Estimate the number of iterations of Newton's method needed to find a root of f(x)=cos(x)−x to within 10^(−100).

Izabelle Lowery 2022-10-12 Answered
Estimate the number of iterations of Newton's method needed to find a root of f ( x ) = cos ( x ) x to within 10 100 .
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Answers (1)

Adalyn Pitts
Answered 2022-10-13 Author has 15 answers
The idea behind the reasoning is the quadratic convergence of Newton's algorithm (if the zero is simple).
When you are near the zero α, an iteration takes you from α + δ to
( α + δ ) f ( α + δ ) f ( α + δ ) α + δ f ( α ) δ + f ( α ) δ 2 2 f ( α ) + f ( α ) δ α + f ( α ) 2 f ( α ) δ 2 ,
so each step roughly doubles the number of correct digits.
If you start with approximately one correct digit, after seven steps, you have roughly 2 7 = 128 correct digits.
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