Accumulator Error Feedback

From Wikimization

CSUM() in Digital Signal Processing terms: z^-1 is a unit delay, Q is a floating-point quantizer to 64 bits, q_i 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

links

Accuracy and Stability of Numerical Algorithms 2e, ch.4.3, Nicholas J. Higham, 2002

For multiplier error feedback, see:

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