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Ch, a commercial C/C++-based interpreted language with computational array for scientific numerical computation and visualization. [18] APMonitor: APMonitor is a mathematical modeling language for describing and solving representations of physical systems in the form of differential and algebraic equations.
[1] [8] In this method, a time derivative of the dependent variable is added to Laplace’s equation. Finite differences are then used to approximate the spatial derivatives, and the resulting system of equations is solved by MOL. It is also possible to solve elliptical problems by a semi-analytical method of lines. [9]
Linear and non-linear equations. In the case of a single equation, the "solver" is more appropriately called a root-finding algorithm. Systems of linear equations. Nonlinear systems. Systems of polynomial equations, which are a special case of non linear systems, better solved by specific solvers. Linear and non-linear optimisation problems
The conjugate gradient method is often implemented as an iterative algorithm, applicable to sparse systems that are too large to be handled by a direct implementation or other direct methods such as the Cholesky decomposition. Large sparse systems often arise when numerically solving partial differential equations or optimization problems.
In numerical analysis, a multigrid method (MG method) is an algorithm for solving differential equations using a hierarchy of discretizations. They are an example of a class of techniques called multiresolution methods , very useful in problems exhibiting multiple scales of behavior.
This is a list of mathematics-based methods. Adams' method (differential equations) Akra–Bazzi method (asymptotic analysis) Bisection method (root finding) Brent's method (root finding) Condorcet method (voting systems) Coombs' method (voting systems) Copeland's method (voting systems) Crank–Nicolson method (numerical analysis) D'Hondt ...
An extension of this idea is to choose dynamically between different methods of different orders (this is called a variable order method). Methods based on Richardson extrapolation, [14] such as the Bulirsch–Stoer algorithm, [15] [16] are often used to construct various methods of different orders. Other desirable features include:
Explicit methods calculate the state of a system at a later time from the state of the system at the current time, while implicit methods find a solution by solving an equation involving both the current state of the system and the later one.