pyneqsys 0.5.7

Package for numerically solving symbolically defined systems of non-linear equations.

pyneqsys provides a convenience class for representing and solving non-linear equation systems from symbolic expressions (provided e.g. with the help of SymPy).

The numerical root finding is perfomed using either:

• scipy: scipy.optimize.root

• mpmath (arbitrary precision): mpmath.calculus.optimization.MDNewton

• kinsol (from SUNDIALS): pykinsol.solve

• nleq2 (ZIB library free for academic use): pynleq2.solve

• levmar (Levenberg-Marquardt): levmar.levmar

In addition to offering a unified interface to different solvers, pyneqsys can also derive the Jacobian analytically (when using pyneqsys.SymbolicSys). This is useful since doing so manually is widely recognized as both tedious and error prone.

The symbolic representation is usually in the form of SymPy expressions, but the user may choose another symbolic back-end (see sym).

In addition to deriving the Jacobian analytically the symbolic representation can for example apply row-reduce. This is usful for when you have a overdetermined system ( formed from e.g. applying conservation laws) and want to solve the system by root-finding rather than using a least-square optimization of e.g. Levenberg-Marquardt style.

Last, but not the least having a symbolic representation of your system of equations allows you to generate publication quality latex representations of your equations (through SymPy’s latex printer) from a single source‒no more error prone hand-rewriting of the same equations in another format for presentation!

Documentation

Autogenerated API documentation for latest stable release is found here: https://bjodah.github.io/pyneqsys/latest (and the development version for the current master branch is found here: http://hera.physchem.kth.se/~pyneqsys/branches/master/html).

Installation

Simplest way to install pyneqsys and its dependencies is through the conda package manager:

$ conda install -c bjodah pyneqsys pytest
$ pytest --pyargs pyneqsys

Optional dependencies

If you used conda to install pyneqsys you can skip this section. But if you use pip you may want to know that the default installation of pyneqsys only requires SciPy:

$ pip install pyneqsys
$ pytest --pyargs pyneqsys -rs

The above command should finish without errors but with some skipped tests. The reason for why some tests are skipped should be because missing optional solvers. To install the optional solvers you will first need to install third party libraries for the solvers and then their python bindings. The 3rd party requirements are as follows:

• pykinsol (requires SUNDIALS ==2.7.0)

• levar

• mpmath

if you want to see what packages need to be installed on a Debian based system you may look at this Dockerfile.

If you manage to install all three external libraries you may install pyneqsys with the option “all”:

$ pip install pyneqsys[all]
$ pytest --pyargs pyneqsys -rs

now there should be no skipped tests. If you try to install pyneqsys on a machine where you do not have root permissions you may find the flag --user helpful when using pip. Also if there are multiple versions of python installed you may want to invoke python for an explicit version of python, e.g.:

$ python3.6 -m pip install --user pyneqsys[all]

see setup.py for the exact list of requirements.

Using Docker

If you have Docker installed, you may use it to host a jupyter notebook server:

$ ./scripts/host-jupyter-using-docker.sh . 8888

the first time you run the command some dependencies will be downloaded. When the installation is complete there will be a link visible which you can open in your browser. You can also run the test suite using the same docker-image:

$ ./scripts/host-jupyter-using-docker.sh . 0

there will be one skipped test (due to symengine missing in this pip installed environment) and quite a few instances of RintimeWarning.

Examples

Example reformulated from SciPy documentation:

>>> from pyneqsys.symbolic import SymbolicSys
>>> neqsys = SymbolicSys.from_callback(
...     lambda x: [(x[0] - x[1])**3/2 + x[0] - 1,
...                (x[1] - x[0])**3/2 + x[1]], 2)
>>> x, info = neqsys.solve([1, 0])
>>> assert info['success']
>>> print(x)
[0.8411639 0.1588361]

here we did not need to enter the jacobian manually, SymPy did that for us. For expressions containing transcendental functions we need to provide a “backend” keyword arguemnt to enable symbolic derivation of the jacobian:

>>> import math
>>> def powell(x, params, backend=math):
...     A, exp = params[0], backend.exp
...     return A*x[0]*x[1] - 1, exp(-x[0]) + exp(-x[1]) - (1 + A**-1)
>>> powell_sys = SymbolicSys.from_callback(powell, 2, 1, names='x0 x1'.split())
>>> x, info = powell_sys.solve([1, 1], [1000.0])
>>> assert info['success']
>>> print(', '.join(['%.6e' % _ for _ in sorted(x)]))
1.477106e-04, 6.769996e+00

pyneqsys also allows the user to solve a system of equations for a span of values for a parameter, and optionally plot the result vs. the varied value:

>>> import matplotlib.pyplot as plt
>>> import numpy as np
>>> x0_varied, x0_idx = np.linspace(1e3, 3e3), 0
>>> all_x, all_info = powell_sys.solve_and_plot_series(x, [1000.0], x0_varied, x0_idx)
>>> plt.savefig('example.png')

For more examples look see examples/, and rendered jupyter notebooks here: http://hera.physchem.kth.se/~pyneqsys/branches/master/examples

Run notebooks using binder

Using only a web-browser (and an internet connection) it is possible to explore the notebooks here: (by the courtesy of the people behind mybinder)

Citing

If you make use of pyneqsys in e.g. academic work you may cite the following peer-reviewed publication:

Depending on what underlying solver you are using you should also cite the appropriate paper (you can look at the list of references in the JOSS article). If you need to reference, in addition to the paper, a specific point version of pyneqsys (for e.g. reproducibility) you can get per-version DOIs from the zendodo archive:

Licensing

The source code is Open Source and is released under the simplified 2-clause BSD license. See LICENSE for further details.

Contributing

Contributors are welcome to suggest improvements at https://github.com/bjodah/pyneqsys (see further details here).

Author

Björn I. Dahlgren, contact:

• gmail address: bjodah