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►In what follows this is accomplished in two ways: i) via the Lagrangeinterpolation of §3.3(i) ; and ii) by constructing a pointwise continued fraction, or PWCF, as follows:
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►Comparisons of the precisions of Lagrange and PWCF interpolations to obtain the derivatives, are shown in Figure 18.40.2.
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►►►Figure 18.40.2: Derivative Rule inversions for carried out via Lagrange and PWCF interpolations.
…For the derivative rule Lagrangeinterpolation (red points) gives digits in the central region, while PWCF interpolation (blue points) gives .
Magnify
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►If , then is the Lagrangeinterpolation polynomial for the set (§3.3(i)).
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►For many applications a spline function is a more adaptable approximating tool than the Lagrangeinterpolation polynomial involving a comparable number of parameters; see §3.3(i), where a single polynomial is used for interpolating
on the complete interval .
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►The nodes
are prescribed, and the weights
and error term
are found by integrating the product of the Lagrangeinterpolation polynomial of degree and .
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