Chapter 3 Interpolation and polyno(dal Approximation
Chapter 3 Interpolation and polynomial Approximation
The Taylor polynomial p(x=1++0.5x which approximates f(x =e over [-1, 1] 2.5 y=p(x) 0.5 -08-06-04-02 04 06 0.8 Figure 1.1
The Chebyshev approximation q(x =1+.129772x+0.532042x for f(x=e over 1-1, 1] 15 08-06-04-02 02 04 06 08 Figure 1.2
The graph of the collocation polynomial that passes through(1, 2), (2, 1),(3, 5), (4, 6), and(5, 1) 15 Figure 1.3
3.1 Taylor Series and Calculation d Functions
3.1 Taylor Series and Calculation of Functions
able 1. 1 Taylor Series Expansions for Some Common Functions sin(a)=a + + for all a 6 cos( )=1 ×y for all e=1+x ×乙3 +,+,+ TOr 2!3!4! In(1+.)=a + 1<x<1 arctan( =a + 1<x<1 +2=1++(p-1)2 +… for la|<1
Table 1.2 partial sums s used to Determine e Sn=1++3+…+ n012345678901 2.5 2.666666666666.. 2.708333333 2.716666666666 2.718805555555. 2.71825:968254 2.718278769841. 2.718281525573. 2.718281801146 2.718281826199. 2.718281828286 13 2.718281828447. 14 2.718281828458. 15 2.718281828459
Theorem 1. 1(Taylor Polynomial Approximation). Assume that f E CN+[a, bI and o∈[a, bl is a fixed value.lfg∈[a,bl,then f(r)=PN(r)+EN() (11 where PN(a)is a polynomial that can be used to approximate f(r) f()≈PN(r)=∑ N f(k)(sol(a-o) k! The error term EN() has the form N+1 EN(a) N+1 (N+1 for some value c=c(r) that lies between a and a
Example 1.1 Show why 15 terms are all that are needed to obtain the 13 digit approximation e=2.718281828459 in Table 1. 2
Figure 1.3 The graphs of y=e and y=P, (x)=1+x 4 y=P,(X) 15 -0.5 0.5 1.5 X
