Taylor（）函数总结

Taylor（）函数总结

Taylor展开式公式的具体形式见百度百科：https://baike.baidu.com/item/泰勒公式

• 麦克劳林展开：(到第五项)

  syms x
  T1 = taylor(exp(x))
  T2 = taylor(sin(x))
  T3 = taylor(cos(x))
  
  T1 =
  x^5/120 + x^4/24 + x^3/6 + x^2/2 + x + 1
  
  T2 =
  x^5/120 - x^3/6 + x
  
  T3 =
  x^4/24 - x^2/2 + 1
  

  注：我们可以使用sympref函数来调整多项式的输出顺序：

  sympref('PolynomialDisplayStyle','ascend');
  T1
  T2
  T3
  T1 =
  1 + x + x^2/2 + x^3/6 + x^4/24 + x^5/120
  
  T2 =
  x - x^3/6 + x^5/120
  
  T3 =
  1 - x^2/2 + x^4/24
  
  

  如果不需要反序输出调回default即可

  sympref('default');
  

  • 自行确定x0

    有两种办法：

    方法一：

 syms x
 T = taylor(log(x), x, 'ExpansionPoint', 1)
 T =
 x - (x - 1)^2/2 + (x - 1)^3/3 - (x - 1)^4/4 + (x - 1)^5/5 - 1

​ 方法二：

T = taylor(acot(x), x, 1)
T =
pi/4 - x/2 + (x - 1)^2/4 - (x - 1)^3/12 + (x - 1)^5/40 + 1/2

• 展开项数的确定

 syms x;
 f=exp(x);
 taylor(f,x,'Order',20)
 ans =

x^7/5040 + x^6/720 + x^5/120 + x^4/24 + x^3/6 + x^2/2 + x + 1

用代码来总结一下：

 syms x;
 y=exp(x)+log10(x);
 T=taylor(y,x,'Expansionpoint',1,'Order',20)
 T =

exp(1) + (exp(1)/479001600 - 1/(12*log(10)))*(x - 1)^12 + (exp(1)/87178291200 - 1/(14*log(10)))*(x - 1)^14 + (exp(1)/6227020800 + 1/(13*log(10)))*(x - 1)^13 + (exp(1)/20922789888000 - 1/(16*log(10)))*(x - 1)^16 + (exp(1)/1307674368000 + 1/(15*log(10)))*(x - 1)^15 + (exp(1)/6402373705728000 - 1/(18*log(10)))*(x - 1)^18 + (exp(1) + 1/log(10))*(x - 1) + (exp(1)/355687428096000 + 1/(17*log(10)))*(x - 1)^17 + (exp(1)/2 - 1/(2*log(10)))*(x - 1)^2 + (exp(1)/6 + 1/(3*log(10)))*(x - 1)^3 + (exp(1)/24 - 1/(4*log(10)))*(x - 1)^4 + (exp(1)/120 + 1/(5*log(10)))*(x - 1)^5 + (exp(1)/720 - 1/(6*log(10)))*(x - 1)^6 + (exp(1)/5040 + 1/(7*log(10)))*(x - 1)^7 + (exp(1)/40320 - 1/(8*log(10)))*(x - 1)^8 + (exp(1)/362880 + 1/(9*log(10)))*(x - 1)^9 + (exp(1)/3628800 - 1/(10*log(10)))*(x - 1)^10 + (exp(1)/39916800 + 1/(11*log(10)))*(x - 1)^11 + (exp(1)/121645100408832000 + 1/(19*log(10)))*(x - 1)^19