Newton's Method
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Tap a term to see what it means, or .
Calculate
Example: the positive root of x² − 2 = 0 is √2 ≈ 1.41421356. Starting from x₀ = 1, how many steps does the method take to get all those digits right? Change the coefficients of ax² + bx + c, the initial guess and the number of iterations.
Step by step
What each term means
- Next Approximation (xₙ₊₁)
- Where the tangent to the curve at xₙ crosses the x axis. Near a simple root, the number of correct digits roughly doubles each step.
- Current Approximation (xₙ)
- The previous step's guess; at the start, the initial guess x₀. A bad guess may lead to another root, or nowhere.
- Function Value (f(xₙ))
- How far the function still is from zero. At the root it is zero, and the method stops moving.
- Derivative (f′(xₙ))
- The slope of the tangent at xₙ. Dividing f(xₙ) by it says how far to go for the tangent to touch the axis. If it is zero, the tangent is horizontal and the method fails.