毕业论文-Hermite插值的若干问题研究.doc

烟台大学毕业论文（设计） 烟 台 大 学毕 业 论 文（设 计）Hermite插值的若干问题研究Study of Hermite interpolation problems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Abstract This paper is detailed to discuss some problems of the Hermite interpolation, mainly introduced the double Hermite interpolation in the specific application of practical problems, and through several examples establish Hermite interpolation polynomial method, two point three times of Hermite interpolation and its remainder term, Hermite interpolation formula of the related problems. And the program calculation to compare several methods. By derivation and proof that cubic Hermite interpolation has been high, high Runge phenomenon may occur againKeywords: Double Hermite interpolation polynomial, the uniqueness theorem, the theorem of the error目 录1、Hermite插值概述 1.1 Hermite插值定义.1 1.2 Hermite插值公式的推导.2 1.3 几个重要的定理42、两点三次插值及其余项 2.1 两点三次插值 .32.2 两点三次插值的余项42.3 三次埃尔米特插值多项式62.4 二重Hermite插值多式73、重节点插商与Hermmite插值 3.1 重节点插商.104、例题及其解答125、参考文献176、附录187、谢辞2051 Hermite插值概述1.1 Hermite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m1-1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条件有N+1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：为，上充分光滑函数，对给定的插值定节，及相应的重数标号和时，若有满足则称为关于节点及重数标号的Hermite插值多项式。1.2 Hermite插值公式的推导1、Hermite插值问题求一个次数不大于n+r+1的代数多项式,满足： -(1)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2、Hermite插值公式的推导(建立Hermite插值多项式的方法)令-(2)其中和都是次待定多项式，并且它们满足一下条件： -(3) -(4) 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)。1.求解 由条件(3)知是的二重零点 且由条件(3)知是的零点 （1）当时具有如下形式： -(5) 其中，是待定系数由条件（3）知即 由上述两式解得： 将A,B代入式(5)，得 -(6)其中， （2）当时，具有如下形式： -(7)由条件（3）知将C代入式(7)，得 -(8)其中， 综合(1)(2)得到即式(6),(8)2.求解 由条件(4)知是的二重零点，且由条件(4)知 是的零点。当时，具有如下形式： -(9)由条件（4）知将D代入式(9)，得 -(10)其中， 由式(2)(6)(8)(10)所表示的多项式称为Hermite插值多项式其中由式(6)(8)(10)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：1.3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插值多项式误差为这里定理 2（唯一性定理） Hermite插值问题式（1）的解H(x) 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，且满足于是必含有因式和故的次数至少为，矛盾。证毕定理 3（Hermite插值余项定理）Hermite插值公式的余项为其中， 是插值区间内的某一点证明：引进辅助函数由条件（1）知 即有个单根和r+1个二重根由Rolle定理知，在内至少有n+r+2个零点，至少有n+r+1个零点依此类推可知：在内至少有一个零点因此即得 若，则响应的Hermite差值多项式为 -(11)其中余项公式为：特别当时，插值条件为：由此得三次Hermite插值多项式：-(12) 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2.1 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： 应用四个插值基函数表示，设的插值基函数为希望插值系数与Lagrange插值一样简单重新假设其中 可知 是的二重零点，即可假设由 可得 . Lagrange插值基函数即 类似可得 将以上结果代入得两个节点的三次Hermite插值公式 2.2两点三次Hermite插值的余项两点三次Hermite插值的误差为 均为的二重零点，因此可设：其中待定构造辅助函数 均是二重根 因此至少有5个零点连续使用4次Rolle定理，可得，至少存在一点，使得 即所以,两点三次Hermite插值的余项为 以上分析都能成立吗？当在上存在时，上述余项公式成立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, 在xi上的函数值和一阶导数值分别为和, 求三次多项式, 使其满足：称为三次埃尔米特插值多项式。 误差估计定理4 设f(x)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） 设 则当时,余项有如下估计式（误差限） 二重Hermite插值多项式常用的Hermite插值为mi=2 的情况，即给定的插值节点均为二重节点，更具体些 ，及插值节点，若有满足，就称为关于节点的二重Hermite插值多项式。上存在且连续时，上述余项公式成立Hermite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 ，在节点1,2处的导数值为求的两点三次插值多项式，及在处的函数值解： 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n+1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的近似值,并估计其截断误差.解： x121144 f(x)1112 f (x)1/221/24 得由 可求得 5 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, 2008.2 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, 1983.3 徐树方、高立、张平方, 数值线性代数， 北京大学出版社, 2000.4 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, 2006.5 张德丰,，Matlab数值分析与应用,，国防工业出版社, 2007.6 周品、何正风， Matlab数值分析,，机械工业出版社, 2009.7 韩丹夫、吴庆标，数值计算方法，浙江大学出版社，2006.6.8 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xx 2013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！A large group of tea merchants on camels and horses from Northwest Chinas Shaanxi province pass through a stop on the ancient Silk Road, Gansus Zhangye city during their journey to Kazakhstan, May 5, 2015. The caravan, consisting of more than 100 camels, three horse-drawn carriages and four support vehicles, started the trip from Jingyang county in Shaanxi on Sept 19, 2014. It will pass through Gansu province and Xinjiang Uygur autonomous region, and finally arrive in Almaty, formerly known as Alma-Ata, the largest city in Kazakhstan, and Dungan in Zhambyl province. The trip will cover about 15,000 kilometers and take the caravan more than one year to complete. The caravan is expected to return to Jingyang in March 2016. Then they will come back, carrying specialty products from Kazakhstan A small art troupe founded six decades ago has grown into a household name in the Inner Mongolia autonomous region. In the 1950s, Ulan Muqir Art Troupe was created by nine young musicians, who toured remote villages on horses and performed traditional Mongolian music and dances for nomadic families. The 54-year-old was born in Tongliao, in eastern Inner Mongolia and joined the troupe in 1975.He says there are 74 branch troupes across Inner Mongolia and actors give around 100 shows every year to local nomadic people. I can still recall the days when I toured with the troupe in the early 80s. We sat on the back of pickup trucks for hours. The sky was blue, and we couldnt help but sing the folk songs, Nasun says. The vastness of Inner Mongolia and the lack of entertainment options for people living there, made their lives lonely. The nomadic people were very excited about our visits, Nasun recalls. We didnt have a formal stage. The audience just sat on the grass. Usually, the performances became a big party with local people joining in. For him, the rewarding part about touring isnt just about sharing art with nomadic families but also about gaining inspiration for the music and dance. Ulan Muqir literally translates as red burgeon, and todays performers of the troupe still tour the regions villages and entertain nomadic families, but their fame has spread around the world. On May 16 and 17, nearly 100 singers and dancers from the troupe performed at Beijings Poly Theater. Their show, titled Ulan Muqir on the Grassland, depicted the history and development of the art troupe. Being from the region allowed me to embrace the culture of Inner Mongolia and being a member of the troupe showed me where I belonged, Nasun, the art troupes president, who is also a renowned tenor, tells China Daily. During a tour in 1985, he went to a village and met an elderly local man, who told him a story about his friendship with a solider from Shenyang, capital of Northeast Chinas Liaoning province, decades ago. The solider gave the old man a handmade saddle when they bid farewell. The story inspired Nasun to write Carved Saddle, a song that later became one of his most popular numbers. Now, every year, Nasun recruits young singers and dancers for the troupe. The troupe has also designed a new repertoire, which is mostly based on the daily lives of Mongolian people, especially the lives of nomadic families, and has combined contemporary musical elements with folk songs of the region. Haimu, a 25-year-old khoomei (a local variant of overtone singing) singer, joined the troupe three years ago. Along with a six-member band, he performs fast songs and soft ones that he writesall while playing the horse-head fiddle.Although I learned the piano since childhood and grew up listening to various kinds of music, to me, the folk music of Inner Mongolia is the root, he says. Performing in remote villages is pleasant. I feel at home on the boundless grasslands, and the warm people there make me feel fulfilled. The first round of spring auction season in Beijing ended last week, but it failed to create much spring in the art market. Although two pieces of Chinese painting fetched more than 100 million yuan, the decline in trading volume and sale rate showed a downturn this year. In the “Grand View: Chinese Painting Highlight” session at China Guardian 2015 spring auctions, Pan Tianshous representative work Eagle, Rock and Flora hit a record auction price of 279 million yuan, while Li Kerans masterwork Jinggang Mountain fetched 126.5 million yuan, an unexpected high in recent years. However, the trading volume fells to 1.87 billion yuan from 2.22 billion yuan in the same period the year before. The Huangchen 2015 Spring Auctions, which recorded 42.5 million yuan in total sales, experienced the same. The section number went down to 5 from 12 compared to last year. According to expert Shao Jianwu, the art market did not attract much excitement this year due to the booming stock market and the persistent problems of forgery and fake deals. The two pieces of Chinese painting notched up high price this spring due to their own value not due to a revival in the market. The successful transaction of two works with a hammer price of more than 100 million yuan once again confirms that valuable and rare works of high quality always earn the long-lasting high price. Chinese painting and calligraphy is still the pillar of the market. However, the overall sales rate has dropped slightly, which reflects the demand for the ordinary auction goods has weakened,” said the president of China Guardians Hu Yanyan. Besides, the entry of new buyers also makes certain changes to the direction of purchasing. In a word, the overall market trend is very unpredictable. Although China Guardian 2015 Spring Auctions has achieved remarkable results, it does not mean that the market has walked out of the predicament. We still have to wait for the big economic boom to invigorate the art market.” Even so, the hot pursuit of some special items appeared this spring. A special session of China Guardian Spring Auctions titled “Fine GiltBronze Buddhist Images” achieved a great deal with total sales volume of 41.89 million yuan and 92 percent sale rate. The 5th Shamar Rinpoche Statue from 16-17th century of Tibet was sold for 5.06 million yuan. The Beijing Chengxuan Auctions featured almost 2,800 items of coins and stamps in three sessions with good sale rate. The Huachen Auctions also set a special session of photographs under the condition of large-scale decline of auction sessions. There was a palpable dull thud of disappointment that accompanied the return of the imperial entourage of Zhen Huan to her homeland. It followed a couple years of hushed excitement as Chinese fans were fed tidbits about their proud concubine who was supposed to conquer the high ground of the North American market. Zhen Huan is, of course, the title character of The Legend of Zhen Huan, a 2011 television series that swept China off its feet and later took other Asian countries by storm. Two years ago, it was reported that HBO, a premium cable service headquartered in the United States, was going to air it in North America after some modification. Now, a condensed version that provides English subtitles but no dubbing has finally been made available on Netflix for online streaming. This version, highly anticipated as a milestone in Chinas cultural foray overseas, has been widely panned by its home audience. Retitled Empresses in the Palace, the American version has been shortened from its original 76 episodes at 45 minutes each, to six 90-minute episodes. The quick pacing threw off many native viewers, who are accustomed to a more leisurely daytime-soap-style narrative rhythm. (Chinese TV stations would run two or three episodes every day.) I did not finish the full-length version and found the truncated one not difficult to follow. Whats lost, I believe, are the interesting setups and pauses that illuminate the Chinese art of storytelling. Much of the plot is still there. It is the flavor that was sacrificed. The American edition uses the framework of the Empress Dowager in her senior years reminiscing at the beginning and the end of each episode, hinting at whats to come and recapping the key points. This device, not used in the original, is culturally understandable but artistically mediocre. What puzzles me is the two new songs for the opening and end credits. They were written in English, but sung by Chinese with an uncomfortable accent. They were obviously designed to appeal to an English-speaking base, but do not jibe with the Chinese dialogue. Speaking of the dialogue, the English translation, picked apart by some Chinese, is too literal for my taste. I can imagine a typical American hit by a flurry of royal ranks, addresses and greetings, even multiple names and titles for the same person. The first half hour must be a swamp to wade through, very much like my experience of getting through a Tolstoy tome with its endless inflections of names transliterated into lengthy Chinese. I see the choice of verbatim translation as an effort for conveying exotica. It is fairly competent, with no error that I could detect, but fails to rise above words or capture the essence of the language. A cultural product usually crosses over to a foreign territory first by an emphasis on the commonalities. But whether inside or outside China, the temptation to sell it for the differences is just too great. Sure, the sumptuous sets and costumes are a big attraction, but the narrative technique has becomehow shall I put it?a bit anglicized, which is necessary for cultural export. Judging by the responses, this legend, which, contrary to the claim of the English trailer, is totally fictitious, has departed from China but not yet landed on American shores. I am a big fan of Ralph Waldo Emersons essay titled Self-Reliance. If youve not yet read it, I strongly encourage you to do so. In this essay, the venerable Emerson talks about eschewing the trappings of society and finding ones own path. He urges the reader to find rectitude and lead a moral life. He professes that, only by being self reliant, as opposed to relying on the government and being dictated to by society, can one begin leading a decent and purposeful life. He avers that such a life is the only life worth living. I ag