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An error estimate for fixed point computation of b^x
#1
Ok, we want to compute the lower fixed point of for, as usual, .

We do that by iterating:



while iterating we can not directly compute the difference but we can compute and we ask our self how close must come to 1 such that



First we translate the difference into a quotient:


is decreasing. Now lets compute


To make an estimate we want to know whether , is decreasing for . To decide this we differentiate:


for



This is true for because then .

So we know that is strictly increasing for small enough . So we get
iff
for , equivalently:

iff
iff
The right side can be further simplified:



The condition is satisfied for , i.e. on the right side is a constant . So during the iteration we can decrease according to without fear that becomes to small and the iteration does not stop.

Proposition. Let , let be the lower fixed point of , let and then for each :
iff .
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