The Convergence Of Cubic Spline Interpolation And The Error Estimations Of Certain Generalized Cubic Spline Interpolation

In scientific research and engineering, we often encounter the problemof calculating the value of function. However, functional representation isreally complex, and sometimes there is even no specific analytic formula. Forthis reason, we can construct a suitable easier function to replace theunknown function according to the observed data. This is interpolationmethod. The widely applied interpolation method is one of the importantmethods of approximation of fuction.First, this paper presents a brief introduction of interpolation method, themost commonly used cubic spline interpolation and its solution, and theconvergence of cubic spline inerpolation. Second, this paper makesdiscussion about the convergence of cubic spline interpolation in somespecial conditions. Last, this paper also gives derivation about the problem ofthe error estimations of certain generalized cubic spline interpolation.This article, on the one hand is the summary and overview of somerelated knowledge and conclusions, on the other hand is the deduction andimprovement of some related conclusions under some special conditions.Main contents are as follows:I. Introduction of interpolationII. Introduction of cubic spline interpolation and its solutionIII. Convergence of cubic spline interpolationWe name a = x₀ < x₁ < x₂ <…… < x_n = bas graduation of interval[ a , b ]: Δ_nDefined ashi = xi ? xi ?1 , ? n = max hi , |?n = mi ? aj =x1 hhijAnd use ( )nS ? x to represent cubic spline interpolation founction f ( x ) thatsatisfies boundary condition S ??( x0 ) = S ??( xn),S ????( x0 ) = S ????( xn) about graduation? n. If f ( x0 ) = f ( xn), then ( )nS x is cubic cycle spline interpolationfounction.So about graduation sequence ? n that satisfies ? n?ú 0,and aboutwhat founction family f ( x ), convergence of ( )nS ? x can be garanteed? Wehave:Theorem: suppose f ( x ) ?ê C[ a , b], and f ( x0 ) = f ( xn), ( )nS ? x is cubic cyclespline interpolation founction about ? n, for every graduation sequence ? nthatsatisfies ? n?ú 0,lim ( ) ( ) 0n ?ú?T S ?nx ? f x=the necessary and sufficient condition is f ( x ) ?ê Lip1, and when f ( x ) ?ê Lip1,then( ) ( )54S ? n x ? f x ?ü k? n.IV. the Error Estimations of Certain Generalized Cubic SplineInterpolationAbout certain interval[0,1] graduation ? n :0= x0 < x1 < x2 < < xi < xi +1< < xn= 1, generalized spline founction family132[ , ] [0,1]0( ) | ( ) ( ) ( )nxi xi ij jjL s x s x |á ?x s x C? ? ? +?== ????? = ?? ?ê?????interpolation problem:s ? ( xi ) = f ( xi ) (i = 0,1,2, , n ) s ??? ( xi ) = f ??( xi ) (i = 0, n),In the condition that ? j ( x ) ?ê C[ 40,1]( j= 0,1,2,3) and0 1 2 30 1 2 30 1 2 30 1 2 3( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) 0 [0,1]( ) ( ) ( ) ( )( ) ( ) ( ) ( )x x x xx x x xD x xx x x xx x x x? ? ? ?? ? ? ?? ? ? ?? ? ? ????? ?? ??= ???????? ???? ?????ù ?ê?????????? ?????? ??????When ? n?ú 0, the interpolation problem has unique solution, and haserror estimation:Theorem: suppose f ( x ) ?ê C[ r0,1](1 ?ü r?ü 3), then for q = 0,1, we have3 1( ) ( ) ( ) 4 (4) ( )0 0q ( ) q ( ) r q ( r , )q jpj ps? x f x M ? w f M ? ??f= =? ?ü ? ? + ? ??? ?? i ?????.The above mentioned theorems give approximation order of cubicgeneralized spline interpolation about founctions with different soothingdegree. But the coefficient of the above mentioned estimation is not given, soit is not suitable to make specific error estimation about given graduation.Therefore, through derivation, in conditions weaker than the abovementioned theorems, we get error estimation with specific coefficient of cubicgeneralized spline interpolation:Theorem: suppose ? j ( x ) ?ê C[ 40,1]( j = 0,1,2,3), D ( x) ?ù 0, then for sufficientlysmall?, we haveM? , makef ( x ) ?ê C*[0,1], then s? ( x ) ? f ( x ) ?ü M ???? f ? 4+ ??? 1 + ?????w( f, ?)??? .f ( x ) ?ê C*[r0,1], then ( ) ( ) ( ) 4( )0( ) ( ) ( , )pp r j p r prjs? x f x M ? f ? ?w f=? ?ü ??? ?? ? +? ????( r = 1,2 0?ü p ?ü r)3 2f ( x ) ?ê C[ 0,1] ?é C*[0,1],then ( ) ( ) 3 ( )4 30p ( ) p ( ) jp p( , )js? x f x M ? f ? ?w f=? ?ü ??? ?? ? +? ??????????( p = 0,1,2)4 2f ( x ) ?ê C[ 0,1] ?é C*[0,1], then ( ) ( ) 4 ( )40p ( ) p ( ) jpjs? x f x M ? f?=? ?ü ?? ?( p = 0,1,2).As the application of general result, for exponent spline interpolation(? j ( x ) = e jxj= 0,1,2,3), we have:Theorem: suppose 1?< 6,When f ( x ) ?ê C*[20,1], for p = 0,1,2, we have( )( ) ( ) ( ) ( ) 68 52 19 2 50( , ) 2 ( , )211s? p x ? f px ?ü ??? f ?? + f ???? ? + ?w f ???? ? ??? ? ? p + w f????? ??pWhen f ( x ) ?ê C[ 30 ,1] ?é C*[20,1], for p = 0,1,2, we have( ) ( ) ( ) ( ) 68 52 19 50 4 1( , )311 4s? p x ? f px ?ü ??? f ?? + f ???? + f ?????? ??? ? ? p + w f??????? ??p;When f ( x ) ?ê C[ 40 ,1] ?é C*[20,1], for p = 0,1,2, we have( ) ( ) ( ) ( ) 68 52 1950 1(4)411 8s? p x ? f px ?ü ??? ??? f ?? + f ???? + f ?????????+ f ??? ??p.