# 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 often takes about twice as long, % but produces more accurate sums when the number of % elements is large. -David Gleich % % Also see SUM. % % Example: % 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

Sorting is not integral above because the commented Example
(inspired by Higham) would then display false positive results.

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 = y - (s_hat - s_hat_old); %calculate parentheses first 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

Further Remarks on Reducing Truncation Errors, William Kahan, 1964

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