https://github.com/bjodah/accupy
:dart: Accurate sums and dot products for Python.
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:dart: Accurate sums and dot products for Python.
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Accurate sums and (dot) products for Python.
[](https://pypi.org/project/accupy) [](https://pypi.org/pypi/accupy/) [](https://doi.org/10.5281/zenodo.1185173) [](https://github.com/nschloe/accupy) [](https://pypistats.org/packages/accupy) [](https://discord.gg/hnTJ5MRX2Y) [](https://github.com/nschloe/accupy/actions?query=workflow%3Aci) [](https://codecov.io/gh/nschloe/accupy) [](https://github.com/psf/black) ### Sums Summing up values in a list can get tricky if the values are floating point numbers; digit cancellation can occur and the result may come out wrong. A classical example is the sum ``` 1.0e16 + 1.0 - 1.0e16 ``` The actual result is `1.0`, but in double precision, this will result in `0.0`. While in this example the failure is quite obvious, it can get a lot more tricky than that. accupy provides ```python p, exact, cond = accupy.generate_ill_conditioned_sum(100, 1.0e20) ``` which, given a length and a target condition number, will produce an array of floating point numbers that is hard to sum up. Given one or two vectors, accupy can compute the condition of the sum or dot product via ```python accupy.cond(x) accupy.cond(x, y) ``` accupy has the following methods for summation: - `accupy.kahan_sum(p)`: [Kahan summation](https://en.wikipedia.org/wiki/Kahan_summation_algorithm) - `accupy.fsum(p)`: A vectorization wrapper around [math.fsum](https://docs.python.org/3/library/math.html#math.fsum) (which uses Shewchuck's algorithm [[1]](#references) (see also [here](https://code.activestate.com/recipes/393090/))). - `accupy.ksum(p, K=2)`: Summation in K-fold precision (from [[2]](#references)) All summation methods sum the first dimension of a multidimensional NumPy array. Let's compare them. #### Accuracy comparison (sum)  As expected, the naive [sum](https://docs.python.org/3/library/functions.html#sum) performs very badly with ill-conditioned sums; likewise for [`numpy.sum`](https://docs.scipy.org/doc/numpy/reference/generated/numpy.sum.html) which uses pairwise summation. Kahan summation not significantly better; [this, too, is expected](https://en.wikipedia.org/wiki/Kahan_summation_algorithm#Accuracy). Computing the sum with 2-fold accuracy in `accupy.ksum` gives the correct result if the condition is at most in the range of machine precision; further increasing `K` helps with worse conditions. Shewchuck's algorithm in `math.fsum` always gives the correct result to full floating point precision. #### Runtime comparison (sum)   We compare more and more sums of fixed size (above) and larger and larger sums, but a fixed number of them (below). In both cases, the least accurate method is the fastest (`numpy.sum`), and the most accurate the slowest (`accupy.fsum`). ### Dot products accupy has the following methods for dot products: - `accupy.fdot(p)`: A transformation of the dot product of length _n_ into a sum of length _2n_, computed with [math.fsum](https://docs.python.org/3/library/math.html#math.fsum) - `accupy.kdot(p, K=2)`: Dot product in K-fold precision (from [[2]](#references)) Let's compare them. #### Accuracy comparison (dot) accupy can construct ill-conditioned dot products with ```python x, y, exact, cond = accupy.generate_ill_conditioned_dot_product(100, 1.0e20) ``` With this, the accuracy of the different methods is compared.  As for sums, `numpy.dot` is the least accurate, followed by instanced of `kdot`. `fdot` is provably accurate up into the last digit #### Runtime comparison (dot)   NumPy's `numpy.dot` is _much_ faster than all alternatives provided by accupy. This is because the bookkeeping of truncation errors takes more steps, but mostly because of NumPy's highly optimized dot implementation. ### References 1. [Richard Shewchuk, _Adaptive Precision Floating-Point Arithmetic and Fast Robust Geometric Predicates_, J. Discrete Comput. Geom. (1997), 18(305), 305363](https://doi.org/10.1007/PL00009321) 2. [Takeshi Ogita, Siegfried M. Rump, and Shin'ichi Oishi, _Accurate Sum and Dot Product_, SIAM J. Sci. Comput. (2006), 26(6), 19551988 (34 pages)](https://doi.org/10.1137/030601818) ### Dependencies accupy needs the C++ [Eigen library](http://eigen.tuxfamily.org/index.php?title=Main_Page), provided in Debian/Ubuntu by [`libeigen3-dev`](https://packages.ubuntu.com/search?keywords=libeigen3-dev). ### Installation accupy is [available from the Python Package Index](https://pypi.org/project/accupy/), so with ``` pip install accupy ``` you can install. ### Testing To run the tests, just check out this repository and type ``` MPLBACKEND=Agg pytest ```
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- Name: Bjorn
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- Repositories: 48
- Profile: https://github.com/bjodah