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r'''
# What is pyerrors?
`pyerrors` is a python package for error computation and propagation of Markov chain Monte Carlo data.
It is based on the **gamma method** [arXiv:hep-lat/0306017](https://arxiv.org/abs/hep-lat/0306017). Some of its features are:
- **automatic differentiation** as suggested in [arXiv:1809.01289](https://arxiv.org/abs/1809.01289) (partly based on the [autograd](https://github.com/HIPS/autograd) package)
- **treatment of slow modes** in the simulation as suggested in [arXiv:1009.5228](https://arxiv.org/abs/1009.5228)
- coherent **error propagation** for data from **different Markov chains**
- **non-linear fits with x- and y-errors** and exact linear error propagation based on automatic differentiation as introduced in [arXiv:1809.01289](https://arxiv.org/abs/1809.01289)
- **real and complex matrix operations** and their error propagation based on automatic differentiation (Cholesky decomposition, calculation of eigenvalues and eigenvectors, singular value decomposition...)
It is based on the gamma method [arXiv:hep-lat/0306017](https://arxiv.org/abs/hep-lat/0306017). Some of its features are:
- automatic differentiation for exact liner error propagation as suggested in [arXiv:1809.01289](https://arxiv.org/abs/1809.01289) (partly based on the [autograd](https://github.com/HIPS/autograd) package).
- treatment of slow modes in the simulation as suggested in [arXiv:1009.5228](https://arxiv.org/abs/1009.5228).
- coherent error propagation for data from different Markov chains.
- non-linear fits with x- and y-errors and exact linear error propagation based on automatic differentiation as introduced in [arXiv:1809.01289](https://arxiv.org/abs/1809.01289).
- real and complex matrix operations and their error propagation based on automatic differentiation (Matrix inverse, Cholesky decomposition, calculation of eigenvalues and eigenvectors, singular value decomposition...).
There exist similar publicly available implementations of gamma method error analysis suites in
- [Fortran](https://gitlab.ift.uam-csic.es/alberto/aderrors)
- [Julia](https://gitlab.ift.uam-csic.es/alberto/aderrors.jl)
- [Python](https://github.com/mbruno46/pyobs)
There exist similar publicly available implementations of gamma method error analysis suites in [Fortran](https://gitlab.ift.uam-csic.es/alberto/aderrors), [Julia](https://gitlab.ift.uam-csic.es/alberto/aderrors.jl) and [Python](https://github.com/mbruno46/pyobs).
## Basic example