# Accumulator Error Feedback

### From Wikimization

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In practice, input sorting can sometimes achieve more accurate summation. | In practice, input sorting can sometimes achieve more accurate summation. | ||

- | Sorting became integral to | + | Sorting became integral to modifications to Kahan's algorithm, such as those from Knuth and Priest. |

But the very same accuracy dependence on input data prevails. | But the very same accuracy dependence on input data prevails. | ||

## Revision as of 21:55, 22 February 2018

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.

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 **(**defined as 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.

In practice, input sorting can sometimes achieve more accurate summation. Sorting became integral to modifications to Kahan's algorithm, 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

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