Accumulator Error Feedback
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(inspired by Higham) would then display false positive results.<br> | (inspired by Higham) would then display false positive results.<br> | ||
In practice, input sorting | In practice, input sorting | ||
- | should begin the <tt>csum()</tt> function to achieve the most accurate summation | + | should begin the <tt>csum()</tt> function to achieve the most accurate summation: |
<pre> | <pre> | ||
function s_hat = csum(x) | function s_hat = csum(x) |
Revision as of 21:50, 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.
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
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