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    Compute the fourth root of 20 using Newton-Raphson interation

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    Give the Newton-Raphson iteration scheme for computing the pth power of a number.

    Specialize this to computing the fourth root of 20. You can take the initial guess to be 2.

    At each iteration, state the number of correct digits.

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    Solution Preview

    Given y, we want to compute x = y^p. This means that x satisfies the equation:

    x^(1/p) = y ----->

    x^q - y = 0

    where q = 1/p

    We can then apply Newton-Raphson iteration. In general, given an equation f(x) = 0 and

    initial guess x_0, this yields the sequence of approximants to the solution of:

    x_{n+1} = x_n - ...

    Solution Summary

    The given answer assumes familiarity with Newton-Raphson iteration. It gives a detailed answer using the Newton-Raphson formula.