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Prove that the iterative scheme

$$X_{r+1} = g(X_r) = e^{X_r^{2}-2X_r}$$

with a suitable starting point, converges to the root in $[0.4,0.6]$, by showing that $g$ is a contraction mapping on this interval.

Compute the root in $[0.4,0.6]$ to two decimal places

Why the iterative scheme would not be appropriate to use to try to find the larger root?

Empiricist
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keos
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1 Answers1

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$g(x) = e^{x^2 - 2x}$

$|g'(x)| = |(2x - 1)e^{x^2 - 2x}| \leq 0.2 \times e^{-0.64} < 1$

By Banach fixed point theorem, the fixed-point iteration converges with any initial guess in $[0.4, 0.6]$.

If the root is larger, then $|g'(x)|$ can be very large in a neighborhood of the root. The fixed-point iteration may fail.

Empiricist
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