Accumulator Error Feedback
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| - | + | [[Image:Gleich.jpg|thumb|right|429px|CSUM() in Digital Signal Processing terms: | |
| + | z<sup>-1</sup> is a unit delay, Q is a floating-point quantizer to 64 bits, | ||
| + | q<sub>i</sub> represents error due to quantization (additive by definition). <br>-Jon Dattorro]] | ||
| + | <pre> | ||
| + | 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 | ||
| + | % | ||
| + | % See also SUM | ||
| + | % | ||
| + | % Example: | ||
| + | % v=rand(1e7,1); | ||
| + | % sum1 = sum(v); | ||
| + | % sum2 = csum(v); | ||
| + | % fprintf('sum1 = %18.16e\nsum2 = %18.16e\n', sum1, sum2); | ||
| + | |||
| + | s_hat=0; y=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 | ||
| + | </pre> | ||
| + | |||
| + | === links === | ||
| + | [http://www.google.com/books?id=FJyBjjtHREQC&dq=Accuracy+and+Stability+of+Numerical+Algorithms&printsec=frontcover&source=bn#PPA92,M1 Accuracy and Stability of Numerical Algorithms, Nicholas J. Higham, 1996] | ||
| + | |||
| + | For multiplier error feedback, see: | ||
| + | |||
| + | [http://www.stanford.edu/~dattorro/HiFi.pdf Implementation of Recursive Digital Filters for High-Fidelity Audio] | ||
| + | |||
| + | [http://www.stanford.edu/~dattorro/CorrectionsHiFi.pdf Comments on the above...] | ||
Revision as of 16:24, 5 March 2009
-Jon Dattorro
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
%
% See also SUM
%
% Example:
% v=rand(1e7,1);
% sum1 = sum(v);
% sum2 = csum(v);
% fprintf('sum1 = %18.16e\nsum2 = %18.16e\n', sum1, sum2);
s_hat=0; y=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
links
Accuracy and Stability of Numerical Algorithms, Nicholas J. Higham, 1996
For multiplier error feedback, see:
Implementation of Recursive Digital Filters for High-Fidelity Audio