毕业论文-hermite插值的若干问题研究

烟台大学毕业论文（设计）HERMITE插值的若干问题研究STUDYOFHERMITEINTERPOLATIONPROBLEMS申请学位学士院系数学与信息科学学院专业信息与计算科学专业姓名学号2XXXXXXXX6指导老师XXXX（副教授）2013年05月25日完成地点烟台大学摘要本文主要是具体讨论HERMITE插值的若干问题，主要介绍了二重HERMITE插值在具体应用中出现的实际问题，并通过几个例子说明建立HERMITE插值多项式的方法、两点三次HERMITE插值及其余项、HERMITE插值公式的相关问题。并编程计算来比较几种方法。通过推导和证明得知三次HERMITE插值已经较高，再高可能发生RUNGE现象。关键词二重HERMITE插值多项式，唯一性定理，误差定理ABSTRACTTHISPAPERISDETAILEDTODISCUSSSOMEPROBLEMSOFTHEHERMITEINTERPOLATION,MAINLYINTRODUCEDTHEDOUBLEHERMITEINTERPOLATIONINTHESPECIFICAPPLICATIONOFPRACTICALPROBLEMS,ANDTHROUGHSEVERALEXAMPLESESTABLISHHERMITEINTERPOLATIONPOLYNOMIALMETHOD,TWOPOINTTHREETIMESOFHERMITEINTERPOLATIONANDITSREMAINDERTERM,HERMITEINTERPOLATIONFORMULAOFTHERELATEDPROBLEMSANDTHEPROGRAMCALCULATIONTOCOMPARESEVERALMETHODSBYDERIVATIONANDPROOFTHATCUBICHERMITEINTERPOLATIONHASBEENHIGH,HIGHRUNGEPHENOMENONMAYOCCURAGAINKEYWORDSDOUBLEHERMITEINTERPOLATIONPOLYNOMIAL,THEUNIQUENESSTHEOREM,THETHEOREMOFTHEERROR目录1、HERMITE插值概述11HERMITE插值定义112HERMITE插值公式的推导213几个重要的定理42、两点三次插值及其余项21两点三次插值322两点三次插值的余项423三次埃尔米特插值多项式624二重HERMITE插值多式73、重节点插商与HERMMITE插值31重节点插商104、例题及其解答125、参考文献176、附录187、谢辞201HERMITE插值概述11HERMITE插值的定义理论背景许多实际插值问题中，为使插值函数能更好地和原来的函数重合，不但要求二者在节点上函数值相等，而且还要求相切，对应的导数值也相等，甚至要求高阶导数也相等。这类插值称作切触插值，或埃尔米特HERMITE插值。满足这种要求的插值多项式就是埃尔米特插值多项式。埃尔米特插值是另一类插值问题，这类插值在给定的节点处，不但要求插值多项式的函数值与被插函数的函数值相同。同时还要求在节点处，插值多项式的一阶直至指定阶的导数值，也与被插函数的相应阶导数值相等。HERMITE插值在不同的节点，提出的差值条件个数可以不同，若在某节点XI，要求插值函数多项式的函数值，一阶导数值，直至M11阶导数值均与被插函数的函数值相同及相应的导数值相等。我们称XI为MI重插值点节,因此，HERMITE插值应给出两组数，一组为插值点节点，另一组为相应的重数标号。若，这就说明了给出的插值0NIX0NI01NIIMN条件有N1个，为了保证插值多项式的存在唯一性，这时的HERMITE插值多项式应在上求得，于是可作如下定义NP为，上充分光滑函数，对给定的插值定节，及相应的重数标号F,AB0NIX和时，若有满足0NIM1IINNHXP1,0,0,LIIIFLMN则称为关于节点及重数标号的HERMITE插值多项式。HXF0NIX0NI12HERMITE插值公式的推导1、HERMITE插值问题求一个次数不大于NR1的代数多项式,满足HX1,012IIXFN,IIHR称以上的插值问题为HERMITE插值问题注意式1包含个条件，所以能够确定次数不大于的代数多项2NR1NR2、HERMITE插值公式的推导建立HERMITE插值多项式的方法令200NRKKKKHXHFXHFX其中和都是次待定多项式，并且它们满足0,1KHX,1KR1NR一下条件3,0,0,1KIIIHXIKNR4,0,0KIIXIKIHR显然满足条件3,4的多项式2的次数不大于次，且满足插值条件1。1NR1求解,1KXN由条件3知是的二重零点0,IRIKHX且由条件3知是的零点2,IXIK（1）当时具有如下形式KRKH22220111KKKRRNHXABXXXXX（）（）（）5其中，是待定系数01RNIIIRK,AB由条件（3）知,0KHX即201RNKIKIIRABXX22001012101RNRNRKIKIKKJKIKIJIRIRNNKKIKIJRIRJXXXXXAB由上述两式解得012NJJRKKJRIIIRKXX201KRNKIIIRAXB将A,B代入式5，得610,1KKNKRKNKRHXXLLXLX其中，0NIKIKXL0RIKRIKXL0NKNIKILXX01RKRIKILX（2）当时，具有如下形式1RKH7201RNKIIIRKXCXX由条件（3）知1KHX201RNKIKIIRCXX将C代入式7，得8,2,RKKNWXHLRN其中，0RRIIX0RRKKIIX0NIKIKXL综合12得到即式6,8,1KHN2求解0,KX由条件4知是的二重零点，且由条件4知,IRIKHX是的零点。,12,IXRNKHX当时，具有如下形式0KRKH900NRKIIIKXDX由条件（4）知1KH00001NRNRRKIKIKIKIJJIIJIJXXXX将D代入式9，得10,1KKNKRHLR其中，0NIKIKXL0RIKRIKLX由式26810所表示的多项式称为HERMITE插值多项式其中由式6810所表示的多项式称为HERMITE插值基函数HERMITE插值多项式的余项为212120,NNJJRXFHXAB13几个重要的定理定理1（误差定理）若，则为关于上节点的二重HERMITE插值多项式误差为2,NFCABFX,AB0NIX222121NNNNFRFHWX这里0101,NNMIXXMAX定理2（唯一性定理）HERMITE插值问题式（1）的解HX存在而且唯一证明存在性已由上面推导，下证唯一性反证法，设插值问题式1有两个不同的解令，并且其次为次数不大于的多项式，且满足12GXHX1NR0,1,IG于是必含有因式和X2,1IXR,2,IXRN故的次数至少为，矛盾。证毕GNR定理3（HERMITE插值余项定理）HERMITE插值公式的余项为2NRNRFFXHWX其中，是插值区间内的某一点,AB证明引进辅助函数NRWTFTFHTFXH由条件（1）知010NFXXFX010RXFX即有个单根和R1个二重根TNR12,RN,R由ROLLE定理知，在内至少有NR2个零点，至少有NR1个零点TF（）,AB“FT依此类推可知在内至少有一个零点2NRT,因此220NRNRFFXHW即得2NRFFXHW若，则响应的HERMITE差值多项式为RN1100NNKKKKHXHFXHFX其中221,01,KKNKXXLXNH余项公式为2,NNFFHWAB特别当时，插值条件为1RN,01IIIIXFHXF由此得三次HERMITE插值多项式120010011HXHFXFHFHF201012X201012XX2100XHX2011XHX多项式（12）常用作分段低次插值，称为分段三次HERMITE插值2两点三次HERMITE插值及其余项21两点三次HERMITE插值先考虑只有两个节点的插值问题设在节点处的函数值为，在节点处的一阶导数值为，两个FX01,X01,Y01,X01,Y节点最高可以用3次HERMITE多项式，作为插值函数应满足插值条件3HX3H30XY1XY3应用四个插值基函数表示，设的插值基函数为3HX3HX,0,23IHX30123XAHA希望插值系数与LAGRANGE插值一样简单重新假设30101HXYXYX其中01X01X0X01X10X01X0X01X11可知是的二重零点，即可假设X0201XAXB由01X0X可得3012AX023011BXX22001132301011XXAXBXX201011LAGRANGE插值基函数2201011XXLX即22010101XL类似可得210101XXLX220001LX220111XLX将以上结果代入30101HXYXYX得两个节点的三次HERMITE插值公式30101HXYXYX2222100011LLYXLYXL2200101011001XXXYY22两点三次HERMITE插值的余项两点三次HERMITE插值的误差为33RXFHX0III33IIIXFX,1I均为的二重零点，因此可设01,X3RX22301RXKX其中待定KX构造辅助函数22301TFHTXTTX0IIIIIXK22301IIIIIFXX0,1I均是二重根22301XFHK因此至少有5个零点T连续使用4次ROLLE定理，可得，至少存在一点，使得01,X40即44FKX所以,两点三次HERMITE插值的余项为422301FRXXX01X以上分析都能成立吗当在上存在时，上述余项公式成立4FX01,三次埃尔米特插值多项式设是区间A,B上的实函数,是A,B上相异两点,且X0X1,YFX01,X在XI上的函数值和一阶导数值分别为和,F01IIYFIIMF求三次多项式,使其满足3H30,1IIHXYM称为三次埃尔米特插值多项式。3X误差估计定理4设FX在包含X0、X1的区间A,B内存在四阶导数，则当XA,B时有余项（且与X有关）42233011RXFHXFXX（）,AB设则当时,0144MAXM1余项有如下估计式（误差限）438MRXH二重HERMITE插值多项式常用的HERMITE插值为MI2的情况，即给定的插值节点均为二重节点，更具体些0NIX，及插值节点，若有满足2,FXCAB0NIX2121NNHP21NIIHXF，就称为关于节点的二重HERMITE插21,01,NIIHFX21NFX0I值多项式。上存在且连续时，上述余项公式成立01,XHERMITE插值优点分段线性插值的算法简单，计算量小，然而从整体上看，逼近函数不够光滑，在节点处，逼近函数的左右导数不相等。HERMITE插值的逼近函数与被逼近函数不仅在插值节点上取相同的函数值，而且逼近函数与被逼近函数在插值节点上去相同的若干阶导数值。HERMITE插值法结合了函数的导数值，使得插值的精度更为提高。HERMITE插值具有少节点得到高次插值多项式的特点HERMITE插值插值多项式灵活多样HERMITE插值在节点一定的条件下，可以多种构造插值条件3重节点插商与HERMITE插值4例题及解答例题1（两点三次插值多项式）已知在节点1,2处的函数值为FX12,3FF，在节点1,2处的导数值为FX10,21F求的两点三次插值多项式，及在处的函数值FFX5,7解30101HXYHXYHX2201011002X2210100XYYXX22232131HXX3279X315265FH79101,201,301,Y作为多项式插值,三次已是较高的次数，次数再高就有可能发生RUNGE现象,因此，对有N1节点的插值问题，我们可以使用分段两点三次HERMITE插值例2（误差估计）已知及其一阶导数的数据见下表,用埃尔米特插值公式计算12FX的近似值,并估计其截断误差15解231424XXHX214112XX2122233312194651XXX214得312518035H由47216FX可求得X121144FX1112FX1/221/24233152419860R5参考文献1蒋尔雄、赵风光、苏仰峰,数值逼近第二版,，复旦大学出版社,20082陈传璋、金福临、朱学炎、欧阳光中,，数学分析上、下册,高等教育出版社出版,19833徐树方、高立、张平方,数值线性代数，北京大学出版社,20004萧树铁、姜启源、张立平、何青、高立,，大学数学数学实验,高等教育出版社,20065张德丰,，MATLAB数值分析与应用,，国防工业出版社,20076周品、何正风，MATLAB数值分析,，机械工业出版社,20097韩丹夫、吴庆标，数值计算方法，浙江大学出版社，200668欧阳洁，数值分析，高等教育出版社6附录7致谢四年的努力学习，三个月的精心准备，毕业论文终于到了划句号的时候，在此之际，我首先要向在论文协作中给予我悉心关怀、鼓励和指导的陈传军老师致以深深的敬意和谢意。在大学期间陈老师教授过我数值逼近和微分方程数值解法两门课程，共四个学期。无论是在授课指导我写论文过程中，陈老师教学态度认真，为人和蔼可亲，在设计思想方面，更有着独特的见解，给我的设计注入了许多专业的思想，激发了我的灵感。这些都使我受益匪浅，并终身难忘。十分庆幸在即将离开烟台大学之际，又能跟随陈老师学习。毕业设计和毕业论文写作的经历和收获，值得一生回味。同时，我还要感谢烟台大学数学与信息科学学院的曾经给我授课的各位老师，这是他们的传道、授业、解惑，让我学到了专业知识，为这次毕业设计和毕业论文的完成打下了坚实的基础，并且我从他们身上学到了如何求知治学、如何为人处世。各位老师对我的教育和影响，将使我终身受益。我还要感谢同学和朋友的关心和帮助，烟台大学对我多年的培养，这些都是我这次完成毕业设计和毕业论文的不可或缺的因素。谨向我的父母和家人表示诚挚的谢意，使他们不变的支持、无微不至的关怀，促使了我的前进。没有他们就没有我，我的点滴成就来自于他们。我知道，这篇论文还存在缺点和不足之处，但是它是一个起点，在此基础上，我会更加努力，在以后的研究生学习中，不断提高自己。由于涉及知识范围广且本人水平有限，本文不免有错漏和不妥之处，望大家批评指正，不吝赐教最后，衷心感谢在百忙之中评阅论文的各位老师、专家、教授烟台大学毕业论文设计评审表（指导教师用）题目HERMITE插值的若干问题研究学生姓名XXX学号20XXXXXX16专业信息与计算科学指导教师姓名XXXXX职称副教授所学专业计算数学是否同意参加答辩得分评语指导教师（签字）年月日烟台大学毕业论文设计评审表评阅人用题目HERMITE插值的若干问题研究学生姓名XXXXX学号XXXXXXXX6专业信息与计算科学指导教师姓名XXXX职称副教授所学专业计算数学是否同意参加答辩得分评语评阅人（签字）年月日HERMITE插值的若干问题研究姓名XXX导师陈XX2013年05月25日完成地点烟台大学烟台大学毕业论文（设计）任务书院（系）数学与信息科学学院姓名XXXX学号2XXXXXXXXX毕业届别2013专业信息与计算科学毕业论文（设计）题目HERMITE插值的若干问题研究指导教师XXX学历博士职称副教授所学专业计算数学具体要求主要内容、基本要求、主要参考资料等1主要内容本文主要介绍HERMITE插值，包括它的定义、二重HERMITE插值多项式、唯一性定理、误差定理。通过两个典型的HERMITE插值的例子，说明建立HERMITE插值多项式的方法。并列举常用的两点三次插值。最后简单讨论了HERMITE插值的优缺点。2基本要求论文的推导、论证等要有独创性前后连贯通顺论文想的清,说的明白,想的深,说的透,做到深入简出,言简意赅3主要参考资料数学分析、数值逼近、数值线性代数、微分方程数值解法等进度安排第一阶段810周收集、阅读参考资料，熟悉相关理论知识第二阶段1112周总体构思、列出提纲、完成初稿第三阶段1315周修改完善论文,按要求书写论文并完成定稿第四阶段16周论文装订,完成论文答辩指导教师（签字）年月日院（系）意见教学院长（主任）（签字）年月日备注您好，为你提供优秀的毕业论文参考资料，请您删除以下内容，O_O谢谢ALARGEGROUPOFTEAMERCHANTSONCAMELSANDHORSESFROMNORTHWESTCHINASSHAANXIPROVINCEPASSTHROUGHASTOPONTHEANCIENTSILKROAD,GANSUSZHANGYECITYDURINGTHEIRJOURNEYTOKAZAKHSTAN,MAY5,2015THECARAVAN,CONSISTINGOFMORETHAN100CAMELS,THREEHORSEDRAWNCARRIAGESANDFOURSUPPORTVEHICLES,STARTEDTHETRIPFROMJINGYANGCOUNTYINSHAANXIONSEPT19,2014ITWILLPASSTHROUGHGANSUPROVINCEANDXINJIANGUYGURAUTONOMOUSREGION,ANDFINALLYARRIVEINALMATY,FORMERLYKNOWNASALMAATA,THELARGESTCITYINKAZAKHSTAN,ANDDUNGANINZHAMBYLPROVINCETHETRIPWILLCOVERABOUT15,000KILOMETERSANDTAKETHECARAVANMORETHANONEYEARTOCOMPLETETHECARAVANISEXPECTEDTORETURNTOJINGYANGINMARCH2016THENTHEYWILLCOMEBACK,CARRYINGSPECIALTYPRODUCTSFROMKAZAKHSTANASMALLARTTROUPEFOUNDEDSIXDECADESAGOHASGROWNINTOAHOUSEHOLDNAMEINTHEINNERMONGOLIAAUTONOMOUSREGIONINTHE1950S,ULANMUQIRARTTROUPEWASCREATEDBYNINEYOUNGMUSICIANS,WHOTOUREDREMOTEVILLAGESONHORSESANDPERFORMEDTRADITIONALMONGOLIANMUSICANDDANCESFORNOMADICFAMILIESTHE54YEAROLDWASBORNINTONGLIAO,INEASTERNINNERMONGOLIAANDJOINEDTHETROUPEIN1975HESAYSTHEREARE74BRANCHTROUPESACROSSINNERMONGOLIAANDACTORSGIVEAROUND100SHOWSEVERYYEARTOLOCALNOMADICPEOPLE“ICANSTILLRECALLTHEDAYSWHENITOUREDWITHTHETROUPEINTHEEARLY80SWESATONTHEBACKOFPICKUPTRUCKSFORHOURSTHESKYWASBLUE,ANDWECOULDNTHELPBUTSINGTHEFOLKSONGS,“NASUNSAYSTHEVASTNESSOFINNERMONGOLIAANDTHELACKOFENTERTAINMENTOPTIONSFORPEOPLELIVINGTHERE,MADETHEIRLIVESLONELY“THENOMADICPEOPLEWEREVERYEXCITEDABOUTOURVISITS,“NASUNRECALLS“WEDIDNTHAVEAFORMALSTAGETHEAUDIENCEJUSTSATONTHEGRASSUSUALLY,THEPERFORMANCESBECAMEABIGPARTYWITHLOCALPEOPLEJOININGIN“FORHIM,THEREWARDINGPARTABOUTTOURINGISNTJUSTABOUTSHARINGARTWITHNOMADICFAMILIESBUTALSOABOUTGAININGINSPIRATIONFORTHEMUSICANDDANCEULANMUQIRLITERALLYTRANSLATESAS“REDBURGEON“,ANDTODAYSPERFORMERSOFTHETROUPESTILLTOURTHEREGIONSVILLAGESANDENTERTAINNOMADICFAMILIES,BUTTHEIRFAMEHASSPREADAROUNDTHEWORLDONMAY16AND17,NEARLY100SINGERSANDDANCERSFROMTHETROUPEPERFORMEDATBEIJINGSPOLYTHEATERTHEIRSHOW,TITLEDULANMUQIRONTHEGRASSLAND,DEPICTEDTHEHISTORYANDDEVELOPMENTOFTHEARTTROUPE“BEINGFROMTHEREGIONALLOWEDMETOEMBRACETHECULTUREOFINNERMONGOLIAANDBEINGAMEMBEROFTHETROUPESHOWEDMEWHEREIBELONGED,“NASUN,THEARTTROUPESPRESIDENT,WHOISALSOARENOWNEDTENOR,TELLSCHINADAILYDURINGATOURIN1985,HEWENTTOAVILLAGEANDMETANELDERLYLOCALMAN,WHOTOLDHIMASTORYABOUTHISFRIENDSHIPWITHASOLIDERFROMSHENYANG,CAPITALOFNORTHEASTCHINASLIAONINGPROVINCE,DECADESAGOTHESOLIDERGAVETHEOLDMANAHANDMADESADDLEWHENTHEYBIDFAREWELLTHESTORYINSPIREDNASUNTOWRITECARVEDSADDLE,ASONGTHATLATERBECAMEONEOFHISMOSTPOPULARNUMBERSNOW,EVERYYEAR,NASUNRECRUITSYOUNGSINGERSANDDANCERSFORTHETROUPETHETROUPEHASALSODESIGNEDANEWREPERTOIRE,WHICHISMOSTLYBASEDONTHEDAILYLIVESOFMONGOLIANPEOPLE,ESPECIALLYTHELIVESOFNOMADICFAMILIES,ANDHASCOMBINEDCONTEMPORARYMUSICALELEMENTSWITHFOLKSONGSOFTHEREGIONHAIMU,A25YEAROLDKHOOMEIALOCALVARIANTOFOVERTONESINGINGSINGER,JOINEDTHETROUPETHREEYEARSAGOALONGWITHASIXMEMBERBAND,HEPERFORMSFASTSONGSANDSOFTONESTHATHEWRITESALLWHILEPLAYINGTHEHORSEHEADFIDDLE“ALTHOUGHILEARNEDTHEPIANOSINCECHILDHOODANDGREWUPLISTENINGTOVARIOUSKINDSOFMUSIC,TOME,THEFOLKMUSICOFINNERMONGOLIAISTHEROOT,“HESAYS“PERFORMINGINREMOTEVILLAGESISPLEASANTIFEELATHOMEONTHEBOUNDLESSGRASSLANDS,ANDTHEWARMPEOPLETHEREMAKEMEFEELFULFILLED“THEFIRSTROUNDOFSPRINGAUCTIONSEASONINBEIJINGENDEDLASTWEEK,BUTITFAILEDTOCREATEMUCHSPRINGINTHEARTMARKETALTHOUGHTWOPIECESOFCHINESEPAINTINGFETCHEDMORETHAN100MILLIONYUAN,THEDECLINEINTRADINGVOLUMEANDSAL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