Accumulator Error Feedback

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Input sorting, in descending order of absolute value, achieves more accurate summation whereas ascending order reliably fails.
Input sorting, in descending order of absolute value, achieves more accurate summation whereas ascending order reliably fails.
-
Sorting is not integral to Kahan's algorithm here because it would defeat reversal of the input sequence in the example.
+
Sorting is not integral to Kahan's algorithm above because it would defeat reversal of the input sequence in the example.
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),

Revision as of 14:18, 25 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 + 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, cannot 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 of absolute value, achieves more accurate summation whereas ascending order reliably fails. Sorting is not integral to Kahan's algorithm above because it would defeat reversal of the input sequence in the example. 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:

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