6 The relative error bound of every ( backwards stable ) summation method by a fixed algorithm in fixed precision (i.e.
2 In double precision, this corresponds to an n of roughly 1016, much larger than most sums.
Support by libraries edit In general, built-in "sum" functions in computer languages typically provide no guarantees that a particular summation algorithm will be employed, much less Kahan summation.Alternatives edit Although Kahan's algorithm achieves O ( 1 ) displaystyle O(1) error growth for summing n numbers, only slightly worse O ( log n ) displaystyle O(log n) growth can be achieved by pairwise summation : one recursively divides the set of numbers into.For higher dimensions (than 1 Million we expect to get a significant gain on runtime)."C for Numerical Computing".However, note that matrix multiplication is associative, and therefore the result would be correct as long as the proper ordering were enforced, as in the binary tree reduction technique.The following shows how the first round of reduction is performed.However, simply increasing the precision of the calculations is not practical in general; if input is already double precision, few systems supply quadruple precision and if they did, input could then be quadruple precision."Bringing skeletons out of the closet: a pragmatic manifesto for skeletal parallel programming".Strategy is to use this shared memory by copying values from global area into a same Work-Group : then, this latter is divided into two parts (with size " stride and we add elements of each subgroup side by side with an offset equal.C (10005.9 - 10003.1) -.75987 This extracts whatever went.Displaystyle frac E_nS_nleq left2varepsilon O(nvarepsilon 2)rightfrac sum limits _i1nx_ileftsum limits _i1nx_iright.7 Another alternative is to use arbitrary precision arithmetic, which in principle need no rounding at all with a cost of much greater computational effort.Y.71828 -.0415900 The shortfall from the previous stage gets included.Actually, one has to call " clEnqueueNDRangeKernel " function in this loop and use after " clSetKernelArg " with the new array as argument.2 An ill-conditioned summation problem is one in which this ratio is large, and in this case even compensated summation can have a large relative error.Suppose that one is summing n values x i, for i 1,.,.With a plain summation, each incoming value would be aligned with sum and many low order digits lost (by truncation reduction scpi pinel or rounding).Zeitschrift für Angewandte Mathematik und Mechanik (in German).
2 3 4, this type of operator is commonly used in parallel programming to reduce the elements of an array into a single result.
Below the results of this evaluation plotted with Matlab script plot_performances_sumGPU.