# Accumulator Error Feedback

(Difference between revisions)
 Revision as of 15:46, 2 March 2009 (edit) (Redirecting to PageRank)← Previous diff Revision as of 15:24, 5 March 2009 (edit) (undo)Next diff → Line 1: Line 1: - #REDIRECT [[PageRank]] + [[Image:Gleich.jpg|thumb|right|429px|CSUM() in Digital Signal Processing terms: + z-1 is a unit delay, Q is a floating-point quantizer to 64 bits, + qi represents error due to quantization (additive by definition).
-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
+                                                            %
+                                                            %
+                                                            % 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 === + [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 15:24, 5 March 2009

CSUM() in Digital Signal Processing terms: z-1 is a unit delay, Q is a floating-point quantizer to 64 bits, qi represents error due to quantization (additive by definition).
-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
%
%
% 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
```