Error margins

[Albert Chan]

(10-27-2024 02:51 PM)Thomas Klemm Wrote:  You might consider dual numbers:
Automatic differentiation using dual numbers
Here's an example on how this could be used for a circuit of two parallel resistors.

FYI, this only work with + * (parallel resistance intervals work due to luck)
For - /, we need to flip the sign of 2nd number, to maximize intervals
Other changes may need to adapt to other functions.

lua> R1 = D.new(3,4e-5) ≡ 3 ± 4*ε
lua> R2 = D.new(5,6e-5) ≡ 5 ± 6*ε
lua> pprint(R1-R2) -- BAD
{ -2, -1.9999999999999998e-05 }
lua> pprint(R1/R2) -- BAD
{ 0.6, 8e-07 }

lua> R2[2] = -R2[2] -- flip sign
lua> pprint(R1-R2) -- OK
{ -2, 1e-04 }
lua> pprint(R1/R2) -- OK
{ 0.6, 1.52e-05 }

Note, last is not true interval, but if ε is small, it is good enough.
Interval math, center is affected by ε term. Dual does not do that.

R1/R2 = (3-4ε)/(5+6ε) .. (3+4ε)/(5-6ε) ≈ 0.6000000001824 ± 1.520000000219e-5

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For more complicated expression, however, end-points may not work.
For f(x±eps), we really need to consider the whole shape.

a = min(f(x±eps))
b = max(f(x±eps))
f(x±eps) = (b+a)/2 ± (b-a)/2

Worse, calculations cannot not be "chained".
If we apply g on top, f(x±eps) calculations should be tossed.

a = min(g(f(x±eps)))
b = max(g(f(x±eps)))
g(f(x±eps)) = (b+a)/2 ± (b-a)/2

Of course, we can ignore all this, and just use interval end-points.
But, resulting intervals may not be right. Garbage in, Garbage out.