Accumulator Error Feedback

From Wikimization

(Difference between revisions)
Jump to: navigation, search
(links)
Line 18: Line 18:
%
%
% % Matlab csum() example:
% % Matlab csum() example:
-
% clear all
+
% clear all
-
% % v = sort(randn(13e6,1),'descend'); %better when sorted
+
% csumv = 0;
-
% v = randn(13e6,1);
+
% while ~csumv
-
% rsumv = abs(sum(v) - sum(v(end:-1:1)));
+
% v = randn(13e6,1);
-
% disp(['rsumv = ' num2str(rsumv,'%18.16f')]);
+
% rsumv = abs(sum(v) - sum(v(end:-1:1)));
-
% csumv = abs(csum(v) - csum(v(end:-1:1)));
+
% disp(['rsumv = ' num2str(rsumv,'%18.16f')]);
-
% disp(['csumv = ' num2str(csumv,'%18.16f')]);
+
% csumv = abs(csum(v) - csum(v(end:-1:1)));
-
% % vsumv = sum(vpa(v)) - sum(vpa(v(end:-1:1))); %vpa toolbox 32GB RAM
+
% disp(['csumv = ' num2str(csumv,'%18.16e')]);
-
% % disp(['vsumv = ' char(vsumv)])
+
% end
 +
[~, idx] = sort(abs(x),'descend');
 +
x = x(idx);
s_hat=0; e=0;
s_hat=0; e=0;
for i=1:numel(x)
for i=1:numel(x)
Line 35: Line 37:
e = (s_hat_old - s_hat) + y; %calculate difference first (Higham)
e = (s_hat_old - s_hat) + y; %calculate difference first (Higham)
end
end
 +
return
</pre>
</pre>

Revision as of 15:29, 5 October 2017

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
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
%
% Also see SUM.
%
% % Matlab csum() example:
% clear all
% csumv = 0;
% while ~csumv
%    v = randn(13e6,1);
%    rsumv = abs(sum(v) - sum(v(end:-1:1)));
%    disp(['rsumv = ' num2str(rsumv,'%18.16f')]);
%    csumv = abs(csum(v) - csum(v(end:-1:1)));
%    disp(['csumv = ' num2str(csumv,'%18.16e')]);
% end

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

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

Personal tools