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牛顿多项式(英语:Newton Polynomial)是数值分析中一种用于插值的多项式,以英格兰数学家暨物理学家牛顿命名。
给定包含
个数据点的集合
。
如果对于
,满足
,那么应用牛顿插值公式所得到的牛顿插值多项式为
![{\displaystyle N(x):=\sum _{j=0}^{k}a_{j}n_{j}(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7bb15443c96172a7ef235cb973010feab25e428)
其中每个
为牛顿基本多项式(或称插值基函数),其表达式为
![{\displaystyle n_{j}(x):=\prod _{i=0}^{j-1}(x-x_{i})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/227efa64e8b5301108a7a5da5cbbb86dcf59af04)
其中
,并且
。
系数
,而
表示差商。
差商表(高阶差商是两个低一阶差商的差商)
|
阶差商 |
阶差商 |
阶差商 |
阶差商 |
![{\displaystyle \ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b8619532e44ee1ccae3ab03405a6885260d09ed) |
阶差商
|
![{\displaystyle x_{0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf) |
![{\displaystyle f[x_{0}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8684dad2a9bd05dfa2a2556965bb3d82b0616431) |
|
|
|
|
|
![{\displaystyle x_{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8788bf85d532fa88d1fb25eff6ae382a601c308) |
![{\displaystyle f[x_{1}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3fe3bc79fe72985198a9f8c7c6ca75d1c00dcf1) |
![{\displaystyle f[x_{0},x_{1}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a31f8458da9ea4ec290a96134207b45c5d7a4f7c) |
|
|
|
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![{\displaystyle x_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7af1b928f06e4c7e3e8ebfd60704656719bd766) |
![{\displaystyle f[x_{2}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7ce16ec7c1226872e3ee3a60c35e7667daab378) |
![{\displaystyle f[x_{1},x_{2}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d16b06171f52c855c9ac0c05d47cb6be5fb30e40) |
![{\displaystyle f[x_{0},x_{1},x_{2}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d46b58779485963bbee5306d1f8dcd8550e5a86b) |
|
|
|
![{\displaystyle x_{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/766d09a498699be10e276ad49145c921f8cbe335) |
![{\displaystyle f[x_{3}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b782040953b2db0bc3108f8d90972d46be3ed43) |
![{\displaystyle f[x_{2},x_{3}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae2d7f2d8ecb1b59657da15ba30c0b599fa7ca7c) |
![{\displaystyle f[x_{1},x_{2},x_{3}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1381c903e45e87baed129d821406f7035ccf883c) |
![{\displaystyle f[x_{0},x_{1},x_{2},x_{3}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22c97723f190624c9bcfaa90c9ace00dfe10311f) |
|
|
![{\displaystyle \ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b8619532e44ee1ccae3ab03405a6885260d09ed) |
![{\displaystyle \ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b8619532e44ee1ccae3ab03405a6885260d09ed) |
![{\displaystyle \ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b8619532e44ee1ccae3ab03405a6885260d09ed) |
![{\displaystyle \ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b8619532e44ee1ccae3ab03405a6885260d09ed) |
![{\displaystyle \ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b8619532e44ee1ccae3ab03405a6885260d09ed) |
![{\displaystyle \ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b8619532e44ee1ccae3ab03405a6885260d09ed) |
|
![{\displaystyle x_{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d2b88c64c76a03611549fb9b4cf4ed060b56002) |
![{\displaystyle f[x_{k}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/068ffa6f95a0dd5df1cc3e859ec8635a193de831) |
![{\displaystyle f[x_{k-1},x_{k}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a2b4156a5b63df490e477942478c138dd5920b6) |
![{\displaystyle f[x_{k-2},x_{k-1},x_{k}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/485cd20523471021cfda6710bcb76dc2290f8648) |
![{\displaystyle f[x_{k-3},x_{k-2},x_{k-1},x_{k}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5945aa10854d14a83fc859064b914e78a6d2780b) |
![{\displaystyle \ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b8619532e44ee1ccae3ab03405a6885260d09ed) |
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因此,牛顿多项式可以写作:
![{\displaystyle N(x)=[y_{0}]+[y_{0},y_{1}](x-x_{0})+\cdots +[y_{0},\ldots ,y_{k}](x-x_{0})(x-x_{1})\cdots (x-x_{k-1})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee525923c8be242f4e437763dfeb8e9f5a14d805)