Mathematics / Free calculator

Newton–Raphson root finder

Find a real root using an explicitly supplied derivative and inspect residuals at every iteration.

01 Your inputs

Use x, + − * / ^, sin, cos, exp, ln, sqrt. Angles are in radians. Use explicit multiplication for clarity.

Entered function
f(x)=x2−2
Entered derivative
f′(x)=2·x

1e-14 to 0.1. Arrows adjust by a factor of 10; you can also type any value in range. A residual tolerance is not a guaranteed bound on root error.

1–80 whole iterations.

Calculated on your device. Inputs are not transmitted or stored.

02 Your result

Enter your values and select Calculate.

The method

xn+1=xn−f(xn)f′(xn)

A worked example

For f(x) = x² − 2, f′(x) = 2x, and x₀ = 1, Newton’s method converges to √2 ≈ 1.4142135624.

Assumptions & limits

Real scalar functions with the restricted expression parser. Tolerances are absolute. Stopping by residual does not guarantee a small root error for flat or poorly conditioned functions. Discontinuities, poles, and convergence failures require independent review. The supplied derivative is not automatically verified. A derivative smaller than 10⁻¹⁴ max(1, |x|) stops iteration with a warning; stagnation or exhausted iterations are reported, never labeled convergence. Full double precision between operations; displayed values use up to 10 decimal places. Numerical outputs above 10¹⁰⁰ and nonfinite function evaluations are rejected.

Calculation source: Math Workbench: Newton–Raphson method.

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