中图科萝技术大学数学系 University of Science and Technology of China DEPARTMENT OF MATHEMATICS 例：(-1,2)，(0,0)，(2,1)，(3,3) (x-0(x-2(x-3) =1-0y-1-2M-3 4(x)= (x+1(x-2)(x-3) (0+10(0-2)0-3) 4()=x+1x-0x-3) 4)=+1x-0x-2) (2+1)2-0(2-3) (3+1)3-03-2) 8(x)=21(x)+0L,(x)+1L2(x)+3L(x)
数 学 系 University of Science and Technology of China DEPARTMENT OF MATHEMATICS 例： (−1,2),(0,0),(2,1),(3,3) ( 1 0)( 1 2)( 1 3) ( 0)( 2)( 3) ( ) 0 − − − − − − − − − = x x x l x (3 1)(3 0)(3 2) ( 1)( 0)( 2) ( ) 3 + − − + − − = x x x l x (2 1)(2 0)(2 3) ( 1)( 0)( 3) ( ) 2 + − − + − − = x x x l x (0 1)(0 2)(0 3) ( 1)( 2)( 3) ( ) 1 + − − + − − = x x x l x ( ) 2 ( ) 0 ( ) 1 ( ) 3 ( ) 0 1 2 3 g x = l x + l x + l x + l x
中图 科亨技术大学数学系 University of Science and Technology of China DEPARTMENT OF MATHEMATICS 算法：1= (x-xo)(x-x-1)(x-x+1).(x-xn) (x,-xo).(x,-x;-1)(x-x+1)(x,-xn) fx=0.0 for(i=0;i<-n;i++) tmp=1.0; for(j-0;j<i;j++) tmp=tmp*(x-x[j])/(x[i]-x[j]) for(j=i+1:j<=n;j++) tmp=tmp*(x-x[j])/(x[i]-x[j]); fx=fx+tmp*y[i]; } return fx;
数 学 系 University of Science and Technology of China DEPARTMENT OF MATHEMATICS 算法： fx=0.0 for(i=0;i<=n;i++) { tmp=1.0; for(j=0;j<i;j++) tmp=tmp*(x-x[j])/(x[i]-x[j]); for(j=i+1;j<=n;j++) tmp=tmp*(x-x[j])/(x[i]-x[j]); fx=fx+tmp*y[i]; } return fx; 0 1 1 0 1 1 ( ) ( )( ) ( ) ( ) ( )( ) ( ) i i n i i i i i i i n x x x x x x x x l x x x x x x x x − + − + − − − − = − − − −
中图科兰技术大空数学案 University of Science and Technology of China DEPARTMENT OF MATHEMATICS Lab02 Lagrange插值 对数)十5列构造插值，并求 m5.0.10 插值节点取为： 四=-5+91=01-N Chebyshev.点 2i+1 (2)x,=-5c0 2N+2 π，i=0,1,.N 对N=5,10,20,40比较以上两组节点的结果
数 学 系 University of Science and Technology of China DEPARTMENT OF MATHEMATICS Lab02 Lagrange插值 2 1 ( ) , [ 5,5] 1 f x x x = − + 对函数 构造插值，并求 5 5 max ( ) ( ) max ( ) ( ) , 5, 0, 100 10 i i i x i i f x p x f y p y y i − − − = − = 插值节点取为： 10 5 , 0,1, i x i i N N (1) = − + = 2 1 5cos , 0,1, 2 2 i i x i N N + = − = + (2) 对N=5，10，20，40比较以上两组节点的结果。 Chebyshev点
中图科亨技术大空数学系 University of Science and Technology of China DEPARTMENT OF MATHEMATICS 误差R(x)=f(x)-L(x) 解：fx,)=L,(x,),i=0,n ∴.R,(x)=0,i=0,.n .R (x)=k(x)(x-xo).(x-x,) 求k(x)=? Va,k(a)=? w(t)=f(t)-L(t)-k(a)(t-x).(t-x) 易知w(x)=0,i=0,.n and w(a=0
数 学 系 University of Science and Technology of China DEPARTMENT OF MATHEMATICS 误差 R (x) f (x) L (x) n = − n 解： ( ) ( )( ) ( ) ( ) 0 , 0, ( ) ( ) , 0, n 0 n n i i n i R x k x x x x x R x i n f x L x i n = − − = = = = 求 k(x) = ? ( ) 0, 0, and ( ) 0 ( ) ( ) ( ) ( )( ) ( ) , ( ) ? 0 = = = = − − − − = x i n a t f t L t k a t x t x a k a i n n 设 易知
中固科亨技术大学数学系 University of Science and Technology of China DEPARTMENT OF MATHEMATICS ∴.w(t)有n+2个零点 .35,wm+(5)=0 yw+(5)=f+(5)-k(a(n+1)1 k(a=+ (n+1)月 由a的任意性 :Rx)=56x-x)x-） (n+1)川
数 学 系 University of Science and Technology of China DEPARTMENT OF MATHEMATICS (t) 有n+2个零点 ( 1)! ( ) ( ) ( ) ( ) ( )( 1)! , ( ) 0 ( 1) ( 1) ( 1) ( 1) + = = − + = + + + + n f k a f k a n n n n n ( ) ( ) ( 1)! ( ) ( ) 0 ( 1) n n n x x x x n f R x − − + = + 由a的任意性