XSum
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XSum - SUM with error compensation
The accuracy of the sum of floating point numbers is limited by the truncation error. E.g. SUM([1e16, 1, -1e16]) replies 0 instead of 1 and the error of SUM(RANDN(N, 1)) is about EPS*(N / 10).
Kahan, Knuth, Dekker, Ogita and Rump (and others) have derived some methods to reduce the influence of rounding errors, which are implemented here as fast C-Mex: XSum(RANDN(N, 1), 'Knuth') is exact to all 15 digits.
Y = XSum(X, N, Method)
INPUT:
X: Double array of any size.
N: Dimension to operate on.
Method: String: 'Double', 'Long', 'Kahan', 'Knuth', 'KnuthLong', 'Knuth2'.
OUTPUT:
Y: Double array, equivalent to SUM, but with compensated error depending
on the Method. The high-precision result is rounded to double precision.
METHODS: (speed and accuracy compared to SUM)
- Double: A thread-safe implementation of Matlab's SUM. At least in Matlab 2008a to 2009b the results of the multi-threaded SUM can differ slightly from call to call. 50% faster than single-threaded SUM (MSVC++ 2008), equivalent accuracy.
- Long: Accumulated in a 80 bit long double, if the compiler support this (e.g. LCC v3.8). 3.5 more valid digits, 40% slower.
- Kahan: The local error is subtracted from the next element. 1 to 3 more valid digits, 10% slower.
- Knuth: As if the sum is accumulated in a 128 bit float: about 15 more valid digits. About the same speed (MSVC++2008 compiler). This is suitable for the most real world problems.
- Knuth2: 30 more valid digits as if it is accumulated in a 196 bit float. 60% slower.
- KnuthLong: As Knuth, but using long doubles to get about 21 more valid digits, if supported by the compiler. 2.5 times slower.
Free download from Shareware Connection - XSum - SUM with error compensation
Version: 1.0 | Size: 10 KB | Platform: Matlab, Scripts
Released Date: 19-05-2013 | Rating: 0 | Title: XSum
Author Url: http://www.mathworks.com
Program Info Url: http://www.mathworks.com
Download Url: http://www.mathworks.com/matlabcentral/fx_files/26800/1/XSum.zip
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