Mathematics Branches, Topics, and Sub-Topics

A structured visual guide to the major mathematical areas and their relationships.

Search by code, branch, topic, subtopic, or a keyword from the descriptions.

65Hxx Nonlinear equations

This subtopic introduces the core ideas in nonlinear equations, including foundational concepts, standard methods, and the main questions used to organize the area. Typical uses include building mathematical background, framing related research problems, and supporting applications in neighboring fields where these concepts provide useful structure.

Specific topics

65H04 Roots of polynomial equations

Overview

Roots of polynomial equations. This topic studies numerical methods for nonlinear equations, including root-finding, fixed-point iterations, and convergence analysis.

Related Wikipedia Page

Wikipedia: Newton method

Useful Links

Key Ideas

  • local/global convergence of nonlinear solvers
  • derivative-based and derivative-free methods
  • stability and stopping criteria

Typical Uses

Used to compute zeros and fixed points in nonlinear models.

Applications

  • Engineering design equations
  • Nonlinear PDE discretizations
  • Optimization subproblems

References

Recommended Textbooks

65H05 Numerical computation of solutions to single equations

Overview

Numerical computation of solutions to single equations. This topic studies numerical methods for nonlinear equations, including root-finding, fixed-point iterations, and convergence analysis.

Related Wikipedia Page

Wikipedia: Newton method

Useful Links

Key Ideas

  • local/global convergence of nonlinear solvers
  • derivative-based and derivative-free methods
  • stability and stopping criteria

Typical Uses

Used to compute zeros and fixed points in nonlinear models.

Applications

  • Engineering design equations
  • Nonlinear PDE discretizations
  • Optimization subproblems

References

Recommended Textbooks

65H10 Numerical computation of solutions to systems of equations

Overview

Numerical computation of solutions to systems of equations. This topic studies numerical methods for nonlinear equations, including root-finding, fixed-point iterations, and convergence analysis.

Related Wikipedia Page

Wikipedia: Newton method

Useful Links

Key Ideas

  • local/global convergence of nonlinear solvers
  • derivative-based and derivative-free methods
  • stability and stopping criteria

Typical Uses

Used to compute zeros and fixed points in nonlinear models.

Applications

  • Engineering design equations
  • Nonlinear PDE discretizations
  • Optimization subproblems

References

Recommended Textbooks

65H14 Continuation and homotopy methods for computing solutions of nonlinear equations

Overview

Continuation and homotopy methods for computing solutions of nonlinear equations. This topic studies numerical methods for nonlinear equations, including root-finding, fixed-point iterations, and convergence analysis.

Related Wikipedia Page

Wikipedia: Newton method

Useful Links

Key Ideas

  • local/global convergence of nonlinear solvers
  • derivative-based and derivative-free methods
  • stability and stopping criteria

Typical Uses

Used to compute zeros and fixed points in nonlinear models.

Applications

  • Engineering design equations
  • Nonlinear PDE discretizations
  • Optimization subproblems

References

Recommended Textbooks

65H17 Eigenvalue problems for nonlinear operators

Overview

Eigenvalue problems for nonlinear operators. This topic studies numerical methods for nonlinear equations, including root-finding, fixed-point iterations, and convergence analysis.

Related Wikipedia Page

Wikipedia: Newton method

Useful Links

Key Ideas

  • local/global convergence of nonlinear solvers
  • derivative-based and derivative-free methods
  • stability and stopping criteria

Typical Uses

Used to compute zeros and fixed points in nonlinear models.

Applications

  • Engineering design equations
  • Nonlinear PDE discretizations
  • Optimization subproblems

References

Recommended Textbooks

65H20 Global methods, including homotopy approaches to the numerical solution of nonlinear equations

Overview

Global methods, including homotopy approaches to the numerical solution of nonlinear equations. This topic studies numerical methods for nonlinear equations, including root-finding, fixed-point iterations, and convergence analysis.

Related Wikipedia Page

Wikipedia: Newton method

Useful Links

Key Ideas

  • local/global convergence of nonlinear solvers
  • derivative-based and derivative-free methods
  • stability and stopping criteria

Typical Uses

Used to compute zeros and fixed points in nonlinear models.

Applications

  • Engineering design equations
  • Nonlinear PDE discretizations
  • Optimization subproblems

References

Recommended Textbooks