Accumulator Error Feedback

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=== sorting ===
=== sorting ===
-
In practice, input sorting
+
Sorting is not integral above because the commented Example
 +
(inspired by Higham) would then display false positive results.
 +
But, in practice, input sorting
 +
should begin the <tt>csum()</tt> function to achieve the most accurate summation.
<pre>
<pre>
 +
function s_hat = csum(x)
 +
s_hat=0; e=0;
[~, idx] = sort(abs(x),'descend');
[~, idx] = sort(abs(x),'descend');
x = x(idx);
x = x(idx);
 +
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
</pre>
</pre>
-
should begin the <tt>csum()</tt> 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 complete absence of sorting, <tt>csum()</tt> can be more accurate than conventional summation by orders of magnitude.
Even in complete absence of sorting, <tt>csum()</tt> can be more accurate than conventional summation by orders of magnitude.

Revision as of 20:11, 20 December 2017

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.
%
% 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 Matlab 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

Sorting is not integral above because the commented Example (inspired by Higham) would then display false positive results. But, 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 = (s_hat_old - s_hat) + y;  %calculate difference first (Higham)
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

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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