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This form of simple trend extrapolation helps to direct attention towards the forces, which can change the projected pattern. A more elaborated curve that uses times series analysis can often reveal surprising historical and current data patterns. The qualitative trend analysis is one of the most demanding and creative methods in Futures Studies.
Order of accuracy — rate at which numerical solution of differential equation converges to exact solution; Series acceleration — methods to accelerate the speed of convergence of a series Aitken's delta-squared process — most useful for linearly converging sequences; Minimum polynomial extrapolation — for vector sequences; Richardson ...
Another problem of extrapolation is loosely related to the problem of analytic continuation, where (typically) a power series representation of a function is expanded at one of its points of convergence to produce a power series with a larger radius of convergence. In effect, a set of data from a small region is used to extrapolate a function ...
The diagram opposite shows a 2nd order solution to G A Sod's shock tube problem (Sod, 1978) using the above high resolution Kurganov and Tadmor Central Scheme (KT) with Linear Extrapolation and Ospre limiter. This illustrates clearly demonstrates the effectiveness of the MUSCL approach to solving the Euler equations.
The distributions of a wide variety of physical, biological, and human-made phenomena approximately follow a power law over a wide range of magnitudes: these include the sizes of craters on the moon and of solar flares, [2] cloud sizes, [3] the foraging pattern of various species, [4] the sizes of activity patterns of neuronal populations, [5] the frequencies of words in most languages ...
In numerical analysis, Aitken's delta-squared process or Aitken extrapolation is a series acceleration method used for accelerating the rate of convergence of a sequence. It is named after Alexander Aitken, who introduced this method in 1926. [1] It is most useful for accelerating the convergence of a sequence that is converging linearly.
The trend-cycle component can just be referred to as the "trend" component, even though it may contain cyclical behavior. [3] For example, a seasonal decomposition of time series by Loess (STL) [ 4 ] plot decomposes a time series into seasonal, trend and irregular components using loess and plots the components separately, whereby the cyclical ...
For example, the second-order equation y′′ = −y can be rewritten as two first-order equations: y′ = z and z′ = −y. In this section, we describe numerical methods for IVPs, and remark that boundary value problems (BVPs) require a different set of tools. In a BVP, one defines values, or components of the solution y at more than one ...