Linked from

The 50 pages that link to Newton's method, each with the reason it gives.

• Isaac NewtonBroader topic: Newton devised this method as a practical tool for solving equations.

• DerivativeRelated: Each update uses the function’s derivative to refine the root estimate.

• Numerical analysisBroader topic: It illustrates how local information can produce rapidly converging numerical solutions.

• Jacobian matrixRelated: For systems of equations, each update solves a linear system built from the Jacobian.

• Gradient descentCompared with: It uses curvature information that ordinary gradient descent ignores.

• Square rootRelated: Applying it to x² − a = 0 provides a fast way to approximate √a.

• DifferentiabilityRelated: The method uses local linearization and can fail when derivatives are unavailable or poorly behaved.

• Fixed pointBroader topic: Its root estimates are fixed points of the iteration function used by the method.

• Hessian matrixRelated: Newton optimization steps depend on solving a system involving the Hessian.

• Tangent lineRelated: Each update uses the tangent’s x-intercept as the next root estimate.

• Banach fixed-point theoremCompared with: Unlike contraction iteration, its convergence is generally local and depends on the starting point.

• Bisection methodCompared with: It can converge rapidly near a root but may fail without a good initial estimate.

• Taylor's theoremRelated: Its local convergence analysis uses Taylor expansions around the root.

• Limit of a sequenceRelated: Convergence of its iterates determines whether the method approaches a root.

• Implicit function theoremRelated: The theorem’s local solvability perspective underlies analysis of parameterized root-finding problems.

• Polynomial rootRelated: It can approximate polynomial roots when exact factorization is unavailable.

• Secant lineRelated: Its secant-method variant replaces derivative information with slopes between iterates.

• Differential calculusRelated: Each update uses a derivative to turn local slope into a root estimate.

• Polynomial equationRelated: It estimates solutions when exact algebraic methods are impractical.

• Inverse function theoremRelated: Local invertibility helps explain why Newton iteration can converge near a regular root.

• Multiplicity of a rootRelated: Repeated roots often make its convergence slower than for simple roots.

• Quadratic formulaCompared with: It can approximate roots of many equations, unlike the quadratic formula, which applies specifically to quadratics.

• Fixed-point iterationRelated: It can be written as fixed-point iteration with map x − f(x)/f′(x).

• KAM theoryRelated: KAM proofs use rapidly convergent Newton-like iterations to correct successive errors.

• Root-finding algorithmRelated: Its fast local convergence contrasts with its sensitivity to starting values and derivatives.

• Cubic equationRelated: It approximates cubic roots when exact radical expressions are impractical.

• Interior-point methodRelated: Newton steps commonly solve the linearized systems that determine interior-point updates.

• Iterative methodBroader topic: Its repeated tangent-based updates can rapidly refine an approximate root.

• Monotone sequenceRelated: Under suitable conditions, monotonicity can show its iterates approach a root without overshooting.

• Positive-definite matrixRelated: A positive-definite Hessian makes the local quadratic model strictly convex and its Newton system well behaved.

• Second derivative testRelated: Optimization variants use second derivatives to locate stationary points, then classify them.

• Complex dynamicsRelated: Its complex-plane basins form a direct computational example of dynamical behavior.

• Linear approximationRelated: Each update solves the linear approximation to the function at the current estimate.

• Secant methodCompared with: The secant method replaces Newton’s derivative with a difference quotient.

• Synthetic divisionCompared with: Unlike synthetic division, it approximates roots rather than dividing by a chosen linear factor.

• Zero of a functionRelated: It can rapidly refine an approximate zero when suitable derivatives and starting values are available.

• Descartes' rule of signsRelated: After root counts suggest relevant positive or negative roots, Newton's method can approximate them.

• Differentiable functionRelated: Its update step relies on the derivative to linearize the root-finding problem.

• Hensel's lemmaRelated: Hensel lifting is a p-adic analogue of Newton iteration, with divisibility controlling each correction.

• Kepler's equationRelated: It commonly solves Kepler's equation when no elementary closed-form solution is available.

• Numerical optimizationRelated: Optimization variants use curvature information to make more informed steps.

• Second derivativeRelated: Its optimization variant uses second derivatives to refine steps toward a minimum.

• Binomial seriesRelated: Binomial expansions can provide local approximations for powers used in numerical calculations.

• Cube rootRelated: Applied to x³ − a = 0, it rapidly approximates the cube root of a.

• Jia XianCompared with: It offers a later, structurally different approach to numerical root finding.

• Lagrange inversion theoremCompared with: It approximates solutions numerically rather than extracting exact inverse-series coefficients.

• Nth rootRelated: Applying it to xⁿ − a = 0 approximates an nth root numerically.

• Shooting methodRelated: A sensitivity estimate can guide corrections to the guessed initial slope.

• Square root of 2Related: Applied to x² − 2 = 0, it rapidly approximates the positive root.

• Equation solvingRelated: It finds approximate solutions when exact algebraic methods are impractical.