# Accumulator Error Feedback

### From Wikimization

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

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

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