Welcome to Kaczmarz Algorithms’s documentation!¶
Kaczmarz Algorithms¶
Variants of the Kaczmarz algorithm for solving linear systems in Python.
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.