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.
Worked Example: Finding √2 Step by Step
Take f(x) = x² − 2, whose positive root is √2. Newton’s update is xn+1 = xn − f(xn)/f′(xn), which simplifies to xn+1 = (xn + 2/xn)/2. Starting from x0 = 1:
| n | xn | f(xn) |
|---|---|---|
| 0 | 1.000000 | −1 |
| 1 | 1.500000 | 0.250000 |
| 2 | 1.416667 | 0.006944 |
| 3 | 1.414216 | 0.000006 |
| 4 | 1.414214 | 0.000000 |
After four steps the root is correct to six decimal places. Notice how the number of correct digits roughly doubles at each step. That fast convergence is the reason the method is so widely used, and it is also the first thing to check when a question asks you to justify an answer.
Try it with the calculator above: enter x² − 2 and a starting guess of 1, and compare the table it produces with the one here.
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 →