 An error estimate for fixed point computation of b^x - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Computation (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=8) +--- Thread: An error estimate for fixed point computation of b^x (/showthread.php?tid=173) An error estimate for fixed point computation of b^x - bo198214 - 05/31/2008 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 .