Newton's division-free reciprocal

MediumCalculus~5m

Some hardware has fast multipliers but no divider, so 1/a1/a is computed by Newton's method on a function whose root is 1/a1/a — and the resulting iteration must contain no division. Derive that iteration, then compute one step for a=3a=3 starting from the guess x0=0.3x_0 = 0.3. Give x1x_1 (to 4 decimals).

Show hints (2)+
  1. Use f(x)=1/xaf(x)=1/x - a so the root is 1/a1/a; its derivative is 1/x2-1/x^2.
  2. The Newton step simplifies to xn+1=xn(2axn)x_{n+1}=x_n(2-a x_n) — no division. Plug a=3a=3, x0=0.3x_0=0.3.

Answer

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0.33 (± 0.0001)

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Asked at: DRW, Two Sigma

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