How to use the Bisection Method Calculator
- 1. Enter f(x)Type the function, for example x^3 - x - 1.
- 2. Choose a and bPick two points where f(a) and f(b) have opposite signs.
- 3. Read the rootEach row halves the interval until it is smaller than your tolerance.
How bisection works
If f is continuous and f(a) and f(b) have opposite signs, there is a root between them. Bisection takes the midpoint c = (a + b)/2, checks the sign of f(c), and keeps the half where the sign changes. After n steps the interval is (b − a)/2ⁿ wide, so you can predict the number of steps before you start.
It is slower than Newton-Raphson but it never fails once the root is bracketed, which makes it a good first method.
Worked example
Solve x³ − x − 1 = 0 on [1, 2]. f(1) = −1 and f(2) = 5, so a root lies between them.
Step 1: c = 1.5, f(1.5) = 0.875 > 0, so the root is in [1, 1.5]. Step 2: c = 1.25, f(1.25) = −0.297 < 0, so it is in [1.25, 1.5]. Step 3: c = 1.375, f = 0.225, so [1.25, 1.375]. Step 4: c = 1.3125. Continuing to a tolerance of 0.0001 takes 14 steps and gives x ≈ 1.3248; the exact root is 1.324718.
Choosing the interval and tolerance
Make a quick table of f(x) at whole numbers and look for a sign change. If f(a) and f(b) have the same sign the tool says so, because there may be no root or an even number of roots in between. Each extra decimal place costs about 3.3 more steps. Once you have a rough root, the Newton-Raphson calculator refines it in a few steps.
Frequently asked questions
How many iterations will bisection take?
About log₂((b − a)/tolerance). For an interval of 1 and tolerance 0.0001 that is 14 steps. The tool shows this number.
Why does it say the root is not bracketed?
f(a) and f(b) have the same sign. Pick a and b so that one value is positive and the other negative.
Is bisection the same as the interval halving method?
Yes. Bisection, interval halving and Bolzano's method are the same thing.
Is this calculator free?
Yes. You enter your name and WhatsApp number once when you see your result, and other Uncodemy tools then open without the form.