kaczmarz-algorithms 0.6.0

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 package.

>>> import kaczmarz

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

Inspecting the Kaczmarz iterates

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

>>> A = [[1, 0, 0],
...      [0, 1, 0],
...      [0, 0, 1]]
>>> b = [1, 1, 1]
>>> x0 = [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, use the ik attribute of the iterates. For example,

>>> iterates = kaczmarz.Cyclic.iterates(A, b, x0)
>>> for xk in iterates:
...     print("Row used:", iterates.ik)
Row used: -1
Row used: 0
Row used: 1
Row used: 2

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

Optional arguments

The solve() and iterates() functions take optional arguments of maxiter and tol to specify a limit on the number of iterations and the desired accuracy of the solution respectively.

Creating your own selection strategy

To implement a selection strategy of your own, inherit from kaczmarz.Base and implement the _select_row_index() method. 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 = len(A)
...         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("Row used:", iterates.ik)
...     print("Iterate:", xk)
Row used: -1
Iterate: [0. 0. 0.]
Row used: 2
Iterate: [0. 0. 1.]
Row used: 1
Iterate: [0. 1. 1.]
Row used: 0
Iterate: [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.