(2) Parabolic interpolationUsed para - curve y = L,(x) approximat e curve y = f(x),L, (x) is called the two difference polynomial ,used basic function method we can get:L,(x)= yk-lk-I(x)+ yklk(x)+ yk+1k+1(x).If basic function lk-i(x)、 Ik(x) and Ik+(x) are quadratic function,and in the node need :1(i=j)(i,j=k-l,k,k+l)1 (x) =?0(i±j)上页下页返圆
上页 下页 返回 (2) Parabolic interpolation ( ) ( ) ( ) ( ) . : ( ) d the two difference polynomial , para - curve ( ) approximat e curve ( ) 2 1 1 1 1 2 2 L x y l x y l x y l x used basic function method we can get L x is calle Used y L x y f x = k− k− + k k + k+ k+ = = ， ( 1 1) 0 ( ) 1 ( ) ( ) and in the node need ( ) ( ) ( ) quadratic function, 1 1 = − + = = − + i j k k k i j i j l x If basic function l x l x and l x are i j k k k ， ， ， ：
the basic functions of meeting the conditions is easy to get,for example lk-, (x), because it has two zero pointsX and xk+1' so can be exp ressed :Ik-1(x) = A(x - x,)(x - xk+1), A is a indetermin ed coefficien t,can be got by the condition Ik-I(xk-) = 1,1A=SO(Xk-1 - X)(xk-1 - Xk+1)(x - x)(x - Xk+1)(k-1(x) (Xk-1 - X,)(xk-1 - Xk+1)(x - xk-1)(x - X+1)(x - xk-1)(x -x)the same to, l,(x) =lk+1(x)(xk - Xk-1)(xk - Xk+1)(Xk+1 -Xk-1)(Xk+1 -Xk)上页下页返圆
上页 下页 返回 . ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( ) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 k k k k k k k k k k k k k k k k k k k k k x x x x x x x x l x x x x x x x x x the same t o l x x x x x x x x x l x − − − − = − − − − = − − − − = + − + − + − + − + − − + + − ， ， ， so x x x x A can be got by the condition l x l x A x x x x A is a x x so can be ressed for example l x because it has two k k k k k k k k k k k k ， ， ， ， ， ( )( ) 1 ( ) 1 ( ) ( )( ), indetermin ed coefficien t, and exp : ( ) zero points the basic functions of meeting the conditions is easy to get 1 1 1 1 1 1 1 1 1 − − + − − − + + − − − = = = − −
二, Lagrange interpolating polynomialUse nth interpolation polynomial y= L,(x) to approch y= f(x) ,or use basisfunction theory to get: L,(x) =Zyl(x) ,called Lagrange interpolationpolymial.In order to structure L,(x) ,firstly define the nth interpolatingbasis function.Definition 1 If nth polynomial 1,(x) (j = 0, 1, ..., n) on the n+1 nodes<x<..<x,[1, k=j;(j, k=0, ,.,n),call the n+1 nthmeets the condition 1(x)=)[0,k±j.polynomial l,(x) (j = 0, ,..,n) nth interpolating basis function.n-l,n-2 are the linear and 2th interpolating basis function.Use thesimilar method to get the nth interpolating basis function.(x-x).(x-x-(x-x+).(x-xm)1 (x) :(k =0, 1, ", n)上页X-x)..(x-Xk-(x-Xk+1).(x-xm)下页返圆
上页 下页 返回 y L (x) = n y = f (x) = = n k Ln x yk l k x 0 ( ) ( ) 二、Lagrange interpolating polynomial Use nth interpolation polynomial to approch ,or use basis function theory to get: ,called Lagrange interpolation polymial.In order to structure ,firstly define the nth interpolating basis function. L (x) n Definition 1 If nth polynomial on the n+1 nodes meets the condition ,call the n+1 nth polynomial nth interpolating basis function. l (x) ( j 0 1 , n) j = ， n x x x 0 1 ( 0 1 , ) 0 , . 1, ( ) j k n k j k j l x j k ， ， ； = = = l (x) ( j 0 1 , n) j = ， n=1,n=2 are the linear and 2th interpolating basis function.Use the similar method to get the nth interpolating basis function. ( 0 1 , ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) 0 1 1 0 1 1 k n x x x x x x x x x x x x x x x x l x k k k k k k n k k n k = ， − − − − − − − − = − + − +
So,the Lagrange interploting polymial isnnnx-xL,(x)=Zyel(x)=)DykXk-Xjk=0k=0j=0j+kIntroduce the sign On+i(x)=(x-xo)(x-xi)..(x-xn), and afterdifferentiating for x,make x = x ,get0n+1(x) =(x-x)...(x -Xk-1)(x -xk+1)...(x -xn),Thus,the Lagrange interploting polymial can be exchanged ton(x)0.ZykL,(x) =上页(x-xk)0n+1(xk)k=0下页返圆
上页 下页 返回 n+1 (x) = (x − x0 )(x − x1 )(x − xn )， k x = x k n k n k n j k j k j j n k k y x x x x L (x) y l (x) ( ) 0 0 0 = = = − − = = = + + − = n k k n k n n k x x x x L x y 0 1 1 ( ) ( ) ( ) ( ) So,the Lagrange interploting polymial is Introduce the sign and after differentiating for x,make ,get ( ) ( ) ( )( ) ( ), n 1 xk = xk − x0 xk − xk 1 xk − xk 1 xk − xn + − + Thus,the Lagrange interploting polymial can be exchanged to
There is the following theorem on the existence and uniqueness ofthe interpolating polynomial:Theorem 1 In the polynomial set Hn ,which's order don't exceed n.the interpolating polynomial L,(x,)=y, (j=0, ,., n) is unique when itmeets the condition L,(x)e HnNote : If the polynomial's order is not limited into n,theinterpolating polynomial is not uniqueFor example P(x)= L,(x)+ p(x)II(x-x) i is also an interpolatingpolynomial,among which p(x) can be any polynomial.上页下页返圆
上页 下页 返回 Hn L (x ) y ( j 0 1 n) n j = j = ，， Note：If the polynomial's order is not limited into n,the interpolating polynomial is not unique. There is the following theorem on the existence and uniqueness of the interpolating polynomial: Theorem 1 In the polynomial set ,which's order don't exceed n, the interpolating polynomial is unique when it meets the condition n Hn L (x) For example is also an interpolating polynomial,among which can be any polynomial. = = + − n i n i P x L x p x x x 0 ( ) ( ) ( ) ( ) p(x)