Overview
Newton's Method approximates a function near the current point with a tangent or quadratic model, then jumps to the model's root or optimum as the next iterate.
FoundationSource checked
Newton's Method
Finds a root or optimum by repeatedly using local derivative information to form a better approximation.
Quick Facts
Compact metadata for implementation and discovery.
- Category
- Optimization
- Difficulty
- Intermediate
- Family
- Second-Order Optimization
- Tier
- Foundation
- Status
- Verified
- Review
- Source Checked
- Sources
- 1
- Formulas
- 1
- Relationships
- 22
- Code example
- Not ready
- Visualization
- Ready
- Aliases
- Newton-Raphson Method
Core Concepts
derivativelocal linearizationiterationconvergence
Quick facts
Complexity
Runtime and memory behavior at a glance.
Best
O(t)
Average
O(t)
Worst
O(t)
Space
O(1)
When to Use
- Use it when derivatives are available and a good initial guess is known.
- Use it when fast local convergence is worth the cost of derivative calculations.
- Use damped or safeguarded variants for difficult nonlinear problems.
When Not to Use
- Avoid it when derivatives are unreliable or expensive and simpler methods are sufficient.
- Avoid it with poor initial guesses unless safeguards are present.
- Avoid it near flat derivatives or singular Hessians without fallback behavior.
What It Does
- Uses derivative information
- Updates an estimate with local curvature
Use Cases
- Finding roots
- Optimizing smooth functions
- Solving nonlinear equations
Inputs
- Function
- Derivative information
- Initial estimate
Outputs
- Improved estimate
- Convergence result
Key Formulas (1)
Newton update
xnext = x − f(x) / f'(x)
x_next = x - f(x) / f'(x)Uses the local tangent line to jump toward a root.
Pseudocode
- 1.Choose an initial estimate
- 2.Evaluate the function and derivative
- 3.Compute the Newton update
- 4.Replace the estimate with the updated value
- 5.Repeat until the change is small
Canonical Example
- Input
- objective = minimize (x - 3)^2, start = x = 0
- Output
- x moves toward 3
- Explanation
- Newton's Method: The algorithm repeatedly improves a candidate solution according to the objective.
Implementation Notes
- Check both step size and residual when deciding convergence.
- Use damping, line search, or bracketing safeguards when jumps are too aggressive.
- Set maximum iterations and report nonconvergence explicitly.
Common Pitfalls
- Dividing by a derivative that is zero or nearly zero.
- Assuming convergence from any starting point.
- Stopping on a small step even though the residual is still large.
Interactive Mechanism Walkthrough
Iterative Convergence & State TrajectoryShow tangent lines pointing from one estimate to the next.
Step 1 of 5: Choose an initial estimate
tangent-updateNotes
Newton's method can converge very quickly near a solution, but that local behavior depends on smoothness and the starting point.
Sources (1)
What each reference supports for this entry.
Numerical Optimization
BookOptimization reference covering Newton methods and second-order updates.
Open sourceExplore Next
Good next jumps from the relationship graph, ranked for useful discovery.
AlternativesBisection Method
Numerical Methods · Foundation
Bisection is slower but more robust when a sign bracket is available.
Compare with thisAlternative toGradient Descent
Optimization · Foundation
Both optimize iteratively, but Newton's Method uses curvature information.
Compare with thisOften paired withImplicit Integration
Simulation · Specialized
Implicit solvers often use Newton-style nonlinear solves.
Compare with thisPrerequisite ofBFGS
Optimization · Specialized
BFGS approximates Newton-style curvature without explicitly forming the Hessian.
Compare with thisRelationships
View full relationship pagePrerequisite ofBFGS
BFGS approximates Newton-style curvature without explicitly forming the Hessian.
Alternative toGradient Descent
Both optimize iteratively, but Newton's Method uses curvature information.
Conceptual neighborSecant Method
Both use local derivative information to improve iterative estimates.
Related Algorithms
Tags
OptimizationIterativeRoot FindingNumerical Methods