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Trying to use Newton’Raphson method to approximate the roots of f(x) = 1/x - D, which would be x = 1/D. This gives x_n+1 = x_n(2-D*x_n).

What would be a good initial guess for this? I saw on Wikipedia that x_0 = 48/17 - 32*D/17 works, but I don’t understand where this approximation comes from, and I don’t see how its useful since it only applies for 0.5 < D < 1. I’m thinking of approximating 1/x with a Geometric Series, but the interval of convergence for that is also limited.

Mailbox
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1 Answers1

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Compare the blue line to the green curve.

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  • I’ve graphed this myself before, I understand that it IS a good approximation. I just don’t understand where it comes from, or how that approximation could potentially be improved—especially since it is only accurate for 0.5 < x < 1. – Mailbox Apr 14 '22 at 14:05
  • @Mailbox: there is no need to precisely justify this approximation (could be least-squares or minmax error), it suffices that it is "good enough" (even the constant 1.33 could be good). The interval (0.5, 1] is on purpose, as it is easy to reduce to this range. Trying to find an approximation in a larger range would be a mistake. –  Apr 14 '22 at 14:36