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7 Gebrochenrationale Funktionen

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7.2 Interpolationspolynome

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7.2.1 Interpolationspolynome nach Newton

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Rekursionsformel:

$c_{i}$	$=$	$γ_{0 i}$
$γ_{i, i}$	$=$	$f (x_{i})$
$γ_{i, k}$	$=$	$\frac{γ_{i + 1, k} - γ_{i, k - 1}}{x_{k} - x_{i}}$

Beispiel 7 - 1:

	$c_{0}$	$=$	$γ_{00}$	$=$	$f (x_{0})$
$i = 0, k = 1$	$c_{1}$	$=$	$γ_{01}$	$=$	$\frac{γ_{1, 1} - γ_{0, 0}}{x_{1} - x_{0}}$	$=$	$\frac{y_{1} - y_{0}}{x_{1} - x_{0}}$

$i = 0, k = 2$	$c_{2}$	$=$	$γ_{02}$	$=$	$\frac{γ_{1, 2} - γ_{0, 1}}{x_{2} - x_{1}}$	$=$	$\frac{\frac{γ_{2, 2} - γ_{1, 1}}{x_{2} - x_{1}} - \frac{γ_{1, 1} - γ_{0, 0}}{x_{1} - x_{0}}}{x_{2} - x_{0}}$

						$=$	$\frac{\frac{y_{2} - y_{1}}{x_{2} - x_{1}} - \frac{y_{1} - y_{0}}{x_{1} - x_{0}}}{x_{2} - x_{0}}$

Rechenschema:

$x_{0}$	$y_{0}$
		$γ_{01} = \frac{y_{1} - y_{0}}{x_{1} - x_{0}}$
$x_{1}$	$y_{1}$		$γ_{02} = \frac{γ_{12} - γ_{01}}{x_{2} - x_{0}}$
		$γ_{12} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$		$γ_{03} = \frac{γ_{13} - γ_{02}}{x_{3} - x_{0}}$
$x_{2}$	$y_{2}$		$γ_{13} = \frac{γ_{23} - γ_{12}}{x_{3} - x_{1}}$
		$γ_{23} = \frac{y_{3} - y_{2}}{x_{3} - x_{2}}$
$x_{3}$	$y_{3}$

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