对于一个函数 f ( x ) f(x) f(x)在区间 [ a , b ] [a,b] [a,b]上有定义，且已知在点 a < = x 0 < x 1 < ⋅ ⋅ ⋅ < x n < = b a<=x_0<x_1<···<x_n<=b a<=x0<x1<⋅⋅⋅<xn<=b上的值 y 0 , y 1 , ⋅ ⋅ ⋅ , y n y_0,y_1,···,y_n y0,y1,⋅⋅⋅,yn. 若存在一个函数 P ( x ) = a 0 + a 1 x + ⋅ ⋅ ⋅ + a n x P(x)=a_0+a_1x+···+a_nx P(x)=a0+a1x+⋅⋅⋅+anx.使得： P ( x i ) = y i P(x_i) = y_i P(xi)=yi 我们就说 P ( x ) 为 f ( x ) P(x)为f(x) P(x)为f(x)的插值函数。对于如何求出这个插值函数，我们有多种方法，我们先设 n = 1 n=1 n=1时的简单情况，假定给定区间 [ x k , x k + 1 ] [x_k,x_{k+1}] [xk,xk+1],及插值端点值 y k = f ( x k ) , y k + 1 = f ( x k + 1 ) y_k=f(x_k),y_{k+1}=f(x_{k+1}) yk=f(xk),yk+1=f(xk+1),要求线性多项式 L 1 ( x ) L_1(x) L1(x)满足： L 1 ( x k ) = y k , L 1 ( x k + 1 ) = y k + 1 L_1(x_k)=y_k,L_1(x_{k+1})=y_{k+1} L1(xk)=yk,L1(xk+1)=yk+1 上式的几何意义是通过两点 ( x k , y k ) , ( x k + 1 , y k + 1 ) (x_k,y_k),(x_{k+1},y_{k+1}) (xk,yk),(xk+1,yk+1)的直线，用公式表达它们的几何意义： L 1 ( x ) = y k + y k + 1 − y k x k + 1 − x k ( x − x k ) ( 点 斜 式 ) L 1 ( x ) = y k x k + 1 − x x k 1 − x k + y k + 1 x − x k x k + 1 − x k ( 两 点 式 ) L_1(x) = y_k+\frac{y_{k+1}-y_k}{x_{k+1}-x_k}(x-x_k) (点斜式) \\ L_1(x) = y_k\frac{x_{k+1}-x}{x_{k_1}-x_k}+y_{k+1}\frac{x-x_k}{x_{k+1}-x_k} (两点式) L1(x)=yk+xk+1−xkyk+1−yk(x−xk)(点斜式)L1(x)=ykxk1−xkxk+1−x+yk+1xk+1−xkx−xk(两点式) 由以上两式可以看出 L 1 ( 1 x ) L_1(1x) L1(1x)是由两个线性函数： l k ( x ) = y k + 1 − y k x k + 1 − x k ， l k + 1 = x − x k x k + 1 − x k l_k(x)=\frac{y_{k+1}-y_k}{x_{k+1}-x_k}，l_{k+1}=\frac{x-x_k}{x_{k+1}-x_k} lk(x)=xk+1−xkyk+1−yk，lk+1=xk+1−xkx−xk 线性组合得到的，其系数分别为 y k y_k yk及 y k + 1 y_{k+1} yk+1,即： L 1 ( x ) = y 1 l k ( x ) + y k + 1 l k + 1 ( x ) . L_1(x)=y_1l_k(x)+y_{k+1}l_{k+1}(x). L1(x)=y1lk(x)+yk+1lk+1(x). 我们称 l k ( x ) l_k(x) lk(x)及 k k + 1 k_{k+1} kk+1为线性插值基函数， L ( x ) L(x) L(x)为一次插值多项式，同理当 n = 2 n=2 n=2时我们也可以构造出二次插值多项式，此外我们可以把它推广到 n n n，所以就得到了 n n n次插值多项式，如下公式； l k ( x ) = ( x − x 0 ) ⋅ ⋅ ⋅ ( x − x k − 1 ) ( x − x k + 1 ) ⋅ ⋅ ⋅ ( x − x n ) ( x k − x 0 ) ⋅ ⋅ ⋅ ( x k − x k − 1 ) ( x k − x k + 1 ) ⋅ ⋅ ⋅ ( x k − x n ) L n ( x ) = ∑ k = 1 n y k l k ( x ) L n ( x j ) ∑ k = 1 n y k l k ( x j ) y j , j = 0 , 1 , ⋅ ⋅ ⋅ , n . ( 1 ) l_k(x) = \frac{(x-x_0)···(x-x_{k-1})(x-x_{k+1})···(x-x_{n})}{(x_k-x_0)···(x_k-x_{k-1})(x_k-x_{k+1})···(x_k-x_{n})} \\ L_n(x)=\sum_{k=1}^ny_kl_k(x)\\ L_n(x_j)\sum_{k=1}^ny_kl_k(x_j) y_j , j= 0,1,···,n. (1) lk(x)=(xk−x0)⋅⋅⋅(xk−xk−1)(xk−xk+1)⋅⋅⋅(xk−xn)(x−x0)⋅⋅⋅(x−xk−1)(x−xk+1)⋅⋅⋅(x−xn)Ln(x)=k=1∑nyklk(x)Ln(xj)k=1∑nyklk(xj)yj,j=0,1,⋅⋅⋅,n.(1) 我们得到的(1)式被称为拉格朗日插值多项式。
