# Bisection method

A few steps of the bisection method applied over the starting range [a1;b1]. The bigger red dot is the root of the function.

The bisection method in mathematics is a root-finding method that repeatedly bisects an interval and then selects a subinterval in which a root must lie for further processing. It is a very simple and robust method, but it is also relatively slow. Because of this, it is often used to obtain a rough approximation to a solution which is then used as a starting point for more rapidly converging methods.[1] The method is also called the interval halving method,[2] the binary search method,[3] or the dichotomy method.[4]

## The method

The method is applicable for numerically solving the equation f(x) = 0 for the real variable x, where f is a continuous function defined on an interval [ab] and where f(a) and f(b) have opposite signs. In this case a and b are said to bracket a root since, by the intermediate value theorem, the continuous function f must have at least one root in the interval (a, b).

At each step the method divides the interval in two by computing the midpoint c = (a+b) / 2 of the interval and the value of the function f(c) at that point. Unless c is itself a root (which is very unlikely, but possible) there are now only two possibilities: either f(a) and f(c) have opposite signs and bracket a root, or f(c) and f(b) have opposite signs and bracket a root.[5] The method selects the subinterval that is guaranteed to be a bracket as the new interval to be used in the next step. In this way an interval that contains a zero of f is reduced in width by 50% at each step. The process is continued until the interval is sufficiently small.

Explicitly, if f(a) and f(c) have opposite signs, then the method sets c as the new value for b, and if f(b) and f(c) have opposite signs then the method sets c as the new a. (If f(c)=0 then c may be taken as the solution and the process stops.) In both cases, the new f(a) and f(b) have opposite signs, so the method is applicable to this smaller interval.[6]

The input for the method is a continuous function f, an interval [a, b], and the function values f(a) and f(b). The function values are of opposite sign (there is at least one zero crossing within the interval). Each iteration performs these steps:

1. Calculate c, the midpoint of the interval, c = 0.5 * (a + b).
2. Calculate the function value at the midpoint, f(c).
3. If convergence is satisfactory (that is, a - c is sufficiently small, or f(c) is sufficiently small), return c and stop iterating.
4. Examine the sign of f(c) and replace either (a, f(a)) or (b, f(b)) with (c, f(c)) so that there is a zero crossing within the new interval.

When implementing the method on a computer, there can be problems with finite precision, so there are often additional convergence tests or limits to the number of iterations. Although f is continuous, finite precision may preclude a function value ever being zero. For f(x) = x − π, there will never be a finite representation of x that gives zero. Floating point representations also have limited precision, so at some point the midpoint of [ab] will be either a or b.

### Algorithm

The method may be written in pseudocode as follows:[7]

INPUT: Function f, endpoint values a, b, tolerance TOL, maximum iterations NMAX
CONDITIONS: a < b, either f(a) < 0 and f(b) > 0 or f(a) > 0 and f(b) < 0
OUTPUT: value which differs from a root of f(x)=0 by less than TOL

N ← 1
While NNMAX # limit iterations to prevent infinite loop
c ← (a + b)/2 # new midpoint
If f(c) = 0 or (ba)/2 < TOL then # solution found
Output(c)
Stop
EndIf
NN + 1 # increment step counter
If sign(f(c)) = sign(f(a)) then ac else bc # new interval
EndWhile
Output("Method failed.") # max number of steps exceeded

## Example: Finding the root of a polynomial

Suppose that the bisection method is used to find a root of the polynomial

${\displaystyle f(x)=x^{3}-x-2\,.}$

First, two numbers ${\displaystyle a}$ and ${\displaystyle b}$ have to be found such that ${\displaystyle f(a)}$ and ${\displaystyle f(b)}$ have opposite signs. For the above function, ${\displaystyle a=1}$ and ${\displaystyle b=2}$ satisfy this criterion, as

${\displaystyle f(1)=(1)^{3}-(1)-2=-2}$

and

${\displaystyle f(2)=(2)^{3}-(2)-2=+4\,.}$

Because the function is continuous, there must be a root within the interval [1, 2].

In the first iteration, the end points of the interval which brackets the root are ${\displaystyle a_{1}=1}$ and ${\displaystyle b_{1}=2}$, so the midpoint is

${\displaystyle c_{1}={\frac {2+1}{2}}=1.5}$

The function value at the midpoint is ${\displaystyle f(c_{1})=(1.5)^{3}-(1.5)-2=-0.125}$. Because ${\displaystyle f(c_{1})}$ is negative, ${\displaystyle a=1}$ is replaced with ${\displaystyle a=1.5}$ for the next iteration to ensure that ${\displaystyle f(a)}$ and ${\displaystyle f(b)}$ have opposite signs. As this continues, the interval between ${\displaystyle a}$ and ${\displaystyle b}$ will become increasingly smaller, converging on the root of the function. See this happen in the table below.

Iteration ${\displaystyle a_{n}}$ ${\displaystyle b_{n}}$ ${\displaystyle c_{n}}$ ${\displaystyle f(c_{n})}$
1 1 2 1.5 −0.125
2 1.5 2 1.75 1.6093750
3 1.5 1.75 1.625 0.6660156
4 1.5 1.625 1.5625 0.2521973
5 1.5 1.5625 1.5312500 0.0591125
6 1.5 1.5312500 1.5156250 −0.0340538
7 1.5156250 1.5312500 1.5234375 0.0122504
8 1.5156250 1.5234375 1.5195313 −0.0109712
9 1.5195313 1.5234375 1.5214844 0.0006222
10 1.5195313 1.5214844 1.5205078 −0.0051789
11 1.5205078 1.5214844 1.5209961 −0.0022794
12 1.5209961 1.5214844 1.5212402 −0.0008289
13 1.5212402 1.5214844 1.5213623 −0.0001034
14 1.5213623 1.5214844 1.5214233 0.0002594
15 1.5213623 1.5214233 1.5213928 0.0000780

After 13 iterations, it becomes apparent that there is a convergence to about 1.521: a root for the polynomial.

## Analysis

The method is guaranteed to converge to a root of f if f is a continuous function on the interval [a, b] and f(a) and f(b) have opposite signs. The absolute error is halved at each step so the method converges linearly, which is comparatively slow.

Specifically, if c1 = a+b/2 is the midpoint of the initial interval, and cn is the midpoint of the interval in the nth step, then the difference between cn and a solution c is bounded by[8]

${\displaystyle |c_{n}-c|\leq {\frac {|b-a|}{2^{n}}}.}$

This formula can be used to determine in advance the number of iterations that the bisection method would need to converge to a root to within a certain tolerance. The number of iterations needed, n, to achieve a given error (or tolerance), ε, is given by: ${\displaystyle n=\log _{2}\left({\frac {\epsilon _{0}}{\epsilon }}\right)={\frac {\log \epsilon _{0}-\log \epsilon }{\log 2}},}$

where ${\displaystyle \epsilon _{0}={\text{initial bracket size}}=b-a.}$

Therefore, the linear convergence is expressed by ${\displaystyle \epsilon _{n+1}={\text{constant}}\times \epsilon _{n}^{m},\ m=1.}$

## References

1. ^ Burden & Faires 1985, p. 31
2. ^ "Archived copy". Archived from the original on 2013-05-19. Retrieved 2013-11-07.
3. ^ Burden & Faires 1985, p. 28
4. ^ "Dichotomy method - Encyclopedia of Mathematics". www.encyclopediaofmath.org. Retrieved 2015-12-21.
5. ^ If the function has the same sign at the endpoints of an interval, the endpoints may or may not bracket roots of the function.
6. ^ Burden & Faires 1985, p. 28 for section
7. ^ Burden & Faires 1985, p. 29. This version recomputes the function values at each iteration rather than carrying them to the next iterations.
8. ^ Burden & Faires 1985, p. 31, Theorem 2.1
• Burden, Richard L.; Faires, J. Douglas (1985), "2.1 The Bisection Algorithm", Numerical Analysis (3rd ed.), PWS Publishers, ISBN 0-87150-857-5