Comparison With Other Root-finding Methods
The secant method does not require that the root remain bracketed like the bisection method does, and hence it does not always converge. The false position method (or regula falsi) uses the same formula as the secant method. However, it does not apply the formula on and, like the secant method, but on and on the last iterate such that and have a different sign. This means that the false position method always converges.
The recurrence formula of the secant method can be derived from the formula for Newton's method
by using the finite difference approximation
- .
If we compare Newton's method with the secant method, we see that Newton's method converges faster (order 2 against α ≈ 1.6). However, Newton's method requires the evaluation of both and its derivative at every step, while the secant method only requires the evaluation of . Therefore, the secant method may occasionally be faster in practice. For instance, if we assume that evaluating takes as much time as evaluating its derivative and we neglect all other costs, we can do two steps of the secant method (decreasing the logarithm of the error by a factor α² ≈ 2.6) for the same cost as one step of Newton's method (decreasing the logarithm of the error by a factor 2), so the secant method is faster. If however we consider parallel processing for the evaluation of the derivative, Newton's method proves its worth, being faster in time, though still spending more steps.
Read more about this topic: Secant Method
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