Accumulator Error Feedback

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%
%
% Example:
% Example:
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% clear all; clc
% csumv=0; rsumv=0;
% csumv=0; rsumv=0;
% n = 100e6;
% n = 100e6;
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=== sorting ===
=== sorting ===
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Sorting is not integral above because the commented Example
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In practice, input sorting can sometimes achieve more accurate summation.
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(inspired by Higham) would then display false positive results.<br>
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Compensated sum accuracy is quite data dependent.
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In practice, input sorting
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Substituting a sine wave of randomized frequency, instead of a random number sequence input,
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should begin the <tt>csum()</tt> function to achieve the most accurate summation:
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can make compensated summation fail to produce more accurate results than a simple sum.
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<pre>
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Sorting became integral to later algorithms, such as those from Knuth and Priest.
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function s_hat = csum(x)
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But the very same accuracy dependence on input data prevails.
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s_hat=0; e=0;
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[~, idx] = sort(abs(x),'descend');
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x = x(idx);
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for i=1:numel(x)
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s_hat_old = s_hat;
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y = x(i) + e;
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s_hat = s_hat_old + y;
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e = y - (s_hat - s_hat_old); %calculate parentheses first
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end
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return
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</pre>
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Even in complete absence of sorting, <tt>csum()</tt> can be more accurate than conventional summation by orders of magnitude.
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=== links ===
=== links ===

Revision as of 19:42, 29 January 2018

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. qi represents error due to quantization (additive by definition).
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.
qi 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:
% clear all; clc
% 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

In practice, input sorting can sometimes achieve more accurate summation. Compensated sum accuracy is quite data dependent. Substituting a sine wave of randomized frequency, instead of a random number sequence input, can make compensated summation fail to produce more accurate results than a simple sum. Sorting became integral to later algorithms, such as those from Knuth and Priest. But the very same accuracy dependence on input data prevails.

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

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