In numerical analysis, the secant method is a root-finding algorithm that uses a succession of roots of secant lines to better approximate a root of a function f. The secant method can be thought of as a finite-difference approximation of Newton's method. However, the method was developed independently of Newton's method and predates it by over 3000 years.
The secant method is defined by the recurrence relation
As can be seen from the recurrence relation, the secant method requires two initial values, x0 and x1, which should ideally be chosen to lie close to the root.
Derivation of the method
Starting with initial values x0 and x1, we construct a line through the points (x0, f(x0)) and (x1, f(x1)), as demonstrated in the picture above. In slope–intercept form, this line has the equation
We find the root of this line – the value of x such that y = 0 – by solving the following equation for x:
The solution is
We then use this new value of x as x2 and repeat the process, using x1 and x2 instead of x0 and x1. We continue this process, solving for x3, x4, etc., until we reach a sufficiently high level of precision (a sufficiently small difference between xn and xn−1):
The iterates of the secant method converge to a root of , if the initial values and are sufficiently close to the root. The order of convergence is φ, where
This result only holds under some technical conditions, namely that be twice continuously differentiable and the root in question be simple (i.e., with multiplicity 1).
If the initial values are not close enough to the root, then there is no guarantee that the secant method converges. There is no general definition of "close enough", but the criterion has to do with how "wiggly" the function is on the interval . For example, if is differentiable on that interval and there is a point where on the interval, then the algorithm may not converge.
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
The secant method can be interpreted as a method in which the derivative is replaced by an approximation and is thus a quasi-Newton method.
If we compare Newton's method with the secant method, we see that Newton's method converges faster (order 2 against φ ≈ 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 φ² ≈ 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.
Broyden's method is a generalization of the secant method to more than one dimension.
The following graph shows the function f in red and the last secant line in bold blue. In the graph, the x intercept of the secant line seems to be a good approximation of the root of f.
A computational example
The secant method is applied to find a root of the function f(x) = x2 − 612. Here is an implementation in the MATLAB language (from calculation, we expect that the iteration converges at x = 24.7386):
f=@(x) x^2 - 612; x(1)=10; x(2)=30; for i=3:7 x(i) = x(i-1) - (f(x(i-1)))*((x(i-1) - x(i-2))/(f(x(i-1)) - f(x(i-2)))); end root=x(7)
- Papakonstantinou, J., The Historical Development of the Secant Method in 1-D, retrieved 2011-06-29
- Kaw, Autar; Kalu, Egwu (2008), Numerical Methods with Applications (1st ed.).
- Allen, Myron B.; Isaacson, Eli L. (1998). Numerical analysis for applied science. John Wiley & Sons. pp. 188–195. ISBN 978-0-471-55266-6.