Accumulator Error Feedback
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(Difference between revisions)
Line 34: | Line 34: | ||
y = x(i) + e; | y = x(i) + e; | ||
s_hat = s_hat_old + y; | s_hat = s_hat_old + y; | ||
- | e = (s_hat_old | + | e = y - (s_hat - s_hat_old); |
end | end | ||
return | return |
Revision as of 20:15, 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. % % % 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 = y - (s_hat - s_hat_old); end return
sorting
Sorting is not integral above because the commented Example
(inspired by Higham) would then display false positive results.
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 = y - (s_hat - s_hat_old); %calculate parentheses first 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
Further Remarks on Reducing Truncation Errors, William Kahan, 1964
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