Accumulator Error Feedback

From Wikimization

csum() in Digital Signal Processing terms: z^-1 is a unit delay,
Q is a 64-bit floating-point quantizer. Algebra represents neither a sequence of instructions or algorithm. It is only meant to remind that an imperfect accumulator introduces noise into a series.
q_i represents error due to quantization (additive by definition).

function s_hat = csum(x)
% CSUM Sum of elements using a compensated summation algorithm.
%
% This Matlab code implements Kahan's compensated 
% summation algorithm (1964) which often takes about twice as long, 
% but produces more accurate sums when the number of 
% elements is large. -David Gleich
%
% Also see SUM.
%
% Example:
% csumv=0;  rsumv=0;
% n = 100e6;
% t = ones(n,1);
% while csumv <= rsumv
%    v = randn(n,1);
%
%    rsumv = abs((t'*v - t'*v(end:-1:1))/sum(v));
%    disp(['rsumv = ' num2str(rsumv,'%1.16f')]);
%
%    csumv = abs((csum(v) - csum(v(end:-1:1)))/sum(v));
%    disp(['csumv = ' num2str(csumv,'%1.16e')]);
% end

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 = y - (s_hat - s_hat_old); 
end
return

sorting

Sorting is not integral above because the commented Example (inspired by Higham) would then display false positive results.
In practice, input sorting should begin the csum() function to achieve the most accurate summation:

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

Even in complete absence of sorting, csum() can be more accurate than conventional summation by orders of magnitude.

links

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

Further Remarks on Reducing Truncation Errors, William Kahan, 1964

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