1.2 Introduction to Interpolation
1.2 Introduction to Interpolation
Let us return to the topic of using a polynomial to calculate approximations to a known function. In Section 1. 1 we saw that the fifth-degree Taylor polynomial for f(r)=ln(1+) 23 24 25 T()=-0+ + .1 2345
Table 1. 4 Values of the Taylor Polynomial T(r) of Degree 5, and the Function In(1+.) and the Error In(1+r)-T(a)on[0, 1 Taylor polynomialFunction E1 rror ln(1+x)ln(1+x)-T() 0.00000 0.0000000 0.0000000 0.18233067 0.18232156 0.0000911 0.4 0.33698133 0.33647224 0.00050906 0.6 0.47515200 0.47000363 0.00514837 0.8 0.61380267 0.58778666 0.0260160 1.0 0.78333333 0.69314718 0.09018615
Example 1.5. Consider the function f(a)=In(1 +r) and the polynomial P()=0.02957026x5-02895295x4+0.28249626x 04890755472+0.991075x based on the six nodes Ck=k 5 for k=0, 1, 2, 3, 4, and 5
he following are empirical descriptions of the approximation P(c)a In(1+a) 1. P(ck)=f(rk)at each node(see Table 1. 5) 2. The maximum error on the interval [-0.1, 1. 1]occurs at =-0.1 and error<0.00026334 for -0 1<2<1. 1(see Figure 1.10) Hence the graph of y=P(a)would appear identical to that of y= In(1+a)(see Figure 1.9). 3. The maximum error on the interval 0, 1] occurs at =0.06472456 and eror0.0005r0≤≤1( (see Figure.10