalar unary plus operations (`pos`).

    See also
    --------
    - wraps: [`vector.vecpos`](https://goessl.github.io/vector/dense/#vector.dense.vector_space.vecpos)
    """
    return vecpos(p)

polyneg(p)

Return the negation.

\[ -p \]
Complexity

For a polynomial of degree \(n\) there will be

  • \(n+1\) scalar negations (neg).
See also
Source code in poly\standard\arithmetic.py 
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def polyneg(p):
    """Return the negation.

    $$
        -p
    $$

    Complexity
    ----------
    For a polynomial of degree $n$ there will be

    - $n+1$ scalar negations (`neg`).

    See also
    --------
    - wraps: [`vector.vecneg`](https://goessl.github.io/vector/dense/#vector.dense.vector_space.vecneg)
    """
    return vecneg(p)

polyadd(*ps)

Return the sum.

\[ p_0 + p_1 + \cdots \]
Complexity

For two polynomials of degrees \(n\) & \(m\) there will be

  • \(\min\{n, m\}+1\) scalar additions (add).
See also
References
Source code in poly\standard\arithmetic.py 
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def polyadd(*ps):
    r"""Return the sum.

    $$
        p_0 + p_1 + \cdots
    $$

    Complexity
    ----------
    For two polynomials of degrees $n$ & $m$ there will be

    - $\min\{n, m\}+1$ scalar additions (`add`).

    See also
    --------
    - for constant or monomial summand: [`polyaddc`][poly.standard.polyaddc]
    - wraps: [`vector.vecadd`](https://goessl.github.io/vector/dense/#vector.dense.vector_space.vecadd)

    References
    ----------
    - `numpy` equivalent: [`numpy.polynomial.polynomial.polyadd`](https://numpy.org/doc/stable/reference/generated/numpy.polynomial.polynomial.polyadd.html)
    """
    return vecadd(*ps)

polyaddc(p, c, n=0)

Return the sum with a monomial.

\[ p + cx^n \]

More efficient than polyadd(p, polymono(n, c)).

Complexity

There will be

  • one scalar addition (add) if \(n\le\deg p\) or
  • one unary plus operations (pos) otherwise.
See also
Source code in poly\standard\arithmetic.py 
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def polyaddc(p, c, n=0):
    r"""Return the sum with a monomial.

    $$
        p + cx^n
    $$

    More efficient than `polyadd(p, polymono(n, c))`.

    Complexity
    ----------
    There will be

    - one scalar addition (`add`) if $n\le\deg p$ or
    - one unary plus operations (`pos`) otherwise.

    See also
    --------
    - for polynomial summand: [`polyadd`][poly.standard.polyadd]
    - wraps: [`vector.vecaddc`](https://goessl.github.io/vector/dense/#vector.dense.vector_space.vecaddc)
    """
    return vecaddc(p, c, i=n)

polysub(p, q)

Return the difference.

\[ p - q \]
Complexity

For two polynomials of degrees \(n\) & \(m\) there will be

  • \(\min\{n, m\}+1\) scalar subtractions (sub) &
  • \(\begin{cases}m-n&m\ge n\\0&m\le n\end{cases}\) negations (neg).
See also
References
Source code in poly\standard\arithmetic.py 
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def polysub(p, q):
    r"""Return the difference.

    $$
        p - q
    $$

    Complexity
    ----------
    For two polynomials of degrees $n$ & $m$ there will be

    - $\min\{n, m\}+1$ scalar subtractions (`sub`) &
    - $\begin{cases}m-n&m\ge n\\0&m\le n\end{cases}$ negations (`neg`).

    See also
    --------
    - for constant or monomial subtrahend: [`polysubc`][poly.standard.polysubc]
    - wraps: [`vector.vecsub`](https://goessl.github.io/vector/dense/#vector.dense.vector_space.vecsub)

    References
    ----------
    - `numpy` equivalent: [`numpy.polynomial.polynomial.polysub`](https://numpy.org/doc/stable/reference/generated/numpy.polynomial.polynomial.polysub.html)
    """
    return vecsub(p, q)

polysubc(p, c, n=0)

Return the difference with a monomial.

\[ p - cx^n \]

More efficient than polysub(p, polymono(n, c)).

Complexity

There will be

  • one scalar subtraction (sub) if \(n\le\deg p\) or
  • one scalar negation (neg) otherwise.
See also
Source code in poly\standard\arithmetic.py 
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def polysubc(p, c, n=0):
    r"""Return the difference with a monomial.

    $$
        p - cx^n
    $$

    More efficient than `polysub(p, polymono(n, c))`.

    Complexity
    ----------
    There will be

    - one scalar subtraction (`sub`) if $n\le\deg p$ or
    - one scalar negation (`neg`) otherwise.

    See also
    --------
    - for polynomial subtrahend: [`polysub`][poly.standard.polysub]
    - wraps: [`vector.vecsubc`](https://goessl.github.io/vector/dense/#vector.dense.vector_space.vecsubc)
    """
    return vecsubc(p, c, i=n)

polyscalarmul(p, a)

Return the product.

\[ pa \]
Complexity

For a polynomial of degree \(n\) there will be

  • \(n+1\) scalar multiplications (mul).
See also
Source code in poly\standard\arithmetic.py 
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def polyscalarmul(p, a):
    r"""Return the product.

    $$
        pa
    $$

    Complexity
    ----------
    For a polynomial of degree $n$ there will be

    - $n+1$ scalar multiplications (`mul`).

    See also
    --------
    - for polynomial factor: [`polymul`][poly.standard.polymul]
    - wraps: [`vector.vecmul`](https://goessl.github.io/vector/dense/#vector.dense.vector_space.vecmul)
    """
    return vecmul(p, a)

polyscalarrmul(a, p)

Return the product.

\[ ap \]
Complexity

For a polynomial of degree \(n\) there will be

  • \(n+1\) scalar multiplications (rmul).
See also
Source code in poly\standard\arithmetic.py 
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def polyscalarrmul(a, p):
    r"""Return the product.

    $$
        ap
    $$

    Complexity
    ----------
    For a polynomial of degree $n$ there will be

    - $n+1$ scalar multiplications (`rmul`).

    See also
    --------
    - for polynomial factor: [`polymul`][poly.standard.polymul]
    - wraps: [`vector.vecrmul`](https://goessl.github.io/vector/dense/#vector.dense.vector_space.vecrmul)
    """
    return vecrmul(a, p)

polyscalartruediv(p, a)

Return the true quotient.

\[ \frac{p}{a} \]
Complexity

For a polynomial of degree \(n\) there will be

  • \(n+1\) scalar true divisions (truediv).
See also
Source code in poly\standard\arithmetic.py 
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def polyscalartruediv(p, a):
    r"""Return the true quotient.

    $$
        \frac{p}{a}
    $$

    Complexity
    ----------
    For a polynomial of degree $n$ there will be

    - $n+1$ scalar true divisions (`truediv`).

    See also
    --------
    - wraps: [`vector.vectruediv`](https://goessl.github.io/vector/dense/#vector.dense.vector_space.vectruediv)
    """
    return vectruediv(p, a)

polyscalarfloordiv(p, a)

Return the floor quotient.

\[ \sum_k\left\lfloor\frac{a_k}{a}\right\rfloor x^k \]
Complexity

For a polynomial of degree \(n\) there will be

  • \(n+1\) scalar floor divisions (floordiv).
See also
Source code in poly\standard\arithmetic.py 
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def polyscalarfloordiv(p, a):
    r"""Return the floor quotient.

    $$
        \sum_k\left\lfloor\frac{a_k}{a}\right\rfloor x^k
    $$

    Complexity
    ----------
    For a polynomial of degree $n$ there will be

    - $n+1$ scalar floor divisions (`floordiv`).

    See also
    --------
    - included in: [`polyscalardivmod`][poly.standard.polyscalardivmod]
    - wraps: [`vector.vecfloordiv`](https://goessl.github.io/vector/dense/#vector.dense.vector_space.vecfloordiv)
    """
    return vecfloordiv(p, a)

polyscalarmod(p, a)

Return the remainder.

\[ \sum_k\left(a_k \mod a \right)x^k \]
Complexity

For a polynomial of degree \(n\) there will be

  • \(n+1\) scalar modulos (mod).
See also
Source code in poly\standard\arithmetic.py 
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def polyscalarmod(p, a):
    r"""Return the remainder.

    $$
        \sum_k\left(a_k \mod a \right)x^k
    $$

    Complexity
    ----------
    For a polynomial of degree $n$ there will be

    - $n+1$ scalar modulos (`mod`).

    See also
    --------
    - included in: [`polyscalardivmod`][poly.standard.polyscalardivmod]
    - wraps: [`vector.vecmod`](https://goessl.github.io/vector/dense/#vector.dense.vector_space.vecmod)
    """
    return vecmod(p, a)

polyscalardivmod(p, a)

Return the floor quotient and remainder.

\[ \sum_k\left\lfloor\frac{a_k}{a}\right\rfloor x^k, \ \sum_k\left(a_k \mod a \right)x^k \]
Complexity

For a polynomial of degree \(n\) there will be

  • \(n+1\) scalar divmods (divmod).
See also
Source code in poly\standard\arithmetic.py 
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def polyscalardivmod(p, a):
    r"""Return the floor quotient and remainder.

    $$
        \sum_k\left\lfloor\frac{a_k}{a}\right\rfloor x^k, \ \sum_k\left(a_k \mod a \right)x^k
    $$

    Complexity
    ----------
    For a polynomial of degree $n$ there will be

    - $n+1$ scalar divmods (`divmod`).

    See also
    --------
    - combines: [`polyscalarfloordiv`][poly.standard.polyscalarfloordiv] & [`polyscalarmod`][poly.standard.polyscalarmod]
    - wraps: [`vector.vecdivmod`](https://goessl.github.io/vector/dense/#vector.dense.vector_space.vecdivmod)
    """
    return vecdivmod(p, a)

polymul(*ps, method='naive', one=1)

Return the product.

\[ p_0 p_1 \cdots \]

Available methods are

See also
References
Source code in poly\standard\arithmetic.py 
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def polymul(*ps, method='naive', one=1):
    r"""Return the product.

    $$
        p_0 p_1 \cdots
    $$

    Available methods are

    - [`naive`][poly.standard.polymul_naive] &
    - [`karatsuba`][poly.standard.polymul_karatsuba].

    See also
    --------
    - implementations: [`polymul_naive`][poly.standard.arithmetic.polymul_naive],
    [`polymul_karatsuba`][poly.standard.arithmetic.polymul_karatsuba]
    - for scalar factor: [`polyscalarmul`][poly.standard.arithmetic.polyscalarmul]
    - for monomial factor: [`polymulx`][poly.standard.arithmetic.polymulx]

    References
    ----------
    - `numpy` equivalent: [`numpy.polynomial.polynomial.polymul`](https://numpy.org/doc/stable/reference/generated/numpy.polynomial.polynomial.polymul.html)
    """
    ps = tuple(map(tuple, ps))
    match method:
        case 'naive':
            return reduce_default(polymul_naive, ps, default=(one,))
        case 'karatsuba':
            return reduce_default(polymul_karatsuba, ps, default=(one,))
        case _:
            raise ValueError('Invalid method')

polymul_naive(p, q)

Return the product.

\[ pq \]

Uses naive multiplication and summation.

q must be a sequence.

Complexity

For two polynomials of degrees \(n\) & \(m\) there will be

  • \(\begin{cases}nm&n\ge1\land m\ge1\\0&n\le0\lor m\le0\end{cases}\) scalar additions (add) &
  • \((n+1)(m+1)\) scalar mutliplications (mul).
See also
Source code in poly\standard\arithmetic.py 
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def polymul_naive(p, q):
    r"""Return the product.

    $$
        pq
    $$

    Uses naive multiplication and summation.

    `q` must be a sequence.

    Complexity
    ----------
    For two polynomials of degrees $n$ & $m$ there will be

    - $\begin{cases}nm&n\ge1\land m\ge1\\0&n\le0\lor m\le0\end{cases}$ scalar additions (`add`) &
    - $(n+1)(m+1)$ scalar mutliplications (`mul`).

    See also
    --------
    - for any implementation: [`polymul`][poly.standard.arithmetic.polymul]
    - other implementations:
    [`polymul_karatsuba`][poly.standard.arithmetic.polymul_karatsuba]
    """
    if not q:
        return () #polyzero

    r, sentinel = [], object()
    for i, pi in enumerate(p):
        r.extend([sentinel] * (i+len(q) - len(r)))
        for j, qj in enumerate(q):
            if r[i+j] is sentinel:
                r[i+j] = pi * qj
            else:
                r[i+j] += pi * qj
    return tuple(r)

polymul_karatsuba(p, q)

Return the product.

\[ p q \]

Uses the Karatsuba algorithm.

Both arguments must be sequences.

TODO: complexity

See also
References
Source code in poly\standard\arithmetic.py 
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def polymul_karatsuba(p, q):
    """Return the product.

    $$
        p q
    $$

    Uses the Karatsuba algorithm.

    Both arguments must be sequences.

    TODO: complexity

    See also
    --------
    - for any implementation: [`polymul`][poly.standard.arithmetic.polymul]
    - other implementations:
    [`polymul_naive`][poly.standard.arithmetic.polymul_naive]

    References
    ----------
    - [Wikipedia - Karatsuba algorithm](https://en.wikipedia.org/wiki/Karatsuba_algorithm)
    """
    if not p or not q:
        return ()

    p, q = sorted((q, p), key=len) #p shorter, q longer
    ql, qu = q[:len(p)], q[len(p):]
    rl, ru = _polymul_karatsuba(p, ql), polymul_naive(p, qu)
    #return vecadd(rl, (0,)*len(p)+ru)
    return rl[:len(p)] + polyadd(rl[len(p):], ru)

polymulx(p, n=1, zero=0)

Return the product with a monomial.

\[ px^n \]

More efficient than polymul(p, polymonom(n)).

Complexity

There are no scalar arithmetic operations.

See also
References
Source code in poly\standard\arithmetic.py 
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def polymulx(p, n=1, zero=0):
    """Return the product with a monomial.

    $$
        px^n
    $$

    More efficient than `polymul(p, polymonom(n))`.

    Complexity
    ----------
    There are no scalar arithmetic operations.

    See also
    --------
    - for polynomial factor: [`polymul`][poly.standard.arithmetic.polymul]
    - wraps: [`vector.vecrshift`](https://goessl.github.io/vector/dense/#vector.dense.utility.vecrshift)

    References
    ----------
    - `numpy` equivalent: [`numpy.polynomial.polynomial.polymulx`](https://numpy.org/doc/stable/reference/generated/numpy.polynomial.polynomial.polymulx.html)
    """
    return vecrshift(p, n, zero=zero)

polypow(p, n, method='naive')

Return the polynomial p raised to the nonnegative n-th power.

\[ p^n \]

p must be a sequence.

Available methods are

TODO: mod parameter

See also
References
Source code in poly\standard\arithmetic.py 
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def polypow(p, n, method='naive'):
    """Return the polynomial `p` raised to the nonnegative `n`-th power.

    $$
        p^n
    $$

    `p` must be a sequence.

    Available methods are 

    - [`naive`][poly.standard.polypow_naive] &
    - [`binary`][poly.standard.polypow_binary].

    TODO: mod parameter

    See also
    --------
    - implementations: [`polypow_naive`][poly.standard.arithmetic.polypow_naive],
    [`polypow_binary`][poly.standard.polypow_binary]
    - for sequence of powers: [`polypows`][poly.standard.arithmetic.polypows]

    References
    ----------
    - `numpy` equivalent: [`numpy.polynomial.polynomial.polypow`](https://numpy.org/doc/stable/reference/generated/numpy.polynomial.polynomial.polypow.html)
    """
    match method:
        case 'naive':
            return polypow_naive(p, n)
        case 'binary':
            return polypow_binary(p, n)
        case _:
            raise ValueError('Invalid method')

polypow_naive(p, n, one=1)

Return the polynomial p raised to the nonnegative n-th power.

\[ p^n \]

Uses repeated multiplication.

p must be a sequence.

Complexity

For a polynomial of degree \(n\) and exponent \(k\) there will be

  • \(\begin{cases}\frac{n^2k(k-1)}{2}&n\ge0\\0&n\leq0\end{cases}\) scalar additions (add) &
  • \(\begin{cases}\frac{(nk+2)(n+1)(k-1)}{2}&k>0\\0&k=0\end{cases}\) scalar multiplications (mul).
See also
Source code in poly\standard\arithmetic.py 
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def polypow_naive(p, n, one=1):
    r"""Return the polynomial `p` raised to the nonnegative `n`-th power.

    $$
        p^n
    $$

    Uses repeated multiplication.

    `p` must be a sequence.

    Complexity
    ----------
    For a polynomial of degree $n$ and exponent $k$ there will be

    - $\begin{cases}\frac{n^2k(k-1)}{2}&n\ge0\\0&n\leq0\end{cases}$ scalar additions (`add`) &
    - $\begin{cases}\frac{(nk+2)(n+1)(k-1)}{2}&k>0\\0&k=0\end{cases}$ scalar multiplications (`mul`).

    See also
    --------
    - for any implementation: [`polypow`][poly.standard.arithmetic.polypow]
    - other implementations:
    [`polypow_binary`][poly.standard.arithmetic.polypow_binary]
    - uses: [`polypows`][poly.standard.arithmetic.polypows]
    """
    return next(polypows(p, start=n, one=one))

polypow_binary(p, n, one=1)

Return the polynomial p raised to the nonnegative n-th power.

\[ p^n \]

Uses exponentiation by squaring.

p must be a sequence.

Complexity

For a polynomial of degree \(n\) and exponent \(k\) let

\[ \begin{aligned} k &= \sum_jb_j2^j \\ L &= \operatorname{bitlength}(k) \\ w &= \operatorname{popcount}(k) \end{aligned} \]

where \(j_0\) is the least significant \(1\)-bit position of \(k\), \(\operatorname{bitlength}(k)\) is the number of bits of the binary representation of \(k\) and \(\operatorname{popcount}(k)\) is the number of \(1\)-bits.

Further define

\[ \begin{aligned} A(k) &= \frac{4^L-1}{3}+\sum_{\substack{i<j \\ b_i=b_j=1}}2^{i+j} \\ B(k) &= 2(2^L-1)+k-2^{j_0}+\sum_{j\mid b_j=1}2^j\operatorname{popcount}(k\gg(j+1)) \\ C(k) &= L+w-1. \end{aligned} \]
L = k.bit_length()
w = k.bit_count()
C = L + w - 1
B = 2*(2**L-1) + k-2**((k&-k).bit_length()-1) + sum(2**j * (k>>(j+1)).bit_count() for j in range(L) if ((k>>j)&0x01)==0x01)
A = (4**L-1)//3 + sum(2**(i+j) for j in range(L) for i in range(j) if ((k>>j)&0x01)==((k>>i)&0x01)==0x01)

Then there will be

  • \(A(k)n^2\) scalar additions (add) &
  • \(A(k)n^2+B(k)n+C(k)\) scalar multiplications (mul).
See also
References
Source code in poly\standard\arithmetic.py 
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def polypow_binary(p, n, one=1):
    r"""Return the polynomial `p` raised to the nonnegative `n`-th power.

    $$
        p^n
    $$

    Uses exponentiation by squaring.

    `p` must be a sequence.

    Complexity
    ----------
    For a polynomial of degree $n$ and exponent $k$ let

    $$
        \begin{aligned}
            k &= \sum_jb_j2^j \\
            L &= \operatorname{bitlength}(k) \\
            w &= \operatorname{popcount}(k)
        \end{aligned}
    $$

    where $j_0$ is the least significant $1$-bit position of $k$,
    $\operatorname{bitlength}(k)$ is the number of bits of the binary
    representation of $k$ and $\operatorname{popcount}(k)$ is the number of
    $1$-bits.

    Further define

    $$
        \begin{aligned}
            A(k) &= \frac{4^L-1}{3}+\sum_{\substack{i<j \\ b_i=b_j=1}}2^{i+j} \\
            B(k) &= 2(2^L-1)+k-2^{j_0}+\sum_{j\mid b_j=1}2^j\operatorname{popcount}(k\gg(j+1)) \\
            C(k) &= L+w-1.
        \end{aligned}
    $$

    ```python
    L = k.bit_length()
    w = k.bit_count()
    C = L + w - 1
    B = 2*(2**L-1) + k-2**((k&-k).bit_length()-1) + sum(2**j * (k>>(j+1)).bit_count() for j in range(L) if ((k>>j)&0x01)==0x01)
    A = (4**L-1)//3 + sum(2**(i+j) for j in range(L) for i in range(j) if ((k>>j)&0x01)==((k>>i)&0x01)==0x01)
    ```

    Then there will be

    - $A(k)n^2$ scalar additions (`add`) &
    - $A(k)n^2+B(k)n+C(k)$ scalar multiplications (`mul`).

    See also
    --------
    - for any implementation: [`polypow`][poly.standard.arithmetic.polypow]
    - other implementations:
    [`polypow_naive`][poly.standard.arithmetic.polypow_naive]

    References
    ----------
    - [Wikipedia - Exponentiation by squaring](https://en.wikipedia.org/wiki/Exponentiation_by_squaring)
    - Sequence $C(k)$: [A056791](https://oeis.org/A056791)
    """
    if n == 0:
        return (one,)
    r = None
    while n:
        if n % 2 == 1:
            r = polymul_naive(r, p) if r is not None else p
        p = polymul_naive(p, p)
        n //= 2
    return r

polypows(p, start=0, one=1)

Yield the powers.

\[ (p^n)_{n\in\mathbb{N}_0} = (1, p, p^2, \dots) \]

Uses iterative multiplication to calculate powers consecutively.

p must be a sequence.

Notes

Was first .evaluation.polycoms in analogy to polyvals but then the submodules arithmetic & evaluation would not be separable (.evaluation.polycoms uses .arithmetic.polymul and .arithmetic.polypow uses .evaluation.polycoms).

See also
Source code in poly\standard\arithmetic.py 
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def polypows(p, start=0, one=1):
    r"""Yield the powers.

    $$
        (p^n)_{n\in\mathbb{N}_0} = (1, p, p^2, \dots)
    $$

    Uses iterative multiplication to calculate powers consecutively.

    `p` must be a sequence.

    Notes
    -----
    Was first `.evaluation.polycoms` in analogy to
    [`polyvals`][poly.standard.polyvals] but then the submodules `arithmetic` &
    `evaluation` would not be separable (`.evaluation.polycoms` uses
    `.arithmetic.polymul` and `.arithmetic.polypow` uses
    `.evaluation.polycoms`).

    See also
    --------
    - used by: [`polypow_naive`][poly.standard.arithmetic.polypow_naive], [`polycom_iterative`][poly.standard.polycom_iterative]
    - for scalar arguments: [`polyvals`][poly.standard.polyvals]
    """
    if start <= 0:
        yield (one,)
    yield (q := reduce_default(polymul_naive, repeat(p, start), initial=MISSING, default=p))
    while True:
        q = polymul_naive(q, p)
        yield q

calculus

polyder(p, k=1)

Return the derivative.

\[ p^{(k)} \]
Complexity

For the \(k\)-th derivative of a polynomial of degree \(n\) there will be

  • \(\begin{cases}n-k+1&k\le n\\0&k>n\end{cases}\) scalar multiplications with integers (rmul).
Notes

For monomials:

\[ \begin{aligned} \frac{d}{dx}x^n &= nx^{n-1} \\ \frac{d^2}{dx^2}x^n &= n(n-1)x^{n-2} \\ &\vdots \\ \frac{d^k}{dx^k}x^n &= n(n-1)\cdots(n-k+1)x^{n-k} \\ &= (n)_kx^{n-k} \\ &= \frac{n!}{(n-k)!}x^{n-k} \\ &= {}_nP_kx^{n-k} \end{aligned} \]

And for polynomials:

\[ \begin{aligned} \frac{d^k}{dx^k}p(x) &= \frac{d^k}{dx^k}\sum_{l=0}^np_lx^l \\ &= \sum_{l=k}^np_l\,{}_lP_kx^{l-k} \\ &= \sum_{l=0}^{n-k}p_{l+k}\,{}_{l+k}P_kx^l \end{aligned} \]

Where \((n)_k\) is the Falling factorial and \({}_nP_k\) is the number of k-permutations of n with \((n)_k=\frac{n!}{(n-k)!}={}_nP_k\) (falling factorials are used because their definition appears in the derivation; permutations are used because a fast implementation is provided by math.perm).

References
Source code in poly\standard\calculus.py 
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def polyder(p, k=1):
    r"""Return the derivative.

    $$
        p^{(k)}
    $$

    Complexity
    ----------
    For the $k$-th derivative of a polynomial of degree $n$ there will be

    - $\begin{cases}n-k+1&k\le n\\0&k>n\end{cases}$ scalar multiplications with integers (`rmul`).

    Notes
    -----
    For monomials:

    $$
        \begin{aligned}
            \frac{d}{dx}x^n &= nx^{n-1} \\
            \frac{d^2}{dx^2}x^n &= n(n-1)x^{n-2} \\
            &\vdots \\
            \frac{d^k}{dx^k}x^n &= n(n-1)\cdots(n-k+1)x^{n-k} \\
            &= (n)_kx^{n-k} \\
            &= \frac{n!}{(n-k)!}x^{n-k} \\
            &= {}_nP_kx^{n-k}
        \end{aligned}
    $$

    And for polynomials:

    $$
        \begin{aligned}
            \frac{d^k}{dx^k}p(x) &= \frac{d^k}{dx^k}\sum_{l=0}^np_lx^l \\
            &= \sum_{l=k}^np_l\,{}_lP_kx^{l-k} \\
            &= \sum_{l=0}^{n-k}p_{l+k}\,{}_{l+k}P_kx^l
        \end{aligned}
    $$

    Where $(n)_k$ is the [Falling factorial](https://en.wikipedia.org/wiki/Falling_and_rising_factorials#Properties)
    and ${}_nP_k$ is the [number of k-permutations of n](https://en.wikipedia.org/wiki/Permutation#k-permutations_of_n)
    with $(n)_k=\frac{n!}{(n-k)!}={}_nP_k$ (falling factorials are
    used because their definition appears in the derivation; permutations are
    used because a fast implementation is provided by `math.perm`).

    References
    ----------
    - [more_itertools.polynomial_derivative](https://more-itertools.readthedocs.io/en/stable/api.html#more_itertools.polynomial_derivative)
    - `numpy` equivalent: [`numpy.polynomial.polynomial.polyder`](https://numpy.org/doc/stable/reference/generated/numpy.polynomial.polynomial.polyder.html)
    - [Wikipedia - Potenzregel - Höhere Ableitung einer Potenzfunktion mit natürlichem Exponenten](https://de.wikipedia.org/wiki/Potenzregel#H%C3%B6here_Ableitung_einer_Potenzfunktion_mit_nat%C3%BCrlichem_Exponenten)
    """
    return vechadamard((perm(l+k, k) for l in count()), islice(p, k, None))

polyantider(p, c=0, b=0)

Return the antiderivative.

\[ \int_b^xp(y)\,\mathrm{d}y+c \]

TODO: Higher antiderivatives, complexity

Notes

Let

\[ I_{b, c}[f](x) = \int_b^xf(y)\,\mathrm{d}y+c. \]

Then we have for monomials:

\[ \begin{aligned} I_{b,c}[\cdot^n](x) &= \int_b^xy^n\,\mathrm{d}y+c \\ &= \left.\frac{y^{n+1}}{n+1}\right|_{y=b}^x+c \\ &= \frac{x^{n+1}}{n+1}-\frac{b^{n+1}}{n+1}+c \end{aligned} \]

For polynomials:

\[ \begin{aligned} I_{b,c}[p](x) &= I_{b,c}\left[\sum_ka_k\cdot^k\right](x) \\ &= \int_b^x\sum_ka_ky^k\,\mathrm{d}y+c \\ &= \sum_ka_k\left(\frac{x^{k+1}}{k+1}-\frac{b^{k+1}}{k+1}\right)+c \\ &= \sum_ka_{k-1}\left(\frac{x^k}{k}-\frac{b^k}{k}\right)+c \\ &= \sum_k\frac{a_{k-1}}{k}x^k-\sum_k\frac{a_{k-1}}{k}b^k+c \end{aligned} \]
Notes

Integration is called antiderivative (antider) instead of integrate (int) to avoid keyword collisions.

References
Source code in poly\standard\calculus.py 
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def polyantider(p, c=0, b=0):
    r"""Return the antiderivative.

    $$
        \int_b^xp(y)\,\mathrm{d}y+c
    $$

    TODO: Higher antiderivatives, complexity

    Notes
    -----
    Let

    $$
        I_{b, c}[f](x) = \int_b^xf(y)\,\mathrm{d}y+c.
    $$

    Then we have for monomials:

    $$
        \begin{aligned}
            I_{b,c}[\cdot^n](x) &= \int_b^xy^n\,\mathrm{d}y+c \\
            &= \left.\frac{y^{n+1}}{n+1}\right|_{y=b}^x+c \\
            &= \frac{x^{n+1}}{n+1}-\frac{b^{n+1}}{n+1}+c
        \end{aligned}
    $$

    For polynomials:

    $$
        \begin{aligned}
            I_{b,c}[p](x) &= I_{b,c}\left[\sum_ka_k\cdot^k\right](x) \\
            &= \int_b^x\sum_ka_ky^k\,\mathrm{d}y+c \\
            &= \sum_ka_k\left(\frac{x^{k+1}}{k+1}-\frac{b^{k+1}}{k+1}\right)+c \\
            &= \sum_ka_{k-1}\left(\frac{x^k}{k}-\frac{b^k}{k}\right)+c \\
            &= \sum_k\frac{a_{k-1}}{k}x^k-\sum_k\frac{a_{k-1}}{k}b^k+c
        \end{aligned}
    $$

    Notes
    -----
    Integration is called antiderivative (`antider`)
    instead of integrate (`int`) to avoid keyword collisions.

    References
    ----------
    - `numpy` equivalent: [`numpy.polynomial.polynomial.polyint`](https://numpy.org/doc/stable/reference/generated/numpy.polynomial.polynomial.polyint.html)
    """
    P = vechadamardtruediv(p, count(1))
    return vecrshift(P, 1, zero=c-b*polyval_iterative(P, b) if b else c)

conversion

polysympify(p, x=x)

Return the coefficient iterable p as a sympy.Poly.

Source code in poly\standard\conversion.py 
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def polysympify(p, x=x):
    """Return the coefficient iterable `p` as a [`sympy.Poly`](https://docs.sympy.org/latest/modules/polys/reference.html#sympy.polys.polytools.Poly)."""
    return sp.Poly.from_list(tuple(reversed(tuple(p))), x)

polyunsympify(p)

Return sympy.Poly(p) as a coefficient tuple.

Source code in poly\standard\conversion.py 
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def polyunsympify(p):
    """Return [`sympy.Poly(p)`](https://docs.sympy.org/latest/modules/polys/reference.html#sympy.polys.polytools.Poly) as a coefficient `tuple`."""
    return tuple(reversed(p.all_coeffs()))