# Accumulator Error Feedback

(Difference between revisions)
Jump to: navigation, search
 Revision as of 20:42, 29 January 2018 (edit)← Previous diff Revision as of 20:46, 29 January 2018 (edit) (undo)Next diff → Line 9: Line 9: % CSUM Sum of elements using a compensated summation algorithm. % CSUM Sum of elements using a compensated summation algorithm. % % - % This Matlab code implements Kahan's compensated + % This Matlab code implements Kahan's compensated summation algorithm (1964) - % summation algorithm (1964) which often takes about twice as long, + % which takes about twice as long as sum() - % but produces more accurate sums when the number of + % but produces more accurate sums when number of elements is large. - % elements is large. -David Gleich + % -David Gleich % % % Also see SUM. % Also see SUM. Line 42: Line 42: === sorting === === sorting === - In practice, input sorting can sometimes achieve more accurate summation. Compensated sum accuracy is quite data dependent. Compensated sum accuracy is quite data dependent. - Substituting a sine wave of randomized frequency, instead of a random number sequence input, + Substituting a unit sine wave at a random frequency, instead of a random number sequence input, can make compensated summation fail to produce more accurate results than a simple sum. can make compensated summation fail to produce more accurate results than a simple sum. + + In practice, input sorting can sometimes achieve more accurate summation. Sorting became integral to later algorithms, such as those from Knuth and Priest. Sorting became integral to later algorithms, such as those from Knuth and Priest. But the very same accuracy dependence on input data prevails. But the very same accuracy dependence on input data prevails.

## Revision as of 20:46, 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).
```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 takes about twice as long as sum()
% but produces more accurate sums when 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

Compensated sum accuracy is quite data dependent. Substituting a unit sine wave at a random frequency, instead of a random number sequence input, can make compensated summation fail to produce more accurate results than a simple sum.

In practice, input sorting can sometimes achieve more accurate summation. Sorting became integral to later algorithms, such as those from Knuth and Priest. But the very same accuracy dependence on input data prevails.

### links

For multiplier error feedback, see: