泰勒公式的理解和快速推导，格雷戈里-牛顿插值公式的简单推导
泰勒公式的理解和快速推导，格雷戈里-牛顿插值公式的简单推导。其实，泰勒公式本质就是用多项幂级数逼近一个单变量函数，也可以说是使任何单变量函数都可以展开成幂级数。假设我们在取点时，特意使任意相邻两点之间的水平间距都相等。佩亚诺型余项和拉格朗日余项。泰勒级数和泰勒展开式，根据上述推导，实际上当我们已知无数个点，即时，泰勒公式具有无穷项，它就像幂级数,称之为泰勒级数。但是在得到公式的过程中，我们却没有去讨论级数的是否收敛。泰勒展开式是特殊的泰勒级数，或者说是收敛情况下的泰勒级数，其具有收敛区间，在收敛区间内，和一定存在。麦克劳林公式
泰勒公式的理解和简单推导，格雷戈里-牛顿插值公式的简单推导
前言：重在记录，可能出错。
可以直接看常用函数的泰勒公式的快速推导
一、格雷戈里-牛顿插值公式的简单推导
1.插值：函数的图像一般由点组成，那么我们可以通过已知的点来找另一个函数近似等于原函数。很明显，已知点越多，就越逼近原函数。
其实，泰勒公式本质就是用多项幂级数逼近一个单变量函数，也可以说是使任何单变量函数都可以展开成幂级数。
牛顿插值：这个一般会在大学的计算方法、数值分析等课程中学到。牛顿插值是插值方法的一种。
接下来，开始牛顿插值的无敌推导：
(1).如果只有一个点\({A \left( x\mathop{{}}\nolimits_{{0}},f \left( x\mathop{{}}\nolimits_{{0}} \left) \right) \right. \right. }\)，那么我们找的近似函数即为\({f \left( x \left) =f \left( x\mathop{{}}\nolimits_{{0}} \right) \right. \right. }\)，代入点A符合。
(2).如果有两个点\({A \left( x\mathop{{}}\nolimits_{{0}},f \left( x\mathop{{}}\nolimits_{{0}} \left) \left) \text{、}B \left( x\mathop{{}}\nolimits_{{1}},f \left( x\mathop{{}}\nolimits_{{1}} \left) \right) \right. \right. \right. \right. \right. \right. }\)，我们在\({f \left( x \left) =f \left( x\mathop{{}}\nolimits_{{0}} \right) \right. \right. }\)的基础上再加上一项，代入点A、B均符合即可。设近似函数为:
\[{\begin{array}{*{20}{l}}{f \left( x \left) =f \left( x\mathop{{}}\nolimits_{{0}} \left) +C \left( x \right) \right. \right. \right. \right. }\\{C \left( x \left) =f \left( x \left) -f \left( x\mathop{{}}\nolimits_{{0}} \right) \right. \right. \right. \right. }\\{C \left( x\mathop{{}}\nolimits_{{0}} \left) =f \left( x\mathop{{}}\nolimits_{{0}} \left) -f \left( x\mathop{{}}\nolimits_{{0}} \left) =0=a\mathop{{}}\nolimits_{{1}} \left( x-x\mathop{{}}\nolimits_{{0}} \left) \text{，}\text{写}\text{出}\text{大}\text{概}\text{形}\text{式}\right. \right. \right. \right. \right. \right. \right. \right. }\\{C \left( x\mathop{{}}\nolimits_{{1}} \left) =f \left( x\mathop{{}}\nolimits_{{1}} \left) -f \left( x\mathop{{}}\nolimits_{{0}} \left) =a\mathop{{}}\nolimits_{{1}} \left( x\mathop{{}}\nolimits_{{1}}-x\mathop{{}}\nolimits_{{0}} \right) \right. \right. \right. \right. \right. \right. }\\{a\mathop{{}}\nolimits_{{1}}=\frac{{f \left( x\mathop{{}}\nolimits_{{1}} \left) -f \left( x\mathop{{}}\nolimits_{{0}} \right) \right. \right. }}{{x\mathop{{}}\nolimits_{{1}}-x\mathop{{}}\nolimits_{{0}}}}}\\{C \left( x \left) =\frac{{f \left( x\mathop{{}}\nolimits_{{1}} \left) -f \left( x\mathop{{}}\nolimits_{{0}} \right) \right. \right. }}{{x\mathop{{}}\nolimits_{{1}}-x\mathop{{}}\nolimits_{{0}}}}\left( x-x\mathop{{}}\nolimits_{{0}} \right) \right. \right. }\end{array}}
\]
因此，用两个点找的近似函数为：
\[f \left( x \left) =f \left( x\mathop{{}}\nolimits_{{0}} \left) +\frac{{f \left( x\mathop{{}}\nolimits_{{1}} \left) -f \left( x\mathop{{}}\nolimits_{{0}} \right) \right. \right. }}{{x\mathop{{}}\nolimits_{{1}}-x\mathop{{}}\nolimits_{{0}}}} \left( x-x\mathop{{}}\nolimits_{{0}} \right) \right. \right. \right. \right.
\]
(3).同理，三个点：
\[{\begin{array}{*{20}{l}}{{\left. {\begin{array}{*{20}{l}}{{C \left( x\mathop{{}}\nolimits_{{0}} \left) =\right. \right. }f \left( x\mathop{{}}\nolimits_{{0}} \left) -f \left( x\mathop{{}}\nolimits_{{0}} \left) -\frac{{f \left( x\mathop{{}}\nolimits_{{1}} \left) -f \left( x\mathop{{}}\nolimits_{{0}} \right) \right. \right. }}{{x\mathop{{}}\nolimits_{{1}}-x\mathop{{}}\nolimits_{{0}}}}\left( x\mathop{{}}\nolimits_{{0}}-x\mathop{{}}\nolimits_{{0}} \left) =0\right. \right. \right. \right. \right. \right. }\\{C \left( x\mathop{{}}\nolimits_{{1}} \left) =f \left( x\mathop{{}}\nolimits_{{1}} \left) -f \left( x\mathop{{}}\nolimits_{{0}} \left) -\frac{{f \left( x\mathop{{}}\nolimits_{{1}} \left) -f \left( x\mathop{{}}\nolimits_{{0}} \right) \right. \right. }}{{x\mathop{{}}\nolimits_{{1}}-x\mathop{{}}\nolimits_{{0}}}}\left( x\mathop{{}}\nolimits_{{1}}-x\mathop{{}}\nolimits_{{0}} \left) =0\right. \right. \right. \right. \right. \right. \right. \right. }\end{array}} \right\} }}\\{C \left( x \left) =a\mathop{{}}\nolimits_{{2}} \left( x-x\mathop{{}}\nolimits_{{0}} \left) \left( x-x\mathop{{}}\nolimits_{{1}} \left) \text{，}\text{写}\text{出}\text{大}\text{概}\text{形}\text{式}\right. \right. \right. \right. \right. \right. }\\{C \left( x\mathop{{}}\nolimits_{{2}} \left) =f \left( x\mathop{{}}\nolimits_{{2}} \left) -f \left( x\mathop{{}}\nolimits_{{0}} \left) -\frac{{f \left( x\mathop{{}}\nolimits_{{1}} \left) -f \left( x\mathop{{}}\nolimits_{{0}} \right) \right. \right. }}{{x\mathop{{}}\nolimits_{{1}}-x\mathop{{}}\nolimits_{{0}}}}\left( x\mathop{{}}\nolimits_{{2}}-x\mathop{{}}\nolimits_{{0}} \left) =a\mathop{{}}\nolimits_{{2}} \left( x\mathop{{}}\nolimits_{{2}}-x\mathop{{}}\nolimits_{{0}} \left) \left( x\mathop{{}}\nolimits_{{2}}-x\mathop{{}}\nolimits_{{1}} \right) \right. \right. \right. \right. \right. \right. \right. \right. \right. \right. }\\{a\mathop{{}}\nolimits_{{2}}=\frac{{f \left( x\mathop{{}}\nolimits_{{2}} \left) -f \left( x\mathop{{}}\nolimits_{{0}} \left) -\frac{{f \left( x\mathop{{}}\nolimits_{{1}} \left) -f \left( x\mathop{{}}\nolimits_{{0}} \right) \right. \right. }}{{x\mathop{{}}\nolimits_{{1}}-x\mathop{{}}\nolimits_{{0}}}}\left( x\mathop{{}}\nolimits_{{2}}-x\mathop{{}}\nolimits_{{0}} \right) \right. \right. \right. \right. }}{{ \left( x\mathop{{}}\nolimits_{{2}}-x\mathop{{}}\nolimits_{{0}} \left) \left( x\mathop{{}}\nolimits_{{2}}-x\mathop{{}}\nolimits_{{1}} \right) \right. \right. }}=\frac{{\frac{{f \left( x\mathop{{}}\nolimits_{{2}} \left) -f \left( x\mathop{{}}\nolimits_{{0}} \right) \right. \right. }}{{x\mathop{{}}\nolimits_{{2}}-x\mathop{{}}\nolimits_{{0}}}}-\frac{{f \left( x\mathop{{}}\nolimits_{{1}} \left) -f \left( x\mathop{{}}\nolimits_{{0}} \right) \right. \right. }}{{x\mathop{{}}\nolimits_{{1}}-x\mathop{{}}\nolimits_{{0}}}}}}{{ \left( x\mathop{{}}\nolimits_{{2}}-x\mathop{{}}\nolimits_{{1}} \right) }}}\end{array}}
\]
因此，用三个点找的近似函数为：
\[{f \left( x \left) =f \left( x\mathop{{}}\nolimits_{{0}} \left) +\frac{{f \left( x\mathop{{}}\nolimits_{{1}} \left) -f \left( x\mathop{{}}\nolimits_{{0}} \right) \right. \right. }}{{x\mathop{{}}\nolimits_{{1}}-x\mathop{{}}\nolimits_{{0}}}}\left( x-x\mathop{{}}\nolimits_{{0}} \left) +\frac{{\frac{{f \left( x\mathop{{}}\nolimits_{{2}} \left) -f \left( x\mathop{{}}\nolimits_{{0}} \right) \right. \right. }}{{x\mathop{{}}\nolimits_{{2}}-x\mathop{{}}\nolimits_{{0}}}}-\frac{{f \left( x\mathop{{}}\nolimits_{{1}} \left) -f \left( x\mathop{{}}\nolimits_{{0}} \right) \right. \right. }}{{x\mathop{{}}\nolimits_{{1}}-x\mathop{{}}\nolimits_{{0}}}}}}{{x\mathop{{}}\nolimits_{{2}}-x\mathop{{}}\nolimits_{{1}}}} \left( x-x\mathop{{}}\nolimits_{{0}} \left) \left( x-x\mathop{{}}\nolimits_{{1}} \right) \right. \right. \right. \right. \right. \right. \right. \right. }
\]
(4).总结，若有n+1个点：
\[\color{red}{a\mathop{{}}\nolimits_{{n}}=\frac{{f \left( x\mathop{{}}\nolimits_{{n}} \left) -a\mathop{{}}\nolimits_{{0}}-a\mathop{{}}\nolimits_{{1}} \left( x\mathop{{}}\nolimits_{{n}}-x\mathop{{}}\nolimits_{{0}} \left) - \cdots -a\mathop{{}}\nolimits_{{n-1}} \left( x\mathop{{}}\nolimits_{{n}}-x\mathop{{}}\nolimits_{{0}} \left) \left( x\mathop{{}}\nolimits_{{n}}-x\mathop{{}}\nolimits_{{1}} \left) \cdots \left( x\mathop{{}}\nolimits_{{n}}-x\mathop{{}}\nolimits_{{n-2}} \right) \right. \right. \right. \right. \right. \right. \right. \right. }}{{ \left( x\mathop{{}}\nolimits_{{n}}-x\mathop{{}}\nolimits_{{0}} \left) \left( x\mathop{{}}\nolimits_{{n}}-x\mathop{{}}\nolimits_{{1}} \left) \cdots \left( x\mathop{{}}\nolimits_{{n}}-x\mathop{{}}\nolimits_{{n-1}} \right) \right. \right. \right. \right. }}}
\]
近似函数为：
\[\color{red}{f \left( x \left) =a\mathop{{}}\nolimits_{{0}}+a\mathop{{}}\nolimits_{{1}}\left( x-x\mathop{{}}\nolimits_{{0}} \left) +a\mathop{{}}\nolimits_{{2}} \left( x-x\mathop{{}}\nolimits_{{0}} \left) \left( x-x\mathop{{}}\nolimits_{{1}} \left) + \cdots +a\mathop{{}}\nolimits_{{n}} \left( x-x\mathop{{}}\nolimits_{{0}} \left) \left( x-x\mathop{{}}\nolimits_{{1}} \left) \cdots \left( x-x\mathop{{}}\nolimits_{{n-1}} \right) \right. \right. \right. \right. \right. \right. \right. \right. \right. \right. \right. \right. }
\]
注：其中\({a\mathop{{}}\nolimits_{{0}} \cdots a\mathop{{}}\nolimits_{{n}}}\)均为常数，为了通俗一些，就不再说牛顿插值中的差商了，毕竟重点不在这儿。
这就是简化通俗版的非常出名的格雷戈里-牛顿插值公式了。
二、泰勒公式的理解和简单推导
1.泰勒公式：
(1).假设我们在取点时，特意使任意相邻两点之间的水平间距都相等，等于\({ \Delta x}\)，即\({x\mathop{{}}\nolimits_{{n+1}}-x\mathop{{}}\nolimits_{{n}}= \Delta x}\)。此时，再看\({a\mathop{{}}\nolimits_{{0}} \cdots a\mathop{{}}\nolimits_{{n}}}\)：
\[{\begin{array}{*{20}{l}}{a\mathop{{}}\nolimits_{{1}}=\frac{{f \left( x\mathop{{}}\nolimits_{{1}} \left) -f \left( x\mathop{{}}\nolimits_{{0}} \right) \right. \right. }}{{x\mathop{{}}\nolimits_{{1}}-x\mathop{{}}\nolimits_{{0}}}}=\frac{{f \left( x\mathop{{}}\nolimits_{{0}}+ \Delta x \left) -f \left( x\mathop{{}}\nolimits_{{0}} \right) \right. \right. }}{{ \Delta x}}}\\{\begin{array}{*{20}{l}}{a\mathop{{}}\nolimits_{{2}}=\frac{{f \left( x\mathop{{}}\nolimits_{{2}} \left) -f \left( x\mathop{{}}\nolimits_{{0}} \left) \text{ }-\text{ }\frac{{f \left( x\mathop{{}}\nolimits_{{1}} \left) -f \left( x\mathop{{}}\nolimits_{{0}} \left) \text{ }\text{ }\text{ }\text{ }\right. \right. \right. \right. }}{{x\mathop{{}}\nolimits_{{1}}-x\mathop{{}}\nolimits_{{0}}}}\text{ } \left( x\mathop{{}}\nolimits_{{2}}-x\mathop{{}}\nolimits_{{0}} \right) \right. \right. \right. \right. }}{{ \left( x\mathop{{}}\nolimits_{{2}}-x\mathop{{}}\nolimits_{{0}} \left) \text{ } \left( x\mathop{{}}\nolimits_{{2}}-x\mathop{{}}\nolimits_{{1}} \right) \right. \right. }}}\\{\text{ }\text{ }\text{ }\text{ }=\frac{{f \left( x\mathop{{}}\nolimits_{{0}}+2 \Delta x \left) -f \left( x\mathop{{}}\nolimits_{{0}} \left) \text{ }-\text{ }\frac{{f \left( x\mathop{{}}\nolimits_{{0}}+ \Delta x \left) \text{ }-\text{ }f \left( x\mathop{{}}\nolimits_{{0}} \left) \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\right. \right. \right. \right. }}{{ \Delta x}}\text{ } \left( 2 \Delta x \right) \right. \right. \right. \right. }}{{2 \Delta x\text{ } \Delta x}}}\\{\text{ }\text{ }\text{ }\text{ }=\frac{{f \left( x\mathop{{}}\nolimits_{{0}}+2 \Delta x \left) \text{ }-\text{ }2f \left( x\mathop{{}}\nolimits_{{0}}+ \Delta x \left) \text{ }+\text{ }f \left( x\mathop{{}}\nolimits_{{0}} \right) \right. \right. \right. \right. }}{{2 \Delta x\text{ } \Delta x}}}\\{\text{ }\text{ }\text{ }\text{ }=\frac{{\frac{{f \left( x\mathop{{}}\nolimits_{{0}}+2 \Delta x \left) \text{ }-\text{ }f \left( x\mathop{{}}\nolimits_{{0}}+ \Delta x \left) \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\right. \right. \right. \right. }}{{ \Delta x}}\text{ }-\text{ }\frac{{ \left( f \left( x\mathop{{}}\nolimits_{{0}}+ \Delta x \left) \text{ }-\text{ }f \left( x\mathop{{}}\nolimits_{{0}} \left) \right) \right. \right. \right. \right. }}{{ \Delta x}}}}{{2 \Delta x}}}\end{array}}\end{array}}
\]
注：同样，为了通俗一些，就不再说差分了，毕竟重点不在这儿。
当$ { \Delta x \to 0}$时，我们发现(⊙ₒ⊙)!!
\(x\mathop{{}}\nolimits_{{0}}=x\mathop{{}}\nolimits_{{1}}=x\mathop{{}}\nolimits_{{2}}=x\mathop{{}}\nolimits_{{3}}= \cdots =x\mathop{{}}\nolimits_{{n}}\)
并且\({n!}a\mathop{{}}\nolimits_{{n}}\)就是是函数在$ {x=x\mathop{{}}\nolimits_{{0}}}$处导数的定义。
$ a\mathop{{}}\nolimits_{{1}}\text{是}f \left( x\mathop{{}}\nolimits_{{0}} \left) \text{的}\text{一}\text{阶}\text{导}，2!a\mathop{{}}\nolimits_{{2}}\text{是}f \left( x\mathop{{}}\nolimits_{{0}} \left) \text{的}\text{二}\text{阶}\text{导}， \cdots ，n!a\mathop{{}}\nolimits_{{n}}\text{是}f \left( x\mathop{{}}\nolimits_{{0}} \left) \text{的}n\text{阶}\text{导}\right. \right. \right. \right. \right. \right. $
根据规律，我们大胆得到公式：
\[\color{red}{\begin{array}{*{20}{l}}{{f \left( x \left) =f \left( x\mathop{{}}\nolimits_{{0}} \right) +{f \prime } \left( x\mathop{{}}\nolimits_{{0}} \left) \left( x-x\mathop{{}}\nolimits_{{0}} \left) +\frac{{{f''} \left( x\mathop{{}}\nolimits_{{0}} \right) }}{{2!}} \left( x-x\mathop{{}}\nolimits_{{0}} \left) \mathop{{}}\nolimits^{{2}}+ \cdots +\frac{{f\mathop{{}}\nolimits^{{ \left( n \right) }} \left( x\mathop{{}}\nolimits_{{0}} \right) }}{{n!}} \left( x-x\mathop{{}}\nolimits_{{0}} \left) \mathop{{}}\nolimits^{{n}}+R\mathop{{}}\nolimits_{{n+1}} \left( x \left) \right. \right. \right. \right. \right. \right. \right. \right. \right. \right. \right. \right. }}\\{R\mathop{{}}\nolimits_{{n+1}} \left( x \left) \text{是}\text{余}\text{项}\right. \right. }\end{array}}
\]
这就是n阶泰勒公式
(2).佩亚诺型余项和拉格朗日余项：
佩亚诺型余项：泰勒公式的定性形式:用于求未定式极限及估计无穷小阶数等问题
$ {{{R\mathop{{}}\nolimits_{{n+1}} \left( x \left) = {\rm O} \left( \left( x-x\mathop{{}}\nolimits_{{0}} \left) \mathop{{}}\nolimits^{{n}} \right) \right. \right. \right. \right. }}}$
拉格朗日余项：泰勒公式的定量形式：用于求近似计算函数值等问题
$ R\mathop{{}}\nolimits_{{n+1}} \left( x \left) =\frac{{f\mathop{{}}\nolimits^{{ \left( n+1 \right) }}\text{ }\text{ }\text{ } \left( \xi \right) }}{{ \left( n+1 \left) !\right. \right. }} \left( x-x\mathop{{}}\nolimits_{{0}} \left) \mathop{{}}\nolimits^{{n+1}}， \xi \text{介}\text{于}x\mathop{{}}\nolimits_{{0}}\text{和}x\text{之}\text{间}\right. \right. \right. \right. $
(3).泰勒级数和泰勒展开式：
根据上述推导，实际上当我们已知无数个点，即$ {n \to \infty }$时，泰勒公式具有无穷项，它就像幂级数,称之为泰勒级数。但是在得到公式的过程中，我们却没有去讨论级数的是否收敛。这里需要补充的是，泰勒展开式：
\[\color{red}{{{f \left( x \left) =f \left( x\mathop{{}}\nolimits_{{0}} \right) +{f \prime } \left( x\mathop{{}}\nolimits_{{0}} \left) \left( x-x\mathop{{}}\nolimits_{{0}} \left) +\frac{{{f '' } \left( x\mathop{{}}\nolimits_{{0}} \right) }}{{2!}} \left( x-x\mathop{{}}\nolimits_{{0}} \left) \mathop{{}}\nolimits^{{2}}+ \cdots +\frac{{f\mathop{{}}\nolimits^{{ \left( n \right) }} \left( x\mathop{{}}\nolimits_{{0}} \right) }}{{n!}} \left( x-x\mathop{{}}\nolimits_{{0}} \left) \mathop{{}}\nolimits^{{n}}+ \cdots \text{ }\right. \right. \right. \right. \right. \right. \right. \right. \right. \right. }}}
\]
泰勒展开式是特殊的泰勒级数，或者说是收敛情况下的泰勒级数，其具有收敛区间，在收敛区间内，和一定存在。
(4).麦克劳林公式：泰勒公式的一种特殊形式。
当\({x\mathop{{}}\nolimits_{{0}}=0}\)时，泰勒公式变为：
\[\color{red}{{f \left( x \left) =f \left( 0 \right) +{f \prime } \left( 0 \left) x+\frac{{{f''} \left( 0 \right) }}{{2!}}x\mathop{{}}\nolimits^{{2}}+ \cdots +\frac{{f\mathop{{}}\nolimits^{{ \left( n \right) }} \left( 0 \right) }}{{n!}}x\mathop{{}}\nolimits^{{n}}+R\mathop{{}}\nolimits_{{n+1}} \left( x \left) \right. \right. \right. \right. \right. \right. }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }R\mathop{{}}\nolimits_{{n+1}} \left( x \left) \text{是}\text{余}\text{项}\right. \right. }
\]
