How many iterations are required to obtain full IEEE double precision accuracy for Secant Method?

Consider the rate of convergence in the Secant method is given by e(k) = c*e(k-1)^1.618.

Given an initial error e(0) = 2^−1, what is the number of iterations k required to obtain full IEEE double precision accuracy of ε = 2^−53 ?

I calculate the value of k analytically by

e(k) = e(0)^(1.618^k) = ε

The calculation gives k ≈ 8.2509. Thus, the number of iterations is 9. Is it correct?

[Matt J]

You're assuming c=1?