算法:记插值节点为x算法记插值节点为0,x1,···,xn,f(x的各阶差商的各阶差商为f0,f1,f2,···,fn(1s←fn(2计算s←fk+s*(x-xk(k=n-1,n-2,···,0计算(3N(x=s根据代数插值存在唯一性定理,根据代数插值存在唯一性定理n次牛顿插值公式恒等于n次拉格朗日插值公式误差余项也相等,次拉格朗日插值公式,误差余项也相等等于次拉格朗日插值公式误差余项也相等,即f(n+1(ξRn=f[x,x0,x1,⋯,xn]ωn+1(xRn(x=ωn+1(x(n+1!f(n+1(ξf[x,x0,x1,⋯,xn]=(n+1!16/18例:1234567推导计算公式P(n=∑k3k=1n193610022544178482764**********/237/261/291/2127/234561/41/41/40019P(n=1+8(n−1+(n−1(n−2+3(n−1(n−2(n−321+(n−1(n−2(n−3(n−4417/18P(n=1+(n-1(8+(n-2(19/2+(n-3(3+(n-4/4=1+(n–1(8+(n–2(19/2+(n–3(n/4+2=1+(n–1(8+(n–2(19/2+n2/4+5n/4–6=1+(n–1(8+(n–2(n2/4+5n/4+7/2=1+(n–1(n3/4+3n2/4+n+1=n4/4+n3/2+n2/4nn22n22(n+2n+1=(n+1=[(n+1]2=44218/18
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