Enter a polynomial and a starting guess, and instantly see every Newton-Raphson iteration converge on the root.
Newton-Raphson Method Calculator
How It’s Calculated
Newton-Raphson refines a guess using the tangent line at that point: xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ). Geometrically, it follows the tangent down to where it crosses the x-axis, and treats that crossing as the next, better guess. Repeated enough times, xₙ converges rapidly toward an actual root — provided the initial guess is reasonably close and f'(xₙ) never hits exactly 0 along the way.
Worked Example
For f(x) = x³ − 4, starting at x₀ = 1: f(1) = −3, f'(1) = 3, so x₁ = 1 − (−3/3) = 2. f(2) = 4, f'(2) = 12, so x₂ = 2 − 4/12 ≈ 1.667. Continuing this converges toward the cube root of 4, ≈ 1.5874.
Frequently Asked Questions
Q: How do I pick a good initial guess?
Anywhere reasonably close to where you expect the root — checking that f(x) changes sign nearby (as in the Bisection Method) is a reliable way to narrow down a starting point first.
Q: What if f'(xₙ) becomes 0 during iteration?
The tangent line is horizontal and never crosses the x-axis, so the next step is undefined — Newton-Raphson fails from that point and needs a different starting guess.
Newton-Raphson is one of the core numerical methods across all four VTU Engineering Mathematics semesters. See how Daniel Classes teaches VTU Engineering Mathematics →