To find approximate sqrta we can use Newton's method to approximately solve the equation x^2−a=0 for x, starting from some rational x_0. Newton's method in general is only locally convergent, so we have to be careful with initialization.Show that in this case, the method always converges to something if x_0≠0.

zementj5 2022-09-25 Answered
To find approximate a we can use Newton's method to approximately solve the equation x 2 a = 0 for x, starting from some rational x 0 .
Newton's method in general is only locally convergent, so we have to be careful with initialization.
Show that in this case, the method always converges to something if x 0 0.
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Answers (1)

Dillon Levy
Answered 2022-09-26 Author has 12 answers
The most convenient method to analyze this iteration is to explore the fraction
θ n = x n a x n + a
as you will then find that
θ n + 1 = θ n 2 θ n = θ 0 2 n   x n = 1 + θ 0 2 n 1 θ 0 2 n a = a + 2 θ 0 2 n 1 θ 0 2 n a
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