Accumulator Error Feedback

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% See also SUM
% See also SUM
%
%
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% Matlab example:
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% clear all
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% v=rand(1e7,1);
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% % v=sort(randn(13e6,1),'descend'); %better when sorted
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% sum1 = sum(v);
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% v=randn(13e6,1);
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% sum2 = csum(v);
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% rsumv = abs(sum(v) - sum(v(end:-1:1)));
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% fprintf('sum1 = %18.16e\nsum2 = %18.16e\n', sum1, sum2);
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% disp(['rsumv = ' num2str(rsumv,'%18.16f')]);
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% csumv = abs(csum(v) - csum(v(end:-1:1)));
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% disp(['csumv = ' num2str(csumv,'%18.16f')]);
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% % vsumv = sum(vpa(v)) - sum(vpa(v(end:-1:1))); %vpa toolbox 32GB RAM
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% % disp(['vsumv = ' char(vsumv)])
s_hat=0; e=0;
s_hat=0; e=0;
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=== links ===
=== links ===
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[http://servidor.demec.ufpr.br/CFD/bibliografia/Higham_2002_Accuracy%20and%20Stability%20of%20Numerical%20Algorithms.pdf Accuracy and Stability of Numerical Algorithms, ch.4.3, Nicholas J. Higham, 1996]
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[http://servidor.demec.ufpr.br/CFD/bibliografia/Higham_2002_Accuracy%20and%20Stability%20of%20Numerical%20Algorithms.pdf Accuracy and Stability of Numerical Algorithms 2e, ch.4.3, Nicholas J. Higham, 2002]
For multiplier error feedback, see:
For multiplier error feedback, see:

Revision as of 18:20, 25 September 2017

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
%
%  clear all
% % v=sort(randn(13e6,1),'descend');             %better when sorted
%  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,'%18.16f')]);
% % vsumv = sum(vpa(v)) - sum(vpa(v(end:-1:1))); %vpa toolbox 32GB RAM
% % disp(['vsumv = ' char(vsumv)])

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

links

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

For multiplier error feedback, see:

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