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Calculation Machine

Newton Raphson Calculator

One Newton step at a time, with convergence check.

Reviewed by the Calculation Machine team · How we check our maths

Your values

ⓘ Live results

Results
Next estimate xₙ₊₁
2.25
Step size
0.25
Relative change
11.111111%
Converging?
Yes — step is shrinking
Estimated correct digits gained
0.6

Quick answer

Newton's method improves a root estimate with xₙ₊₁ = xₙ − f(xₙ) ÷ f′(xₙ), roughly doubling the number of correct digits each step.

The formula

xₙ₊₁ = xₙ − f(xₙ) ÷ f′(xₙ)

Worked example

Worked example

Using the values the calculator loads with.

Inputs

  • Current estimate xₙ2
  • f(xₙ)-1
  • f′(xₙ)4

Result

  • Next estimate xₙ₊₁2.25
  • Step size0.25
  • Relative change11.111111%
  • Converging?Yes — step is shrinking
  • Estimated correct digits gained0.6

How to use the Newton Raphson Calculator

  1. 1Enter your current estimate of the root.
  2. 2Enter the function value and derivative at that point.
  3. 3Feed the new estimate back in and repeat until the step size is tiny.

Frequently asked questions

When does Newton's method fail?

Near a zero derivative, at inflection points, or with a poor starting guess — it can oscillate or shoot off entirely.

How many iterations do I need?

Usually 4 to 6 from a decent start, because the method converges quadratically.

About this calculator