XSum 1.0

XSum - SUM with error compensationThe 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.

Requirements:	No special requirements
Platforms:	Matlab
Keyword:	2008a,  2009b,  Differ,  Faster,  Matlab,  Matlabs,  Multithreaded,  Results,  Slightly
Users rating:	0/10