Accumulator Error Feedback

CSUM() in Digital Signal Processing terms: z-1 is a unit delay, Q is a floating-point quantizer to 64 bits, qi represents error due to quantization (additive by definition).
-Jon Dattorro
```function s_hat = 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 roundoff 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. -David Gleich
%
% Also see SUM.
%
% % Matlab csum() example:
% clear all
% csumv = 0;  rsumv = 0;
% while csumv <= rsumv
%    v = randn(13e6,1);
%    rsumv = abs(sum(v) - sum(v(end:-1:1)));
%    disp(['rsumv = ' num2str(rsumv,'%18.16f')]);
%    csumv = abs(csum(v) - csum(v(end:-1:1)));
%    disp(['csumv = ' num2str(csumv,'%18.16e')]);
% end

[~, idx] = sort(abs(x),'descend');
x = x(idx);
s_hat=0; e=0;
for i=1:numel(x)
s_hat_old = s_hat;
y = x(i) + e;
s_hat = s_hat_old + y;
e = (s_hat_old - s_hat) + y;  %calculate difference first (Higham)
end
return
```