Accumulator Error Feedback

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[[Image:Gleich.jpg|thumb|right|429px|CSUM() in Digital Signal Processing terms:
[[Image:Gleich.jpg|thumb|right|429px|CSUM() in Digital Signal Processing terms:
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z<sup>-1</sup> is a unit delay, Q is a floating-point quantizer to 64 bits,
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z<sup>-1</sup> is a unit delay, Q a floating-point quantizer to 64 bits,
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q<sub>i</sub> represents error due to quantization (additive by definition). &nbsp;&nbsp;<math>-</math> Jon Dattorro]]
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q<sub>i</sub> represents error due to quantization (additive by definition).
 +
Algebra represents neither a sequence of instructions or algorithm.
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It is only meant to remind that an imperfect accumulator introduces noise into a series.]]
<pre>
<pre>
function s_hat = csum(x)
function s_hat = csum(x)
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% Also see SUM.
% Also see SUM.
%
%
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% % Matlab csum() example:
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% % Matlab csum() Example:
% clear all
% clear all
% csumv=0; rsumv=0;
% csumv=0; rsumv=0;
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</pre>
</pre>
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=== notes ===
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=== sorting ===
In practice, input sorting
In practice, input sorting
<pre>
<pre>
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That is not presented here because the commented Example (inspired by Higham) would then display false positive results.
That is not presented here because the commented Example (inspired by Higham) would then display false positive results.
Even in absence of sorting, <tt>csum()</tt> is more accurate than conventional summation by orders of magnitude.
Even in absence of sorting, <tt>csum()</tt> is more accurate than conventional summation by orders of magnitude.
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Equations in the Figure represent neither a sequence of instructions or algorithm.
 
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The are meant simply to remind us that an imperfect accumulator introduces noise into a series.
 
=== links ===
=== links ===

Revision as of 22:35, 28 November 2017

CSUM() in Digital Signal Processing terms:  z-1 is a unit delay, Q a floating-point quantizer to 64 bits,  qi represents error due to quantization (additive by definition).   Algebra represents neither a sequence of instructions or algorithm. It is only meant to remind that an imperfect accumulator introduces noise into a series.
CSUM() in Digital Signal Processing terms: z-1 is a unit delay, Q a floating-point quantizer to 64 bits, qi represents error due to quantization (additive by definition). Algebra represents neither a sequence of instructions or algorithm. It is only meant to remind that an imperfect accumulator introduces noise into a series.
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')]);
%    [~, idx] = sort(abs(v),'descend'); 
%    x = v(idx);
%    csumv = abs(csum(x) - csum(x(end:-1:1)));
%    disp(['csumv = ' num2str(csumv,'%18.16e')]);
% 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 = (s_hat_old - s_hat) + y;  %calculate difference first (Higham)
end
return

sorting

In practice, input sorting

[~, idx] = sort(abs(x),'descend'); 
x = x(idx);

should begin the csum() subroutine to achieve the most accurate summation. That is not presented here because the commented Example (inspired by Higham) would then display false positive results. Even in absence of sorting, csum() is more accurate than conventional summation by orders of magnitude.

links

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

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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