# Accumulator Error Feedback

### From Wikimization

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% 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 | + | |

% summation algorithm (1964) which often takes about twice as long, | % summation algorithm (1964) which often takes about twice as long, | ||

% but produces more accurate sums when the number of | % but produces more accurate sums when the number of | ||

Line 57: | Line 53: | ||

y = x(i) + e; | y = x(i) + e; | ||

s_hat = s_hat_old + y; | s_hat = s_hat_old + y; | ||

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

end | end | ||

return | return |

## Revision as of 19:13, 29 January 2018

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. % % % 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.

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