Revision as of 01:44, 24 April 2014

This article is about a root-finding algorithm. See all root-finding algorithms

Definition

The secant method is a root-finding algorithm that makes successive point estimates for the value of a root of a continuous function. In general, the secant method is not guaranteed to converge towards a root, but under some conditions, it does. A slight variant of this method, called the false position method, functions very similarly to the bisection method.

=Iterative process

The secant method requires two initial guesses for the root, say $x_{0}$ and $x_{1}$ . For $n \geq 2$ , we define $x_{n}$ as the following affine combination of the previous two guesses $x_{n - 1}$ and $x_{n - 2}$

$x_{n} = \frac{x_{n - 2} f (x_{n - 1}) - x_{n - 1} f (x_{n - 2})}{f (x_{n - 1}) - f (x_{n - 2})}$

Geometrically, this can be interpreted as follows: we make a line through the points Failed to parse (syntax error): {\displaystyle (x_{n-2},f(x_[n-2})} and $(x_{n - 1}, f (x_{n - 1})$ in the $(x, f (x))$ -plane, and define $x_{n}$ as the $x$ -coordinate of the intersection of this line with the $x$ -axis.

@@ Line 2: / Line 2: @@
 ==Definition==
-The '''secant method''' is a root-finding algorithm.
+The '''secant method''' is a root-finding algorithm that makes successive point estimates for the value of a root of a [[continuous function]]. In general, the secant method is not guaranteed to converge towards a root, but under some conditions, it does. A slight variant of this method, called the [[false position method]], functions very similarly to the [[bisection method]].
+==Iterative process=
+The secant method requires ''two'' initial guesses for the root, say <math>x_0</math> and <math>x_1</math>. For <math>n \ge 2</math>, we define <math>x_n</math> as the following affine combination of the previous two guesses <math>x_{n-1}</math> and <math>x_{n-2}</math>
+<math>x_n = \frac{x_{n-2}f(x_{n-1}) - x_{n-1}f(x_{n-2})}{f(x_{n-1}) - f(x_{n-2})}</math>
+Geometrically, this can be interpreted as follows: we make a line through the points <math>(x_{n-2},f(x_[n-2})</math> and <math>(x_{n-1},f(x_{n-1})</math> in the <math>(x,f(x))</math>-plane, and define <math>x_n</math> as the <math>x</math>-coordinate of the intersection of this line with the <math>x</math>-axis.