kaczmarz-algorithms 0.3.0

Variants of the Kaczmarz algorithm for solving linear systems.

Kaczmarz Algorithms

Variants of the Kaczmarz algorithm for solving linear systems in Python.

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Installation

To install Kaczmarz Algorithms, run this command in your terminal:

    $ pip install -U kaczmarz-algorithms

This is the preferred method to install Kaczmarz Algorithms, as it will always install the most recent stable release.

If you don't have pip installed, these installation instructions can guide you through the process.

Usage

First, import the kaczmarz and numpy packages.

>>> import kaczmarz
>>> import numpy as np

Solving a system of equations

To solve the system of equations 3 * x0 + x1 = 9 and x0 + 2 * x1 = 8 using the Kaczmarz algorithm with the cyclic selection rule, use the kaczmarz.Cyclic.solve function.

>>> A = np.array([[3, 1],
...               [1, 2]])
>>> b = np.array([9, 8])
>>> x = kaczmarz.Cyclic.solve(A, b)
>>> x
array([2., 3.])

Check that the solution is correct:

>>> np.allclose(A @ x, b)
True

Inspecting the Kaczmarz iterates

To access the iterates of the Kaczmarz algorithm with the cyclic selection rule, use the kaczmarz.Cyclic.iterates function.

>>> A = np.array([[1, 0, 0],
...               [0, 1, 0],
...               [0, 0, 1]])
>>> b = np.array([1, 1, 1])
>>> x0 = np.array([0, 0, 0])  # Initial iterate
>>> for xk in kaczmarz.Cyclic.iterates(A, b, x0):
...     xk
array([0., 0., 0.])
array([1., 0., 0.])
array([1., 1., 0.])
array([1., 1., 1.])

Inspecting the rows/equations used

To access the row index used at each iteration of the Kaczmarz algorithm with the cyclic selection rule, use the ik attribute of the kaczmarz.Cyclic.iterates iterable.

>>> iterates = kaczmarz.Cyclic.iterates(A, b, x0)
>>> for xk in iterates:
...     print("After projecting onto equation {}: {}".format(iterates.ik, xk))
After projecting onto equation -1: [0. 0. 0.]
After projecting onto equation 0: [1. 0. 0.]
After projecting onto equation 1: [1. 1. 0.]
After projecting onto equation 2: [1. 1. 1.]

The initial value of iterates.ik is -1, since no projections have been performed yet at the start of the algorithm.

Creating your own selection strategy

To implement a selection strategy of your own, inherit from kaczmarz.Base and implement the select_row_index function. For example, to implement a strategy which uses of the equations of your system in reverse cyclic order:

>>> class ReverseCyclic(kaczmarz.Base):
...     def __init__(self, A, *args, **kwargs):
...         super().__init__(A, *args, **kwargs)
...         self.n_rows = A.shape[0]
...         self.row_index = None
...
...     def select_row_index(self, xk):
...         if self.row_index is None:
...             self.row_index = self.n_rows
...         self.row_index = (self.row_index - 1) % self.n_rows
...         return self.row_index

Your new class will inherit solve and iterates class methods which work the same way as kaczmarz.Cyclic.solve and kaczmarz.Cyclic.iterates described above.

>>> iterates = ReverseCyclic.iterates(A, b, x0)
>>> for xk in iterates:
...     print("After projecting onto equation {}: {}".format(iterates.ik, xk))
After projecting onto equation -1: [0. 0. 0.]
After projecting onto equation 2: [0. 0. 1.]
After projecting onto equation 1: [0. 1. 1.]
After projecting onto equation 0: [1. 1. 1.]

For information about the optional arguments of solve and iterates, as well as the other selection strategies available other than Cyclic, see readthedocs.io.

Citing

If you use our code in an academic setting, please consider citing our code. You can find the appropriate DOI for whichever version you are using on zenodo.org.

Development

See CONTRIBUTING.md for information related to developing the code.