finitediff

finitediff containts two implementations (Fortran 90 and C++) version of Begnt Fornberg’s formulae for generation of finite difference weights on aribtrarily spaced one dimensional grids. The finite difference weights can be used for optimized inter-/extrapolation data series for up to arbitrary derivative order. Python bindings are provided.

Capabilities

finitediff currently provides callbacks for estimation of derivatives or interpolation either at a single point or over an array (available from the Python bindings).

The user may also manually generate the corresponding weights. (see calculate_weights)

Documentation

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

Examples

Generating finite difference weights is simple using C++11:

#include "finitediff_templated.hpp"
#include <vector>
#include <string>
#include <iostream>

int main(){
    const unsigned max_deriv = 2;
    std::vector<std::string> labels {"0th derivative", "1st derivative", "2nd derivative"};
    std::vector<double> x {0, 1, -1, 2, -2};  // Fourth order of accuracy
    auto coeffs = finitediff::generate_weights(x, max_deriv);
    for (unsigned deriv_i = 0; deriv_i <= max_deriv; deriv_i++){
        std::cout << labels[deriv_i] << ": ";
        for (unsigned idx = 0; idx < x.size(); idx++){
            std::cout << coeffs[deriv_i*x.size() + idx] << " ";
        }
        std::cout << std::endl;
    }
}

$ cd examples/
$ g++ -std=c++11 demo.cpp -I../include
$ ./a.out
Zeroth derivative (interpolation): 1 -0 0 0 -0
First derivative: -0 0.666667 -0.666667 -0.0833333 0.0833333
Second derivative: -2.5 1.33333 1.33333 -0.0833333 -0.0833333

and of course using the python bindings:

>>> from finitediff import get_weights
>>> import numpy as np
>>> c = get_weights(np.array([-1., 0, 1]), 0, maxorder=1)
>>> np.allclose(c[:, 1], [-.5, 0, .5])
True

see the examples/ directory for more examples.

Installation

The simplest way to install finitediff is to use conda package manager:

$ conda install -c bjodah finitediff pytest

alternatively, you may also use pip:

$ python -m pip install --user finitediff

(you can skip the --user flag if you have got root permissions), to run the tests you need pytest too:

$ python -m pip install --user --upgrade pytest
$ python -m pytest --pyargs finitediff

Dependencies

You need either a C++ or a Fortran 90 compiler. On debian based linux systems you may install one by issuing:

$ sudo apt-get install gfortran g++

See setup.py for optional (Python) dependencies.

Notes

There is a git subtree under finitediff, update through:

git subtree pull --prefix finitediff/external/newton_interval newton_interval master --squash

where the repo “newton_interval” is https://github.com/bjodah/newton_interval.git

First time you need to add it:

git subtree add --prefix finitediff/external/newton_interval git://github.com/bjodah/newton_interval master

References

The algortihm is a Fortran 90 rewrite of:

http://dx.doi.org/10.1137/S0036144596322507

@article{fornberg_classroom_1998,
  title={Classroom note: Calculation of weights in finite difference formulas},
  author={Fornberg, Bengt},
  journal={SIAM review},
  volume={40},
  number={3},
  pages={685--691},
  year={1998},
  publisher={SIAM}
  doi={10.1137/S0036144596322507}
}

Which is based on an article of the same author:

http://dx.doi.org/10.1090/S0025-5718-1988-0935077-0

@article{fornberg_generation_1988,
  title={Generation of finite difference formulas on arbitrarily spaced grids},
  author={Fornberg, Bengt},
  journal={Mathematics of computation},
  volume={51},
  number={184},
  pages={699--706},
  year={1988}
  doi={10.1090/S0025-5718-1988-0935077-0}
}