Concepts inN-body simulations: The performance of some integrators
N-body problem
The -body problem is the problem of predicting the motion of a group of celestial objects that interact with each other gravitationally. Solving this problem has been motivated by the need to understand the motion of the Sun, planets and the visible stars. Its first complete mathematical formulation appeared in Isaac Newton's Principia (the -body problem in general relativity is considerably more difficult).
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Integrator
An integrator is a device to perform the mathematical operation known as integration, a fundamental operation in calculus. The integration function is often part of engineering and scientific calculations. Mechanical integrators are used in such applications as metering of water flow or electric power. Electronic analog integrators were the basis of analog computers.
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Round-off error
For the acrobatic movement, roundoff, see Roundoff. 50x40px This article does not cite any references or sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. A round-off error, also called rounding error, is the difference between the calculated approximation of a number and its exact mathematical value.
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Symplectic integrator
In mathematics, a symplectic integrator (SI) is a numerical integration scheme for a specific group of differential equations related to classical mechanics and symplectic geometry. Symplectic integrators form the subclass of geometric integrators which, by definition, are canonical transformations. They are widely used in molecular dynamics, discrete element methods, accelerator physics, and celestial mechanics.
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N-body simulation
An N-body simulation is a simulation of a dynamical system of particles, usually under the influence of physical forces, such as gravity. In cosmology, they are used to study processes of non-linear structure formation such as the process of forming galaxy filaments and galaxy halos from dark matter in physical cosmology. Direct N-body simulations are used to study the dynamical evolution of star clusters.
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Truncation
In mathematics and computer science, truncation is the term for limiting the number of digits right of the decimal point, by discarding the least significant ones. For example, consider the real numbers 5.6341432543653654 32.438191288 −6.3444444444444 To truncate these numbers to 4 decimal digits, we only consider the 4 digits to the right of the decimal point.
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Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). One of the earliest mathematical writings is a Babylonian tablet from the Yale Babylonian Collection(YBC 7289), which gives a sexagesimal numerical approximation of, the length of the diagonal in a unit square.
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Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus. Given a function f of a real variable x and an interval [a, b] of the real line, the definite integral is defined informally to be the area of the region in the xy-plane bounded by the graph of f, the x-axis, and the vertical lines x = a and x = b, such that areas above the axis add to the total, and the area below the x axis subtract from the total.
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