Accumulator Error Feedback

From Wikimization

(Difference between revisions)

Revision as of 15:43, 18 February 2018

csum() in Digital Signal Processing terms: z^-1 is a unit delay,
Q is a 64-bit floating-point quantizer.

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 takes about twice as long as sum() but 
% produces more accurate sums when number of elements is large. 
%                                                -David Gleich
% Also see SUM.
%
% Example:
% clear all; clc
% csumv=0;  rsumv=0;
% n = 100e6;
% t = ones(n,1);
% while csumv <= rsumv
%    v = randn(n,1);
%
%    rsumv = abs((t'*v - t'*v(end:-1:1))/sum(v));
%    disp(['rsumv = ' num2str(rsumv,'%1.16f')]);
%
%    csumv = abs((csum(v) - csum(v(end:-1:1)))/sum(v));
%    disp(['csumv = ' num2str(csumv,'%1.16e')]);
% 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

summing

ones(1,n)*v and sum(v) produce different results in Matlab 2017b with vectors having only a few hundred entries.

sorting

Floating-point compensated summation accuracy is data dependent. Substituting a unit sinusoid at arbitrary frequency, instead of a random number sequence input, can make compensated summation fail to produce more accurate results than a simple sum.

In practice, input sorting can sometimes achieve more accurate summation. Sorting became integral to later algorithms, such as those from Knuth and Priest. But the very same accuracy dependence on input data prevails.

references

Accuracy and Stability of Numerical Algorithms 2e, ch.4.3, Nicholas J. Higham, 2002

Further Remarks on Reducing Truncation Errors, William Kahan, 1964

XSum() Matlab program - Fast Sum with Error Compensation, Jan Simon, 2014

For fixed-point multiplier error feedback, see:

Implementation of Recursive Digital Filters for High-Fidelity Audio

Comments on Implementation of Recursive Digital Filters for High-Fidelity Audio

Retrieved from "http://www.convexoptimization.com/wikimization/index.php/Accumulator_Error_Feedback"

@@ Line 1: / Line 1: @@
-[[Image:Gleich.jpg|thumb|right|429px|CSUM() in Digital Signal Processing terms:
+[[Image:Gleich.jpg|thumb|right|429px|<tt>csum()</tt> in Digital Signal Processing terms:
-z<sup>-1</sup> is a unit delay, Q is a floating-point quantizer to 64 bits,
+z<sup>-1</sup> is a unit delay,<br>Q is a 64-bit floating-point quantizer.
-q<sub>i</sub> represents error due to quantization (additive by definition).  <br>-Jon Dattorro]]
+]]
 <pre>
-function s_hat=csum(x)
+function s_hat = csum(x)
 % CSUM Sum of elements using a compensated summation algorithm.
 %
-% For large vectors, the native sum command in Matlab does
+% This Matlab code implements
-% not appear to use a compensated summation algorithm which
+% Kahan's compensated summation algorithm (1964)
-% can cause significant roundoff errors.
+% which takes about twice as long as sum() but
+% produces more accurate sums when number of elements is large.
+%                                                -David Gleich
+% Also see SUM.
 %
-% This code implements a variant of Kahan's compensated
+% Example:
-% summation algorithm which often takes about twice as long,
+% clear all; clc
-% but produces more accurate sums when the number of
+% csumv=0;  rsumv=0;
-% elements is large. -David Gleich
+% n = 100e6;
+% t = ones(n,1);
+% while csumv <= rsumv
+%    v = randn(n,1);
 %
-% See also SUM
+%    rsumv = abs((t'*v - t'*v(end:-1:1))/sum(v));
+%    disp(['rsumv = ' num2str(rsumv,'%1.16f')]);
 %
-% Matlab Example:
+%    csumv = abs((csum(v) - csum(v(end:-1:1)))/sum(v));
-%   v=rand(1e7,1);
+%    disp(['csumv = ' num2str(csumv,'%1.16e')]);
-%   sum1 = sum(v);
+% end
-%   sum2 = csum(v);
-%   fprintf('sum1 = %18.16e\nsum2 = %18.16e\n', sum1, sum2);
-s_hat=0; y=0; e=0;
+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)
+   e = y - (s_hat - s_hat_old);
 end
+return
 </pre>
-=== links ===
+=== summing ===
-[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]
+<tt>ones(1,n)*v</tt>&nbsp; and &nbsp;<tt>sum(v)</tt>&nbsp; produce different results in Matlab 2017b with vectors having only a few hundred entries.
-For multiplier error feedback, see:
+=== sorting ===
+Floating-point compensated summation accuracy is data dependent.
+Substituting a unit sinusoid at arbitrary frequency, instead of a random number sequence input,
+can make compensated summation fail to produce more accurate results than a simple sum.
-[http://www.stanford.edu/~dattorro/HiFi.pdf Implementation of Recursive Digital Filters for High-Fidelity Audio]
+In practice, input sorting can sometimes achieve more accurate summation.
+Sorting became integral to later algorithms, such as those from Knuth and Priest.
+But the very same accuracy dependence on input data prevails.
-[http://www.stanford.edu/~dattorro/CorrectionsHiFi.pdf Comments on the above...]
+=== references ===
+[http://servidor.demec.ufpr.br/CFD/bibliografia/Higham_2002_Accuracy%20and%20Stability%20of%20Numerical%20Algorithms.pdf Accuracy and Stability of Numerical Algorithms 2e, ch.4.3, Nicholas J. Higham, 2002]
+[http://www.convexoptimization.com/TOOLS/Kahan.pdf Further Remarks on Reducing Truncation Errors, William Kahan, 1964]
+[http://www.mathworks.com/matlabcentral/fileexchange/26800-xsum XSum() Matlab program - Fast Sum with Error Compensation, Jan Simon, 2014]
+For fixed-point multiplier error feedback, see:
+[http://ccrma.stanford.edu/~dattorro/HiFi.pdf Implementation of Recursive Digital Filters for High-Fidelity Audio]
+[http://ccrma.stanford.edu/~dattorro/CorrectionsHiFi.pdf Comments on Implementation of Recursive Digital Filters for High-Fidelity Audio]

Accumulator Error Feedback

From Wikimization

Revision as of 15:43, 18 February 2018

summing

sorting

references

Views

Personal tools

Navigation

Search

Toolbox