View source for Secant method

{{Use American English|date = March 2019}}
{{Short description|Root-finding method}}
[[Image:Secant method.svg|thumb|300px|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.]]

In [[numerical analysis]], the '''secant method''' is a [[root-finding algorithm]] that uses a succession of [[Root of a function|roots]] of [[secant line]]s to better approximate a root of a [[Function (mathematics)|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.<ref>{{Citation |last=Papakonstantinou |first=J. |title=The Historical Development of the Secant Method in 1-D |url=http://citation.allacademic.com/meta/p_mla_apa_research_citation/2/0/0/0/4/p200044_index.html | accessdate = 2011-06-29}}</ref>

==The method==
The secant method is defined by the [[recurrence relation]]
:<math>
x_n
 = x_{n-1} - f(x_{n-1}) \frac{x_{n-1} - x_{n-2}}{f(x_{n-1}) - f(x_{n-2})}
 = \frac{x_{n-2} f(x_{n-1}) - x_{n-1} f(x_{n-2})}{f(x_{n-1}) - f(x_{n-2})}.
</math>

As can be seen from the recurrence relation, the secant method requires two initial values, ''x''<sub>0</sub> and ''x''<sub>1</sub>, which should ideally be chosen to lie close to the root.

==Derivation of the method==
Starting with initial values {{math|''x''<sub>0</sub>}} and {{math|''x''<sub>1</sub>}}, we construct a line through the points {{math|(''x''<sub>0</sub>, ''f''(''x''<sub>0</sub>))}} and {{math|(''x''<sub>1</sub>, ''f''(''x''<sub>1</sub>))}}, as shown in the picture above. In slope–intercept form, the equation of this line is

:<math>y = \frac{f(x_1) - f(x_0)}{x_1 - x_0}(x - x_1) + f(x_1).</math>
    
The root of this linear function, that is the value of {{mvar|x}} such that {{math|''y'' {{=}} 0}} is

:<math>x = x_1 - f(x_1) \frac{x_1 - x_0}{f(x_1) - f(x_0)}.</math>

We then use this new value of {{mvar|x}} as {{math|''x''<sub>2</sub>}} and repeat the process, using {{math|''x''<sub>1</sub>}} and {{math|''x''<sub>2</sub>}} instead of {{math|''x''<sub>0</sub>}} and {{math|''x''<sub>1</sub>}}. We continue this process, solving for {{math|''x''<sub>3</sub>}}, {{math|''x''<sub>4</sub>}}, etc., until we reach a sufficiently high level of precision (a sufficiently small difference between {{math|''x''<sub>''n''</sub>}} and {{math|''x''<sub>''n''−1</sub>}}):

:<math>
\begin{align}
x_2 & = x_1 - f(x_1) \frac{x_1 - x_0}{f(x_1) - f(x_0)}, \\[6pt]
x_3 & = x_2 - f(x_2) \frac{x_2 - x_1}{f(x_2) - f(x_1)}, \\[6pt]
& \,\,\,\vdots \\[6pt]
x_n & = x_{n-1} - f(x_{n-1}) \frac{x_{n-1} - x_{n-2}}{f(x_{n-1}) - f(x_{n-2})}.
\end{align}
</math>

==Convergence==

The iterates <math>x_n</math> of the secant method converge to a root of <math>f</math>, if the initial values <math>x_0</math> and <math>x_1</math> are sufficiently close to the root. The [[order of convergence]] is ''φ'', where
:<math>\varphi = \frac{1+\sqrt{5}}{2} \approx 1.618</math>
is the [[golden ratio]]. In particular, the convergence is superlinear, but not quite [[quadratic convergence|quadratic]].

This result only holds under some technical conditions, namely that <math>f</math> 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 <math>[x_0, x_1]</math>. For example, if <math>f</math> is differentiable on that interval and there is a point where <math>f' = 0</math> 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 {{lang|la|regula falsi}}) uses the same formula as the secant method. However, it does not apply the formula on <math>x_{n-1}</math> and <math>x_{n-2}</math>, like the secant method, but on <math>x_{n-1}</math> and on the last iterate <math>x_k</math> such that <math>f(x_k)</math> and <math>f(x_{n-1})</math> 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]]
:<math>x_n = x_{n-1} - \frac{f(x_{n-1})}{f'(x_{n-1})}</math>
by using the [[finite-difference]] approximation
:<math>f'(x_{n-1}) \approx \frac{f(x_{n-1}) - f(x_{n-2})}{x_{n-1} - x_{n-2}}.</math>
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''&nbsp;≈&nbsp;1.6). However, Newton's method requires the evaluation of both <math>f</math> and its derivative <math>f'</math> at every step, while the secant method only requires the evaluation of <math>f</math>. Therefore, the secant method may occasionally be faster in practice. For instance, if we assume that evaluating <math>f</math> 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 ''φ''<sup>2</sup>&nbsp;≈&nbsp;2.6) for the same cost as one step of Newton's method (decreasing the logarithm of the error by a factor&nbsp;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==

[[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''.

[[Image:Secant method example code result.svg|center]]

==Computational example==
Below, the secant method is implementation in the [[Python (programming language)|Python]] programming language. 

It is then applied to find a root of the function {{math|''f''(''x'') {{=}} ''x''<sup>2</sup> − 612}} with initial points <math>x_0 = 10</math> and <math>x_1 = 30</math>

<source lang="python3">
def secant_method(f, x0, x1, iterations):
    xs = [x0, x1] + iterations * [None] # allocate x's vector

    for i, x in enumerate(xs):
        if i < iterations:
            x1 = xs[i+1]
            xs[i+2] = x1 - f(x1) * (x1 - x) / (f(x1) - f(x)) # fill component xs[i+2]
    
    return xs[-1] # return the now filled last entry


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

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

print("Root: {}".format(root)) # Root: 24.738633748750722
</source>

==Notes==
{{reflist}}

== See also ==
* [[False position method]]

==References==
* {{citation | last1=Kaw | first1=Autar | last2=Kalu | first2=Egwu | year=2008 | title=Numerical Methods with Applications | edition=1st | url=http://www.autarkaw.com/books/numericalmethods/index.html }}.
* {{cite book| title=Numerical analysis for applied science|first1=Myron B. |last1=Allen |first2=Eli L. |last2=Isaacson | pages=188–195| isbn=978-0-471-55266-6| year=1998| publisher=[[John Wiley & Sons]]| url=https://books.google.com/books?id=PpB9cjOxQAQC}}

==External links==
* [http://numericalmethods.eng.usf.edu/topics/secant_method.html Secant Method] Notes, PPT, Mathcad, Maple, Mathematica, Matlab at [http://numericalmethods.eng.usf.edu Holistic Numerical Methods Institute]
* {{MathWorld|urlname=SecantMethod|title=Secant Method}}

{{Root-finding algorithms}}

[[Category:Root-finding algorithms]]