数学 における差商 （さしょう、英 : divided differences 差分商 [ 1] 差分商 をとる操作を再帰的 に繰り返すことで与えられる。歴史的には対数表や三角函数表の計算に用いられ、チャールズ・バベッジ の階差機関 （初期の機械式計算機 ）はこれを実装するものとして設計された[ 2]
差商はニュートン補間 における補間多項式 の計算に用いることができる。
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{\textstyle (x_{0},y_{0}),\ldots ,(x_{n},y_{n})}
前進差商 は
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{\displaystyle {\begin{aligned}{}[y_{\nu }]&:=y_{\nu }\quad (\nu \in \{0,\ldots ,n\}),\\[5pt][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 }}}\quad (\nu \in \{0,\ldots ,n-j\},\ j\in \{1,\ldots ,n\})\end{aligned}}}
後退差商 は
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{\displaystyle {\begin{aligned}{}[y_{\nu }]&:=y_{\nu }\quad (\nu \in \{0,\ldots ,n\}),\\[5pt][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}}}\quad (\nu \in \{j,\ldots ,n\},\ j\in \{1,\ldots ,n\})\end{aligned}}}
以下本項では主に前進差商のみを扱い、それを単に差商と呼ぶ。
節点（の x -座標）が等間隔に並んでいるときには、前進差分 （あるいは後退・中央の各有限差分）の定める差分商 によって差商を記述することができる。この場合は、より一般の差商よりも計算が楽になる。注目すべきは、差商の分母が既知であるならば、差分商から差商を容易に回復することができることである。
すなわち、上と同じ節点のデータが与えられ、適当な数 h > 0xν = x 0 + νh (ν = 0, …, n ) となっているとき、前進差分は
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{\displaystyle {\begin{aligned}\Delta ^{(0)}y_{i}&:=y_{i},\\\Delta ^{(k)}y_{i}&:=\Delta ^{(k-1)}y_{i+1}-\Delta ^{(k-1)}y_{i}\quad (k\geq 1)\end{aligned}}}
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{\displaystyle f[x_{0},x_{1},\ldots ,x_{k}]={\frac {1}{k!h^{k}}}\Delta ^{(k)}f(x_{0})}
[ 3]
隣り合う二項から次の階数の項が次々に作られる様子は、パスカルの三角形 の類似で、差商を三角形状に並べるとわかりよい。
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{\begin{matrix}x_{0}&y_{0}=[y_{0}]&&&\\&&[y_{0},y_{1}]&&\\x_{1}&y_{1}=[y_{1}]&&[y_{0},y_{1},y_{2}]&\\&&[y_{1},y_{2}]&&[y_{0},y_{1},y_{2},y_{3}]\\x_{2}&y_{2}=[y_{2}]&&[y_{1},y_{2},y_{3}]&\\&&[y_{2},y_{3}]&&\\x_{3}&y_{3}=[y_{3}]&&&\end{matrix}}
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{\begin{matrix}y_{0}&&&\\&\Delta y_{0}&&\\y_{1}&&\Delta ^{2}y_{0}&\\&\Delta y_{1}&&\Delta ^{3}y_{0}\\y_{2}&&\Delta ^{2}y_{1}&\\&\Delta y_{2}&&\\y_{3}&&&\end{matrix}}
節点が函数 f を用いて
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{\textstyle (x_{0},f(x_{0})),\ldots ,(x_{n},f(x_{n}))}
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{\displaystyle {\begin{aligned}f[x_{\nu }]&:=f(x_{\nu }),\qquad \nu \in \{0,\ldots ,n\},\\[5pt]f[x_{\nu },\ldots ,x_{\nu +j}]&:={\frac {f[x_{\nu +1},\ldots ,x_{\nu +j}]-f[x_{\nu },\ldots ,x_{\nu +j-1}]}{x_{\nu +j}-x_{\nu }}}\quad (\nu \in \{0,\ldots ,n-j\},\ j\in \{1,\ldots ,n\})\end{aligned}}}
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{\displaystyle [x_{0},\ldots ,x_{n}]f,\quad [x_{0},\ldots ,x_{n};f],\quad D[x_{0},\ldots ,x_{n}]f}
具体的に書き下せば:
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{\displaystyle {\begin{aligned}f[x_{0}]&=f(x_{0})\\f[x_{0},x_{1}]&={\frac {f(x_{0})}{(x_{0}-x_{1})}}+{\frac {f(x_{1})}{(x_{1}-x_{0})}}\\f[x_{0},x_{1},x_{2}]&={\frac {f(x_{0})}{(x_{0}-x_{1})\cdot (x_{0}-x_{2})}}+{\frac {f(x_{1})}{(x_{1}-x_{0})\cdot (x_{1}-x_{2})}}+{\frac {f(x_{2})}{(x_{2}-x_{0})\cdot (x_{2}-x_{1})}}\\f[x_{0},x_{1},x_{2},x_{3}]&={\frac {f(x_{0})}{(x_{0}-x_{1})\cdot (x_{0}-x_{2})\cdot (x_{0}-x_{3})}}+{\frac {f(x_{1})}{(x_{1}-x_{0})\cdot (x_{1}-x_{2})\cdot (x_{1}-x_{3})}}+{\frac {f(x_{2})}{(x_{2}-x_{0})\cdot (x_{2}-x_{1})\cdot (x_{2}-x_{3})}}+{\frac {f(x_{3})}{(x_{3}-x_{0})\cdot (x_{3}-x_{1})\cdot (x_{3}-x_{2})}}\end{aligned}}}
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{\displaystyle f[x_{0},\dots ,x_{n}]=\sum _{j=0}^{n}{\frac {f(x_{j})}{\prod _{k=0,\ldots n;k\neq j}(x_{j}-x_{k})}}}
多項式函数
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{\textstyle q(\xi )=(\xi -x_{0})\cdots (\xi -x_{n})}
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{\displaystyle f[x_{0},\dots ,x_{n}]=\sum _{j=0}^{n}{\frac {f(x_{j})}{q'(x_{j})}}}
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{\displaystyle f[x_{0},\dots ,x_{n}]=\sum _{j=0}^{n}\lim _{x\to x_{j}}\left[{\frac {f(x_{j})(x-x_{j})}{\prod _{k=0}^{n}(x-x_{k})}}\right]}
線型性 :
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{\displaystyle {\begin{aligned}&(f+g)[x_{0},\dots ,x_{n}]=f[x_{0},\dots ,x_{n}]+g[x_{0},\dots ,x_{n}]\\&(\lambda \cdot f)[x_{0},\dots ,x_{n}]=\lambda \cdot f[x_{0},\dots ,x_{n}]\end{aligned}}}
ライプニッツの法則 :
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{\displaystyle (f\cdot g)[x_{0},\dots ,x_{n}]=f[x_{0}]\cdot g[x_{0},\dots ,x_{n}]+f[x_{0},x_{1}]\cdot g[x_{1},\dots ,x_{n}]+\dots +f[x_{0},\dots ,x_{n}]\cdot g[x_{n}]}
対称性:
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{\displaystyle f[x_{0},\dots ,x_{n}]=f[x_{\sigma (0)},\dots ,x_{\sigma (n)}]\quad (\sigma \in S_{n})}
差商に対する平均値の定理 より:
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{\displaystyle f[x_{0},\dots ,x_{n}]={\frac {f^{(n)}(\xi )}{n!}}\quad (\exists \xi \in (\min x_{k},\max x_{k}))}
次数 n − 1Bスプライン B n −1
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{\displaystyle f[x_{0},\ldots ,x_{n}]={\frac {1}{n!}}\int _{x_{0}}^{x_{n}}f^{(n)}(t)B_{n-1}(t)\,dt}
f (n ) f の n -階導函数である。
これを差商のペアノ形 (Peano form ) と言い、B n −1ペアノ核 （英語版 ） ジゼッペ・ペアノ に因む。
考えている節点が集積しているならば、ほとんどゼロに近い値での割り算が生じ、相対誤差 が大きくなるから、数値計算はおぼつかない。しかしその場合も、差分商 を微分商 で（あるいはその逆に）近似することはできる:
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{\displaystyle {\frac {f(y)-f(x)}{y-x}}\approx f'(x)\quad (x\approx y).}
この近似はテイラーの定理
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{\displaystyle f(y)=f(x)+f'(x)\cdot (y-x)+f''(x)\cdot {\frac {(y-x)^{2}}{2!}}+f'''(x)\cdot {\frac {(y-x)^{3}}{3!}}+\dotsb }
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{\displaystyle {\frac {f(y)-f(x)}{y-x}}=f'(x)+f''(x)\cdot {\frac {y-x}{2!}}+f'''(x)\cdot {\frac {(y-x)^{2}}{3!}}+\dotsb }
y − x の奇数冪の項を消すことができる:
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{\displaystyle {\frac {f(y)-f(x)}{y-x}}={\frac {f(m+h)-f(m-h)}{2\cdot h}}=f'(m)+f'''(m)\cdot {\frac {h^{2}}{3!}}+\dotsb .}
原理的には、テイラー級数および任意の函数項級数 （英語版 ） 冪函数 に関する無限和で、函数にその差商を対応させる写像 f ↦ f [x 0 , …, xn ]線型汎函数 であるから、基底函数にこの汎函数を適用すればよい。
通常の冪函数 pn (x ) ≔ xn
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{\displaystyle f=f(0)\cdot p_{0}+f'(0)\cdot p_{1}+{\frac {f''(0)}{2!}}\cdot p_{2}+{\frac {f'''(0)}{3!}}\cdot p_{3}+\dotsc }
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{\displaystyle f[x_{0},\dots ,x_{n}]=f(0)\cdot p_{0}[x_{0},\dots ,x_{n}]+f'(0)\cdot p_{1}[x_{0},\dots ,x_{n}]+{\frac {f''(0)}{2!}}\cdot p_{2}[x_{0},\dots ,x_{n}]+{\frac {f'''(0)}{3!}}\cdot p_{3}[x_{0},\dots ,x_{n}]+\dotsb }
n 項は多項式の次数よりも高階の差分だから消えており、それ以降の項も以下のように知ることができる:
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{\displaystyle {\begin{cases}p_{j}[x_{0},\dots ,x_{n}]=0&(j<n)\\[3pt]p_{n}[x_{0},\dots ,x_{n}]=1&\\[3pt]p_{n+1}[x_{0},\dots ,x_{n}]&=x_{0}+\dots +x_{n}\\[8pt]p_{n+m}[x_{0},\dots ,x_{n}]&=\displaystyle \sum _{0\leq a_{1}\leq a_{2}\leq \dots \leq a_{m}\leq n}\prod _{j\in \{a_{1},a_{2},\ldots ,a_{m}\}}x_{j}.\\\end{cases}}}
多項式の差商はライプニッツの法則の恩恵を受けられる点で興味を持たれる。
行列 J を
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{\displaystyle J={\begin{pmatrix}x_{0}&1&0&0&\cdots &0\\0&x_{1}&1&0&\cdots &0\\0&0&x_{2}&1&&0\\\vdots &\vdots &&\ddots &\ddots &\\\vdots &\vdots &&&\ddots &1\\0&0&0&0&\cdots &x_{n}\end{pmatrix}}}
x 0 , …, xn 恒等写像 f (x ) = x Jn は冪函数 xn に対する差商が全て現れる。差商の線型性により、多項式 p に対する多項式函数を J に適用 したものは、
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{\displaystyle {\begin{aligned}p(J)&=a_{0}\cdot I+a_{1}\cdot J+\dots +a_{n}\cdot J^{n}\\[5pt]&={\begin{pmatrix}p[x_{0}]&p[x_{0},x_{1}]&p[x_{0},x_{1},x_{2}]&\ldots &p[x_{0},\dots ,x_{n}]\\0&p[x_{1}]&p[x_{1},x_{2}]&\ldots &p[x_{1},\dots ,x_{n}]\\[10pt]\vdots &\ddots &\ddots &\ddots &\vdots \\[10pt]0&\ldots &0&0&p[x_{n}]\end{pmatrix}}\end{aligned}}}
オピッツの公式 と呼ばれる[ 4] [ 5]
多項式 p の次数を無限大に飛ばして形式冪級数 とすれば、テイラー展開可能な函数 f に対しても、その差商を J を用いた行列表示
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{\displaystyle T_{f}(x_{0},\dots ,x_{n})=f(J)={\begin{pmatrix}f[x_{0}]&f[x_{0},x_{1}]&f[x_{0},x_{1},x_{2}]&\ldots &f[x_{0},\dots ,x_{n}]\\0&f[x_{1}]&f[x_{1},x_{2}]&\ldots &f[x_{1},\dots ,x_{n}]\\[20pt]\vdots &\vdots &\vdots &\ddots &\vdots \\0&0&0&\ldots &f[x_{n}]\end{pmatrix}}}
節点 x 0 , …, xn J はジョルダン細胞であり、ジョルダン標準形 を考えることでスカラー函数を行列函数 に一般化することができる。
^ 世界大百科事典『差分商 』 - コトバンク
^ Isaacson, Walter (2014). The Innovators . Simon & Schuster. p. 20. ISBN 978-1-4767-0869-0 ^ Burden, Richard L.; Faires, J. Douglas (2011). Numerical Analysis (9th ed.). p. 129 ^ de Boor, Carl , Divided Differences , Surv. Approx. Theory 1 (2005), 46–69, [1] ^ Opitz, G. Steigungsmatrizen , Z. Angew. Math. Mech. (1964), 44, T52–T54
Louis Melville Milne-Thomson (2000) [1933]. The Calculus of Finite Differences . American Mathematical Soc.. Chapter 1: Divided Differences. ISBN 978-0-8218-2107-7 Myron B. Allen; Eli L. Isaacson (1998). Numerical Analysis for Applied Science . John Wiley & Sons. Appendix A. ISBN 978-1-118-03027-1 Ron Goldman (2002). Pyramid Algorithms: A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling . Morgan Kaufmann. Chapter 4:Newton Interpolation and Difference Triangles. ISBN 978-0-08-051547-2
![{\displaystyle {\begin{aligned}{}[y_{\nu }]&:=y_{\nu }\quad (\nu \in \{0,\ldots ,n\}),\\[5pt][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 }}}\quad (\nu \in \{0,\ldots ,n-j\},\ j\in \{1,\ldots ,n\})\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7adab842d600ccb3dc24ae843c1e99b261511e78)
![{\displaystyle {\begin{aligned}{}[y_{\nu }]&:=y_{\nu }\quad (\nu \in \{0,\ldots ,n\}),\\[5pt][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}}}\quad (\nu \in \{j,\ldots ,n\},\ j\in \{1,\ldots ,n\})\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e84eb97156d2ffd4872de15c2a5e169d00f960e9)
![{\displaystyle f[x_{0},x_{1},\ldots ,x_{k}]={\frac {1}{k!h^{k}}}\Delta ^{(k)}f(x_{0})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a329caf76ef94a24c1137518eb08869f536ad9af)
![{\begin{matrix}x_{0}&y_{0}=[y_{0}]&&&\\&&[y_{0},y_{1}]&&\\x_{1}&y_{1}=[y_{1}]&&[y_{0},y_{1},y_{2}]&\\&&[y_{1},y_{2}]&&[y_{0},y_{1},y_{2},y_{3}]\\x_{2}&y_{2}=[y_{2}]&&[y_{1},y_{2},y_{3}]&\\&&[y_{2},y_{3}]&&\\x_{3}&y_{3}=[y_{3}]&&&\end{matrix}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/214ccc8b79db893acee51880fcaa47eb92b20129)
![{\displaystyle {\begin{aligned}f[x_{\nu }]&:=f(x_{\nu }),\qquad \nu \in \{0,\ldots ,n\},\\[5pt]f[x_{\nu },\ldots ,x_{\nu +j}]&:={\frac {f[x_{\nu +1},\ldots ,x_{\nu +j}]-f[x_{\nu },\ldots ,x_{\nu +j-1}]}{x_{\nu +j}-x_{\nu }}}\quad (\nu \in \{0,\ldots ,n-j\},\ j\in \{1,\ldots ,n\})\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb356244238fceb55eafba4ece02d3c314457acd)
![{\displaystyle [x_{0},\ldots ,x_{n}]f,\quad [x_{0},\ldots ,x_{n};f],\quad D[x_{0},\ldots ,x_{n}]f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff9c095eb9794999b7b20d93bfc06337814672ad)
![{\displaystyle {\begin{aligned}f[x_{0}]&=f(x_{0})\\f[x_{0},x_{1}]&={\frac {f(x_{0})}{(x_{0}-x_{1})}}+{\frac {f(x_{1})}{(x_{1}-x_{0})}}\\f[x_{0},x_{1},x_{2}]&={\frac {f(x_{0})}{(x_{0}-x_{1})\cdot (x_{0}-x_{2})}}+{\frac {f(x_{1})}{(x_{1}-x_{0})\cdot (x_{1}-x_{2})}}+{\frac {f(x_{2})}{(x_{2}-x_{0})\cdot (x_{2}-x_{1})}}\\f[x_{0},x_{1},x_{2},x_{3}]&={\frac {f(x_{0})}{(x_{0}-x_{1})\cdot (x_{0}-x_{2})\cdot (x_{0}-x_{3})}}+{\frac {f(x_{1})}{(x_{1}-x_{0})\cdot (x_{1}-x_{2})\cdot (x_{1}-x_{3})}}+{\frac {f(x_{2})}{(x_{2}-x_{0})\cdot (x_{2}-x_{1})\cdot (x_{2}-x_{3})}}+{\frac {f(x_{3})}{(x_{3}-x_{0})\cdot (x_{3}-x_{1})\cdot (x_{3}-x_{2})}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6bc279b3f26167a6bd4665a416bcc281c82ccf1)
![{\displaystyle f[x_{0},\dots ,x_{n}]=\sum _{j=0}^{n}{\frac {f(x_{j})}{\prod _{k=0,\ldots n;k\neq j}(x_{j}-x_{k})}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a75a1a70f2b2ce65481ace666de7b61fc663a789)
![{\displaystyle f[x_{0},\dots ,x_{n}]=\sum _{j=0}^{n}{\frac {f(x_{j})}{q'(x_{j})}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d705ece3e5b49985062d6e443f2643f300cadcc)
![{\displaystyle f[x_{0},\dots ,x_{n}]=\sum _{j=0}^{n}\lim _{x\to x_{j}}\left[{\frac {f(x_{j})(x-x_{j})}{\prod _{k=0}^{n}(x-x_{k})}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/308e8b6e5bc029f8e2892cbde2f70688a0e51ab2)
![{\displaystyle {\begin{aligned}&(f+g)[x_{0},\dots ,x_{n}]=f[x_{0},\dots ,x_{n}]+g[x_{0},\dots ,x_{n}]\\&(\lambda \cdot f)[x_{0},\dots ,x_{n}]=\lambda \cdot f[x_{0},\dots ,x_{n}]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9be546dd277ae763e3ba5ee3932bb428ea22174)
![{\displaystyle (f\cdot g)[x_{0},\dots ,x_{n}]=f[x_{0}]\cdot g[x_{0},\dots ,x_{n}]+f[x_{0},x_{1}]\cdot g[x_{1},\dots ,x_{n}]+\dots +f[x_{0},\dots ,x_{n}]\cdot g[x_{n}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/534095e5ec06e0a714829f32de106d9c65c62b9e)
![{\displaystyle f[x_{0},\dots ,x_{n}]=f[x_{\sigma (0)},\dots ,x_{\sigma (n)}]\quad (\sigma \in S_{n})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a55d7e0952e31a0868f6404449ed027d0d567d18)
![{\displaystyle f[x_{0},\dots ,x_{n}]={\frac {f^{(n)}(\xi )}{n!}}\quad (\exists \xi \in (\min x_{k},\max x_{k}))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e6f3292ee79d3b2dd630613ef5aca02433e7659)
![{\displaystyle f[x_{0},\ldots ,x_{n}]={\frac {1}{n!}}\int _{x_{0}}^{x_{n}}f^{(n)}(t)B_{n-1}(t)\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ae40d8692899cbc0fdcb156a32205e51c00250b)
![{\displaystyle f[x_{0},\dots ,x_{n}]=f(0)\cdot p_{0}[x_{0},\dots ,x_{n}]+f'(0)\cdot p_{1}[x_{0},\dots ,x_{n}]+{\frac {f''(0)}{2!}}\cdot p_{2}[x_{0},\dots ,x_{n}]+{\frac {f'''(0)}{3!}}\cdot p_{3}[x_{0},\dots ,x_{n}]+\dotsb }](https://wikimedia.org/api/rest_v1/media/math/render/svg/68fb1eeabfaf8b8ffd60f97c93121ef5a3bc2c80)
![{\displaystyle {\begin{cases}p_{j}[x_{0},\dots ,x_{n}]=0&(j<n)\\[3pt]p_{n}[x_{0},\dots ,x_{n}]=1&\\[3pt]p_{n+1}[x_{0},\dots ,x_{n}]&=x_{0}+\dots +x_{n}\\[8pt]p_{n+m}[x_{0},\dots ,x_{n}]&=\displaystyle \sum _{0\leq a_{1}\leq a_{2}\leq \dots \leq a_{m}\leq n}\prod _{j\in \{a_{1},a_{2},\ldots ,a_{m}\}}x_{j}.\\\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2cf80153de407d5ff16e8f114f2a01806cf92bb3)
![{\displaystyle {\begin{aligned}p(J)&=a_{0}\cdot I+a_{1}\cdot J+\dots +a_{n}\cdot J^{n}\\[5pt]&={\begin{pmatrix}p[x_{0}]&p[x_{0},x_{1}]&p[x_{0},x_{1},x_{2}]&\ldots &p[x_{0},\dots ,x_{n}]\\0&p[x_{1}]&p[x_{1},x_{2}]&\ldots &p[x_{1},\dots ,x_{n}]\\[10pt]\vdots &\ddots &\ddots &\ddots &\vdots \\[10pt]0&\ldots &0&0&p[x_{n}]\end{pmatrix}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/561cca04360c3e214137a4f4d028a26af04bd9c5)
![{\displaystyle T_{f}(x_{0},\dots ,x_{n})=f(J)={\begin{pmatrix}f[x_{0}]&f[x_{0},x_{1}]&f[x_{0},x_{1},x_{2}]&\ldots &f[x_{0},\dots ,x_{n}]\\0&f[x_{1}]&f[x_{1},x_{2}]&\ldots &f[x_{1},\dots ,x_{n}]\\[20pt]\vdots &\vdots &\vdots &\ddots &\vdots \\0&0&0&\ldots &f[x_{n}]\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bcb37eba944f72511ad0641966bce8050bade77)