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Given k + 1 {\displaystyle k+1} data points:
( x 0 , y 0 ) , ( x 1 , y 1 ) , … , ( x k , y k ) {\displaystyle (x_{0},y_{0}),(x_{1},y_{1}),\dots ,(x_{k},y_{k})}
The forward divided differences are defined as:
The backward divided differences are defined as:
data points:
![{\displaystyle [y_{\nu }]:=y_{\nu },\qquad \nu \in \{0,\ldots ,k\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0b559f584afa31efe09d04b4be0e091807e66c6)
![{\displaystyle [y_{\nu },\ldots ,y_{\nu +j}]:={\frac {[y_{\nu +1},\ldots ,y_{\nu +j}]-[y_{\nu },\ldots ,y_{\nu +j-1}]}{x_{\nu +j}-x_{\nu }}},\qquad \nu \in \{0,\ldots ,k-j\},\ j\in \{1,\ldots ,k\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3fbf429b65bad77486a9d90314d4af6e5edf0c1)
![{\displaystyle [y_{\nu },\ldots ,y_{\nu -j}]:={\frac {[y_{\nu },\ldots ,y_{\nu -j+1}]-[y_{\nu -1},\ldots ,y_{\nu -j}]}{x_{\nu }-x_{\nu -j}}},\qquad \nu \in \{j,\ldots ,k\},\ j\in \{1,\ldots ,k\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9ca4b8abdebfbfaf509c340f8aa07f1508e8c8b)