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**Polynomial and Rational Extrapolation **

In Chapter , we used extrapolation as a tool to estimate the discretization errors of high - order Runge - Kutta methods. We learned that these error estimates are "asymptotically correct;"thus, they converge at the same rate as actual discretization errors. Indeed, using local extrapolation, we added these error estimates to the computed solution to obtain a higher - order approximation. In this chapter, we study the use of low - order methods with repeated extrapolation to obtain high - order methods satisfying prescribed accuracy criteria. As in previous chapters, we'll consider the scalar IVP

.

Now the order of the approximation is increased by in each successive column of the extrapolation tableau If the trapezoidal rule is solved exactly at each step then it has an expansion in powers of h

Extrapolation with rational functions is also possible Section II

The basic idea is to approximate z t h

by a rational function R h

and then evaluate R

Bulirsch and Stoer

derived a scheme where Rqi h

is dened as the rational approximation that interpolates z t h

at h hi hi hiq for h hi hi hiq The values of Rqi Rqi

can be obtained from the following recursion

when the method has an error expansion in powers of h The computation of ac cording to

b

is equivalent to interpolating z t h

by a function of the form

There are several enhancements to the basic extrapolation approach Some of these

follow

A variableorder extrapolation algorithm could choose the order ie the number of columns in the tableau

adaptively Error estimates can be computed as the dif ference between the rst subdiagonal element and the diagonal or by the dierence between two successive diagonal elements Extrapolation methods can be written as RungeKutta methods if the step size and order of the extrapolation are xed In this case convergence stability and error bounds follow from the results of Chapter for general onestep methods Implicit methods can be used in combination with extrapolation to solve sti problems A survey of the state of the art was written by Deuhard who prefers polynomial extrapolation to rational extrapolation because rational extrapolation lacks translation invariance rational extrapolation can impose restrictions on the base step size and polynomial extrapolation is slightly more ecient

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