LaTeX
x_{k+1}=x_k-[\nabla^2f(x_k)]^{-1}\nabla f(x_k)Variables
- ∇f: gradient
- ∇²f: Hessian
- x_k: current iterate
How to use this formula
Uses gradient and Hessian information to form a local quadratic step.
Important notes
- The Hessian must be invertible or regularized.
- A line search or trust region can improve robustness.
Quick example
For a positive-definite quadratic, Newton’s method reaches the minimizer in one exact step.
Applicability, worked calculation, and verification
Assumptions and domain checks
- The Hessian must be invertible or regularized.
- Preserve matrix order and verify dimension compatibility; multiplication and inversion are not generally commutative or always defined.
- Specify the objective, constraints, feasible domain, and any convexity or differentiability assumptions used to justify the result.
Worked example
For a positive-definite quadratic, Newton’s method reaches the minimizer in one exact step.
Common mistakes
- Do not reorder factors or mix incompatible dimensions when using Newton Optimization Step; matrix operations are order-sensitive.
- Do not treat a stationary point as a global optimum without checking constraints, boundaries, and the required optimality conditions.
Continue the workflow
Use Newton Optimization Step in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
x_{k+1}=x_k-[\nabla^2f(x_k)]^{-1}\nabla f(x_k). - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-23
Automated quality check: Kept noindex until the missing evidence is supplied.
Formula references
- Digital Library of Mathematical FunctionsNational Institute of Standards and Technology — Definitions, notation, identities, and reference material for mathematical functions.
Frequently asked questions
What is the Newton Optimization Step used for?
Uses gradient and Hessian information to form a local quadratic step.
Can I copy this formula as LaTeX?
Yes. Copy x_{k+1}=x_k-[\nabla^2f(x_k)]^{-1}\nabla f(x_k) or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.