This is a list of
numerical analysis topics.
General
Error
Error analysis (mathematics)
Elementary and special functions
Numerical linear algebra
Numerical linear algebra — study of numerical algorithms for linear algebra problems
Basic concepts
- Types of matrices appearing in numerical analysis:
- Algorithms for matrix multiplication:
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Matrix decompositions:
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Matrix splitting — expressing a given matrix as a sum or difference of matrices
Solving systems of linear equations
Eigenvalue algorithms
Eigenvalue algorithm — a numerical algorithm for locating the eigenvalues of a matrix
Other concepts and algorithms
Interpolation and approximation
Interpolation — construct a function going through some given data points
Polynomial interpolation
Polynomial interpolation — interpolation by polynomials
Spline interpolation
Spline interpolation — interpolation by piecewise polynomials
Trigonometric interpolation
Trigonometric interpolation — interpolation by trigonometric polynomials
Other interpolants
Approximation theory
Approximation theory
Miscellaneous
Finding roots of nonlinear equations
- See
#Numerical linear algebra for linear equations
Root-finding algorithm — algorithms for solving the equation f(x) = 0
- General methods:
- Methods for polynomials:
- Analysis:
-
Numerical continuation — tracking a root as one parameter in the equation changes
Optimization
Mathematical optimization — algorithm for finding maxima or minima of a given function
Basic concepts
Linear programming
Linear programming (also treats integer programming) — objective function and constraints are linear
Convex optimization
Convex optimization
Nonlinear programming
Nonlinear programming — the most general optimization problem in the usual framework
- Special cases of nonlinear programming:
- General algorithms:
Optimal control and infinite-dimensional optimization
Optimal control
Infinite-dimensional optimization
Uncertainty and randomness
Theoretical aspects
Applications
Miscellaneous
Numerical quadrature (integration)
Numerical integration — the numerical evaluation of an integral
Numerical methods for ordinary differential equations
Numerical methods for ordinary differential equations — the numerical solution of ordinary differential equations (ODEs)
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Euler method — the most basic method for solving an ODE
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Explicit and implicit methods — implicit methods need to solve an equation at every step
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Backward Euler method — implicit variant of the Euler method
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Trapezoidal rule — second-order implicit method
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Runge–Kutta methods — one of the two main classes of methods for initial-value problems
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Linear multistep method — the other main class of methods for initial-value problems
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General linear methods — a class of methods encapsulating linear multistep and Runge-Kutta methods
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Bulirsch–Stoer algorithm — combines the midpoint method with Richardson extrapolation to attain arbitrary order
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Exponential integrator — based on splitting ODE in a linear part, which is solved exactly, and a nonlinear part
- Methods designed for the solution of ODEs from classical physics:
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Geometric integrator — a method that preserves some geometric structure of the equation
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Symplectic integrator — a method for the solution of Hamilton's equations that preserves the symplectic structure
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Energy drift — phenomenon that energy, which should be conserved, drifts away due to numerical errors
- Other methods for initial value problems (IVPs):
- Methods for solving two-point boundary value problems (BVPs):
- Methods for solving differential-algebraic equations (DAEs), i.e., ODEs with constraints:
- Methods for solving stochastic differential equations (SDEs):
- Methods for solving integral equations:
- Analysis:
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Stiff equation — roughly, an ODE for which unstable methods need a very short step size, but stable methods do not
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L-stability — method is A-stable and stability function vanishes at infinity
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Adaptive stepsize — automatically changing the step size when that seems advantageous
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Parareal -- a parallel-in-time integration algorithm
Numerical methods for partial differential equations
Numerical partial differential equations — the numerical solution of partial differential equations (PDEs)
Finite difference methods
Finite difference method — based on approximating differential operators with difference operators
Finite element methods, gradient discretisation methods
Finite element method — based on a discretization of the space of solutions
gradient discretisation method — based on both the discretization of the solution and of its gradient
Other methods
Techniques for improving these methods
Grids and meshes
Analysis
Applications
Software
For a large list of software, see the
list of numerical-analysis software.
Journals
Researchers
References