Accumulator Error Feedback

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% This Matlab code implements
% This Matlab code implements
% Kahan's compensated summation algorithm (1964)
% Kahan's compensated summation algorithm (1964)
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% which takes about twice as long as sum() but
 
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% produces more accurate sums when number of elements is large.
 
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% -David Gleich
 
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% Also see SUM.
 
%
%
% Example:
% Example:
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can make compensated summation fail to produce more accurate results than a simple sum.
can make compensated summation fail to produce more accurate results than a simple sum.
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In practice, input sorting can sometimes achieve more accurate summation.
+
Input sorting, in descending order, achieves more accurate summation whereas ascending order reliably fails.
 +
Sorting is not integral to Kahan's algorithm here because is would defeat reversal of the input sequence.
Sorting later became integral to modifications of Kahan's algorithm, such as Priest's
Sorting later became integral to modifications of Kahan's algorithm, such as Priest's
([http://servidor.demec.ufpr.br/CFD/bibliografia/Higham_2002_Accuracy%20and%20Stability%20of%20Numerical%20Algorithms.pdf Higham] Algorithm 4.3).
([http://servidor.demec.ufpr.br/CFD/bibliografia/Higham_2002_Accuracy%20and%20Stability%20of%20Numerical%20Algorithms.pdf Higham] Algorithm 4.3).
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But the very same accuracy dependence on input data prevails.
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But the same accuracy dependence on input data prevails.
=== references ===
=== references ===

Revision as of 23:38, 22 February 2018

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

Matlab's variable precision arithmetic (VPA), (vpa(), sym()) from Mathworks' Symbolic Math Toolbox, can neither accurately sum a few hundred entries in quadruple precision. Error creeps up above |2e-16| for sequences with high condition number (heavy cancellation defined as large sum|x|/|sum x|). Use Advanpix Multiprecision Computing Toolbox for MATLAB, preferentially. Advanpix is hundreds of times faster than Matlab VPA. Higham measures speed here: https://nickhigham.wordpress.com/2017/08/31/how-fast-is-quadruple-precision-arithmetic

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.

Input sorting, in descending order, achieves more accurate summation whereas ascending order reliably fails. Sorting is not integral to Kahan's algorithm here because is would defeat reversal of the input sequence. Sorting later became integral to modifications of Kahan's algorithm, such as Priest's (Higham Algorithm 4.3). But the 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

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