What happens if I leave the f'(x) field blank?

The solver automatically calculates the derivative numerically using a central difference method with a step size of h ≈ 1e-7.

What stopping criteria are used to determine convergence?

The iteration stops when |f(x)| falls below the specified tolerance or when step changes satisfy the convergence threshold.

Why does the iteration fail or diverge?

Iterations can fail if the derivative equals zero at a step, if the initial guess is too far from a root, or if the function diverges.

What math functions can I include in the expression?

You can use standard operators (+, -, *, /, ^), parentheses, and functions including sin, cos, exp, ln, and sqrt.

How many iterations can I configure?

You can set the maximum iteration cap between 2 and 200 steps, with a default limit of 50.

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Newton-Raphson Root Finder

Newton-Raphson iteration for f(x) = 0 with analytic or numeric derivative, full iteration table, and explicit convergence/divergence reporting.

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2 examples

x² − 2 = 0 from x₀ = 1 → √2 ≈ 1.4142135624 in 4 iterations

Newton's method on x² − 2 doubles the number of correct digits per step: 1 → 1.5 → 1.4166667 → 1.4142157 → 1.4142135624, where |f| drops below 1e-10. With no derivative supplied, the numeric central difference reproduces the analytic tangent 2x to machine precision.

Newton–Raphson root finding f(x) = x^2 - 2 f'(x): numeric central difference (h ≈ 1e-7) Start: x₀ = 1, tolerance = 1e-10, max iterations = 50 Iterations: k x_k f(x_k) 0 1 -1 1 1.5 0.25 2 1.4166666667 0.0069444444 3 1.4142156863 0.0000060073 4 1.4142135624 0 Converged after 4 iterations Root: x ≈ 1.4142135624 |f(root)| ≈ 4.5e-12

cos(x) − x = 0 → the Dottie number 0.7390851332

The fixed point of cos(x) is found with the analytic derivative −sin(x) − 1: starting from x₀ = 1 the iteration converges quadratically to 0.7390851332151607 in 4 iterations — the Dottie number.

Newton–Raphson root finding f(x) = cos(x) - x f'(x): analytic (-sin(x) - 1) Start: x₀ = 1, tolerance = 1e-10, max iterations = 50 Iterations: k x_k f(x_k) 0 1 -0.4596976941 1 0.7503638678 -0.0189230738 2 0.7391128909 -0.0000464559 3 0.7390851334 -3e-10 4 0.7390851332 0 Converged after 4 iterations Root: x ≈ 0.7390851332 |f(root)| ≈ 0.0e+0

Inputs

Set the required fields, then run the tool.

6 options

ContentPaste or type the main input values.2

f(x) expressiontextRequiredThe function whose root is sought, in the variable x. Supports + - * / ^, parentheses, and functions like exp, sin, cos, sqrt, ln.f'(x) expression (optional)textOptionalOptional analytic derivative; when blank a central difference (h ≈ 1e-7) is used.

SettingsAdjust formats, ranges, numbers, and modes.4

Initial guess x₀numberOptionalStarting point of the iteration.TolerancenumberOptionalConvergence threshold on |f(x)| and on the step size.Max IterationsnumberOptionalHard cap on the number of iterations (2–200).Decimal PlacesnumberOptional

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Newton-Raphson Root Finder

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Use cases

Examples

1. Finding the square root of 2

2. Computing the Dottie number

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