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function s=csum(x)
% CSUM Sum of elements using a compensated summation algorithm
% For large vectors, the native sum command in Matlab does not appear to
% use a compensated summation algorithm which can cause significant round
% off errors.
% This code implements a variant of Kahan's compensated summation algorithm
% which often takes about twice as long, but produces more accurate sums
% when the number of elements is large.
% See also SUM
% Example:
%   v=rand(1e7,1);
%   sum1 = sum(v);
%   sum2 = csum(v);
%   fprintf('sum1 = %18.16e\nsum2 = %18.16e\n', sum1, sum2);

% David Gleich, Stanford University, 2008

s=0; e=0; temp=0; y=0;
for i=1:numel(x)
   temp=s; y=x(i)+e; s=temp+y; e=(temp-s)+y;
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