Secant method

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 secant method predates Newton's method by over 3000 years.[1]

The first two iterations of the secant method. The red curve shows the function f, and the blue lines are the secants. For this particular case, the secant method will not converge to the visible root.

The methodEdit

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 methodEdit

Starting with initial values x0 and x1, we construct a line through the points (x0, f(x0)) and (x1, f(x1)), as shown in the picture above. In slope–intercept form, the equation of this line is

 

The root of this linear function, that is the value of x such that y = 0 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):

 

ConvergenceEdit

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

 

is the golden ratio. In particular, the convergence is superlinear, but not quite quadratic.

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 methodsEdit

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; however, only with a linear order of convergence. Bracketing with a super-linear order of convergence as the secant method can be attained with improvements to the false position method (see Regula falsi § Improvements in regula falsi) such as the ITP method or Illinois method.

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.

GeneralizationsEdit

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.

Computational exampleEdit

Below, the secant method is implemented in the Python programming language.

It is then applied to find a root of the function f(x) = x2 − 612 with initial points   and  

def secant_method(f, x0, x1, iterations):
    """Return the root calculated using the secant method."""
    for i in range(iterations):
        x2 = x1 - f(x1) * (x1 - x0) / float(f(x1) - f(x0))
        x0, x1 = x1, x2
    return x2

def f_example(x):
    return x ** 2 - 612

root = secant_method(f_example, 10, 30, 5)

print("Root: {}".format(root))  # Root: 24.738633748750722

NotesEdit

  1. ^ Papakonstantinou, J., The Historical Development of the Secant Method in 1-D, retrieved 2011-06-29

See alsoEdit

ReferencesEdit

External linksEdit