Newton's method: when it converges, barely converges, and fails entirely
burny_tech · x · 2026-10-01
John D. Cook's article uses Kepler's equation M = E − e sin E to show three faces of Newton's method with the naive guess E = M:
- The Good: for e < 0.5, convergence is guaranteed for all M, with the number of correct digits roughly doubling each step.
- The Bad: at e = 0.991 and M = 0.13π, there's no theoretical guarantee, and the first iterate jumps to E = 5 (violating the known bound E < π) — yet the method still converges to full floating-point precision.
- The Ugly: cited work fixed M = 0.13π and nudged e from 0.991 to 0.993; convergence breaks at certain values just thousandths apart.
Takeaway: Newton's method often works when it shouldn't, and better initial guesses (e.g. Machin's method) extend the good region — but the edge cases remain fragile.
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