cxroots

cxroots is a Python package for finding all the roots of a function, f(z), of a single complex variable within a given contour, C, in the complex plane. It requires only that both:

• f(z) has no roots or poles on C
• f(z) is analytic in the interior of C

The implementation is primarily based on [KB] and evaluates contour integrals involving f(z) and its derivative f'(z) to determine the roots. If f'(z) is not provided then it is approximated using a finite difference method. The roots are further refined using Newton-Raphson if f'(z) is given or Muller's method if not. See the documentation for a more details and a tutorial.

With Python installed you can install cxroots by entering in the terminal/command line

pip install cxroots

Example

from numpy import exp, cos, sin
f = lambda z: (exp(2*z)*cos(z)-1-sin(z)+z**5)*(z*(z+2))**2

from cxroots import Circle
C = Circle(0,3)
roots = C.roots(f)
roots.show()

print(roots)

Multiplicity |               Root              

------------------------------------------------ 2 | -2.000000000000 +0.000000000000i

1 | -0.651114070264 -0.390425719088i 1 | -0.651114070264 +0.390425719088i 3 | 0.000000000000 +0.000000000000i 1 | 0.648578080954 -1.356622683988i 1 | 0.648578080954 +1.356622683988i 1 | 2.237557782467 +0.000000000000i

See also

The Fortran 90 package ZEAL is a direct implementation of [KB].

Citing cxroots

R. Parini. cxroots: A Python module to find all the roots of a complex analytic function within a given contour (2018), https://github.com/rparini/cxroots https://doi.org/10.5281/zenodo.7013117

BibTex:

@misc{cxroots,
  author = {Robert Parini},
  title = {{cxroots: A Python module to find all the roots of a complex analytic function within a given contour}},
      doi = {10.5281/zenodo.7013116},
      url = {https://github.com/rparini/cxroots},
  year = {2018}
}

---

Development

• Install the pre-commit and then run pre-commit install. The pre-commit scripts can also be run manually with pre-commit run --all-files
• The project uses:
  • Black to maintain consistent formatting. It is run as part of the pre-commit hook and is recommended to be run on save in the developer's editor.
  • isort to sensibly order import statements in Python files. It is also run as part of the pre-commit hook.

Release Procedure

Making a release on GitHub with the tag vX.Y.Z will update the documentation on master and push cxroots at the tagged commit to PyPI.

The documentation can be manually generated by running ./create_docs in the docs_src directory.

References

KB

P. Kravanja and M. Van Barel. Computing the Zeros of Analytic Functions. Springer, Berlin, Heidelberg, 2000.