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The equation $ e^{x} - 4x^{2}=0 $ has a root between 4 and 5. Fixed point iteration with iteration function $ \frac{1}{2}e^{\frac{x}{2}} $

(A) diverges (B) converges (C) oscillates (D) converges monotonically

I know that if we use the fixed point iteration scheme $ x_{n+1}= \phi(x_{n}) $ with iteration function $ \phi(x) $ and $ \phi^{'}(x) \lt 1 $ in the neighbourhood of the solution (which also contain the initial approximation $ x_{0} $ ) then the iterations converge to the solution . for $ \phi(x)= \frac{1}{2}e^{\frac{x}{2}} $ , we have $ \phi^{'}(x) \gt 1 $ in the interval $ [4,5] $ so we can't say whether the iterations converge to the solution or not. Then how can we proceed or think of any other way?

Mohit Sharma
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1 Answers1

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As it is not possible to "stumble" upon the solution from outside of $[4,5]$, and we know that $|\phi'(x)|>1$ for all $x$ in $[4,5]$, it follows that the iterations will simply diverge.

$\phi'(x)$ essentially represents the behavior of the iterations near the root, with some notable cases:

  • Oscillation occurs if $\phi'(x)$ is negative over all values.

  • The convergence or divergence is monotone if $\phi'(x)$ is positive over all values.

  • Convergence occurs if $|\phi'(x)|<1$ and you can expect to get $|\phi'(x)|$ times closer to the fixed-point on every iteration.

  • Divergence occurs if $|\phi'(x)|>1$ and you can expect to get $|\phi'(x)|$ times farther from the fixed-point on every iteration.

The only case where it would be uncertain if this converged or not would be if you had both $|\phi'(x)|<1$ and $|\phi'(x)|>1$ near the root. For this example, this is not the case.

Simply Beautiful Art
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