Mathematics · the experiment
Double your correct digits
Find the zero of f(x) = eˣ − 1, starting from x = 0.1. Count what each question asked of the function earns you, up to the Kung–Traub limit.
1 · Two questions per step
Beginner
Newton's method asks two questions at each step: the value f(x) and the slope f′(x). Close to the zero, what does one step do to the number of correct digits?
2 · One step, more questions
Intermediate
With four questions per step, a memoryless method can reach order 8. From x = 0.1, that is one correct digit, how many correct digits after a single step?
A single step from x = 0.1. Pick the number of questions that passes 15 correct digits with as few questions as possible.
3 · Start closer
Expert
Still four questions, so order 8, but starting ten times closer: x = 0.01. How many correct digits after one step?
Four questions per step. Move the starting point closer until a single step passes 30 correct digits.
4 · Ask everything at once
Expert
Six questions, asked in three groups of two: both questions of a group are sent together, without waiting for their answers. What is the highest possible order?
Limit reached
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