Consider the equation $f(x)=0$ where $f$ is a $C^1$ function which has a root $x^*$. We can rewrite the equation as $x=g(x)=x-cf(x)$ for some constant $c$. We are asked to find the values of $c$ for which the iterative method $x_{n+1}=g(x_{n})$ converges to the root $x^*$ for initial $x_0$ close to $x^*$. Also for which values of $c$ do we have a superlinear convergence?
Fixed-Point Iteration: For what constant does it converge?
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1In view of the Banach fixed point theorem, you need the derivative of the fixed point mapping to be less than $1$ in absolute value at $x^*$. – 2017-01-26
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0@Masacroso "Root of a function" is the same thing as "zero of a function". "Root of an equation" does not make sense but that's not what was said. – 2017-01-26
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0@Ian yes, you are right, thank you. – 2017-01-26