# Accumulator Error Feedback

### From Wikimization

(Difference between revisions)

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q<sub>i</sub> represents error due to quantization (additive by definition). <br>-Jon Dattorro]] | q<sub>i</sub> represents error due to quantization (additive by definition). <br>-Jon Dattorro]] | ||

<pre> | <pre> | ||

- | function s_hat=csum(x) | + | function s_hat = csum(x) |

% CSUM Sum of elements using a compensated summation algorithm. | % CSUM Sum of elements using a compensated summation algorithm. | ||

% | % | ||

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% elements is large. -David Gleich | % elements is large. -David Gleich | ||

% | % | ||

- | % | + | % Also see SUM. |

% | % | ||

+ | % % Matlab csum() example: | ||

% clear all | % clear all | ||

- | % % v = sort(randn(13e6,1),'descend'); | + | % % v = sort(randn(13e6,1),'descend'); %better when sorted |

% v = randn(13e6,1); | % v = randn(13e6,1); | ||

% rsumv = abs(sum(v) - sum(v(end:-1:1))); | % rsumv = abs(sum(v) - sum(v(end:-1:1))); |

## Revision as of 19:41, 25 September 2017

function s_hat = csum(x) % CSUM Sum of elements using a compensated summation algorithm. % % For large vectors, the native sum command in Matlab does % not appear to use a compensated summation algorithm which % can cause significant roundoff errors. % % This code implements a variant of Kahan's compensated % summation algorithm 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 % % v = sort(randn(13e6,1),'descend'); %better when sorted % v = randn(13e6,1); % rsumv = abs(sum(v) - sum(v(end:-1:1))); % disp(['rsumv = ' num2str(rsumv,'%18.16f')]); % csumv = abs(csum(v) - csum(v(end:-1:1))); % disp(['csumv = ' num2str(csumv,'%18.16f')]); % % vsumv = sum(vpa(v)) - sum(vpa(v(end:-1:1))); %vpa toolbox 32GB RAM % % disp(['vsumv = ' char(vsumv)]) 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

### links

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

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