nschloe/accupy

Accurate sums and dot products for Python.
Python C++ Makefile Fetching latest commit…
Cannot retrieve the latest commit at this time.
Type Name Latest commit message Commit time
Failed to load latest commit information. .circleci .github/workflows accupy Aug 9, 2019 src test Aug 9, 2019 .bandit Feb 16, 2018 .flake8 Jun 4, 2018 .gitignore LICENSE Feb 28, 2019 MANIFEST.in Makefile Aug 9, 2019 README.md Sep 25, 2019 codecov.yml Feb 16, 2018 setup.py Aug 9, 2019 test_requirements.txt

Accurate sums and (dot) products for Python.

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

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.

accupy has the following methods for summation:

• accupy.kahan_sum(p): Kahan summation

• accupy.fsum(p): A vectorization wrapper around math.fsum (which uses Shewchuck's algorithm  (see also here)).

• accupy.ksum(p, K=2): Summation in K-fold precision (from )

All summation methods sum the first dimension of a multidimensional NumPy array.

Let's compare them.

Accuracy comparison (sum) As expected, the naive sum performs very badly with ill-conditioned sums; likewise for numpy.sum which uses pairwise summation. Kahan summation not significantly better; this, too, is expected.

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

• accupy.kdot(p, K=2): Dot product in K-fold precision (from )

Let's compare them.

Accuracy comparison (dot)

accupy can construct ill-conditioned dot products with

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.

Dependencies

accupy needs the C++ Eigen library, provided in Debian/Ubuntu by libeigen3-dev.

Installation

accupy is available from the Python Package Index, so with

pip install -U accupy