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)=In(1+a)is T(a)=r (119)
Table 1. 4 Values of the Taylor Polynomial T(r) of Degree 5, and the Function In(1 +r) and the Error In(1 +a)-T(r)on[0, 1 Taylor polynomial Function Error T(r) mn(1+)1m(1+x)-T(x) 0.0000000 0.00000000 0.00000000 0.18233067 0.18232156 0.00000911 0.4 0.33698133 0.33647224 0.00050906 0.6 0.47515200 0.47000363 0.00514837 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(x)=0.02957026x5-0.12895295x4+0.28249626x 0.48907554x2+0.99910735x based on the six nodes ck=k /5 for k=0, 1, 2, 3, 4, and 5
The following are empirical descriptions of the approximation P(a)In(1+r) 1. P(ark)=f(ak)at each node(see Table 1.5) 2. The maximum error on the interval [-0.1, 1. 1]occurs at x=-01 and error<0.00026334 for -0 1<s1.1(see Figure 1.10)Hence the graph of y= P(a)would appear identical to that of y=In(1+r)(see Figure 1.9) 3. The maximum error on the interval 0, 1] occurs at 2=0.06472456 and error<0.0002050≤x≤1( see figure1.0