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Comment by gpsarakis | Hacker News Reader

Nice analysis. Hope you don't mind me adding that by omitting terms of the Taylor series you do have some loss of precision, however small. Also, solving linear equation systems may even introduce instability as the following must be preserved: http://en.wikipedia.org/wiki/Diagonally_dominant_matrix

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The point is that when you're considering the Taylor series for a dual number argument, you don't lose any precision, because higher powers of the "imaginary" part of the dual number vanish. The example he gives is

    f(a + be) = f(a) + f'(a)be + 0.5 * f''(a) b^2 e^2 + O(e^3)
              = f(a) + f'(a)be

because e^n=0 for all n>1. This isn't an approximation - it's an exact relationship for dual numbers!

You will lose some precision by using floating point numbers instead of an arbitrary-precision real number type, but this is a limitation of the machine you're working on. The method is exact.

No, the coefficients of the Taylor series are the exact derivatives, assuming the actual arithmetic were exact (it's not, because IEEE 754). There's no loss of precision there.

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