Newton's method

Newton's method is an iterative numerical method for approximating a root of an equation by repeatedly finding where a tangent line crosses the axis.

Newton–Raphson iteration: An iteration that updates an estimate by subtracting the function value divided by its derivative. This is the standard formula behind Newton's method.
Root of a function: An input value at which a function equals zero. Newton's method is designed to approximate these values.
Root-finding algorithm: A procedure for approximating values where a function equals zero. Newton's method is one member of this broad class of numerical algorithms.
Bisection method: A root-finding method that repeatedly halves an interval whose endpoints bracket a root. Bisection converges reliably under a sign-change condition, while Newton's method can fail from a poor starting guess.
Taylor series: A function is represented locally as a sum of terms involving its derivatives. The tangent-line step uses the first-order Taylor approximation near the current estimate.
Tangent line: A line that locally approximates a curve at a point and shares its slope there. The method estimates a root where the current point's tangent meets the horizontal axis.
Nonlinear equation: An equation in which the unknown does not appear only in a linear combination. Newton's method often approximates solutions where algebraic rearrangement cannot isolate the unknown.
Secant method: A root-finding method that estimates derivatives using slopes between successive approximations. The secant method avoids exact derivatives, trading Newton's usual fast local convergence for simpler function information.
Derivative: A measure of a function's instantaneous rate of change, defined through a limiting ratio. The derivative supplies the tangent slope used to calculate each update.
Initial value problem: A differential equation problem specified together with the value of its solution at one point. Despite the similar phrase, Newton's method needs an initial guess for a root, not initial data for a differential equation.

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