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 predated the latter by over 3,000 years. ^[1]

The method[link]

The first two iterations of the secant method. The red curve shows the function f and the blue lines are the secants.

The secant method is defined by the recurrence relation

Failed to parse (Missing texvc executable; please see math/README to configure.): x_n=x_{n-1}-f(x_{n-1})\frac{x_{n-1}-x_{n-2}}{f(x_{n-1})-f(x_{n-2})}

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[link]

Starting with initial values Failed to parse (Missing texvc executable; please see math/README to configure.): x_0

and Failed to parse (Missing texvc executable; please see math/README to configure.): x_1

, we construct a line through the points Failed to parse (Missing texvc executable; please see math/README to configure.): (~x_0,~f(x_0)~)

and Failed to parse (Missing texvc executable; please see math/README to configure.): (~x_1,~f(x_1)~)

, as demonstrated in the picture on the right. In point-slope form, this line has the equation

Failed to parse (Missing texvc executable; please see math/README to configure.): y = \frac{f(x_1)-f(x_0)}{x_1-x_0}(x-x_1) + f(x_1)

We find the root of this line – the value of Failed to parse (Missing texvc executable; please see math/README to configure.): x

such that Failed to parse (Missing texvc executable; please see math/README to configure.): y=0
– by solving the following equation for Failed to parse (Missing texvc executable; please see math/README to configure.): x

Failed to parse (Missing texvc executable; please see math/README to configure.): 0 = \frac{f(x_1)-f(x_0)}{x_1-x_0}(x-x_1) + f(x_1)

The solution is

Failed to parse (Missing texvc executable; please see math/README to configure.): x = x_1 - f(x_1)\frac{x_1-x_0}{f(x_1)-f(x_0)}

We then use this new value of Failed to parse (Missing texvc executable; please see math/README to configure.): x

as Failed to parse (Missing texvc executable; please see math/README to configure.): x_2
and repeat the process using Failed to parse (Missing texvc executable; please see math/README to configure.): x_1
and Failed to parse (Missing texvc executable; please see math/README to configure.): x_2
instead of Failed to parse (Missing texvc executable; please see math/README to configure.): x_0
and Failed to parse (Missing texvc executable; please see math/README to configure.): x_1

. We continue this process, solving for Failed to parse (Missing texvc executable; please see math/README to configure.): x_3 , Failed to parse (Missing texvc executable; please see math/README to configure.): x_4 , etc., until we reach a sufficiently high level of precision (a sufficiently small difference between Failed to parse (Missing texvc executable; please see math/README to configure.): x_n

and Failed to parse (Missing texvc executable; please see math/README to configure.): x_{n-1}

).

Failed to parse (Missing texvc executable; please see math/README to configure.): x_2 = x_1 - f(x_1)\frac{x_1-x_0}{f(x_1)-f(x_0)}

Failed to parse (Missing texvc executable; please see math/README to configure.): x_3 = x_2 - f(x_2)\frac{x_2-x_1}{f(x_2)-f(x_1)}

...

Failed to parse (Missing texvc executable; please see math/README to configure.): x_n = x_{n-1} - f(x_{n-1})\frac{x_{n-1}-x_{n-2}}{f(x_{n-1})-f(x_{n-2})}

Convergence[link]

The iterates Failed to parse (Missing texvc executable; please see math/README to configure.): x_n

of the secant method converge to a root of Failed to parse (Missing texvc executable; please see math/README to configure.): f

, if the initial values Failed to parse (Missing texvc executable; please see math/README to configure.): x_0

and Failed to parse (Missing texvc executable; please see math/README to configure.): x_1
are sufficiently close to the root. The order of convergence is α, where

Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha = \frac{1+\sqrt{5}}{2} \approx 1.618

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

This result only holds under some technical conditions, namely that Failed to parse (Missing texvc executable; please see math/README to configure.): f

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 Failed to parse (Missing texvc executable; please see math/README to configure.): [~x_0,~x_1~] . For example, if Failed to parse (Missing texvc executable; please see math/README to configure.): f

is differentiable on that interval and there is a point where Failed to parse (Missing texvc executable; please see math/README to configure.): f^\prime = 0
on the interval, then the algorithm may not converge.

Comparison with other root-finding methods[link]

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 Failed to parse (Missing texvc executable; please see math/README to configure.): x_{n-1}

and Failed to parse (Missing texvc executable; please see math/README to configure.): x_n

, like the secant method, but on Failed to parse (Missing texvc executable; please see math/README to configure.): x_n

and on the last iterate Failed to parse (Missing texvc executable; please see math/README to configure.): x_k
such that Failed to parse (Missing texvc executable; please see math/README to configure.): f(x_k)
and Failed to parse (Missing texvc executable; please see math/README to configure.): f(x_n)
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

Failed to parse (Missing texvc executable; please see math/README to configure.): x_{n} = x_{n-1} - \frac{f(x_{n-1})}{f^\prime(x_{n-1})}

by using the finite difference approximation

Failed to parse (Missing texvc executable; please see math/README to configure.): f^\prime(x_{n-1}) \approx \frac{f(x_{n-1}) - f(x_{n-2})}{x_{n-1} - x_{n-2}}

. 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 Failed to parse (Missing texvc executable; please see math/README to configure.): f

and its derivative Failed to parse (Missing texvc executable; please see math/README to configure.): f^\prime
at every step, while the secant method only requires the evaluation of Failed to parse (Missing texvc executable; please see math/README to configure.): f

. Therefore, the secant method may occasionally be faster in practice. For instance, if we assume that evaluating Failed to parse (Missing texvc executable; please see math/README to configure.): f

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.

Generalizations[link]

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[link]

The Secant method is applied to find a root of the function f(x)=x²−612. Here is an implementation in the Matlab language.

<source lang=Matlab>

1. 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) </source>

Notes[link]

See also[link]

• False position method

References[link]

• Kaw, Autar; Kalu, Egwu (2008), Numerical Methods with Applications (1st ed.), http://www.autarkaw.com/books/numericalmethods/index.html .
• Allen, Myron B.; Isaacson, Eli L.. Numerical analysis for applied science. John Wiley & Sons. pp. 188–195. ISBN 978-0-471-55266-6. http://books.google.co.uk/books?id=PpB9cjOxQAQC. 

External links[link]

• Java code By Behzad Torkian
• Animations for the secant method
• Secant Method Notes, PPT, Mathcad, Maple, Mathematica, Matlab at Holistic Numerical Methods Institute
• Weisstein, Eric W., "Secant Method" from MathWorld.
• Module for Secant Method by John H. Mathews
• Online root finding of a polynomial-Secant method by Farhad Mazlumi