Bisection Method Calculator

The bisection method is the most dependable way to track down a root of a function, the place where its graph crosses zero, and this calculator runs it for you on any function you supply. Enter a function of x, an interval whose ends you believe straddle the root, and how many steps to take, and it returns the root, the value of the function there, and the size of the final interval, updating as you type. The idea is beautifully simple and almost impossible to break. If a continuous function is negative at one end of an interval and positive at the other, it must pass through zero somewhere in between. The method takes the midpoint, checks the sign of the function there, and keeps whichever half still has a sign change, so the root stays trapped in an interval that is cut exactly in half at every step. Repeat this and the interval shrinks relentlessly, closing in on the root with steady, guaranteed progress: roughly every ten steps improve the accuracy a thousandfold. It is not the fastest technique, methods like Newton's can leap to the answer more quickly, but it is the most robust, never diverging or overshooting as long as the starting interval genuinely brackets a root, which is why it is a favourite for reliability and a first lesson in numerical methods. The calculator checks that your interval really does bracket a root before it starts. That makes the tool genuinely useful for students learning numerical methods, root-finding and convergence and checking homework, and for anyone needing a sure-footed solution to an equation that has no neat algebraic answer. Use x, the caret for powers, and the usual functions. The method and a worked example are explained clearly below.

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1.41421356
approximate root
f(root)-0
Final interval width0

 

How it works

The function must have opposite signs at a and b, so a root lies between them. The calculator takes the midpoint, evaluates the function, and replaces whichever endpoint has the same sign as the midpoint, halving the interval. After the chosen number of steps, the midpoint of the final interval is the approximate root.

Worked example

For f(x) = x squared minus 2 on the interval 1 to 2: f(1) is minus 1 and f(2) is plus 2, so a root is between them. Bisecting repeatedly closes in on the square root of 2, about 1.41421, where the function is zero.

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Frequently asked questions

What is the bisection method?

The bisection method finds a root of a function by repeatedly halving an interval that is known to contain one. At each step it keeps the half where the function changes sign, so the root is trapped in an interval that shrinks by half every time.

What do you need for the bisection method to work?

You need an interval whose endpoints give function values of opposite sign, one positive and one negative. That guarantees a root lies between them for a continuous function, which is what the method relies on to converge.

How accurate is the bisection method?

The interval halves each step, so accuracy improves steadily and predictably. About ten steps shrink the interval by a factor of a thousand. It is slower than some methods but very reliable, always converging when the starting interval brackets a root.