Krylov subspace methods for linear systems
Project description
KryPy
KryPy is a Python 3 module for Krylov subspace methods for the solution of linear algebraic systems. This includes enhanced versions of CG, MINRES and GMRES as well as methods for the efficient solution of sequences of linear systems.
Features
KryPy gives you an easytouse yet flexible interface to Krylov subspace methods for linear algebraic systems. Compared to the implementations in SciPy (or MATLAB), KryPy allows you to supply additional arguments that may help you to tune the solver for the specific problem you want to solve. The additional arguments may also be of interest if you are doing research on Krylov subspace methods.
Some features of KryPy are:
 Userdefined inner products  useful when solving a linear algebraic system whose operator is selfadjoined in a nonEuclidean innerproduct. This way, CG or MINRES can be applied to selfadjoined (but nonsymmetric/nonHermitian) operators easily.
 Full control of preconditioners  the order of applying preconditioners matters. This is why you can supply two left preconditioners (one of whom implicitly changes the inner product and thus has to be positive definite) and one right preconditioner. Take a look at the arguments
M
,Ml
andMr
.  Get the Arnoldi/Lanczos basis and Hessenberg matrix  you want to extract further information from the generated vectors (e.g. recycling)? Just pass the optional argument
store_arnoldi=True
.  Explicitly computed residuals on demand  if you do research on Krylov subspace methods or preconditioners, then you sometimes want to know the explicitly computed residual in each iteration (in contrast to an updated residual which can be obtained implicitly). Then you should pass the optional argument
explicit_residual=True
.  Compute errors  if you have (for research purposes) the exact solution at hand and want to monitor the error in each iteration instead of the residual, you can supply the optional argument
exact_solution=x_exact
to theLinearSystem
.
Usage
Documentation
The documentation is hosted at krypy.readthedocs.org.
Example
The above convergence history is obtained with the following example where the
Gmres
method is used to solve the linear system A*x=b
with the diagonal matrix
A=diag(1e3,2,...,100)
and right hand side b=[1,...,1]
.
import numpy
import krypy
A = numpy.diag([1.0e3] + list(range(2, 101)))
b = numpy.ones(100)
# sol, out = krypy.cg(A, b)
# sol, out = krypy.minres(A, b)
sol, out = krypy.gmres(A, b)
# sol is None if no solution has been found
# out.resnorms the relative residual norms and some more data
# plot residuals
import matplotlib.pyplot as plt
plt.semilogy(out.resnorms)
plt.show()
Of course, this is just a toy example where you would not use GMRES in practice. KryPy can handle arbitrary large matrices  as long as the (hopefully sparse) matrices and the generated basis of the Krylov subspace fit into your memory. ;) Furthermore, in actual applications, you definitely want to adjust Gmres' parameters such as the residual tolerance.
Help
Help can be optained via Python's builtin help system. For example, you can use
the ?
in ipython
:
from krypy.linsys import Gmres
?Gmres
Installing
pip / PyPi
Simply run pip install krypy
.
Ubuntu
There's an Ubuntu PPA with packages for Python 2 and Python 3.
Installing from source
KryPy has the following dependencies:
 NumPy
 SciPy
Development
KryPy is currently maintained by André Gaul. Feel free to contact André. Please submit feature requests and bugs as github issues.
KryPy is developed with continuous integration. Current status:
Distribution
To create a new release

bump the
__version__
number, 
create a Git tag,
$ git tag a v0.3.1 $ git push tags
and

upload to PyPi:
$ make upload
License
KryPy is free software licensed under the MIT License.
References
KryPy evolved from the PyNosh package (Python framework for nonlinear Schrödinger equations; joint work with Nico Schlömer) which was used for experiments in the following publication:
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