Accumulator Error Feedback

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csum() in Digital Signal Processing terms:  z-1 is a unit delay,Q is a 64-bit floating-point quantizer.
csum() in Digital Signal Processing terms: z-1 is a unit delay,
Q is a 64-bit floating-point quantizer.
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); 


ones(1,n)*v  and  sum(v)  produce different results in Matlab 2017b with vectors having only a few hundred entries.

Matlab's variable precision arithmetic, (vpa(), sym()) from Mathworks' Symbolic Math Toolbox, can neither accurately sum a few hundred entries in quadruple precision. Error creeps up above |2e-16| for sequences with high condition number (defined as sum|x|/|sum x|). Use the Advanpix Multiprecision Computing Toolbox for MATLAB preferentially.


Floating-point compensated summation accuracy is data dependent. Substituting a unit sinusoid at arbitrary 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.


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

Further Remarks on Reducing Truncation Errors, William Kahan, 1964

XSum() Matlab program - Fast Sum with Error Compensation, Jan Simon, 2014

For fixed-point multiplier error feedback, see:

Implementation of Recursive Digital Filters for High-Fidelity Audio

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