Definition:Bisection Method

Definition

Let $f$ be a real function such that:

$f$ is continuous over a closed interval $\closedint a b$

$\map f a$ and $\map f b$ are of opposite sign.

The bisection method is an iterative technique for finding an approximation to at least one solution to the equation $\map f x = 0$ to any desired accuracy.

So, we assume that $\map f a \map f b < 0$ and that $a < b$.

We evaluate $c = \dfrac {a + b} 2$, thereby bisecting $\closedint a b$.

We evaluate $\map f c$.

If $\map f c = 0$, then we have a solution to $\map f x = 0$.

Otherwise, $\map f c$ is of opposite sign to either $\map f a$ or $\map f b$.

If $\map f c$ is of opposite sign to $\map f a$, then there exists a solution to $\map f x = 0$ in $\closedint a c$.

If $\map f c$ is of opposite sign to $\map f b$, then there exists a solution to $\map f x = 0$ in $\closedint c b$.

In either case, a closed interval has been constructed of half the length of $\closedint a b$.

This process can be repeated until the interval of interest is arbitrarily small, enabling the solution to be known to whatever accuracy is required.

Examples

Arbitrary Example

Suppose that, using the bisection method, a solution to $\map f x = 0$ has been found in the interval $\closedint {1.717} {1.724}$.

Then the solution is $x = 1.72$ to $3$ significant figures.

Also see

• Results about the bisection method can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): bisection method
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): bisection method