# Accumulator Error Feedback

### From Wikimization

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[[Image:Gleich.jpg|thumb|right|429px|CSUM() in Digital Signal Processing terms: | [[Image:Gleich.jpg|thumb|right|429px|CSUM() in Digital Signal Processing terms: | ||

- | z<sup>-1</sup> is a unit delay, Q | + | z<sup>-1</sup> is a unit delay, Q a floating-point quantizer to 64 bits, |

- | q<sub>i</sub> represents error due to quantization (additive by definition). | + | q<sub>i</sub> represents error due to quantization (additive by definition). |

+ | Algebra represents neither a sequence of instructions or algorithm. | ||

+ | It is only meant to remind that an imperfect accumulator introduces noise into a series.]] | ||

<pre> | <pre> | ||

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

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% Also see SUM. | % Also see SUM. | ||

% | % | ||

- | % % Matlab csum() | + | % % Matlab csum() Example: |

% clear all | % clear all | ||

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

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

- | === | + | === sorting === |

In practice, input sorting | In practice, input sorting | ||

<pre> | <pre> | ||

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That is not presented here because the commented Example (inspired by Higham) would then display false positive results. | 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 by orders of magnitude. | 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 22:35, 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

### sorting

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.

### 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