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The zeroeth extrapolation, R ... Here is an example of a computer implementation of the ... lower limit b : upper limit max_steps: maximum steps of the ...
Using an adaptive stepsize is of particular importance when there is a large variation in the size of the derivative. For example, when modeling the motion of a satellite about the earth as a standard Kepler orbit, a fixed time-stepping method such as the Euler method may be sufficient.
Otherwise one can use a "null rule" which has the form of the above quadrature rule, but whose value would be zero for a simple integrand (for example, if the integrand were a polynomial of the appropriate degree). See: Richardson extrapolation (see also Romberg's method) Null rules; Epsilon algorithm
An example of Richardson extrapolation method in two dimensions. In numerical analysis , Richardson extrapolation is a sequence acceleration method used to improve the rate of convergence of a sequence of estimates of some value A ∗ = lim h → 0 A ( h ) {\displaystyle A^{\ast }=\lim _{h\to 0}A(h)} .
A concrete example in the case of a one-step method of order can illustrate the general procedure of extrapolation. With such a method, the calculated approximation for small step sizes ℎ can be easily described by a polynomial of the form
Using Richardson extrapolation, the more accurate Simpson estimate (,) + (,) for six function values is combined with the less accurate estimate (,) for three function values by applying the correction [(,) + (,) (,)] /. So, the obtained estimate is exact for polynomials of degree five or less.
An example of MUSCL type state parabolic-reconstruction. It is possible to extend the idea of linear-extrapolation to higher order reconstruction, and an example is shown in the diagram opposite. However, for this case the left and right states are estimated by interpolation of a second-order, upwind biased, difference equation.
Fitting of a noisy curve by an asymmetrical peak model, with an iterative process (Gauss–Newton algorithm with variable damping factor α).Curve fitting [1] [2] is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, [3] possibly subject to constraints.