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
Table 1.5 Values of the Approximating Polynomial P(a) of Example 1.5 and the Function f(a)=In(1+r)and the Error E()on[-01, 1.1 Approximating Function, Error, E(a) polynomial,P(a)f(r)=ln(1+r)I=In(1+r)-P(r) 0.1 0.10509718 0.10536052 0.00026334 0.0 0.00000000 0.00000000 0.00000000 0.1 0.09528988 0.09531018 0.00002030 0.2 0.18232156 0.18232156 0.00000000 0.3 0.26327015 0.26236426 0.00000589 0.4 0.33647224 0.33647224 0.00000000 0.5 0.40546139 0.40546511 0.00000372 0.6 0.47000363 0.47000363 0.00000000 0.7 0.53063292 0.53062825 0.00000467 0.8 0.58778666 0.58778666 0.00000000 0.9 0.64184118 0.64185389 0.00001271 1.0 0.69314718 0.69314718 0.00000000 1.1 0.74206529 0.74193734 0.00012795
0.8 0.7 06 y=In(1+x) 0.5 04 0.3 0.2 0.1 02 04 0.6 0.8 12 14 Figure 1.8
Figure 12 The graph of the error y=E(X=In(1+x)-P(x). 2.5 y=E(X) 0.5 0.5 03 0.4 0.5 0.6 0.7 0.8
