Evaluating numerical algorithm
The Secant Method is an open root-finding method that approximates derivatives using two recent function evaluations — making it faster than Bisection with no need for analytical derivatives. It achieves super-linear convergence (order ≈ 1.618, the golden ratio).
Find the root of using the Secant Method with , , tolerance = 0.0001.
Calculate and to set up the first secant line:
Plug into the secant formula to find the first new estimate :
Update the pair: the old becomes the new , and becomes the new . Repeat until .
| k | x₀ | x₁ | x₂ (New) | f(x₀) | f(x₁) | Ea (%) |
|---|---|---|---|---|---|---|
| 1 | 2.000000 | 3.000000 | 2.058824 | -1.000000 | 16.000000 | 45.714% |
| 2 | 3.000000 | 2.058824 | 2.081264 | 16.000000 | -0.390800 | 1.078% |
| 3 | 2.058824 | 2.081264 | 2.094824 | -0.390800 | -0.147204 | 0.647% |
| 4 | 2.081264 | 2.094824 | 2.094549 | -0.147204 | 0.003044 | 0.013% |
The Secant Method converged in just 4 iterations, compared to 14 for Bisection on the same problem. This is the power of the -order (≈1.618) convergence rate — each step dramatically improves accuracy! Note: unlike Newton-Raphson, no derivative was required.