Error Control and Extrapolation

Knowledge of the magnitude of the error in the integration formulas presented in Section E.3 is very useful in the estimation of the error as well as in the improvement of the calculated integral. 

The integral I numerically calculated using the spacing h is denoted as lih), and if we denote the exact integral as /exact we then have 

Applying the same integration formula but this time use the spacing of h/P, where PiP 1) is some arbitrary number (usually 2), then we have the formula 

If we equate Eqs. E.27 and E.28 and solve for jUi , which is the error of the integration result using the spacing h, we obtain 

This formula provides the estimate of error incurred by the approximation using the spacing of h. If this error is larger than the prespecified error, the spacing has to be reduced until the prespecified error is satisfied. When this is the case, the better estimate of the approximate integral is simply Eq. E.27 with Ah given by Eq. E.29. This process is called extrapolation. 

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