文档介绍：该【几类插值方法及其应用论文 】是由【帅气的小哥哥】上传分享，文档一共【15】页，该文档可以免费在线阅读，需要了解更多关于【几类插值方法及其应用论文 】的内容，可以使用淘豆网的站内搜索功能，选择自己适合的文档，以下文字是截取该文章内的部分文字，如需要获得完整电子版，请下载此文档到您的设备，方便您编辑和打印。severalgroupnumber,thenwithb±a,=c,-2~3measurement,suchasproceedsofcvaluesareequalandequaltothedesignvalue,,b,andcvalueswhileonhorizontalverticalerrorsformeasurement,Generalinironanglecodebitatmeasurementlevelpointsgriderrors,omethylverticalboxcenterlinedistancefora,,tobverticalboxdistanceforb,listcanmeasuredseveralgroupnumber,thenwithb±a,=c,-2~3measurement,suchasproceedsofcvaluesareequalandequaltothedesignvalue,,b,andcvalueswhileonhorizontalverticalerrorsformeasurement,Generalinironanglecodebitatmeasurementlevelpointsgriderrors,omethylverticalboxcenterlinedistancefora,,tobverticalboxdistanceforb,listcanmeasuredseveralgroupnumber,thenwithb±a,=c,-2~3measurement,suchasproceedsofcvaluesareequalandequaltothedesignvalue,,b,andcvalueswhileonhorizontalverticalerrorsformeasurement,Generalinironanglecodebitatmeasurementlevelpointsgriderrors,omethylverticalboxcenterlinedistancefora,,tobverticalboxdistanceforb,listcanmeasured几类插值方法及其应用王莎20249001S041摘要:在工程应用中,经常会遇到函数的表达式是的,但该表达式却比较复杂难以计算,因此,希望用一个既能反映该函数的特性又便于计算的简单函数来描述它。本文对常见的几种插值方法-插值,插值,插值方法的根本思想、插值函数的构造等进行了详细的介绍。关键字:插值基函数,插值多项式,插值节点。 一·引言实际问题中经常有这样的函数,其在某个区间上有有限个离散点,且这些点对应函数值为,假设想得到其它点的值就必须找一个满足上述条件的函数表达式。这就是下边要讨论的插值函数。二·:将待求的次插值多项式写成另一种表达方,式再利用插值条件确定出插值基函由基函数条件,确定多项式系数,:〔1〕,求满足条件的插值函数。由题可知表示过两点的直线,这个问题是我们所熟悉的,它的解可表为以下对称式severalgroupnumber,thenwithb±a,=c,-2~3measurement,suchasproceedsofcvaluesareequalandequaltothedesignvalue,,b,andcvalueswhileonhorizontalverticalerrorsformeasurement,Generalinironanglecodebitatmeasurementlevelpointsgriderrors,omethylverticalboxcenterlinedistancefora,,tobverticalboxdistanceforb,listcanmeasuredseveralgroupnumber,thenwithb±a,=c,-2~3measurement,suchasproceedsofcvaluesareequalandequaltothedesignvalue,,b,andcvalueswhileonhorizontalverticalerrorsformeasurement,Generalinironanglecodebitatmeasurementlevelpointsgriderrors,omethylverticalboxcenterlinedistancefora,,tobverticalboxdistanceforb,listcanmeasuredseveralgroupnumber,thenwithb±a,=c,-2~3measurement,suchasproceedsofcvaluesareequalandequaltothedesignvalue,,b,andcvalueswhileonhorizontalverticalerrorsformeasurement,Generalinironanglecodebitatmeasurementlevelpointsgriderrors,omethylverticalboxcenterlinedistancefora,,tobverticalboxdistanceforb,listcanmeasured此类一次插值称为线性插值,假设令〔由此可得:〕〕那么有这里的可以看作是满足条件的插值多项式,这两个特殊的插值多项式称作上述问题的插值基函数。〔2〕求过三点的插值函数。为了得到插值多项式先解决一个特殊的二次插值问题。求作二次式,使满足〔2-1〕这个问题是容易求解的,由式〔2-1〕的后两个条件知是的两个零点,因而。:类似可以分别构造出满足条件的插值多项式;其表达式分别为,这样构造出的称作问题〔2〕的插值基函数。设取数据作为组合系数,将插值基函数组合得验证可知,这样构造的满足条件,因而它就是问题〔2〕的解。,121的开方值求。severalgroupnumber,thenwithb±a,=c,-2~3measurement,suchasproceedsofcvaluesareequalandequaltothedesignvalue,,b,andcvalueswhileonhorizontalverticalerrorsformeasurement,Generalinironanglecodebitatmeasurementlevelpointsgriderrors,omethylverticalboxcenterlinedistancefora,,tobverticalboxdistanceforb,listcanmeasuredseveralgroupnumber,thenwithb±a,=c,-2~3measurement,suchasproceedsofcvaluesareequalandequaltothedesignvalue,,b,andcvalueswhileonhorizontalverticalerrorsformeasurement,Generalinironanglecodebitatmeasurementlevelpointsgriderrors,omethylverticalboxcenterlinedistancefora,,tobverticalboxdistanceforb,listcanmeasuredseveralgroupnumber,thenwithb±a,=c,-2~3measurement,suchasproceedsofcvaluesareequalandequaltothedesignvalue,,b,andcvalueswhileonhorizontalverticalerrorsformeasurement,Generalinironanglecodebitatmeasurementlevelpointsgriderrors,omethylverticalboxcenterlinedistancefora,,tobverticalboxdistanceforb,listcanmeasured解:将可表示为由此可得所以将代入上式,求得。(3)推广到一般:函数在n+1个不同点上的函数值分别为求一个次数不超过n的多项式,使其满足:即个不同的点可以决定的一个次多项式。过个不同的点分别决定个次插值基函数。每个插值基多项式满足:;??b.,而在其它个点?由于故有因子:?因其已经是n次多项式,故而仅相差一个常数因子。令:?????????由,可以定出,进而得到:??,thenwithb±a,=c,-2~3measurement,suchasproceedsofcvaluesareequalandequaltothedesignvalue,,b,andcvalueswhileonhorizontalverticalerrorsformeasurement,Generalinironanglecodebitatmeasurementlevelpointsgriderrors,omethylverticalboxcenterlinedistancefora,,tobverticalboxdistanceforb,listcanmeasuredseveralgroupnumber,thenwithb±a,=c,-2~3measurement,suchasproceedsofcvaluesareequalandequaltothedesignvalue,,b,andcvalueswhileonhorizontalverticalerrorsformeasurement,Generalinironanglecodebitatmeasurementlevelpointsgriderrors,omethylverticalboxcenterlinedistancefora,,tobverticalboxdistanceforb,listcanmeasuredseveralgroupnumber,thenwithb±a,=c,-2~3measurement,suchasproceedsofcvaluesareequalandequaltothedesignvalue,,b,andcvalueswhileonhorizontalverticalerrorsformeasurement,Generalinironanglecodebitatmeasurementlevelpointsgriderrors,omethylverticalboxcenterlinedistancefora,,tobverticalboxdistanceforb,listcanmeasured是个次插值根本多项式的线性组合,相应的组合系数是。即:???从而是一个次数不超过n的多项式,且满足?例1求过点的拉格朗日型插值多项式。解用4次插值多项式对5个点插值。???可得?severalgroupnumber,thenwithb±a,=c,-2~3measurement,suchasproceedsofcvaluesareequalandequaltothedesignvalue,,b,andcvalueswhileonhorizontalverticalerrorsformeasurement,Generalinironanglecodebitatmeasurementlevelpointsgriderrors,omethylverticalboxcenterlinedistancefora,,tobverticalboxdistanceforb,listcanmeasuredseveralgroupnumber,thenwithb±a,=c,-2~3measurement,suchasproceedsofcvaluesareequalandequaltothedesignvalue,,b,andcvalueswhileonhorizontalverticalerrorsformeasurement,Generalinironanglecodebitatmeasurementlevelpointsgriderrors,omethylverticalboxcenterlinedistancefora,,tobverticalboxdistanceforb,listcanmeasuredseveralgroupnumber,thenwithb±a,=c,-2~3measurement,suchasproceedsofcvaluesareequalandequaltothedesignvalue,,b,andcvalueswhileonhorizontalverticalerrorsformeasurement,Generalinironanglecodebitatmeasurementlevelpointsgriderrors,omethylverticalboxcenterlinedistancefora,,tobverticalboxdistanceforb,listcanmeasured所以拉格朗日插值多项式的截断误差我们在上用多项式来近似代替函数,其截断误差记作??????当x在插值结点上时下面来估计截断误差:定理1:设函数的阶导数在上连续,?????在上存在;插值节点为:????是次拉格朗日插值多项式;那么对任意有:???????其中,ξ依赖于;证明:由插值多项式的要求:??????设其中是待定系数;固定且severalgroupnumber,thenwithb±a,=c,-2~3measurement,suchasproceedsofcvaluesareequalandequaltothedesignvalue,,b,andcvalueswhileonhorizontalverticalerrorsformeasurement,Generalinironanglecodebitatmeasurementlevelpointsgriderrors,omethylverticalboxcenterlinedistancefora,,tobverticalboxdistanceforb,listcanmeasuredseveralgroupnumber,thenwithb±a,=c,-2~3measurement,suchasproceedsofcvaluesareequalandequaltothedesignvalue,,b,andcvalueswhileonhorizontalverticalerrorsformeasurement,Generalinironanglecodebitatmeasurementlevelpointsgriderrors,omethylverticalboxcenterlinedistancefora,,tobverticalboxdistanceforb,listcanmeasuredseveralgroupnumber,thenwithb±a,=c,-2~3measurement,suchasproceedsofcvaluesareequalandequaltothedesignvalue,,b,andcvalueswhileonhorizontalverticalerrorsformeasurement,Generalinironanglecodebitatmeasurementlevelpointsgriderrors,omethylverticalboxcenterlinedistancefora,,tobverticalboxdistanceforb,listcanmeasured作函数那么且所以在上有个零点,反复使用罗尔中值定理:存在?,使;因是n次多项式,故而?是首项系数为1的n+1次多项式,故有????????于是得所以设那么:易知,线性插值的截断误差为:???????二次插值的截断误差为:????三·插值函数的构造severalgroupnumber,thenwithb±a,=c,-2~3measurement,suchasproceedsofcvaluesareequalandequaltothedesignvalue,,b,andcvalueswhileonhorizontalverticalerrorsformeasurement,Generalinironanglecodebitatmeasurementlevelpointsgriderrors,omethylverticalboxcenterlinedistancefora,,tobverticalboxdistanceforb,listcanmeasuredseveralgroupnumber,thenwithb±a,=c,-2~3measurement,suchasproceedsofcvaluesareequalandequaltothedesignvalue,,b,andcvalueswhileonhorizontalverticalerrorsformeasurement,Generalinironanglecodebitatmeasurementlevelpointsgriderrors,omethylverticalboxcenterlinedistancefora,,tobverticalboxdistanceforb,listcanmeasuredseveralgroupnumber,thenwithb±a,=c,-2~3measurement,suchasproceedsofcvaluesareequalandequaltothedesignvalue,,b,andcvalueswhileonhorizontalverticalerrorsformeasurement,Generalinironanglecodebitatmeasurementlevelpointsgriderrors,omethylverticalboxcenterlinedistancefora,,tobverticalboxdistanceforb,:节点处的函数值或一元函数代数方程,将待求的n次插值多项式改写为具有承袭性的形式,然后根据插值条件或选取初值以求得待定系数,进而求得所要的插值函数。插值与插值相比具有承袭性和易于变动节点的特点。:实践中的许多问题归结为求一元代数方程的根,如果是线性函数,那么它的求根较容易;对非线性方程,只有不高于4次的代数方程有求根公式,经常需求出高于4次?的满足一定精度要求的近似解。??,把在处泰勒展开假设取前两项来近似代替,那么的近似线性方程设0,设其根为,那么的计算公式为=-〔k=0,1,2.....〕这即为牛顿法,上式为牛顿迭代公式,其迭代函数为severalgroupnumber,thenwithb±a,=c,-2~3measurement,suchasproceedsofcvaluesareequalandequaltothedesignvalue,,b,andcvalueswhileonhorizontalverticalerrorsformeasurement,Generalinironanglecodebitatmeasurementlevelpointsgriderrors,omethylverticalboxcenterlinedistancefora,,tobverticalboxdistanceforb,listcanmeasuredseveralgroupnumber,thenwithb±a,=c,-2~3measurement,suchasproceedsofcvaluesareequalandequaltothedesignvalue,,b,andcvalueswhileonhorizontalverticalerrorsformeasurement,Generalinironanglecodebitatmeasurementlevelpointsgriderrors,omethylverticalboxcenterlinedistancefora,,tobverticalboxdistanceforb,listcanmeasuredseveralgroupnumber,thenwithb±a,=c,-2~3measurement,suchasproceedsofcvaluesareequalandequaltothedesignvalue,,b,andcvalueswhileonhorizontalverticalerrorsformeasurement,Generalinironanglecodebitatmeasurementlevelpointsgriderrors,omethylverticalboxcenterlinedistancefora,,tobverticalboxdistanceforb,listcanmeasured我们知道,牛顿法是解非线性方程最著名和最有效的方法之一,在单根附近它比一般的迭代格式有较快的收速度,但也要注意它也有缺点:首先,它对迭代初值选取要求较严,初值选取不好,可能导致吧收敛;其次,它每迭代一次要计算的值,这势必增加可计算量。为回避该问题,常用一个固定的迭代假设干步后再求。这就是下面要讲的简化牛顿法的根本思想。简化牛顿法和下山牛顿法简化牛顿法的公式为(3-1)迭代函数假设。即在根附近成立。那么迭代法〔3-1〕局部收敛。此法显然化简了计算量。牛顿下山法牛顿法的收敛依赖于初值的选取,假设偏离较远,那么牛顿法可能发散。为防止迭代发散,我们对迭代过程在附加一项条件,即具有单调性:〔3-2〕保证函数值稳定下降,然后结合牛顿法加快收敛速度,即可达目的。将牛顿法的计算结果〔3-3〕于前一步的近似值适当加权平均作为新的改进值severalgroupnumber,thenwithb±a,=c,-2~3measurement,suchasproceedsofcvaluesareequalandequaltothedesignvalue,,b,andcvalueswhileonhorizontalverticalerrorsformeasurement,Generalinironanglecodebitatmeasurementlevelpointsgriderrors,omethylverticalboxcenterlinedistancefora,,tobverticalboxdistanceforb,listcanmeasuredseveralgroupnumber,thenwithb±a,=c,-2~3measurement,suchasproceedsofcvaluesareequalandequaltothedesignvalue,,b,andcvalueswhileonhorizontalverticalerrorsformeasurement,Generalinironanglecodebitatmeasurementlevelpointsgriderrors,omethylverticalboxcenterlinedistancefora,,tobverticalboxdistanceforb,listcanmeasuredseveralgroupnumber,thenwithb±a,=c,-2~3measurement,suchasproceedsofcvaluesareequalandequaltothedesignvalue,,b,andcvalueswhileonhorizontalverticalerrorsformeasurement,Generalinironanglecodebitatmeasurementlevelpointsgriderrors,omethylverticalboxcenterlinedistancefora,,tobverticalboxdistanceforb,listcanmeasured〔3-4〕其中称〔〕为下山因子,即为〔3-5〕称为牛顿下山法。选择下山因子时,从开始逐次将减半进行试算。直到满足条件〔3-2〕为止。例2求方程的根。〔1〕解:用newton公式法取==-=-计算得=,=,=。〔2〕改用=,依牛顿法公式一次得===(2)中通过逐次取半进行试算,当时可得==-=-,...时均能使条件成立severalgroupnumber,thenwithb±a,=c,-2~3measurement,suchasproceedsofcvaluesareequalandequaltothedesignvalue,,b,andcvalueswhileonhorizontalverticalerrorsformeasurement,Generalinironanglecodebitatmeasurementlevelpointsgriderrors,omethylverticalboxcenterlinedistancefora,,tobverticalboxdistanceforb,listcanmeasuredseveralgroupnumber,thenwithb±a,=c,-2~3measurement,suchasproceedsofcvaluesareequalandequaltothedesignvalue,,b,andcvalueswhileonhorizontalverticalerrorsformeasurement,Generalinironanglecodebitatmeasurementlevelpointsgriderrors,omethylverticalboxcenterlinedistancefora,,tobverticalboxdistancefor