# Newton-Raphson Root Finder Newton-Raphson iteration for f(x) = 0 with analytic or numeric derivative, full iteration table, and explicit convergence/divergence reporting. > Canonical page: https://elysiatools.com/en/tools/newton-raphson-root-finder - **Category:** Math & Numbers - **Keywords:** newton raphson, newton's method, root finding, nonlinear equation, numerical method, iteration, tangent method, quadratic convergence, dottie number, numerical analysis ## Overview The Newton-Raphson Root Finder solves non-linear equations of the form f(x) = 0 using Newton's tangent method. It supports user-defined algebraic and transcendental functions, accepts optional exact derivative expressions or defaults to central difference numeric derivatives, and outputs a complete step-by-step iteration table with convergence status. ## Inputs - **f(x) expression** (text): The function whose root is sought, in the variable x. Supports + - * / ^, parentheses, and functions like exp, sin, cos, sqrt, ln. - **f'(x) expression (optional)** (text): Optional analytic derivative; when blank a central difference (h ≈ 1e-7) is used. - **Initial guess x₀** (number): Starting point of the iteration. - **Tolerance** (number): Convergence threshold on |f(x)| and on the step size. - **Max Iterations** (number): Hard cap on the number of iterations (2–200). - **Decimal Places** (number) ## When to use - You need to find accurate real roots of non-linear algebraic or transcendental equations. - You want to inspect step-by-step convergence values and function evaluations across iterations. - You want to compute roots without manually calculating derivatives by utilizing automated numerical central differences. ## How it works - Enter your target equation f(x) using mathematical operations (+, -, *, /, ^) and supported functions like exp, sin, cos, sqrt, or ln. - Optionally enter the analytic derivative f'(x), or leave it empty to let the solver compute a numeric central difference approximation (h ≈ 1e-7). - Specify an initial guess (x₀), convergence tolerance threshold, maximum iteration limit, and output decimal precision. - The solver computes successive estimates using x_{k+1} = x_k − f(x_k)/f'(x_k) until convergence criteria are met or the iteration limit is reached. ## Use cases - Solving transcendental equations such as finding fixed points for trigonometric functions like cos(x) = x. - Verifying numerical analysis coursework and homework by reviewing the complete step-by-step iteration table. - Calculating root approximations for non-linear engineering equations where algebraic solutions are impractical. ## Frequently asked questions ### 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? 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