There's a great discussion about summation algorithms in Accuracy and Stability of Numerical Algorithms by Nicholas Higham (2002), chapter 4. The focus is naturally on numerical accuracy, but there are lots of ideas and references for recursive, parallel and distributed summation algorithms. Or have a look at some of the CUDA resources by nVidia on speedy summation algorithms on GPUs.

I would say you really don't need to worry about either speed or accuracy of summation in "normal" circumstances. It becomes an issue only in a few special cases: 1) very large volumes of data (say array size of gigabytes or terabytes) typical in supercomputing and high-performance scientific or financial computing; 2) very strict numerical accuracy requirements (accuracy and speed are often opposites); 3) very resource limited computing environment (like a microcontroller). Of course, if you're summing four billion floating point numbers subject to relative error of the order of machine epsilon on a microcontroller, then you need to worry about these things...

In reply to Re: Best way to sum an array? by vrk
in thread Best way to sum an array? by Anonymous Monk

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