Accurate sums and dot products for Python
Accurate sums and (dot) products for Python.
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
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.
Given one or two vectors, accupy can compute the condition of the sum or dot product via
accupy.cond(x) accupy.cond(x, y)
accupy has the following methods for summation:
accupy.kahan_sum(p): Kahan summation
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)
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
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 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
fdot is provably accurate up into the last digit
Runtime comparison (dot)
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.
accupy is available from the Python Package Index, so with
pip install accupy
you can install.
To run the tests, just check out this repository and type
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