# Accumulator Error Feedback

### From Wikimization

(Difference between revisions)

Line 44: | Line 44: | ||

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

- | + | Sorting is not integral above because the commented Example | |

+ | (inspired by Higham) would then display false positive results. | ||

+ | But, in practice, input sorting | ||

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

<pre> | <pre> | ||

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

+ | s_hat=0; e=0; | ||

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

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

+ | 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 | ||

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

- | should begin the <tt>csum()</tt> 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 complete absence of sorting, <tt>csum()</tt> can be more accurate than conventional summation by orders of magnitude. | Even in complete absence of sorting, <tt>csum()</tt> can be more accurate than conventional summation by orders of magnitude. | ||

## Revision as of 20:11, 20 December 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 Matlab 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

Sorting is not integral above because the commented Example
(inspired by Higham) would then display false positive results.
But, 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 = (s_hat_old - s_hat) + y; %calculate difference first (Higham) 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

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