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Lazy

Vector operations as lazy generators.

>>> from vector import veclbasis
>>> veclbasis(3)
<generator object veclbasis at 0x0123456789ABCDEF>
>>> tuple(veclbasis(3))
(0, 0, 0, 1)

Prefixed by vecl... (vector - lazy).

Functions are generators.

Lazy generator versions of dense.

Different behaviour:

Not implemented lazily as these are consumers:

creation

veclzero()

Zero vector.

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

An empty generator.

Source code in vector\lazy\creation.py 
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def veclzero():
    r"""Zero vector.

    $$
        \vec{0} \qquad \mathbb{K}^0
    $$

    An empty generator.
    """
    yield from ()

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

Return a basis vector.

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

Yields i zeros followed by c.

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

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

    Yields `i` zeros followed by `c`.

    See also
    --------
    - for all basis vectors: [`veclbases`][vector.lazy.creation.veclbases]
    """
    yield from chain(repeat(zero, i), (c,))

veclbases(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\lazy\creation.py 
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def veclbases(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: [`veclbasis`][vector.lazy.creation.veclbasis]
    """
    for i in count(start=start):
        yield veclbasis(i, c=c, zero=zero)

veclrand(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\lazy\creation.py 
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def veclrand(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).
    """
    yield from (random() for _ in range(n))

veclrandn(n, mu=0, sigma=1)

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.

Difference to vecrandn: The vector can't be normalised as it isn't materialised.

Notes

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

Source code in vector\lazy\creation.py 
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def veclrandn(n, mu=0, sigma=1):
    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.

    Difference to [`vecrandn`][vector.dense.vecrandn]:
    The vector can't be normalised as it isn't materialised.

    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).
    """
    yield from (gauss(mu, sigma) for _ in range(n))

utility

vecleq(v, w)

Return if two vectors are equal.

\[ \vec{v}=\vec{w} \qquad \mathbb{K}^m\times\mathbb{K}^n\to\mathbb{B} \]
Source code in vector\lazy\utility.py 
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def vecleq(v, w):
    r"""Return if two vectors are equal.

    $$
        \vec{v}=\vec{w} \qquad \mathbb{K}^m\times\mathbb{K}^n\to\mathbb{B}
    $$
    """
    sentinel = object()
    for vi, wi in zip_longest(v, w, fillvalue=sentinel):
        if wi is sentinel:
            yield not bool(vi)
        elif vi is sentinel:
            yield not bool(wi)
        else:
            yield vi == wi

vecltrim(v, tol=None)

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

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

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

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\lazy\utility.py 
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def vecltrim(v, tol=None):
    r"""Remove all trailing near zero (`abs(v_i)<=tol`) coefficients.

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

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

    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)`.
    """
    t = []
    for x in v:
        t.append(x)
        if (x if tol is None else abs(x)>tol):
            yield from t
            t.clear()

veclrshift(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\lazy\utility.py 
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def veclrshift(v, n, zero=0):
    r"""Shift coefficients up.

    $$
        (v_{i-n})_i \qquad \begin{pmatrix}
            0 \\
            \vdots \\
            0 \\
            v_0 \\
            v_1 \\
            \vdots
        \end{pmatrix} \qquad \mathbb{K}^m\to\mathbb{K}^{m+n}
    $$
    """
    yield from chain(repeat(zero, n), v)

vecllshift(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\lazy\utility.py 
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def vecllshift(v, n):
    r"""Shift coefficients down.

    $$
        (v_{i+n})_i \qquad \begin{pmatrix}
            v_n \\
            v_{n+1} \\
            \vdots
        \end{pmatrix} \qquad \mathbb{K}^m\to\mathbb{K}^{\max\{m-n, 0\}}
    $$
    """
    yield from islice(v, n, None)

hilbert_space

veclconj(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.

Source code in vector\lazy\hilbert_space.py 
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def veclconj(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.
    """
    yield from map(try_conjugate, v)

vector_space

veclpos(v)

Return the identity.

\[ +\vec{v} \qquad \mathbb{K}^n\to\mathbb{K}^n \]
Source code in vector\lazy\vector_space.py 
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def veclpos(v):
    r"""Return the identity.

    $$
        +\vec{v} \qquad \mathbb{K}^n\to\mathbb{K}^n
    $$
    """
    yield from map(pos, v)

veclneg(v)

Return the negation.

\[ -\vec{v} \qquad \mathbb{K}^n\to\mathbb{K}^n \]
Source code in vector\lazy\vector_space.py 
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def veclneg(v):
    r"""Return the negation.

    $$
        -\vec{v} \qquad \mathbb{K}^n\to\mathbb{K}^n
    $$
    """
    yield from map(neg, v)

vecladd(*vs)

Return the sum.

\[ \vec{v}_0+\vec{v}_1+\cdots \qquad \mathbb{K}^{n_0}\times\mathbb{K}^{n_1}\times\cdots\to\mathbb{K}^{\max_i n_i} \]
See also
  • for sum on a single coefficient: vecladdc
Source code in vector\lazy\vector_space.py 
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def vecladd(*vs):
    r"""Return the sum.

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

    See also
    --------
    - for sum on a single coefficient: [`vecladdc`][vector.lazy.vector_space.vecladdc]
    """
    yield from map(partial(sum_default, default=MISSING), group_ordinal(*vs))

vecladdc(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 vecladd(v, veclbasis(i, c)).

See also
  • for sum on more coefficients: vecladd
Source code in vector\lazy\vector_space.py 
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def vecladdc(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 `vecladd(v, veclbasis(i, c))`.

    See also
    --------
    - for sum on more coefficients: [`vecladd`][vector.lazy.vector_space.vecladd]
    """
    v = iter(v)
    yield from islice(chain(v, repeat(zero)), i)
    try:
        yield next(v) + c
    except StopIteration:
        yield +c
    yield from v

veclsub(v, w)

Return the difference.

\[ \vec{v}-\vec{w} \qquad \mathbb{K}^m\times\mathbb{K}^n\to\mathbb{K}^{\max\{m, n\}} \]
See also
  • for difference on a single coefficient: veclsubc
Source code in vector\lazy\vector_space.py 
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def veclsub(v, w):
    r"""Return the difference.

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

    See also
    --------
    - for difference on a single coefficient: [`veclsubc`][vector.lazy.vector_space.veclsubc]
    """
    sentinel = object()
    for vi, wi in zip_longest(v, w, fillvalue=sentinel):
        if wi is sentinel:
            yield vi
        elif vi is sentinel:
            yield -wi
        else:
            yield vi - wi

veclsubc(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 veclsub(v, veclbasis(i, c)).

See also
  • for difference on more coefficients: veclsub
Source code in vector\lazy\vector_space.py 
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def veclsubc(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 `veclsub(v, veclbasis(i, c))`.

    See also
    --------
    - for difference on more coefficients: [`veclsub`][vector.lazy.vector_space.veclsub]
    """
    v = iter(v)
    yield from islice(chain(v, repeat(zero)), i)
    try:
        yield next(v) - c
    except StopIteration:
        yield -c
    yield from v

veclmul(v, a)

Return the product.

\[ \vec{v}a \qquad \mathbb{K}\times\mathbb{K}^n\to\mathbb{K}^n \]
Source code in vector\lazy\vector_space.py 
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def veclmul(v, a):
    r"""Return the product.

    $$
        \vec{v}a \qquad \mathbb{K}\times\mathbb{K}^n\to\mathbb{K}^n
    $$
    """
    yield from map(mul, v, repeat(a))

veclrmul(a, v)

Return the product.

\[ a\vec{v} \qquad \mathbb{K}\times\mathbb{K}^n\to\mathbb{K}^n \]
Source code in vector\lazy\vector_space.py 
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def veclrmul(a, v):
    r"""Return the product.

    $$
        a\vec{v} \qquad \mathbb{K}\times\mathbb{K}^n\to\mathbb{K}^n
    $$
    """
    yield from map(mul, repeat(a), v)

vecltruediv(v, a)

Return the true quotient.

\[ \frac{\vec{v}}{a} \qquad \mathbb{K}^n\times\mathbb{K}\to\mathbb{K}^n \]
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\lazy\vector_space.py 
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def vecltruediv(v, a):
    r"""Return the true quotient.

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

    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.
    """
    yield from map(truediv, v, repeat(a))

veclfloordiv(v, a)

Return the floor quotient.

\[ \left\lfloor\frac{\vec{v}}{a}\right\rfloor \qquad \mathbb{K}^n\times\mathbb{K}\to\mathbb{K}^n \]
Source code in vector\lazy\vector_space.py 
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def veclfloordiv(v, a):
    r"""Return the floor quotient.

    $$
        \left\lfloor\frac{\vec{v}}{a}\right\rfloor \qquad \mathbb{K}^n\times\mathbb{K}\to\mathbb{K}^n
    $$
    """
    yield from map(floordiv, v, repeat(a))

veclmod(v, a)

Return the remainder.

\[ \vec{v} \bmod a \qquad \mathbb{K}^n\times\mathbb{K}\to\mathbb{K}^n \]
Source code in vector\lazy\vector_space.py 
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def veclmod(v, a):
    r"""Return the remainder.

    $$
        \vec{v} \bmod a \qquad \mathbb{K}^n\times\mathbb{K}\to\mathbb{K}^n
    $$
    """
    yield from map(mod, v, repeat(a))

vecldivmod(v, a)

Return the floor quotient and remainder.

\[ \left\lfloor\frac{\vec{v}}{a}\right\rfloor, \ \left(\vec{v} \bmod a\right) \qquad \mathbb{K}^n\times\mathbb{K}\to\mathbb{K}^n\times\mathbb{K}^n \]
Source code in vector\lazy\vector_space.py 
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def vecldivmod(v, a):
    r"""Return the floor quotient and remainder.

    $$
        \left\lfloor\frac{\vec{v}}{a}\right\rfloor, \ \left(\vec{v} \bmod a\right) \qquad \mathbb{K}^n\times\mathbb{K}\to\mathbb{K}^n\times\mathbb{K}^n
    $$
    """
    for vi in v:
        yield divmod(vi, a)

elementwise

veclhadamard(*vs)

Return the elementwise product.

\[ \left((\vec{v}_0)_i\cdot(\vec{v}_1)_i\cdot\cdots\right)_i \qquad \mathbb{K}^{n_0}\times\mathbb{K}^{n_1}\times\cdots\to\mathbb{K}^{\min_i n_i} \]
Source code in vector\lazy\elementwise.py 
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def veclhadamard(*vs):
    r"""Return the elementwise product.

    $$
        \left((\vec{v}_0)_i\cdot(\vec{v}_1)_i\cdot\cdots\right)_i \qquad \mathbb{K}^{n_0}\times\mathbb{K}^{n_1}\times\cdots\to\mathbb{K}^{\min_i n_i}
    $$
    """
    yield from map(partial(prod_default, default=MISSING), zip(*vs))

veclhadamardtruediv(v, w)

Return the elementwise true quotient.

\[ \left(\frac{v_i}{w_i}\right)_i \qquad \mathbb{K}^m\times\mathbb{K}^n\to\mathbb{K}^m \]
Source code in vector\lazy\elementwise.py 
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def veclhadamardtruediv(v, w):
    r"""Return the elementwise true quotient.

    $$
        \left(\frac{v_i}{w_i}\right)_i \qquad \mathbb{K}^m\times\mathbb{K}^n\to\mathbb{K}^m
    $$
    """
    yield from map(truediv, v, chain(w, exception_generator(ZeroDivisionError)))

veclhadamardfloordiv(v, w)

Return the elementwise floor quotient.

\[ \left(\left\lfloor\frac{v_i}{w_i}\right\rfloor\right)_i \qquad \mathbb{K}^m\times\mathbb{K}^n\to\mathbb{K}^m \]
Source code in vector\lazy\elementwise.py 
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def veclhadamardfloordiv(v, w):
    r"""Return the elementwise floor quotient.

    $$
        \left(\left\lfloor\frac{v_i}{w_i}\right\rfloor\right)_i \qquad \mathbb{K}^m\times\mathbb{K}^n\to\mathbb{K}^m
    $$
    """
    yield from map(floordiv, v, chain(w, exception_generator(ZeroDivisionError)))

veclhadamardmod(v, w)

Return the elementwise remainder.

\[ \left(v_i \bmod w_i\right)_i \qquad \mathbb{K}^m\times\mathbb{K}^n\to\mathbb{K}^m \]
Source code in vector\lazy\elementwise.py 
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def veclhadamardmod(v, w):
    r"""Return the elementwise remainder.

    $$
        \left(v_i \bmod w_i\right)_i \qquad \mathbb{K}^m\times\mathbb{K}^n\to\mathbb{K}^m
    $$
    """
    yield from map(mod, v, chain(w, exception_generator(ZeroDivisionError)))

veclhadamarddivmod(v, w)

Return the elementwise floor quotient and remainder.

\[ \left(\left\lfloor\frac{v_i}{w_i}\right\rfloor\right)_i, \ \left(v_i \bmod w_i\right)_i \qquad \mathbb{K}^n\times\mathbb{K}^m\to\mathbb{K}^n\times\mathbb{K}^n \]
Source code in vector\lazy\elementwise.py 
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def veclhadamarddivmod(v, w):
    r"""Return the elementwise floor quotient and remainder.

    $$
        \left(\left\lfloor\frac{v_i}{w_i}\right\rfloor\right)_i, \ \left(v_i \bmod w_i\right)_i \qquad \mathbb{K}^n\times\mathbb{K}^m\to\mathbb{K}^n\times\mathbb{K}^n
    $$
    """
    yield from map(divmod, v, chain(w, exception_generator(ZeroDivisionError)))

veclhadamardmin(*vs, key=None)

Return the elementwise minimum.

\[ \left(\min((\vec{v}_0)_i,(\vec{v}_1)_i,\cdots)\right)_i \qquad \mathbb{K}^{n_0}\times\mathbb{K}^{n_1}\times\cdots\to\mathbb{K}^{\max_i n_i} \]
Source code in vector\lazy\elementwise.py 
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def veclhadamardmin(*vs, key=None):
    r"""Return the elementwise minimum.

    $$
        \left(\min((\vec{v}_0)_i,(\vec{v}_1)_i,\cdots)\right)_i \qquad \mathbb{K}^{n_0}\times\mathbb{K}^{n_1}\times\cdots\to\mathbb{K}^{\max_i n_i}
    $$
    """
    yield from map(partial(min, key=key), group_ordinal(*vs))

veclhadamardmax(*vs, key=None)

Return the elementwise maximum.

\[ \left(\max((\vec{v}_0)_i,(\vec{v}_1)_i,\cdots)\right)_i \qquad \mathbb{K}^{n_0}\times\mathbb{K}^{n_1}\times\cdots\to\mathbb{K}^{\max_i n_i} \]
Source code in vector\lazy\elementwise.py 
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def veclhadamardmax(*vs, key=None):
    r"""Return the elementwise maximum.

    $$
        \left(\max((\vec{v}_0)_i,(\vec{v}_1)_i,\cdots)\right)_i \qquad \mathbb{K}^{n_0}\times\mathbb{K}^{n_1}\times\cdots\to\mathbb{K}^{\max_i n_i}
    $$
    """
    yield from map(partial(max, key=key), group_ordinal(*vs))