Example 4. 11. Compute P3(2.5) in Example 1. 10 using nested multipli cation Using (1.59), we write P3(x)=(-0.1(-4)+0.5)(x-3)-2)(x-1)+5 The values in(1.60)are 0.1 S2=-0.1(2.5-4)+0.5=0.65 S1=0.65(2.5-3)-2=-2.325 S0=-2325(25-1)+5=1.5125 Therefore, P3(2.5)=1.5125
Definition 4. 1(Divided Differences). The divided differences for a func tion f(a) are defined as follows f(℃ f[k1,]=- f[ck-1.ak-f(ck-2, ck-11 k-2,k-1 T一工 f(k2,221,4=12132 The recursive rule for constructing higher-order divided differences is f{k-+-…,x]-f k k and is used to construct the divided differences in table 4. 8