Extended trapezoid formula

Note that the points needed with the extended trapezoid formula when the integration step is halved correspond to the points needed by the central point formula and the previous integration step. 

Certain programs, which implement the extended trapezoid formula, exploit the property ... 

This relation means that in many practical cases, the extended trapezoid formula... 

Since the formula for the error of the extended trapezoid formula shows that the terms depend on the even powers of h, it is opportune to use as the independent variable for the interpolation. 

An alternative technique consists of using as the error estimation the difference between the extrapolation obtained with the extended trapezoid formula and the one obtained with the extended central point formula. 

In the automatic methods based on the extended trapezoid formula, overall integration interval is considered if the error is dissatisfactory, new points uniformly distributed on the interval are inserted. 

The advantage to considering the 13 points as simultaneously present is that we can exploit a series of six calculations of the extended trapezoid formula on which to perform the extrapolation. 

To overcome this problem, the same technique used with the extended trapezoid method implemented in the class Bzzlntegral can be adopted split the original interval into three subintervals where the external ones are very small. The objects of the Bzz Integra IGaussBF class use this device too, but for different reasons. In the first case, it is necessary because the extended trapezoid formulae are close, whereas in the second case, the polynomial approximation can be improved in these delicate intervals. 

Bzz Integra L The interval is split into three subintervals. The lateral subintervals are very small and the Gauss-Kronrod formulae are used. The extended trapezoid formulae based on the extrapolation with 13 points are adopted for the central subinterval. 

Improper integrals of the other types whose problems involve both limits are handled by open formulas that do not require the integrand to be evaluated at its endpoints. One such formula, the extended midpoint rule, is accurate to the same order as the extended trapezoidal rule and is used when the limits of integration are located halfway between tabulated abscissas ... 

Semi-open formulas are used when the problem exists at only one limit. At the closed end of the integration, the weights from the standard closed-type formulas are used and at the open end, the weights from open formulas are used. (Weights for closed and open formulas of various orders of error may be found in standard numerical methods texts.) Given a closed extended trapezoidal rule of one order higher than the preceding formula, i.e.. 

In the extended Goda formula (named here the Goda-Takahashi formula), the wave pressure acting along a vertical wall is assumed to have a trapezoidal distribution both above and below the still water level, while the uplift pressure acting... 

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See also in sourсe #XX -- [ , ]

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