Package: accupy
Accurate sums and (dot) products for Python.SumsSumming 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 sum1.0e16 + 1.0 - 1.0e16The actual result is1.0, but in double precision, this will result in0.0.
While in this example the failure is quite obvious, it can get a lot more
tricky than that. accupy providesp,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 viaaccupy.cond(x)accupy.cond(x,y)accupy has the following methods for summation:accupy.kahan_sum(p):Kahan
summationaccupy.fsum(p): A vectorization wrapper aroundmath.fsum(which
uses Shewchuck's algorithm[1](see alsohere)).accupy.ksum(p, K=2): Summation in K-fold precision (from[2])All summation methods sum the first dimension of a multidimensional NumPy array.Let's compare them.Accuracy comparison (sum)As expected, the naivesumperforms very badly
with ill-conditioned sums; likewise fornumpy.sumwhich uses pairwise summation. Kahan summation not significantly better;this,
too, is
expected.Computing the sum with 2-fold accuracy inaccupy.ksumgives the correct
result if the condition is at most in the range of machine precision; further
increasingKhelps with worse conditions.Shewchuck's algorithm inmath.fsumalways 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 productsaccupy has the following methods for dot products:accupy.fdot(p): A transformation of the dot product of lengthninto a
sum of length2n, computed withmath.fsumaccupy.kdot(p, K=2): Dot product in K-fold precision (from[2])Let's compare them.Accuracy comparison (dot)accupy can construct ill-conditioned dot products withx,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.dotis the least accurate, followed by instanced ofkdot.fdotis provably accurate up into the last digitRuntime comparison (dot)NumPy'snumpy.dotismuchfaster 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.ReferencesRichard Shewchuk,Adaptive Precision Floating-Point Arithmetic and Fast
Robust Geometric Predicates, J. Discrete Comput. Geom. (1997), 18(305),
305–363Takeshi Ogita, Siegfried M. Rump, and Shin'ichi Oishi,Accurate Sum and Dot
Product, SIAM J. Sci. Comput. (2006), 26(6), 1955–1988 (34
pages)Dependenciesaccupy needs the C++Eigen
library, provided in
Debian/Ubuntu bylibeigen3-dev.Installationaccupy isavailable from the Python Package Index, so withpip install accupyyou can install.TestingTo run the tests, just check out this repository and typeMPLBACKEND=Agg pytest