Discussion
Let \(f : \mathbb{R} \to \mathbb{R}\) be a twice continuously differentiable function. The order of convergence of the secant method for finding root of the equation \(f(x) = 0\) is
Answer: A
The order of convergence is \(\frac{1 + \sqrt{5}}{2}\), about 1.618.
For the secant method near a simple root, the errors satisfy \(e_{n+1} \approx C e_{n} e_{n-1}\). Seeking \(|e_{n+1}| \approx A|e_{n}|^{p}\) and substituting gives \(p^{2} = p + 1\), whose positive root is the golden ratio:
\[ p = \frac{1 + \sqrt{5}}{2} \approx 1.618 \]
So the secant method is superlinear but slower than Newton's quadratic method. \(\frac{2}{1 + \sqrt{5}} \approx 0.618\) is the reciprocal, less than 1, which is not an order of convergence; \(\frac{1+\sqrt{5}}{3} \approx 1.08\) and \(\frac{3}{1+\sqrt{5}} \approx 0.93\) do not solve \(p^{2} = p + 1\).
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