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GNU MCSim a simulation and numerical integration package, with fast Monte Carlo and Markov chain Monte Carlo capabilities. ML.NET is a free-software machine-learning library for the C# programming language. [4] [5] NAG Library is an extensive software library of highly optimized numerical-analysis routines for various programming environments.
It is the study of numerical methods that attempt to find approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences like economics, medicine, business and even the arts.
In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm.
For some differential equations, application of standard methods—such as the Euler method, explicit Runge–Kutta methods, or multistep methods (for example, Adams–Bashforth methods)—exhibit instability in the solutions, though other methods may produce stable solutions.
Finite difference methods for heat equation and related PDEs: FTCS scheme (forward-time central-space) — first-order explicit; Crank–Nicolson method — second-order implicit; Finite difference methods for hyperbolic PDEs like the wave equation: Lax–Friedrichs method — first-order explicit; Lax–Wendroff method — second-order explicit
To use a finite difference method to approximate the solution to a problem, one must first discretize the problem's domain. This is usually done by dividing the domain into a uniform grid (see image). This means that finite-difference methods produce sets of discrete numerical approximations to the derivative, often in a "time-stepping" manner.
Numerical methods for ordinary differential equations, such as Runge–Kutta methods, can be applied to the restated problem and thus be used to evaluate the integral. For instance, the standard fourth-order Runge–Kutta method applied to the differential equation yields Simpson's rule from above.
The name is in analogy with quadrature, meaning numerical integration, where weighted sums are used in methods such as Simpson's method or the Trapezoidal rule. There are various methods for determining the weight coefficients, for example, the Savitzky–Golay filter. Differential quadrature is used to solve partial differential equations ...
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