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Dense

Dense, immutable vector operations.

>>> from vector import vecadd
>>> a = (5, 6, 7)
>>> b = [2]
>>> c = range(4)
>>> vecadd(a, b, c)
(7, 7, 9, 3)

Prefixed by vec... (vector).

All functions accept vectors as single exhaustible iterables.

They return vectors as tuples.

The functions are type-independent. However, the coefficients used must support necessary scalar operations. For instance, for vector addition, coefficients must be addable.

For complete type safety a zero argument is available. Default is int(0).

Docstring conventions

Summary

Math notation (vector notation if possible, index notation, domain & codomain)

More information ("More efficient than ...").

Complexity

For a vector of length \(n\) there will be - \(x\) scalar additions (add), ...

Notes

Design choices

See also

Similar functions

References

Wikipedia, numpy, ...

creation

veczero = ()

Zero vector.

\[ \vec{0} \qquad \mathbb{K}^0 \]

An empty tuple: ().

vecbasis(i, c=1, zero=0)

Return a basis vector.

\[ c\vec{e}_i \qquad \mathbb{K}^{i+1} \]

Returns a tuple with i many zeros followed by c.

See also
Source code in vector\dense\creation.py 
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def vecbasis(i, c=1, zero=0):
    r"""Return a basis vector.

    $$
        c\vec{e}_i \qquad \mathbb{K}^{i+1}
    $$

    Returns a tuple with `i` many `zero`s followed by `c`.

    See also
    --------
    - for all basis vectors: [`vecbases`][vector.dense.creation.vecbases]
    """
    return (zero,)*i + (c,)

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

Yield all basis vectors.

\[ \left(\vec{e}_n\right)_{n\in\mathbb{N}_{\geq\text{start}}} = \left(\vec{e}_\text{start}, \vec{e}_{\text{start}+1}, \dots \right) \]
See also
Source code in vector\dense\creation.py 
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def vecbases(start=0, c=1, zero=0):
    r"""Yield all basis vectors.

    $$
        \left(\vec{e}_n\right)_{n\in\mathbb{N}_{\geq\text{start}}} = \left(\vec{e}_\text{start}, \vec{e}_{\text{start}+1}, \dots \right)
    $$

    See also
    --------
    - for single basis vector: [`vecbasis`][vector.dense.creation.vecbasis]
    """
    for i in count(start=start):
        yield vecbasis(i, c=c, zero=zero)

vecrand(n)

Return a random vector of uniform sampled float coefficients.

\[ \vec{v}\sim\mathcal{U}^n([0, 1[) \qquad \mathbb{K}^n \]

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

Notes

Naming like numpy.random, because seems more concise (not random & gauss as in the stdlib).

Source code in vector\dense\creation.py 
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def vecrand(n):
    r"""Return a random vector of uniform sampled `float` coefficients.

    $$
        \vec{v}\sim\mathcal{U}^n([0, 1[) \qquad \mathbb{K}^n
    $$

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

    Notes
    -----
    Naming like [`numpy.random`](https://numpy.org/doc/stable/reference/random/legacy.html),
    because seems more concise (not `random` & `gauss` as in the stdlib).
    """
    return tuple(veclrand(n))

vecrandn(n, normed=True, mu=0, sigma=1, weights=None)

Return a random vector of normal sampled float coefficients.

\[ \vec{v}\sim\mathcal{N}^n(\mu, \sigma) \qquad \mathbb{K}^n \]

The coefficients are sampled from a normal distribution.

Notes

Naming like numpy.random, because seems more concise (not random & gauss as in the stdlib).

Source code in vector\dense\creation.py 
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def vecrandn(n, normed=True, mu=0, sigma=1, weights=None):
    r"""Return a random vector of normal sampled `float` coefficients.

    $$
        \vec{v}\sim\mathcal{N}^n(\mu, \sigma) \qquad \mathbb{K}^n
    $$

    The coefficients are sampled from a normal distribution.

    Notes
    -----
    Naming like [`numpy.random`](https://numpy.org/doc/stable/reference/random/legacy.html),
    because seems more concise (not `random` & `gauss` as in the stdlib).
    """
    v = tuple(veclrandn(n, mu, sigma))
    return vectruediv(v, vecabs(v, weights)) if normed else v

utility

veclen(v)

Return the length (number of set coefficients).

Doesn't handle trailing zeros, use vectrim if needed.

Notes

For generators as they have no length, altough the vector is gone then.

Source code in vector\dense\utility.py 
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def veclen(v):
    """Return the length (number of set coefficients).

    Doesn't handle trailing zeros, use [`vectrim`][vector.dense.utility.vectrim]
    if needed.

    Notes
    -----
    For generators as they have no `len`gth, altough the vector is gone then.
    """
    return sum(1 for _ in v)

veceq(v, w)

Return whether two vectors are equal.

\[ \vec{v}\overset{?}{=}\vec{w} \qquad \mathbb{K}^m\times\mathbb{K}^n\to\mathbb{B} \]
Complexity

For two vectors of lengths \(n\) & \(m\) there will be at most

  • \(\min\{n, m\}\) scalar comparisons (eq) &
  • \(|n-m|\) scalar boolean evaluations (bool).
Source code in vector\dense\utility.py 
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def veceq(v, w):
    r"""Return whether two vectors are equal.

    $$
        \vec{v}\overset{?}{=}\vec{w} \qquad \mathbb{K}^m\times\mathbb{K}^n\to\mathbb{B}
    $$

    Complexity
    ----------
    For two vectors of lengths $n$ & $m$ there will be at most

    - $\min\{n, m\}$ scalar comparisons (`eq`) &
    - $|n-m|$ scalar boolean evaluations (`bool`).
    """
    return all(vecleq(v, w))

vectrim(v, tol=None)

Remove all trailing near zero (abs(v_i)<=tol) coefficients.

\[ \begin{pmatrix} v_0 \\ \vdots \\ v_m \end{pmatrix} \ \text{where} \ m=\max\{\, j\mid |v_j|>\text{tol}\,\}\cup\{-1\} \qquad \mathbb{K}^n\to\mathbb{K}^{\leq n} \]

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

Complexity

For a vector of length \(n\) there will be

  • \(n\) scalar absolute evaluations (abs) &
  • \(n\) scalar comparisons (gt).
Notes
  • Cutting of elements that are abs(v_i)<=tol instead of abs(v_i)<tol to allow cutting of elements that are exactly zero by trim(v, 0) instead of trim(v, sys.float_info.min).
Source code in vector\dense\utility.py 
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def vectrim(v, tol=None):
    r"""Remove all trailing near zero (`abs(v_i)<=tol`) coefficients.

    $$
        \begin{pmatrix}
            v_0 \\
            \vdots \\
            v_m
        \end{pmatrix} \ \text{where} \ m=\max\{\, j\mid |v_j|>\text{tol}\,\}\cup\{-1\} \qquad \mathbb{K}^n\to\mathbb{K}^{\leq n}
    $$

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

    Complexity
    ----------
    For a vector of length $n$ there will be

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

    Notes
    -----
    - Cutting of elements that are `abs(v_i)<=tol` instead of `abs(v_i)<tol` to
    allow cutting of elements that are exactly zero by `trim(v, 0)` instead
    of `trim(v, sys.float_info.min)`.
    """
    r, t = [], []
    for x in v:
        t.append(x)
        if (x if tol is None else abs(x)>tol):
            r.extend(t)
            t.clear()
    return tuple(r)

vecrshift(v, n, zero=0)

Shift coefficients up.

\[ (v_{i-n})_i \qquad \begin{pmatrix} 0 \\ \vdots \\ 0 \\ v_0 \\ v_1 \\ \vdots \end{pmatrix} \qquad \mathbb{K}^m\to\mathbb{K}^{m+n} \]
Source code in vector\dense\utility.py 
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def vecrshift(v, n, zero=0):
    r"""Shift coefficients up.

    $$
        (v_{i-n})_i \qquad \begin{pmatrix}
            0 \\
            \vdots \\
            0 \\
            v_0 \\
            v_1 \\
            \vdots
        \end{pmatrix} \qquad \mathbb{K}^m\to\mathbb{K}^{m+n}
    $$
    """
    return tuple(veclrshift(v, n, zero=zero))

veclshift(v, n)

Shift coefficients down.

\[ (v_{i+n})_i \qquad \begin{pmatrix} v_n \\ v_{n+1} \\ \vdots \end{pmatrix} \qquad \mathbb{K}^m\to\mathbb{K}^{\max\{m-n, 0\}} \]
Source code in vector\dense\utility.py 
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def veclshift(v, n):
    r"""Shift coefficients down.

    $$
        (v_{i+n})_i \qquad \begin{pmatrix}
            v_n \\
            v_{n+1} \\
            \vdots
        \end{pmatrix} \qquad \mathbb{K}^m\to\mathbb{K}^{\max\{m-n, 0\}}
    $$
    """
    return tuple(vecllshift(v, n))

hilbert_space

vecconj(v)

Return the complex conjugate.

\[ \vec{v}^* \qquad \mathbb{K}^n\to\mathbb{K}^n \]

Tries to call a method conjugate on each element. If not found, simply keeps the element as is.

Complexity

For a vector of length \(n\) there will be

  • \(n\) scalar conjugations (conjugate).
Source code in vector\dense\hilbert_space.py 
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def vecconj(v):
    r"""Return the complex conjugate.

    $$
        \vec{v}^* \qquad \mathbb{K}^n\to\mathbb{K}^n
    $$

    Tries to call a method `conjugate` on each element.
    If not found, simply keeps the element as is.

    Complexity
    ----------
    For a vector of length $n$ there will be

    - $n$ scalar conjugations (`conjugate`).
    """
    return tuple(veclconj(v))

vecabs(v, weights=None, conjugate=False, zero=0)

Return the Euclidean/\(\ell_{\mathbb{N}_0}^2\)-norm.

\[ ||\vec{v}||_{\ell_{\mathbb{N}_0}^2}=\sqrt{\sum_iv_i^{(*)}v_i\omega_i} \qquad \mathbb{K}^n\to\mathbb{K}_0^+ \]

Returns the square root of vecabsq.

Complexity

For a vector of length \(n\) there will be

  • \(n\) scalar conjugations (conjugate) (if selected),
  • \(n\)/\(2n\) scalar multiplications (mul) without/with weights,
  • \(\begin{cases}n-1&n\ge1\\0&n\le1\end{cases}\) scalar additions (add) &
  • one ^0.5 call.
See also
  • squared version without square root: vecabsq
Source code in vector\dense\hilbert_space.py 
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def vecabs(v, weights=None, conjugate=False, zero=0):
    r"""Return the Euclidean/$\ell_{\mathbb{N}_0}^2$-norm.

    $$
        ||\vec{v}||_{\ell_{\mathbb{N}_0}^2}=\sqrt{\sum_iv_i^{(*)}v_i\omega_i} \qquad \mathbb{K}^n\to\mathbb{K}_0^+
    $$

    Returns the square root of [`vecabsq`][vector.dense.vecabsq].

    Complexity
    ----------
    For a vector of length $n$ there will be

    - $n$ scalar conjugations (`conjugate`) (if selected),
    - $n$/$2n$ scalar multiplications (`mul`) without/with weights,
    - $\begin{cases}n-1&n\ge1\\0&n\le1\end{cases}$ scalar additions (`add`) &
    - one `^0.5` call.

    See also
    --------
    - squared version without square root: [`vecabsq`][vector.dense.hilbert_space.vecabsq]
    """
    #hypot(*v) doesn't work for complex
    #math.sqrt doesn't work for complex and cmath.sqrt always returns complex
    return vecabsq(v, weights=weights, conjugate=conjugate, zero=zero)**0.5

vecabsq(v, weights=None, conjugate=False, zero=0)

Return the sum of absolute squares.

\[ ||\vec{v}||_{\ell_{\mathbb{N}_0}^2}^2=\sum_iv_i^{(*)}v_i\omega_i \qquad \mathbb{K}^n\to\mathbb{K}_0^+ \]
Complexity

For a vector of length \(n\) there will be

  • \(n\) scalar conjugations (conjugate) (if selected),
  • \(\begin{cases}n-1&n\ge1\\0&n\le1\end{cases}\) scalar additions (add) &
  • \(n\)/\(2n\) scalar multiplications (mul) without/with weights.
Notes

Reasons why it exists:

  • Occurs in math.
  • Most importantly: type independent because it doesn't use sqrt.
References
Source code in vector\dense\hilbert_space.py 
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def vecabsq(v, weights=None, conjugate=False, zero=0):
    r"""Return the sum of absolute squares.

    $$
        ||\vec{v}||_{\ell_{\mathbb{N}_0}^2}^2=\sum_iv_i^{(*)}v_i\omega_i \qquad \mathbb{K}^n\to\mathbb{K}_0^+
    $$

    Complexity
    ----------
    For a vector of length $n$ there will be

    - $n$ scalar conjugations (`conjugate`) (if selected),
    - $\begin{cases}n-1&n\ge1\\0&n\le1\end{cases}$ scalar additions (`add`) &
    - $n$/$2n$ scalar multiplications (`mul`) without/with weights.

    Notes
    -----
    Reasons why it exists:

    - Occurs in math.
    - Most importantly: type independent because it doesn't use `sqrt`.

    References
    ----------
    - <https://docs.python.org/3/library/itertools.html#itertools-recipes>: `sum_of_squares`
    """
    vc, v = tee(v, 2)
    if conjugate:
        vc = veclconj(vc)

    if weights is None:
        return sumprod_default(vc, v, default=zero)
    else:
        return sumprod_default(map(mul, vc, v), weights, default=zero)

vecdot(v, w, weights=None, conjugate=False, zero=0)

Return the inner product.

\[ \left<\vec{v}\mid\vec{w}\right>_{\ell_{\mathbb{N}_0}^2}=\sum_iv_i^{(*)}w_i\omega_i \qquad \mathbb{K}^m\times\mathbb{K}^n\to\mathbb{K} \]
Complexity

For two vectors of lengths \(n\) & \(m\) there will be

  • \(\min\{n, m\}\) scalar conjugations (conjugate) (if selected),
  • \(\min\{n, m\}\)/\(2\min\{n, m\}\) scalar multiplications (mul without/with weights) &
  • \(\begin{cases}\min\{n, m\}-1&n\ge1\land m\ge1\\0&n\le1\lor m\le1\end{cases}\) scalar additions (add).
Source code in vector\dense\hilbert_space.py 
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def vecdot(v, w, weights=None, conjugate=False, zero=0):
    r"""Return the inner product.

    $$
        \left<\vec{v}\mid\vec{w}\right>_{\ell_{\mathbb{N}_0}^2}=\sum_iv_i^{(*)}w_i\omega_i \qquad \mathbb{K}^m\times\mathbb{K}^n\to\mathbb{K}
    $$

    Complexity
    ----------
    For two vectors of lengths $n$ & $m$ there will be

    - $\min\{n, m\}$ scalar conjugations (`conjugate`) (if selected),
    - $\min\{n, m\}$/$2\min\{n, m\}$ scalar multiplications (`mul` without/with weights) &
    - $\begin{cases}\min\{n, m\}-1&n\ge1\land m\ge1\\0&n\le1\lor m\le1\end{cases}$ scalar additions (`add`).
    """
    if conjugate:
        v = veclconj(v)

    if weights is None:
        return sumprod_default(v, w, default=zero)
    else:
        return sumprod_default(map(mul, v, w), weights, default=zero)

vector_space

vecpos(v)

Return the identity.

\[ +\vec{v} \qquad \mathbb{K}^n\to\mathbb{K}^n \]
Complexity

For a vector of length \(n\) there will be

  • \(n\) scalar unary plus operations (pos).
Source code in vector\dense\vector_space.py 
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def vecpos(v):
    r"""Return the identity.

    $$
        +\vec{v} \qquad \mathbb{K}^n\to\mathbb{K}^n
    $$

    Complexity
    ----------
    For a vector of length $n$ there will be

    - $n$ scalar unary plus operations (`pos`).
    """
    return tuple(veclpos(v))

vecneg(v)

Return the negation.

\[ -\vec{v} \qquad \mathbb{K}^n\to\mathbb{K}^n \]
Complexity

For a vector of length \(n\) there will be

  • \(n\) scalar negations (neg).
Source code in vector\dense\vector_space.py 
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def vecneg(v):
    r"""Return the negation.

    $$
        -\vec{v} \qquad \mathbb{K}^n\to\mathbb{K}^n
    $$

    Complexity
    ----------
    For a vector of length $n$ there will be

    - $n$ scalar negations (`neg`).
    """
    return tuple(veclneg(v))

vecadd(*vs)

Return the sum.

\[ \vec{v}_0+\vec{v}_1+\cdots \qquad \mathbb{K}^{n_0}\times\mathbb{K}^{n_1}\times\cdots\to\mathbb{K}^{\max_i n_i} \]
Complexity

For two vectors of lengths \(n\) & \(m\) there will be

  • \(\min\{n, m\}\) scalar additions (add).
See also
  • for sum on a single coefficient: vecaddc
Source code in vector\dense\vector_space.py 
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def vecadd(*vs):
    r"""Return the sum.

    $$
        \vec{v}_0+\vec{v}_1+\cdots \qquad \mathbb{K}^{n_0}\times\mathbb{K}^{n_1}\times\cdots\to\mathbb{K}^{\max_i n_i}
    $$

    Complexity
    ----------
    For two vectors of lengths $n$ & $m$ there will be

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

    See also
    --------
    - for sum on a single coefficient: [`vecaddc`][vector.dense.vector_space.vecaddc]
    """
    return tuple(vecladd(*vs))

vecaddc(v, c, i=0, zero=0)

Return the sum with a basis vector.

\[ \vec{v}+c\vec{e}_i \qquad \mathbb{K}^n\to\mathbb{K}^{\max\{n, i+1\}} \]

More efficient than vecadd(v, vecbasis(i, c)).

Complexity

There will be

  • one scalar addition (add) if \(i\le n\) or
  • one unary plus operations (pos) otherwise.
See also
  • for sum on more coefficients: vecadd
Source code in vector\dense\vector_space.py 
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def vecaddc(v, c, i=0, zero=0):
    r"""Return the sum with a basis vector.

    $$
        \vec{v}+c\vec{e}_i \qquad \mathbb{K}^n\to\mathbb{K}^{\max\{n, i+1\}}
    $$

    More efficient than `vecadd(v, vecbasis(i, c))`.

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

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

    See also
    --------
    - for sum on more coefficients: [`vecadd`][vector.dense.vector_space.vecadd]
    """
    return tuple(vecladdc(v, c, i=i, zero=zero))

vecsub(v, w)

Return the difference.

\[ \vec{v}-\vec{w} \qquad \mathbb{K}^m\times\mathbb{K}^n\to\mathbb{K}^{\max\{m, n\}} \]
Complexity

For two vectors of lengths \(n\) & \(m\) there will be

  • \(\min\{n, m\}\) scalar subtractions (sub) &
  • \(\begin{cases}m-n&m\ge n\\0&m\le n\end{cases}\) negations (neg).
See also
  • for difference on a single coefficient: vecsubc
Source code in vector\dense\vector_space.py 
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def vecsub(v, w):
    r"""Return the difference.

    $$
        \vec{v}-\vec{w} \qquad \mathbb{K}^m\times\mathbb{K}^n\to\mathbb{K}^{\max\{m, n\}}
    $$

    Complexity
    ----------
    For two vectors of lengths $n$ & $m$ there will be

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

    See also
    --------
    - for difference on a single coefficient: [`vecsubc`][vector.dense.vector_space.vecsubc]
    """
    return tuple(veclsub(v, w))

vecsubc(v, c, i=0, zero=0)

Return the difference with a basis vector.

\[ \vec{v}-c\vec{e}_i \qquad \mathbb{K}^n\to\mathbb{K}^{\max\{n, i+1\}} \]

More efficient than vecsub(v, vecbasis(i, c)).

Complexity

There will be

  • one scalar subtraction (sub) if \(i\le n\) or
  • one scalar negation (neg) otherwise.
See also
  • for difference on more coefficients: vecsub
Source code in vector\dense\vector_space.py 
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def vecsubc(v, c, i=0, zero=0):
    r"""Return the difference with a basis vector.

    $$
        \vec{v}-c\vec{e}_i \qquad \mathbb{K}^n\to\mathbb{K}^{\max\{n, i+1\}}
    $$

    More efficient than `vecsub(v, vecbasis(i, c))`.

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

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

    See also
    --------
    - for difference on more coefficients: [`vecsub`][vector.dense.vector_space.vecsub]
    """
    return tuple(veclsubc(v, c, i=i, zero=zero))

vecmul(v, a)

Return the product.

\[ \vec{v}a \qquad \mathbb{K}\times\mathbb{K}^n\to\mathbb{K}^n \]
Complexity

For a vector of length \(n\) there will be

  • \(n\) scalar multiplications (rmul).
Source code in vector\dense\vector_space.py 
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def vecmul(v, a):
    r"""Return the product.

    $$
        \vec{v}a \qquad \mathbb{K}\times\mathbb{K}^n\to\mathbb{K}^n
    $$

    Complexity
    ----------
    For a vector of length $n$ there will be

    - $n$ scalar multiplications (`rmul`).
    """
    return tuple(veclmul(v, a))

vecrmul(a, v)

Return the product.

\[ a\vec{v} \qquad \mathbb{K}\times\mathbb{K}^n\to\mathbb{K}^n \]
Complexity

For a vector of length \(n\) there will be

  • \(n\) scalar multiplications (rmul).
Source code in vector\dense\vector_space.py 
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def vecrmul(a, v):
    r"""Return the product.

    $$
        a\vec{v} \qquad \mathbb{K}\times\mathbb{K}^n\to\mathbb{K}^n
    $$

    Complexity
    ----------
    For a vector of length $n$ there will be

    - $n$ scalar multiplications (`rmul`).
    """
    return tuple(veclrmul(a, v))

vectruediv(v, a)

Return the true quotient.

\[ \frac{\vec{v}}{a} \qquad \mathbb{K}^n\times\mathbb{K}\to\mathbb{K}^n \]
Complexity

For a vector of length \(n\) there will be

  • \(n\) scalar true divisions (truediv).
Notes

Why called truediv instead of div?

  • div would be more appropriate for an absolute clean mathematical implementation, that doesn't care about the language used. But the package might be used for pure integers/integer arithmetic, so both, truediv and floordiv operations have to be provided, and none should be privileged over the other by getting the universal div name.
  • truediv/floordiv is unambiguous, like Python operators.
Source code in vector\dense\vector_space.py 
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def vectruediv(v, a):
    r"""Return the true quotient.

    $$
        \frac{\vec{v}}{a} \qquad \mathbb{K}^n\times\mathbb{K}\to\mathbb{K}^n
    $$

    Complexity
    ----------
    For a vector of length $n$ there will be

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

    Notes
    -----
    Why called `truediv` instead of `div`?

    - `div` would be more appropriate for an absolute clean mathematical
    implementation, that doesn't care about the language used. But the package
    might be used for pure integers/integer arithmetic, so both, `truediv`
    and `floordiv` operations have to be provided, and none should be
    privileged over the other by getting the universal `div` name.
    - `truediv`/`floordiv` is unambiguous, like Python `operator`s.
    """
    return tuple(vecltruediv(v, a))

vecfloordiv(v, a)

Return the floor quotient.

\[ \left\lfloor\frac{\vec{v}}{a}\right\rfloor \qquad \mathbb{K}^n\times\mathbb{K}\to\mathbb{K}^n \]
Complexity

For a vector of length \(n\) there will be

  • \(n\) scalar floor divisions (floordiv).
Source code in vector\dense\vector_space.py 
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def vecfloordiv(v, a):
    r"""Return the floor quotient.

    $$
        \left\lfloor\frac{\vec{v}}{a}\right\rfloor \qquad \mathbb{K}^n\times\mathbb{K}\to\mathbb{K}^n
    $$

    Complexity
    ----------
    For a vector of length $n$ there will be

    - $n$ scalar floor divisions (`floordiv`).
    """
    return tuple(veclfloordiv(v, a))

vecmod(v, a)

Return the remainder.

\[ \vec{v} \bmod a \qquad \mathbb{K}^n\times\mathbb{K}\to\mathbb{K}^n \]
Complexity

For a vector of length \(n\) there will be

  • \(n\) scalar modulos (mod).
Source code in vector\dense\vector_space.py 
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def vecmod(v, a):
    r"""Return the remainder.

    $$
        \vec{v} \bmod a \qquad \mathbb{K}^n\times\mathbb{K}\to\mathbb{K}^n
    $$

    Complexity
    ----------
    For a vector of length $n$ there will be

    - $n$ scalar modulos (`mod`).
    """
    return tuple(veclmod(v, a))

vecdivmod(v, a)

Return the floor quotient and remainder.

\[ \left\lfloor\frac{\vec{v}}{a}\right\rfloor, \ \left(\vec{v} \bmod a\right) \qquad \mathbb{K}^n\times\mathbb{K}\to\mathbb{K}^n\times\mathbb{K}^n \]
Complexity

For a vector of length \(n\) there will be

  • \(n\) scalar divmods (divmod).
Source code in vector\dense\vector_space.py 
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def vecdivmod(v, a):
    r"""Return the floor quotient and remainder.

    $$
        \left\lfloor\frac{\vec{v}}{a}\right\rfloor, \ \left(\vec{v} \bmod a\right) \qquad \mathbb{K}^n\times\mathbb{K}\to\mathbb{K}^n\times\mathbb{K}^n
    $$

    Complexity
    ----------
    For a vector of length $n$ there will be

    - $n$ scalar divmods (`divmod`).
    """
    q, r = [], []
    for vi in v:
        qi, ri = divmod(vi, a)
        q.append(qi)
        r.append(ri)
    return tuple(q), tuple(r)

elementwise

vechadamard(*vs)

Return the elementwise product.

\[ \left((\vec{v}_0)_i\cdot(\vec{v}_1)_i\cdot\cdots\right)_i \qquad \mathbb{K}^{n_0}\times\mathbb{K}^{n_1}\times\cdots\to\mathbb{K}^{\min_i n_i} \]
Complexity

For vectors of lengths \(n_1, n_2, \dots, n_N\) there will be

  • \(\begin{cases}(N-1)\min_in_i&N\ge1\land\min_in_i\ge1\\0&N\le1\lor\min_in_i=0\end{cases}\) scalar multiplications (mul).
Source code in vector\dense\elementwise.py 
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def vechadamard(*vs):
    r"""Return the elementwise product.

    $$
        \left((\vec{v}_0)_i\cdot(\vec{v}_1)_i\cdot\cdots\right)_i \qquad \mathbb{K}^{n_0}\times\mathbb{K}^{n_1}\times\cdots\to\mathbb{K}^{\min_i n_i}
    $$

    Complexity
    ----------
    For vectors of lengths $n_1, n_2, \dots, n_N$ there will be

    - $\begin{cases}(N-1)\min_in_i&N\ge1\land\min_in_i\ge1\\0&N\le1\lor\min_in_i=0\end{cases}$ scalar multiplications (`mul`).
    """
    return tuple(veclhadamard(*vs))

vechadamardtruediv(v, w)

Return the elementwise true quotient.

\[ \left(\frac{v_i}{w_i}\right)_i \qquad \mathbb{K}^m\times\mathbb{K}^n\to\mathbb{K}^m \]
Complexity

For two vectors of lengths \(n\) & \(m\) there will be

  • \(n\) scalar true divisions (truediv).
Source code in vector\dense\elementwise.py 
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def vechadamardtruediv(v, w):
    r"""Return the elementwise true quotient.

    $$
        \left(\frac{v_i}{w_i}\right)_i \qquad \mathbb{K}^m\times\mathbb{K}^n\to\mathbb{K}^m
    $$

    Complexity
    ----------
    For two vectors of lengths $n$ & $m$ there will be

    - $n$ scalar true divisions (`truediv`).
    """
    return tuple(veclhadamardtruediv(v, w))

vechadamardfloordiv(v, w)

Return the elementwise floor quotient.

\[ \left(\left\lfloor\frac{v_i}{w_i}\right\rfloor\right)_i \qquad \mathbb{K}^m\times\mathbb{K}^n\to\mathbb{K}^m \]
Complexity

For two vectors of lengths \(n\) & \(m\) there will be

  • \(n\) scalar floor divisions (floordiv).
Source code in vector\dense\elementwise.py 
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def vechadamardfloordiv(v, w):
    r"""Return the elementwise floor quotient.

    $$
        \left(\left\lfloor\frac{v_i}{w_i}\right\rfloor\right)_i \qquad \mathbb{K}^m\times\mathbb{K}^n\to\mathbb{K}^m
    $$

    Complexity
    ----------
    For two vectors of lengths $n$ & $m$ there will be

    - $n$ scalar floor divisions (`floordiv`).
    """
    return tuple(veclhadamardfloordiv(v, w))

vechadamardmod(v, w)

Return the elementwise remainder.

\[ \left(v_i \bmod w_i\right)_i \qquad \mathbb{K}^m\times\mathbb{K}^n\to\mathbb{K}^m \]
Complexity

For two vectors of lengths \(n\) & \(m\) there will be

  • \(n\) scalar modulos (mod).
Source code in vector\dense\elementwise.py 
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def vechadamardmod(v, w):
    r"""Return the elementwise remainder.

    $$
        \left(v_i \bmod w_i\right)_i \qquad \mathbb{K}^m\times\mathbb{K}^n\to\mathbb{K}^m
    $$

    Complexity
    ----------
    For two vectors of lengths $n$ & $m$ there will be

    - $n$ scalar modulos (`mod`).
    """
    return tuple(veclhadamardmod(v, w))

vechadamarddivmod(v, w)

Return the elementwise floor quotient and remainder.

\[ \left(\left\lfloor\frac{v_i}{w_i}\right\rfloor\right)_i, \ \left(v_i \bmod w_i\right)_i \qquad \mathbb{K}^n\times\mathbb{K}^m\to\mathbb{K}^n\times\mathbb{K}^n \]
Complexity

For two vectors of lengths \(n\) & \(m\) there will be

  • \(n\) scalar divmods (divmod).
Source code in vector\dense\elementwise.py 
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def vechadamarddivmod(v, w):
    r"""Return the elementwise floor quotient and remainder.

    $$
        \left(\left\lfloor\frac{v_i}{w_i}\right\rfloor\right)_i, \ \left(v_i \bmod w_i\right)_i \qquad \mathbb{K}^n\times\mathbb{K}^m\to\mathbb{K}^n\times\mathbb{K}^n
    $$

    Complexity
    ----------
    For two vectors of lengths $n$ & $m$ there will be

    - $n$ scalar divmods (`divmod`).
    """
    q, r = [], []
    for qi, ri in veclhadamarddivmod(v, w):
        q.append(qi)
        r.append(ri)
    return tuple(q), tuple(r)

vechadamardmin(*vs, key=None)

Return the elementwise minimum.

\[ \left(\min((\vec{v}_0)_i,(\vec{v}_1)_i,\cdots)\right)_i \qquad \mathbb{K}^{n_0}\times\mathbb{K}^{n_1}\times\cdots\to\mathbb{K}^{\max_i n_i} \]
Complexity

For two vectors of lengths \(n\) & \(m\) there will be

  • \(\min\{n, m\}\) comparisons (lt).
Source code in vector\dense\elementwise.py 
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def vechadamardmin(*vs, key=None):
    r"""Return the elementwise minimum.

    $$
        \left(\min((\vec{v}_0)_i,(\vec{v}_1)_i,\cdots)\right)_i \qquad \mathbb{K}^{n_0}\times\mathbb{K}^{n_1}\times\cdots\to\mathbb{K}^{\max_i n_i}
    $$

    Complexity
    ----------
    For two vectors of lengths $n$ & $m$ there will be

    - $\min\{n, m\}$ comparisons (`lt`).
    """
    return tuple(veclhadamardmin(*vs, key=key))

vechadamardmax(*vs, key=None)

Return the elementwise maximum.

\[ \left(\max((\vec{v}_0)_i,(\vec{v}_1)_i,\cdots)\right)_i \qquad \mathbb{K}^{n_0}\times\mathbb{K}^{n_1}\times\cdots\to\mathbb{K}^{\max_i n_i} \]
Complexity

For two vectors of lengths \(n\) & \(m\) there will be

  • \(\min\{n, m\}\) comparisons (gt).
Source code in vector\dense\elementwise.py 
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def vechadamardmax(*vs, key=None):
    r"""Return the elementwise maximum.

    $$
        \left(\max((\vec{v}_0)_i,(\vec{v}_1)_i,\cdots)\right)_i \qquad \mathbb{K}^{n_0}\times\mathbb{K}^{n_1}\times\cdots\to\mathbb{K}^{\max_i n_i}
    $$

    Complexity
    ----------
    For two vectors of lengths $n$ & $m$ there will be

    - $\min\{n, m\}$ comparisons (`gt`).
    """
    return tuple(veclhadamardmax(*vs, key=key))