) 中图学技术大荸学系 University of Science and Technology of China DEPARTMENT OF MATHEMATICS 特点: 1.与基函数无关 2.与原函数fx)无关 3.基函数个数与点个数相同
数 学 系 University of Science and Technology of China DEPARTMENT OF MATHEMATICS 1. 与基函数无关 2. 与原函数f(x)无关 3. 基函数个数与点个数相同 特点：
) 中图学技术大荸学系 University of Science and Technology of China DEPARTMENT OF MATHEMATICS 对应于Φ=P(x)=spm{1,x,x2,…x"} XX x,)≠0 0≤j<i≤n Vandermonde行列式 病态
数 学 系 University of Science and Technology of China DEPARTMENT OF MATHEMATICS ( ) {1, , , } n 2 n 对应于 = x = span x x x 则 ( ) 0 0 1 0 0 1 1 0 1 n n i j j i n n n n x x x x x x x x = − Vandermonde行列式 病态
) 中图学技术大荸学系 University of Science and Technology of China DEPARTMENT OF MATHEMATICS 多项式插值的 Lagrange型 ●如何找? 在基函数上下功夫,取基函数为{(x)}=0<P O,≠j 要求l(x)=O= li= 则g(x)=∑l(x)f(x) 0
数 学 系 University of Science and Technology of China DEPARTMENT OF MATHEMATICS 多项式插值的Lagrange型 ⚫ 如何找？ 在基函数上下功夫，取基函数为 n n i x i {l ( )} =0 要求 = = = i j i j l i x j i j 1, 0, ( ) 则 ( ) ( ) ( ) 0 i n i i g x l x f x = =
中图苔技术大荸数学系 University of Science and Technology of China DEPARTMENT OF MATHEMATICS 求{l(x)}o,易知: LOx ax (x-x1-1)(x-x+1)……(x )(x +1 (x-x0)…(x-x2-1)(x-x+1)…(x一xn) )(x 记a,(x)=I(x-x)+l(x) o(x) (x1)(x-x1)
数 学 系 University of Science and Technology of China DEPARTMENT OF MATHEMATICS 求 n i x i l 0 { ( )} = ，易知： ( ) ( ) ( )( ) ( ) i x ai x x0 x xi 1 x xi 1 x xn l = − − − − + − ( ) ( )( ) ( ) 1 i 0 i i 1 i i 1 i n i x x x x x x x x a − − − − = − + 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 − + − + − − − − = − − − − 0 ( ) ( ) n n i i x x x = 记 = − ( ) ( ) ' ( )( ) n i n i i x l x x x x = −
中图苔技术大荸数学系 University of Science and Technology of China DEPARTMENT OF MATHEMATICS ●线性插值 X X XI 0 L(x)=f(x)(x)+f(x1)1(x)
数 学 系 University of Science and Technology of China DEPARTMENT OF MATHEMATICS ⚫线性插值 1 0 0 1 0 1 1 0 ( ) , ( ) x x x x l x x x x x l x − − = − − = ( ) ( ) ( ) ( ) ( ) 1 0 0 1 1 L x = f x l x + f x l x