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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 Summary

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

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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 - ...

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