Secant Method

Topics: Numerical Analysis - Numerical Method

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The secant method is an open numerical method of linear interpolation that can solve equations. It’s similar to the Newton method.

Procedure §

It supposes that $f(x)$ is linear, and draws a secant line that passes through $f(x_0)$ and $f(x_1)$. However, unlike the Newton method and other methods (like the bisection method), the interval $x_0$ and $x_1$ must not necessarily contain a root.

With the help of this image, we can visualise that:

$$\frac{x_1-x_2}{f(x_1)}=\frac{x_1-x_0}{f(x_1)-f(x_0)}$$

…and from there:

$$x_2=x_1-f(x_1)\frac{x_1-x_0}{f(x_1)-f(x_0)}$$

Finally, in general, we have that:

$$x_{k+1}=x_k-f(x_k)\frac{x_{k-1}-x_k}{f(x_{k-1})-f(x_k)}$$

Linearity of $f(x)$

Notice that, if the function was linear, the secant line would intersect the $x$ axis exactly on the root.

However, since $f(x)$ is rarely linear, the intersection of the secant line will not be the root, but it will be close.