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vector

An infinite-dimensional vector package.

>>> from vector import vecadd
>>> vecadd((1, 2), (4, 5, 6))
(5, 7, 6)
>>> 
>>> from vector import Vector
>>> v = Vector((1, 2))
>>> w = Vector((4, 5, 6))
>>> v + w
Vector(5, 7, 6, ...)
>>> 
>>> from vector import vecnpadd
>>> vecnpadd((1, 2), ((3, 4, 5),
...                   (6, 7, 8)))
array([[4, 6, 5],
       [7, 9, 8]])

Installation

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

Usage

This package includes

  • general-purpose functions,
  • a clean class
  • improved numpy-routines to handle infinite-dimensional vectors. It operates on vectors of different lengths, treating them as infinite-dimensional by assuming that all components after the given ones are zero.

Functions

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

All functions accept sequences, most even single exhaustible iterables.

They return vectors as tuples.

The functions are type-independent. However, the data types used must support necessary scalar operations. For instance, for vector addition, components must be addable — this may include operations with padded integer zeros. Empty empty operations return zero vector (e.g. vecsum()==veczero) or integer zeros (e.g. vecdot(veczero, veczero)==int(0)).

Padding is done with int(0).

creation stuff

  • veczero = (): Zero vector.
  • vecbasis(i, c=1): Return the i-th basis vector times c. The retured value is a tuple with i integer zeros followed by c.
  • vecrand(n): Return a random vector of n uniform coefficients in [0, 1[.
  • vecrandn(n, normed=True, mu=0, sigma=1): Return a random vector of n normal distributed coefficients.

sequence stuff

  • veceq(v, w): Return if two vectors are equal.
  • vectrim(v, tol=1e-9): Remove all trailing near zero (<=tol) coefficients.
  • vecround(v, ndigits=None): Round all coefficients to the given precision.

Hilbert space stuff

  • vecabsq(v): Return the sum of absolute squares of the coefficients.
  • vecabs(v): Return the Euclidean/L2-norm.
  • vecdot(v, w): Return the inner product of two vectors without conjugation.

vector space stuff

  • vecadd(*vs): Return the sum of vectors.
  • vecsub(v, w): Return the difference of two vectors.
  • vecmul(a, v): Return the product of a scalar and a vector.
  • vectruediv(v, a): Return the true division of a vector and a scalar.
  • vecfloordiv(v, a): Return the floor division of a vector and a scalar.

Class

The immutable Vector class wraps all the mentioned functions into a tidy package, making them easier to use by enabling interaction through operators.

Its coefficients are internally stored as a tuple in the coef attribute and therefore zero-indexed.

Vector operations return the same type (type(v+w)==type(v)) so the class can easily be extended (to e.g. a polynomial class).

initialisation stuff

  • Vector(i): Create a new vector with the given coefficients or the i-th basis vector if an integer i is given.
  • Vector.rand(n): Create a random vector of n uniform coefficients in [0, 1[.
  • Vector.randn(n, normed=True, mu=0, sigma=1)): Create a random vector of n normal distributed coefficients.
  • Vector.ZERO: Zero vector.
>>> from vector import Vector
>>> Vector((1, 2, 3))
Vector(1, 2, 3, ...)
>>> Vector.gauss(3)
Vector(-0.5613820142699765, -0.028308921297709365, 0.8270724508948077, ...)
>>> Vector(3)
Vector(0, 0, 0, 1, ...)

container and sequence stuff

  • len(v): Return the number of set coefficients.
  • v[key]: Return the indexed coefficient or coefficients. Not set coefficients default to 0.
  • iter(v): Return an iterator over the set coefficients.
  • v == w: Return if of same type with same coefficients.
  • v << i: Return a vector with coefficients shifted to lower indices.
  • v >> i: Return a vector with coefficients shifted to higher indices.
  • v.trim(tol=1e-9): Remove all trailing near zero (abs<=tol) coefficients.
  • v.round(ndigits=None): Round all coefficients to the given precision.

Hilbert space stuff

  • v.absq(): Return the sum of absolute squares of the coefficients.
  • abs(v): Return the Euclidean/L2-norm. Return the square root of vecabsq.
  • v @ w: Return the inner product of two vectors without conjugation.

vector space stuff

  • v + w: Return the vector sum.
  • v - w: Return the vector difference.
  • v * a: Return the scalar product.
  • v / a: Return the scalar true division.
  • v // a: Return the scalar floor division.

numpy-routines

>>> from vector import vecnpadd
>>> vecnpadd((1, 2), ((3, 4, 5),
...                   (6, 7, 8)))
array([[4, 6, 5],
       [7, 9, 8]])

numpy-version of the functions are also provided, to many vectors at once. They behave like the ones in numpy.polynomial.polynomial, but also work on 2D-arrays and broadcast to multiple dimensions like the usual numpy operations (but adjust the shapes accordingly).

vecnpzero is np.array([0]) like numpy.polynomial.polynomial.polyzero, not veczero=() (empty tuple, no zero coefficient left) like in the functions and class above.

Padding is done with numpy.int64 zeros.

Creation routines have a dimension argument d. If left to None, the returned values are 1D, so a single vector. If given, the routines return a 2D-array representing mutiple vectors in rows.

creation stuff

  • vecnpzero(d=None): Return d zero vectors. The retured value is a (d, 1)-array of zeros if d is not None or [0] otherwise.
  • vecnpbasis(i, c=1, d=None): Return d many i-th basis vectors times c. The retured value is a (d, i+1)-array if d is not None or (i+1,) otherwise.
  • vecnprand(n, d=None): Return d random vectors of n uniform coefficients in [0, 1[. The retured value is a (d, n)-array if d is not None or (n,) otherwise.
  • vecnprandn(n, normed=True, d=None): Return d random vectors of n normal distributed coefficients. The retured value is a (d, n)-array if d is not None or (n,) otherwise.

sequence stuff

  • vecnpeq(v, w): Return if two vectors are equal.
  • vecnptrim(v, tol=1e-9): Remove all trailing near zero (abs(v_i)<=tol) coefficients.
  • (numpy.round already exists)

Hilbert space stuff

  • vecnpabsq(v): Return the sum of absolute squares of the coefficients.
  • vecnpabs(v): Return the Euclidean/L2-norm.
  • vecnpdot(v, w): Return the inner product of two vectors without conjugation.

vector space stuff

  • vecnpadd(*vs): Return the sum of vectors.
  • vecnpsub(v, w): Return the difference of two vectors.

profiling

For most functions a performance comparison between different approaches has been made. The results can be found in profiling.ipynb.

todo

  • docstrings
  • numpy routines

License (MIT)

Copyright (c) 2022-2024 Sebastian Gössl

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

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

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