Category:Definitions/Bisection Method

This category contains definitions related to Bisection Method.
Related results can be found in Category:Bisection Method.

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.