# Accumulator Error Feedback

### From Wikimization

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x = x(idx); | x = x(idx); | ||

</pre> | </pre> | ||

- | should begin the <tt>csum()</tt> subroutine to achieve | + | should begin the <tt>csum()</tt> subroutine to achieve the most accurate summation. |

- | That is not presented here because the commented | + | That is not presented here because the commented Example (inspired by Higham) would then display false positive results. |

- | Even in absence of sorting, <tt>csum()</tt> is more accurate than conventional summation. | + | Even in absence of sorting, <tt>csum()</tt> is more accurate than conventional summation by orders of magnitude. |

+ | |||

+ | Equations in the Figure represent neither a sequence of instructions or algorithm. | ||

+ | The are meant simply to remind us that an imperfect accumulator introduces noise into a series. | ||

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

## Revision as of 22:23, 28 November 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 % 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

### notes

In practice, input sorting

[~, idx] = sort(abs(x),'descend'); x = x(idx);

should begin the `csum()` subroutine to achieve the most accurate summation.
That is not presented here because the commented Example (inspired by Higham) would then display false positive results.
Even in absence of sorting, `csum()` is more accurate than conventional summation by orders of magnitude.

Equations in the Figure represent neither a sequence of instructions or algorithm. The are meant simply to remind us that an imperfect accumulator introduces noise into a series.

### 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