Newton's method is a root-finding algorithm that uses linear approximation to find a zero of a function. The basic idea is to start with an initial estimate of the zero and iterate using the formula:

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where `f(x)`

is the function and `f'(x)`

is its derivative.

Here's how to implement Newton's method in MATLAB:

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To use this function, you need to define the function `f`

and its derivative `df`

as separate MATLAB functions. For example, to find a zero of the function `f(x) = x^2 - 2`

, you can define:

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Then, you can call the `newton`

function with an initial guess `x0`

and the desired tolerance and maximum number of iterations, like this:

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This will return an estimated zero `x`

and the number of iterations `iter`

required to find it.

Note that Newton's method can fail to converge (i.e., enter an infinite loop) if the initial guess is too far from the zero, or if the function or its derivative are undefined or have a singularity near the zero. Therefore, it's important to choose a reasonable initial guess and to test the function for different inputs.

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