Accumulator Error Feedback

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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
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
%
% See also SUM
%
% Matlab example:
%   v=rand(1e7,1);
%   sum1 = sum(v);
%   sum2 = csum(v);
%   fprintf('sum1 = %18.16e\nsum2 = %18.16e\n', sum1, sum2);

s_hat=0; y=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

links

Accuracy and Stability of Numerical Algorithms, Nicholas J. Higham, 1996

For multiplier error feedback, see:

Implementation of Recursive Digital Filters for High-Fidelity Audio

Comments on Implementation of Recursive Digital Filters for High-Fidelity Audio

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