Chapter 3 Interpolation and polynomial Approximation
Chapter 3 Interpolation and polynomial Approximation
The Taylor polynomial p(x 1+x+0.5x which approximates f(x)=e"over(-1, 11 15 p(x) 0.5 -08-06-04-02 04 0.6 08
The Chebyshev approximation q(x=1+1.129772X+0. 532042x for f(x=e over[-1, 1]1 25 15 y=q(x) 05 08-06-04-02 06
The graph of the collocation polynomial that passes through(1, 2), (2, 1),(3, 5), (4, 6), and(5. 1) 15 35 re 1.3
3. 1 Taylor series and calculation of Functions
3.1 Taylor Series and Calculation of Functions
Table 1.1 Taylor Series Expansions for Some Common Functions Sin( r= ′×25 + or all r 4x6 cos(a=1 + TOr 4!6 e=1+x+-+++ 3!4 n(1+x)=x C乙 3 + 1<x<1 234 arctan(a)=a + 1<x<1 35 +x)2=1+mD-1)2 +… for ac|<1
Table 1.2 Partial sums s used to determine e Sn=1+++…+ n0123456789 2.0 2.5 2.666666666666 2.708333. 2.716666666666 2.718805555555 2.718253968254 2.718278769841 2.718281525573 2.718281801146. 2.718281826199 12 2.718281828286 13 2.718281828447. 14 2.718281828458.. 15 2.718281828459
Theorem 1. 1(Taylor Polynomial Approximation). Assume that f E CN+ a, b ando∈[a, is a fixed value.fr∈[a,],then f(a)=PN(a)+ en(c), where PN(a)is a polynomial that can be used to approximate f(ar) f()≈()=∑ N f()(ro) k! -0 (1.2) The error term EN(a) has the form (N+1 EN(a) (N+1)! for some value c=c(a) that lies between r and To
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 8 4 3 =P1(x) -0.5 0 0.5 15 X
