Newton's Method Root Finder

What Is Newton’s Method? Newton’s Method (also called the Newton-Raphson method) is an iterative algorithm for finding roots of equations — values of x where f(x) = 0. It was described by Isaac Newton around 1669 in England and refined by Joseph Raphson in 1690.

The Formula Starting from an initial guess x_0, each iteration improves the estimate: x_(n+1) = x_n - f(x_n) / f’(x_n). This uses the function value and its derivative at the current point to find a better approximation.

Geometric Interpretation At each step, draw the tangent line to the curve at the current point. Where that tangent line crosses the x-axis is the next estimate. The tangent line is a linear approximation of the curve, and each iteration refines this approximation.

Convergence Newton’s Method converges quadratically near a root — the number of correct digits roughly doubles with each iteration. Starting from a reasonable initial guess, the method typically finds 15+ digits of precision in 5-8 iterations. However, it can fail if the derivative is zero, if the initial guess is too far from the root, or if the function has discontinuities.

Built-In Functions This calculator supports common functions: polynomials (x^2, x^3), trigonometric (sin, cos, tan), exponential (exp, e^x), natural logarithm (ln), and square root (sqrt). The derivative is computed numerically using the central difference method.

Applications Newton’s Method is used everywhere in scientific computing: finding eigenvalues, solving nonlinear systems, optimizing functions, computing square roots, and inverting complicated functions. Most “solve” buttons in scientific calculators use a variant of this method.