How to use the Regula Falsi Method Calculator
- 1. Enter f(x)Type the function, for example x*exp(x) − cos(x).
- 2. Give the intervalChoose a and b so that f(a) and f(b) have opposite signs.
- 3. Read the rootEach row shows a, b, their f values and the new point x.
How regula falsi works
Join the points (a, f(a)) and (b, f(b)) with a straight line. Where it crosses the x-axis is the new estimate:
x = (a f(b) − b f(a)) / (f(b) − f(a)).
Then look at the sign of f(x). If f(a) and f(x) have opposite signs the root lies in [a, x], so b is replaced by x; otherwise a is replaced. Because one end is always on each side, the root stays bracketed and the method cannot run away like the secant method. It is usually faster than bisection, but one end often stays fixed, so convergence is only linear.
Worked example
Find the root of x eˣ = cos x, that is f(x) = x eˣ − cos x, in (0, 1). f(0) = −1 and f(1) = 2.17798. Then x₁ = (0 × 2.17798 − 1 × (−1))/(2.17798 + 1) = 0.31467, and f(0.31467) = −0.51987, so the root lies in (0.31467, 1).
Next x₂ = 0.44673, x₃ = 0.49402, x₄ = 0.50995 and so on. The values settle at x ≈ 0.5177, the answer quoted in most textbooks.
Choosing the interval
Make a quick table of f(x) at a few points and pick two neighbours with opposite signs. For a method that halves the interval every time, see the bisection calculator.
Frequently asked questions
What does regula falsi mean?
It is Latin for 'false position'. A straight line stands in for the curve, giving a 'false' position that is then improved.
Why must f(a) and f(b) have opposite signs?
By the intermediate value theorem a continuous function then has a root between a and b. The method relies on this bracket.
Is it faster than bisection?
Usually yes, because it uses the size of f(a) and f(b), not just their signs. Sometimes one end sticks and it slows down.
When does it stop?
When two successive estimates differ by less than your tolerance, or f(x) is exactly 0.
Is this calculator free?
Yes. You fill a short form once when you see your first result, and other Uncodemy tools then open without the form. Everything is calculated in your browser.