For the following exercises, solve f(x)=0 using the...

For the following exercises, solve $f(x)=0$ using the iteration $x_{n+1}=x_{n}-c f\left(x_{n}\right),$ which differs slightly from Newton's method. Find a $c$ that works and a $c$ that fails to converge, with the exception of $c=0.$
$$f(x)=x^{2}-4 x+3, \quad \text { with } x_{0}=2$$

Key Concepts

Fixed Point Iteration

This is a numerical method for solving equations of the form f(x)=0 by reformulating the problem as finding a fixed point of a function g(x) so that x = g(x). In the given iteration, the update formula x??? = x? - c·f(x?) creates a sequence that ideally converges to a root of f provided that the chosen function g (which depends on c) has a fixed point at the root and appropriate convergence properties.

Convergence and Stability Criteria

For iterative methods to converge to a fixed point, the iteration function must behave contractively near that point. This is typically quantified by the derivative of the iteration function g; convergence requires that the absolute value of g? evaluated at the fixed point is less than 1. If |g?(r)| ? 1 at a root r, then the fixed point is non?attractive, meaning that iterations starting close to r may diverge.

Scaling Parameter in Iterative Methods

The constant c in the iteration x??? = x? - c·f(x?) acts as a scaling factor, modifying the step size of the update. Its proper selection is critical: an appropriate value of c will ensure that the derivative of the corresponding iteration function is small enough (in absolute value) to yield convergence, while a poorly chosen c can make the iterative update unstable and lead to divergence. This idea is closely related to the adjustment made in methods like Newton's method, where the derivative of f is used to determine an optimal step size.

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