# Accumulator Error Feedback

### From Wikimization

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% | % | ||

% Example: | % Example: | ||

+ | % clear all; clc | ||

% csumv=0; rsumv=0; | % csumv=0; rsumv=0; | ||

% n = 100e6; | % n = 100e6; | ||

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=== sorting === | === sorting === | ||

- | + | In practice, input sorting can sometimes achieve more accurate summation. | |

- | + | Compensated sum accuracy is quite data dependent. | |

- | In practice, input sorting | + | Substituting a sine wave of randomized frequency, instead of a random number sequence input, |

- | + | can make compensated summation fail to produce more accurate results than a simple sum. | |

- | + | Sorting became integral to later algorithms, such as those from Knuth and Priest. | |

- | + | But the very same accuracy dependence on input data prevails. | |

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=== links === | === links === |

## Revision as of 19:42, 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. % % 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

In practice, input sorting can sometimes achieve more accurate summation. Compensated sum accuracy is quite data dependent. Substituting a sine wave of randomized frequency, instead of a random number sequence input, can make compensated summation fail to produce more accurate results than a simple sum. 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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