Glad to see "Yes" a couple of times, and am impressed by your speed of reply (as always!) As for the rest, I need some thinking time:-)
David ---------------------------------------- > From: [EMAIL PROTECTED] > To: [email protected] > Date: Thu, 14 Dec 2006 14:44:54 +0000 > Subject: Re: [Prime] Checksum > > On Thursday 14 December 2006 14:05, david eddy wrote: > > Quote from the "Math" page > > << > > There is another error check that is fairly cheap. One property of FFT > > squaring is that: > > (sum of the input FFT values)^2 = (sum of the output IFFT values) > > Since we are using floating point numbers we must change the "equals sign" > > above to "approximately equals". > > > > > > If we were workong to infinite precison, shouldn't the output IFFT values > > be integers? > > Yes. > > > > In which case rounding these values to integers before summing them should > > provide exactly the stringent checksum we want? > > Yes. > > But in any case it's a moot point because the (input) "phase space" values > aren't integers, and adding (FFT run length) strictly positive values > together is likely to cause a loss of precision even if they were. (Run > length = 2^20 is likely to lose about 20 bits of precision in the sum, > compared with the precision of the inputs, if you're thinking in fixed point > terms. When working in floating point, you won't actually get a wraparound > type error like integer overflow, but you do lose the least significant bit > of the mantissa every time you add two values with the same sign and the same > exponent as the exponent of the result has to increase.) > > In fact, as the FFT run length increases, the checksum criterion becomes > increasingly poor as a predictor of accuracy compared with the roundoff error > check criterion, as the imprecision of the checksum is proportional to the > logarithm of the number of elements in the computation whereas the roundoff > error increases much more slowly. > > Also, the "special features" of the transform used for the L-L test give us > the "subtract 2" part of the L-L algorithm for free; this may interfere in > some way with the exact equality mentioned in the basic theory. > > Regards > Brian Beesley > _______________________________________________ > Prime mailing list > [email protected] > http://hogranch.com/mailman/listinfo/prime _________________________________________________________________ Be one of the first to try Windows Live Mail. http://ideas.live.com/programpage.aspx?versionId=5d21c51a-b161-4314-9b0e-4911fb2b2e6d _______________________________________________ Prime mailing list [email protected] http://hogranch.com/mailman/listinfo/prime
