Trapezoidal rule extended

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.. 

The integration was carried out with the extended trapezoidal rule for an integral over function fix)... 

Error of the extended trapezoid rule and the extended middle point rule... 

There are certain features that make the extended trapezoid rule particularly interesting. 

Note that the convergence speed of the extended trapezoid rule is quite slow. Thus, the method has to be used in this form only when the integral does not require a massive computation effort... 

A second usefiil feature in the extended trapezoid rule is in the special form of ( ) its local error. 

Error estimation. By indicating with Th and T2h two applications of the extended trapezoid rule with integration steps h and 2h, respectively, the error with step h is estimated ... 

The Romberg method is much more efficient than the extended trapezoid rule method, but it is unsuitable in its original version since it is less efficient than other alternatives. 

In spite of the modifications introduced by Bulirsch and Stoer, the method derived from the extended trapezoid rule suffers from several shortcomings that make it less efficient than other methods described later, when implemented in a general integration program. 

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