Bisection Method

(Midpoint Method for Equations)

Bisection Method

The bisection method (sometimes called the midpoint method for equations) is a method used to estimate the solution of an equation.

Like the Regula-Falsi Method (and others) we approach this problem by writing the equation in the form f(x) = 0 for some function f(x). This reduces the problem to finding a root for the function f(x).

Like the Regula-Falsi Method the Bisection Method also needs a closed interval [a,b] for which the function f(x) is positive at one endpoint and negative at the other. In other words f(x) must satisfy the condition f(a)f(b) < 0. This means that this algorithm can not be applied to find tangential roots.

There are several advantages that the Bisection method has over the Regula-Falsi Method.

The number of steps required to estimate the root within the desired error can be easily computed before the algorithm is applied. This gives a way to compute how long the algorithm will compute. (Real-time applications)

The way that you get the next point is a much easier computation than how you get the regula-falsi point (rfp).

Bisection Algorithm

The idea for the Bisection Algorithm is to cut the interval [a,b] you are given in half (bisect it) on each iteration by computing the midpoint xmid. The midpoint will replace either a or b depending on if the sign of f(xmid) agrees with f(a) or f(b).

Step 1: Compute xmid = (a+b)/2

Step 2: If sign(f(xmid)) = 0 then end algorithm

else If sign(f(xmid)) = sign(f(a)) then a = xmid

else b = xmid

Step 3: Return to step 1

f(a)

f(b)

a b

root

xmid This shows how the points a, b and xmid are related.

f(x)

Lets apply the Bisection Method to the same function as we did for the Regula-Falsi Method. The equation is: x3-2x-3=0, the function is: f(x)=x3-2x-3.

This function has a root on the interval [0,2]

Iteration a b xmid f(a) f(b) f(xmid)

1 0 2 1 -3 1 -4

2 1 2 1.5 -4 1 -2.262

3 1.5 2 1.75 -2.262 1 -1.140

4 1.75 2 1.875 -1.140 1 -.158

As we mentioned earlier we mentioned that we could compute exactly how many iterations we would need for a given amount of error.

The error is usually measured by looking at the width of the current interval you are considering (i.e. the distance between a and b). The width of the interval at each iteration can be found by dividing the width of the starting interval by 2 for each iteration. This is because each iteration cuts the interval in half. If we let the error we want to achieve err and n be the iterations we get the following:

1log

2

21

2

2

1

1

1

err

abn

err

ababerr

aberr

n

n

n