沈阳工业大学数值分析(2001)2009,2011--2019年考博真题

2011年秋季工学硕士研究生学位课程（数值分析）真题试卷B(总分：28.00，做题时间：90分钟)一、填空题(总题数：6，分数：12.00)1.填空题请完成下列各题，在各题的空处填入恰当的答案。

（分数：2.00）__________________________________________________________________________________________ 解析：2.设｜x｜>>1______（分数：2.00）__________________________________________________________________________________________正确答案：()解析：3.求积分∫ a b f（x）dx的两点Gauss公式为______（分数：2.00）__________________________________________________________________________________________正确答案：()解析：4.设∞ =______，‖A‖ 2 =______．（分数：2.00）__________________________________________________________________________________________正确答案：()解析：5.给定f(x)=x 4，以0为三重节点，2为二重节点的f(x)的Hermite插值多项式为______．（分数：2.00）__________________________________________________________________________________________ 正确答案：(正确答案：x 4)解析：6.己知差分格式r≤______时，该差分格式在L ∞范数下是稳定的．（分数：2.00）__________________________________________________________________________________________正确答案：()解析：二、计算题(总题数：2，分数：4.00)7.给定方程lnx-x 2+4=0，分析该方程存在几个根，并用迭代法求此方程的最大根，精确至3位有效数字．（分数：2.00）__________________________________________________________________________________________ 正确答案：(正确答案：令f(x)=lnx-x 2 +4，则f"(x)= -2x,当x= 时，f"(x)=0. 注意到f(0．01)=-0．6053＜0,f(1)=3＞0,f(3)=-3．9014＜0,而当时，f"(x)＞0，当时，f"(x)＜0，所以方程f(x)=0有两个实根，分别在(0．01，1)和(1，3)内．方程的最大根必在(1，3)内，用Newton迭代格式取x 0 =2，计算得x 1 =2．1980，x 2 =2．1)解析：8.用列主元Gauss（分数：2.00）__________________________________________________________________________________________正确答案：(正确答案：求得x 1 =3，x 2 =1，x 3 =5．)解析：三、综合题(总题数：6，分数：12.00)9.设α，β表示求解方程组．Ax=b的Jacobi迭代法与Gauss-Seidel迭代法收敛的充分必要条件．（分数：2.00）__________________________________________________________________________________________ 正确答案：(正确答案：Jacobi迭代格式的迭代矩阵特征方程为展开得500λ3—15αβλ=0或者λ(500λ2—15αβ)=0，解得λ=0或λ2 = 则Jacobi格式收敛的充要条件为｜αβ｜＜Gauss-Seidel格式迭代矩阵的特征方程为展开得500λ3—15αβλ2 =0或者λ2(500λ-15αβ)=0，解得λ=0或λ则Gauss-Seidel格式收敛的充)解析：10.设x 0，x 1，x 2为互异节点，a，b，m为已知实数．试确定x 0，x 1，x 2的关系，使满足如下三个条件p(x 0 )=a， p"(x 1 )=m，p(x 2 )=b的二次多项式p(x)存在且唯一，并求出这个插值多项式p(x)．（分数：2.00）__________________________________________________________________________________________ 正确答案：(正确答案：由条件p(x 0 )=a，p(x 2 )=b确定一次多项式p 1 (x)，有所以p(x)-P 1(x)=A(x—x 0 )(x—x 2 )，p"(x)=p" 1 (x)+A(x—x 0 +x—x 2 )，p"(x 1+A(2x 1 -x 0 -x 2) 解析：11.求y=｜x｜在[-1，1]上形如c 0 +c 1 x 2的最佳平方逼近多项式．（分数：2.00）__________________________________________________________________________________________ 正确答案：(正确答案：取φ0 (x)=1，φ1 (x)=x 2，则(φ0，φ0)=∫ -11 =2，(φ0，φ1)=∫ -11 x 2)1 x 2，(φ1，φ1)=∫ -1解析：12.已知函数f(x)∈C 3 [0，3]，试确定参数A，B，C，使下面的求积公式数精度尽可能高，并给出此时求积公式的截断误差表达式．（分数：2.00）__________________________________________________________________________________________正确答案：(正确答案：当f(x)=1时左=∫ 03 1dx=3，右=A+B+C，当f(x)=x时左=∫ 03 xdx= ，右=B+2C 当f(x)=x 2时左=∫ 03 x 2 dx=9,右=B+4C．要使公式具有尽可能高的代数精度，则而当f(x)=x 3时，左=∫ 03 x 3)解析：13.给定常微分方程初值问题取正整数n，并记h=a／n，x i =a+ih，0≤i≤n．证明：用梯形公式求解该初值问题所得的数值解为且当h→0时，y n收敛于y(a)．（分数：2.00）__________________________________________________________________________________________正确答案：(正确答案：梯形公式应用于方程有y i+1=y i+ (-y i—y i+1)，即有所以i=1，2，…．当h→0时，n→∞我们有而由方程知解析解y=e -x则y(a)=e -a，所以)解析：14.Ω={0＜x＜3，0＜y＜3)．试用五点差分格式求u(1，1)，u(1，2)，u(2，1)，u(2，2)的近似值．（分数：2.00）__________________________________________________________________________________________正确答案：(正确答案：五点差分格式为根据要求，可取h= ，将(1，1)，(2，1)，(1，2)，(2，2)处的差分格式列成方程组有或者解得u 11=15．8750，u 21=22．6250，u 12=15．8750，u 22 =22．6250．)解析：。

研究生考试命题纸沈阳工业大学 2012 / 2013 学年 第 一 学期课程名称：数值分析 课程编号：000304 任课教师：陈欣 曲绍波 考试形式：闭 卷一、填空（每题3分，共15分）1. 二分法是求解 方程f (x )=0的 根一种方法，其前提是f (x )在有根区间[a ,b ]内单调且 。

2. 设矩阵⎪⎪⎭⎫ ⎝⎛-=0112A ，则1A = 、=2A 、)(A ρ= 。

3. 对于正数a ，使用牛顿法于方程02=-a x 所得到的迭代格式为 ，其收敛阶为 、求110（取x 0=10）的第一个近似值为 。

4. 幂法用来计算实矩阵A 的 特征值及对应的 ，在计算过程中进行“归一化”处理的原因是为了 。

5. 高斯求积公式)33()33()(11f f dx x f +-≈⎰-的代数精度为 ，当区间不是[-1,1]，而是一般区间[a , b ]时，需要做变换 ，使用该公式计算≈⎰311dx x。

二、解答下列各题（每题5分，共10分）1. 请写出经过点A (0,1)，B (2,3)，C (4,5)的拉格朗日插值多项式形式。

说明插值基函数的性质以及拉格朗日插值法的优缺点。

2. 设n 阶可逆矩阵A 已经分解成A =LU ，其中L 下三角矩阵，U 单位上三角矩阵，推导出解线性方程组AX =b 的计算公式。

三、（10分）用不选主元的直接三角分解法解下面线性方程组⎪⎪⎩⎪⎪⎨⎧=+-=-+-=-+-=-342424344343232121x x x x x x x x x x 四、（20分，每题10分）对于线性方程组⎪⎩⎪⎨⎧=++=++=-+9223122321321321x x x x x x x x x 1. 分别写出使用GS 迭代法，SOR 迭代法（ω=1.3）求解的迭代格式，并对初始向量（1，0，0）T ，分别计算第一步近似解向量；2. 分别讨论求解此方程的J —方法和GS —方法的收敛性。

五、（10分）给出函数表如下，用牛顿向前插值公式求f (2.03)的近似值。

沈阳工业大学2019 年博士研究生招生考试题签(请考生将题答在试题紙上，答在题签上无效）Part I Vocabulary (15 points, 0.5 for each)Directions: In this section, there are 30 sentences, each with one word or phrase underlined. Choose the one from the 4 choices marked A, B, C and D that best keeps the meaning of the sentence. Then mark the corresponding letter with a single line on the ANSWER SHEET1.The of AIDS has led to an expansion in research seeking a cureA. InnovationB. selectionC.proliferationD.Conviction2.Most importantly, such an experience helps a heightened sensitivity to other cultures andwill bring about a greater appreciation of one's own culture as wellA.containB.fosterC.discoverD.censor3.The talks between China and Japan were the base of the later agreementsA. InitialB.finalC. terminalD. primitive4.Metals we found in the earl iron, lead , and copper.priseB.containC.concludepose5.Tom give me a new bicycle as for the one he lost.A.substitutpensationC.rewardD.bonus6.Over the past ten years, natural gas production has remained steady, but has risen steadily.A.ConsuinptionB.disposalC.ConditionD.disposition7.The Brons lived in a and comfortably furnished house in the countrysideA.spaciousB.sufficientC. EfficientD. Constant8.David was by his family because he worked with a gang of criminals to rob the commercialbank.A. AbolishedB.diminishedC.desertedD.dismissed9.The talks between the two leaders have reached a stage.A.criticalB.criticismC.desertedD.dismissed10.These goods are for export, though a few of them may be sold on the home market.A. essentiallypletelyC.necessarilyD.remarkably11.We love peace, yet we are not the kind of people to any military threat.A.yield upB.yield toC.yield inD.yleid at12.John used to his teacher's tone and gestures to make his classmates laugh.A.imitateB.stimulateC.initimateD. Resemble13.The professor the translation of a German novel.A.is engaged withB.is busy inC.is occupied withD.is absorbed to14. The findings appears to the laws of physics.A.violentB.ViolateC. V oluntaryD.violet15. It is rather that we do not know how many species there are in the worl D.A.embarrassingB.boringC.demandingD.Violet16. Science is still largely the causes of man kinds of cancer.A.ignorant ofB.ignorant aboutC.ignorant withD.ignorant to17. In the first several years of our economic reform, many military factories have been into civil ones in order to meet the needs of the country.A.changedB.TransformedC.ConvertedD.Altered18. Some plants are so pollution that they can only survive in a perfectly clean environment.A.sensitive toB.sensible toC.sensitive againstD.sensible against19. Don'teat choclate; it will spoil your dinner.A.Desire forB.hunger forC.appetite forD. Eager for20. To help these difficulties, some organizations have opted to group together machines that process parts which have a large set of common characteristics.A.prevailB.overcomeC.overpowerD.Defeat21. Some people find it helpful to have something to do with their hands, like , or playing witha bunch of keys.A.knittingB.BeatingC.meetingD.dining22. Althougu many experts agree that more children are overweight, there is debate over the best ways to the problem.A.pursueB.TackleC.DistinguishD.ultimately23. He has held hundreds of meetings and spent whole weeks resolving these tricky problems, but he has faile D.A.ideologicallyB.personallyC.skepticallyD.Ultimately24. Drider says it is only since Sarkozys government began discussing the matter that she has been insults, harassment and death threats.A.objected toB.inferior toC.confined toD.subject to25. Artists should cultivate their own style rather than always imitating others.A.singleB.IndividualC.privateD.separate26. The new discovery of oil is to the country’s economy.A.SignatureB.significantC.signD. significance27. After the , the employer concluded that she was suitable for the position.A.audienceB.encounterC.interviewD.arrangement28. If businessmen are taxed too much, they will no longer be motivated to work hard, with the result that income from taxation might actually .A. ShrinkB.delayC. DisperseD. Sink29. He hated wandering about and expected to find a position in the Civil Service of government.A.ExcessiveB.AccidentalC.equalD.permanen30. Doctors and researchers have to keep themselves on the latest developments in their sphere of study.A.ConvincedB.IsolatedC.HumiliatedD.updated Part IL Cloze (15 points, 1 for each)Directions: There are 15 blanks in the following passage. For each blank, there are four choices marked A, B. C and D. You should choose the ONE that best fits into the passage. Then mark the corresponding letter on the Answer Sheet with a single line through the center.Predictions of large populations of robots in industry have yet to come true. For a decade ormore, manufacturers of big robots have explained how their machines can make industry more competitive and productive. The market for 31 robots is now oversupplied, and the driving force of the robotics revolution is 32 to be with makers of machines that handle a few kilos at most.Heavy-robot manufacturers are in some difficulty 33 finding customers. They are offering big 34 just to get in the door. There has been a disappointing growth everywhere in the numbers of robots, so have to admit we are either deceiving 35 or that the market is slowly growing, "said John Reekie, Chairman of Colne Robotics. The following things must happen 36 the robotics revolution to occur. We must achieve widespread robot literacy (文化), 37 there has been a computer literacy program, there must be a robot literacy campaign. We must also achieve not just a cut in, but a collate of robot prices. Finally, some ki of artificial intelligence needs to be 38 .Colne makes educational robots and machine tools. It is small 39 with companies like ASEA. But Colne with others and departments in universities such as Surey, Manchester, possesses an advantage 40 the giants. The big companies sell very expensive robots to businesses with expert knowledge in automation. The 41 companies make robots for teaching people, and now they have realized there is a need for small, 42 robots thatThe little companies either bring their educational machines 43 an industrial standard or design from scratch. One technique that they all adopt is to choose _ 44 components where possible. The major cost of making both their models is the electronics, which will fall in price. There is 45 scope for reductions in mechanical costs. The use of standard parts, which are easily replaced, should give these robots mechanical life of something in the order of five years.31. A.small cational C.big D.business32. A.claimed B.called C.said D. told33. A.to B.with C.over D.in34. A.money B.discount C. prices D.costs35. A.ourselves B.customers panies 37. A.as if B.even though C.just as D.although38. A.capable B.possible C. probable D. available39. pared paring C.contrasted D.contrasting40. A.than B.above C.over D. from41. A.other B.big C.small D.robot42. A.expensive B. artificial cational D. low-cost43. A. up to B.to C. in D. about44. A. mechanical B.standard C.cheap D.small45. A. more B.less C. no D. leastPart Ill. Reading Comprehension (30 points, 1.5 for each)Directions: There are four passages in this part. Each of the passages is followed by five questions or unfinished statements. For each of the questions there are four choices marked A, B, C and D. Choose the best one and mark your answer on the ANSWER SHEET with a single line through the center.Passage oneSome pessimistic experts feel that the automobile is bound to fall into disuse. They see a day in the not-too-distant future when all autos will be abandoned and allowed to rust. Other authorities. However, think the auto is here to stay. They hold that the car will remain a leading means of urban travel in the foreseeable future.The motorcar will undoubtedly change significantly over the next 30 years. It should become smaller, safer, nc more economical, and should not be powered by the gasoline engine. The ear of the future should be far more Pollution-free than present types.Regardless of its power source, the auto in the future will still be the main problem in urban traffic congestion ne proposed solution to this problem is the automated highway system.When the auto enters the highway system, a retractable arm will drop from the auto and make contact with a rail, which is similar to those powering subway trains electrically. Once attached to the rail the car will become electrically powered from the system, and control of the vehicle will pass to a central computer. The computer will then monitor all of the car’s movements.The driver will use a telephone to dial instructions about his destination into the system. The computer will calculate the best route, and reserve space for the car all the way to the correct exit from the highway. The driver will then be free to relax and wait for the buzzer that will war him of his coming exit. It is estimated that an automated highway will be able to handle 10,000 vehicles per hour, compared with the 1, 500 to 2,000 vehicles that can be carried by a present-day highway46. One significant improvement in the future car will probably beA. Its power sourceB. its driving systemC. its monitoring systemD. Its seating capacity47. What is the authors main concern?A. How to render automobiles pollution-freB. How to make smaller and safer automobilesC. How to solve the problem of traffic jamsD. How to develop an automated subway system48. What provides autos with electric power in an automated highway system?A. A railB. An engine.C. A retractable armD A computer controlle49. In an automated highway system, all the driver needs to do isA. Keep in the right laneB. Wait to arrive at his destinationC. Keep in constant touch with the computer centerD inform the system of his destination by phone50. What is the authors attitude toward the future of autos?A. EnthusiastB. PessimisticC. Optimistic.D. Cautious.Passage TwoWhile jobs in science, technology, engineering and math (STEM) professions may be plentiful, many teenage unwilling to pursue a long-term career in these fields. According to a new survey, in sixth through twelfth grade felt that careers as doctors and engineers would offer the most job opportur college, but 67 percent were of two minds about whether they would pursue these careers, due to a multitude of difficulties involved.Chief among the challenges is the cost and time it takes to get a degree. According to this survey, 26 percent of respondents felt that the cost and difficulty of pursuing professionally qualifying education in these fields were too high compared with other fields. Additionally, 25 percent of students felt that these career paths were too challenging and involved too much studying.Careers as doctors and engineers were not the only places where students saw opportunities, though. According to the survey, the three career fields offering the greatest opportunity were doctors, engineers and teachers, followed by lawyers and entrepreneurs.“It's encouraging to se that more students see the value of STEM careers like engineering, but clearly STEM) ofessionals and educators can be doing more to support students along this career path, "said Jim Rooney, ASQ chair and quality engineer with ABSG Consulting That is because 25 percent of teenage respondents stated their grades in math and science arent good enough to pursue a future career in those subjects. For once, parents agreed with their children, as 53 percent of parents who responded in a similar survey were worried about the challenges these fields presented to their children. Just over al quarter of parents, 26 percent, also felt that teachers were not preparing their children enough for future careers in STEM fields.Another factor contributing to the non-pursuit of STEM jobs is the growing gender gap in education. Accord to this survey, 30 percent of girls stated that math was their most challenging subject, compared with 19 percent boys. Additionally, 33 percent of girls admitted they felt teachers did not prepare them enough for future careers in STEM careers, compared with just 9 percent of boys.The information in this survey is based on the responses of 713 students and a complementary survey of 327 parents with children between the ages of 10 and 17.51. Many students are reluctant to take up STEM jobs because of the following EXCEPTA. The hard work needed along the career pathB the high tuition they have to pay at collegeC the few work opportunities related to STEMD. The long time taken to get the acadcmic degree52. According to this passage, most students believe that careers as doctors and engineersA. Are really worth pursuingB. Promise easy employmentC. Have nothing to do with mathD. Offer fewer opportunities than lawyers53. According to Paragraph 4, Jim Rooney would like toA. advise students not to take up STEM jobsB. Become a STEM professional himselfC tell STEM professionals to learn moreD see students get more help into STEM careers54. The growing gender gap mentioned in Paragraph 6 refers to the finding thatA. Fewer girls want to engage in STEM careers than boysB. More girls hope to take up STEM than boysC. There are far more male students than female onesD. Boys are more likely to get Stem-related jobs55. This passage is intended toA encourage students to undertake STEM jobB. Describe the difficulty facing STEM studentsC. Introduce the result of a survey on STEM jobsD help college graduates to find STEM jobsPassage ThreeMost people believe the key to happy and productive retirement years is staying healthy. Yet, it's not just about diet and exercise. How you manage your money often determines your level of stress, which, if too high, may lead to chronic health problems like heart diseases.So you'll have to do more than switch to whole wheat and start jogging every day. Your health also depends on feeling good about and in control of your finances. Not a lot of people understand that-and relatively few are doing much about it.In a survey, Principal Financial found that just 48% of American workers are monitoring their spending do from 58% two years ago. Fewer are reevaluating their investments, and while more are using a budget, the rate is still low at just 28%. In other words, they are casual about their money Most American workers see the link between wealth and health as one directional: Staying healthy is the ticket to saving on health care costs and to financial security in retirement. In the survey, 84% said being physically health is good for their financial future and 76% said if they spend money on their health or things they enjoy now they will avoid major health costs later in life.Yet it's not that simple. Good health comes with its own financial demands. Living healthy means living longer, which is wonderful. But without a traditional pension or other source of guaranteed lifetime income how will you pay for all those extra years? You'ii need to save tens of thousands of dollars more than you may have figured in your lifetime-and still there is no guarantee you won't need costly, end-of-life services.None of this is to say that living longer and healthier isnt a blessing, and probably even less a demand on your resources than developing chronic illness and passing away early. But the link between financial health and physical health goes both ways. In the survey, workers were far more likely to rate themselves physically healthy (53%) than financially healthy (31%), suggesting theysee it as a one-way street. But it's also the case that feeling in control of your wealth-no matter how much you have-leads to better health.56. In the first paragraph, the author expresses the belief thatA. Nothing but diet and exercise determines healthB. Diet and exercise have nothing to do with healthC. Health after retirement depends on money aloneD. Wealth also plays a role in staying healthy57. Which of the following statements is true about the workers surveyed?A. They don, t care about their retired life at allB. They believe wealth means a happy retired life.C. They are not in control of their moneyD. They have to save money now for future health58. In the last paragraph, "one-way street"probably means.A. Financial health leads to physical healthB. Physical health leads to financial healthC. financial health equals physical healthD. Financial health repels physical health59. The author is very likely to argue in the last paragraph thatA financial health and physical health are mutually dependentB neither wealth nor health makes any difference to happinesC. Physical health depends on nothing but financial healthD taking care of your wealth will not lead to better health60. The author of this passage intends to argue thatA staying healthy is impossible without accumulating much wealthB being physically healthy can eventually contribute to your wealthC feeling good about your finance is also of health benefitD. A large fortune will increase the chance of deterioration of healthPassage FourThe value a healthy environment provides is invisible in the global cconomy, leading to ecosystem degradation and species loss, according to the Economics of Ecosystems and Biodiversity (TEEB) study released recently. For example, bees and other insects jumping from flower to flower provide an essential service for crop production. Valued at an astonishing $200 billion.As the value is inconspicuous, we tend to ignore it. However, the"broken"financial system must be reformed o that the environment and the valuable services it provides are incorporated into every decision and ansaction, the report concludes. That is vital not only for slowing environmental destruction and species loss, but for modern economics to flourish, Sukhdev, leader of the study, said.While most cost enefit analyses look at natual resoures that can be extracted, such as gold, timber and food, those products only account for one-third of the total economic value provided by the environment, the report said, other "ecosystem services", such as pollination, forests fltering drinking water and wetlands providing flood protection, make up the other two-thirds.Currently, businesses do not pay for the loss of services caused by production or development. Estimated at 2.2 I trillion annually for the top 3, 000 listed companies, those costs are passed onto society, or externalized."We are at a stage now where the rate of loss of ecosystem services and the rate of loss of biodiversity is so severe we can not treat them as mere externalities of economics. Said Sukhde.To internalize environmental values and crops. The TEEB report recommends implementing a variety of financial tools, such as charging for services, creating environmentally friendly markets with eco-labelling and providing financial incentives and subsidies for environmentally friendly businesses.Normal business practices should report negative environmental extemalities and offset their impacts so they have a net zero impact, or even a net positive impact, the report said. And subsidies for industries harming the nvironment, such as fossil fuels, should be phased out.Environmental protection can also save money, according to the TEEB report. For example, New York more than S 6 billion by paying farmers about SI billion to change management of runoff to reduce the water pollution, rather than build a S 6 billion to S& billion waste water treatment plant that would have cost between S 300 million and $500 million a year to operate.But convincing board rooms and consumers across the globe to start paying for things that have been free in the pasl is L guing to be easy Sukhdev said it is not going to happen overnight--new, emerging models will begin to compete with old, lingering ones.“ it could happen, but not in today’s environment,”said Patrick micbaels, a senior fellow in environmental studies at the cato institude. "Right now, people-and it's not just the U.S--people are worried about economic contractions more than they are about environmental protection.”61. The example of bees and other insects in the first paragraph is used to illustrate thatA. Those insects play an essential role in crop productionB. They are essential for the global economy as well as the environmentC a healthy environment provides inconspicuous value in the global economyD insect activities have a crucial impact on global ecosystem and biodiversity62. Why must the"broken" financial system be reformed according to the report?A. The value environment provides should become visible in the global economy.B. Ecosystem services should be considered when making decisions in businessC. The current economic value provided by the environment tends t be ignoredD. To reform the "broken "financial system is vital to slow ecosystem degradation63. What do we learn about the loss of ecosystem services?A. It can be seen as an externality of economicsB. It is estimated at S 2.2 trillion every yearC. It is transferred to society or externalizedD. It is caused and paid for by businesses.64. What does the TEEB report suggest on handling environmental values and costs?A. Carrying out a succession of financial measures.B. Punishing farmers for changing management of runoff.C. Providing financial support for both businesses and farmersD. Increasing subsidies for environmentally disabled industries65. What can we know from what Patrick Michaels said in the last paragrap?A. People are worried more about economy than the environment.B. Nowadays it is hard for people to pay for what used to be freeC New emerging models will take the place of old ones in the endD. Right nowz the global economy is prosperous rather than depressedPart IV Put the following into Chinese. (15 points)My goals in writing this book are twofold.First, I hope to paint a picture of what it was like to serve as president for eight straight years. My conviction is that it is virtually impossible to reach definite conclusions about my presidency for several decades. The passage time allows passions) l, results to clarify, and scholars to compare different approaches. This book contribute to studies on this period in American history.Second, I gave an exhaustive account of the options I weighed and principles I complied with to give you better sense of why I made the decisions I did. This book is primarily based on my recollections. If there inaccuracies, the responsibility is mine.Part V Essay Writing (25 points)Write a composition of about 200 words on the following topic:My Viewpoint on Wide-spread WechatYou are to write in three parts.In the first part, make a description of the popularity of Wechat with examplesIn the second part, illustrate your views on the advantage and disadvantage of Using Wechat In the last part, bring what you have written to a natural conclusion or a summaryMarks will be awarded for content, organization, grammar and appropriateness, Failure to follow the instructions。

2011年秋季工学硕士研究生学位课程（数值分析）真题试卷A(总分28,考试时间90分钟)1. 填空题填空题请完成下列各题，在各题的空处填入恰当的答案。

1. 已知x1=0．724，x2=1．25均为有效数，则｜er(x1x2)｜≤______｜e(x1／x2)｜≤_______．2. 设A=则‖A‖∞=______，cond(A)2=______．3. 超定方程组的最小二乘解为x1=______.x2=_______．4. 用Simpson公式计算积分的近似值为______．5. 设是以0，1，2为节点的三次样条函数，则a=_____，b=_____．2. 计算题1. 给定方程ex-x-2=0，分析此方程有几个实根，并用迭代法求此方程的正根，精确至3位有效数字．2. 用列主元Guass消去法求下列线性方程组的解：3. 给定求解线性方程组Ax=b的迭代格式Bx(k+1)+ωCxk=b，其中试确定ω的值使上述迭代格式收敛．3. 综合题1. 作一个3次多项式H(x)，使得H(a)=b3，H(b)=a3，H"(a)=6b，H"(b)=6a．2. 求函数y(x)=x4在区间[0，1]上的一次最佳一致逼近多项式p(x)．3. 已知函数f(x)∈C4[a，b]，I(f)=∫abf（x）dx1)写出以a，b为二重节点所建立的f(x)的3次Hermite插值多琐式H(x)及插值余项；2)根据f(x)≈H(x)建立一个求解I(f)的数值求积公式IH(x)，并分析该公式的截断误差和代数精度．4. 给定常微分方程初值问题取正整数n，并记h=(b—a)／n，xi=a+ih，0≤i≤n．试确定参数A，B，C，使求解公式yi+1=Ayi+(1-A)yi-1+h[Bf(xi+1，yi+1)+Cf(xi，yi)]的局部截断误差Ri+1的阶数达到最高，指出所达剑的最高阶数并给出局部截断误差表达式．5. 给定如下抛物方程初边值问题：取步长用古典隐格式计算u(x，t)在点处的近似值．。

2009级研究生《数值分析》试卷一.(6分) 已知描述某实际问题的数学模型为xy y x y x u 223),(+=，其中，y x ,由统计方法得到，分别为4,2==y x ,统计方法的误差限为0.01,试求出u 的误差限)(u ε和相对误差限)(u r ε.二.(6分) 已知函数13)(3+=x x f 计算函数)(x f 的2阶均差]2,1,0[f ,和4阶均差]4,3,2,1,0[f .三.(6分)试确定求积公式: )]1(')0('[121)]1()0([21)(10f f f f dx x f -++≈⎰的代数精度.四.(12分) 已知函数122)(23-++=x x x x f 定义在区间[-1,1]上,在空间},,1{)(2x x Span x =Φ上求函数)(x f 的最佳平方逼近多项式.其中,权函数1)(=x ρ,154))(),((,1532))(),((,34))(),((210-==-=x x f x x f x x f ϕϕϕ.五.(16分) 设函数)(x f 满足表中条件:(1) 填写均差计算表(标有*号处不填):(2) 分别求出满足条件)2,1,0(),()(),()(22===k x f x N x f x L k k k k 的 2次 Lagrange 和 Newton 差值多项式.(3) 求出一个四次插值多项式)(4x H ,使其满足表中所有条件.并用多项式降幂形式表示. 六.(16分)(1). 用Romberg 方法计算⎰31dx x ,将计算结果填入下表(*号处不填).(2). 试确定三点 Gauss-Legender 求积公式⎰∑-=≈112)()(k k k x f A dx x f 的Gauss 点k x 与系数k A ,并用三点 Gauss-Legender 求积公式计算积分: ⎰31dx x .七.(14分)(1) 证明方程02ln =--x x 在区间(1,∞)有一个单根.并大致估计单根的取值范围. (2) 写出Newton 迭代公式,并计算此单根的近似值.(要求精度满足: 5110||-+<-k k x x ).八. (12分) 用追赶法求解方程组:⎪⎪⎪⎪⎪⎭⎫ ⎝⎛=⎪⎪⎪⎪⎪⎭⎫ ⎝⎛⎪⎪⎪⎪⎪⎭⎫ ⎝⎛022112111131124321x x x x 的解.九. (12分) 设求解初值问题⎩⎨⎧==00)(),('y x y y x f y 的计算格式为:)],(),([111--+++=n n n n n n y x bf y x af h y y ,假设11)(,)(--==n n n n y x y y x y ,试确定参数b a ,的值,使该计算格式的局部截断误差为二阶,即截断部分为: )(3h o .2008年春季学期数值数学试题一．（10分）设给实数0a >，初值00x >：⑴试建立求1a的Newton 迭代公式，要求在迭代函数中不含除法运算； ⑵证明给定初值0x ，迭代收敛的充分必要条件为020x a<<；⑶该迭代的收敛速度是多少？⑷取00.1x =，计算15的近似值，要求计算迭代三次的值（结果保留5位小数）。