《数值分析》课程PPT教学课件（英文版）Chapter 01 The Solution of Nonlinear Equations 1.2 Introduction to Interpolation

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

Table 1.5 Values of the Approximating Polynomial P(a) of Example 1.5 and the Function f(r)=In(1+r)and the Error E(r)on[-01, 1.1] A approximating Function Error, E(r) polynomial, P(r)I f(r)=In(1+2 (1+x)-P(x) -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 0.74206529 0.74193734 0.00012795

0.8 0.7 0.6 y=n(1+X) 0.5 04 03 0.2 02 04 0.8 12 Figure 1.8

X10 Figure 12 The graph of the error y=E(X=In(1+x-P(x) 15 y=E(X) 0.5 0.5 0.1 0.2 04 0.5 0.6 0.7 0.8 0.9