Pub Date : 2020-02-20DOI: 10.4324/9780367821968-20
N. Newton
of forward Euler To solve dy dt = f (y, t), y(t 0) = y 0 numerically we choose a step length h > 0 and then calculate y 1 = y 0 + hf (y 0 , t 0), t 1 = t 0 + h y 2 = y 1 + hf (y 1 , t 1), t 2 = t 1 + h y 3 = y 2 + hf (y 2 , t 2), t 3 = t 2 + h etc. in which case y i is an approximation to y(t i) at each step. The approximation should get more accurate as h gets smaller. The error at each step will get worse if d 2 y/dt 2 is large and these errors can build up for longer time intervals. 1. We would like to to solve the differential equation dy dt = −10(t − 1)y with the given initial condition: y(0) = e −5. A reference solution to this problem can be plotted in MATLAB by typing the command sequence tt=0:0.01:2; ex=exp(-5*(tt-1).^2);plot(tt,ex,'-k')} (a) Find the analytic solution y(t), and evaluate it at (b) Compute the forward Euler solution by taking seven steps of the method with a step length of h = 0.2. (You will need a calculator to do this.) Generate a table which compare the analytic solution with the numerical solution. At which time t do you find the biggest error ? (c) Compute a more accurate forward Euler solution by taking four steps with a much smaller step length of h = 0.05. Compare this solution with the exact solution y(0.2) and compare the accuracy of this solution with the first time step result obtained in (b). Note that this differential equation is not as benign as it looks; if one wants to solve it over a long time interval, sophisticated numerical methods (like those that are built into MATLAB) are needed. A simple example of a sophisticated method is the implicit Euler method which will be discussed in the lectures. Solution 1. Analytical solution is y(t) = exp(−5(t − 1) 2) Plot of exact solution 1
向前欧拉解决dy dt = f (y, t), y y (t 0) = 0数字我们选择步长h > 0然后计算y 1 = y 0 +高频t (0, 0), t 1 = 0 + h 2 = y 1 +高频(y 1 t 1), t h t 1 + 2 = 3 = y 2 +高频(y 2, t 2), t t 2 + 3 = h等在这种情况下,我是一个近似y (t i)在每一个步骤。当h变小时，近似会变得更精确。如果dy /dt很大，每一步的误差会更大，这些误差会在更长的时间间隔内累积。1. 我们想解微分方程dy dt = - 10(t - 1)y，初始条件为y(0) = e - 5。在MATLAB中可以通过输入命令序列tt= 0:01:2来绘制此问题的参考解;plot(tt,ex，'-k')} (a)找到解析解y(t)，并在(b)处求值。通过步长为h = 0.2的方法进行7步计算前向欧拉解。(你需要一个计算器来做这个。)生成一个比较解析解和数值解的表格。在什么时候你发现最大的错误?(c)采用步长h = 0.05小得多的四步，计算出更精确的正演欧拉解。将该解与精确解y(0.2)进行比较，并将该解的准确性与(b)中获得的第一个时间步结果进行比较。注意，这个微分方程并不像看起来那么温和;如果想在很长一段时间内解决它，就需要复杂的数值方法(比如MATLAB中内置的那些)。一个简单的复杂方法的例子是隐式欧拉法，这将在讲座中讨论。解决方案1。解析解为y(t) = exp(- 5(t - 1) 2)
{"title":"Week 20","authors":"N. Newton","doi":"10.4324/9780367821968-20","DOIUrl":"https://doi.org/10.4324/9780367821968-20","url":null,"abstract":"of forward Euler To solve dy dt = f (y, t), y(t 0) = y 0 numerically we choose a step length h > 0 and then calculate y 1 = y 0 + hf (y 0 , t 0), t 1 = t 0 + h y 2 = y 1 + hf (y 1 , t 1), t 2 = t 1 + h y 3 = y 2 + hf (y 2 , t 2), t 3 = t 2 + h etc. in which case y i is an approximation to y(t i) at each step. The approximation should get more accurate as h gets smaller. The error at each step will get worse if d 2 y/dt 2 is large and these errors can build up for longer time intervals. 1. We would like to to solve the differential equation dy dt = −10(t − 1)y with the given initial condition: y(0) = e −5. A reference solution to this problem can be plotted in MATLAB by typing the command sequence tt=0:0.01:2; ex=exp(-5*(tt-1).^2);plot(tt,ex,'-k')} (a) Find the analytic solution y(t), and evaluate it at (b) Compute the forward Euler solution by taking seven steps of the method with a step length of h = 0.2. (You will need a calculator to do this.) Generate a table which compare the analytic solution with the numerical solution. At which time t do you find the biggest error ? (c) Compute a more accurate forward Euler solution by taking four steps with a much smaller step length of h = 0.05. Compare this solution with the exact solution y(0.2) and compare the accuracy of this solution with the first time step result obtained in (b). Note that this differential equation is not as benign as it looks; if one wants to solve it over a long time interval, sophisticated numerical methods (like those that are built into MATLAB) are needed. A simple example of a sophisticated method is the implicit Euler method which will be discussed in the lectures. Solution 1. Analytical solution is y(t) = exp(−5(t − 1) 2) Plot of exact solution 1","PeriodicalId":363854,"journal":{"name":"Day-by-Day Math Thinking Routines in Kindergarten","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124157941","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-02-20DOI: 10.4324/9780367821968-17
N. Newton
{"title":"Week 17","authors":"N. Newton","doi":"10.4324/9780367821968-17","DOIUrl":"https://doi.org/10.4324/9780367821968-17","url":null,"abstract":"","PeriodicalId":363854,"journal":{"name":"Day-by-Day Math Thinking Routines in Kindergarten","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116364159","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-02-20DOI: 10.4324/9780367821968-16
N. Newton
{"title":"Week 16","authors":"N. Newton","doi":"10.4324/9780367821968-16","DOIUrl":"https://doi.org/10.4324/9780367821968-16","url":null,"abstract":"","PeriodicalId":363854,"journal":{"name":"Day-by-Day Math Thinking Routines in Kindergarten","volume":"65 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126483541","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-02-20DOI: 10.1142/9781938134234_0013
N. Newton
1. Which of the following requires object database technology? a. Storing employee picture on disk b. Retrieving an employee picture c. Retrieving and processing an employee picture together with simpler data about employee d. Displaying an employee picture on a screen. 2. Which of the following supports object oriented concepts? a. SQL 1 b. SQL 2 c. SQL 3 d. All of the above 3. Object in pure object oriented languages hide the implementation details, e.g. attributes and internal procedures, from external world. This properties is called: a. Abstraction b. Encapsulation c. Inheritance d. Polymorphism 4. Which of the object concepts are usually not implemented in today’s object relational DBMS because of the serious performance concerns? a. Data abstraction b. Encapsulation c. Inheritance d. Methods (i.e. procedures) 5. Which of the following do not apply to today’s object relational DBMS? a. Calls to functions that process media data are part of the database language b. Specialized storage structures are provided c. Reliability is of no concern d. Combined access to complex data and relational data is optimized
1. 下列哪项需要对象数据库技术?a.将员工图片保存到磁盘b.提取员工图片c.将员工图片与简单的员工数据一起提取和处理d.将员工图片在屏幕上显示。2. 下面哪一个支持面向对象的概念?a. SQL 1 b. SQL 2 c. SQL 3 d.以上所有3。对象在纯面向对象语言中隐藏了实现细节，例如属性和内部过程，不让外界看到。这些属性被称为:a.抽象b.封装c.继承d.多态性4。由于严重的性能问题，在当今的对象关系DBMS中，哪些对象概念通常没有实现?a.数据抽象b.封装c.继承d.方法(即过程)下列哪一项不适用于今天的对象关系DBMS?a.处理媒体数据的函数调用是数据库语言的一部分b.提供了专门的存储结构c.不考虑可靠性d.优化了复杂数据和关系数据的组合访问
{"title":"Week 13","authors":"N. Newton","doi":"10.1142/9781938134234_0013","DOIUrl":"https://doi.org/10.1142/9781938134234_0013","url":null,"abstract":"1. Which of the following requires object database technology? a. Storing employee picture on disk b. Retrieving an employee picture c. Retrieving and processing an employee picture together with simpler data about employee d. Displaying an employee picture on a screen. 2. Which of the following supports object oriented concepts? a. SQL 1 b. SQL 2 c. SQL 3 d. All of the above 3. Object in pure object oriented languages hide the implementation details, e.g. attributes and internal procedures, from external world. This properties is called: a. Abstraction b. Encapsulation c. Inheritance d. Polymorphism 4. Which of the object concepts are usually not implemented in today’s object relational DBMS because of the serious performance concerns? a. Data abstraction b. Encapsulation c. Inheritance d. Methods (i.e. procedures) 5. Which of the following do not apply to today’s object relational DBMS? a. Calls to functions that process media data are part of the database language b. Specialized storage structures are provided c. Reliability is of no concern d. Combined access to complex data and relational data is optimized","PeriodicalId":363854,"journal":{"name":"Day-by-Day Math Thinking Routines in Kindergarten","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127243376","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-02-20DOI: 10.4324/9780367821968-33
N. Newton
{"title":"Week 33","authors":"N. Newton","doi":"10.4324/9780367821968-33","DOIUrl":"https://doi.org/10.4324/9780367821968-33","url":null,"abstract":"","PeriodicalId":363854,"journal":{"name":"Day-by-Day Math Thinking Routines in Kindergarten","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131038374","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Week 5","authors":"N. Newton","doi":"10.2307/j.ctv550c75.11","DOIUrl":"https://doi.org/10.2307/j.ctv550c75.11","url":null,"abstract":"","PeriodicalId":363854,"journal":{"name":"Day-by-Day Math Thinking Routines in Kindergarten","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126803877","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-02-20DOI: 10.4324/9780367821968-39
N. Newton
{"title":"Week 39","authors":"N. Newton","doi":"10.4324/9780367821968-39","DOIUrl":"https://doi.org/10.4324/9780367821968-39","url":null,"abstract":"","PeriodicalId":363854,"journal":{"name":"Day-by-Day Math Thinking Routines in Kindergarten","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128574422","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-02-20DOI: 10.4324/9780367821968-28
N. Newton
{"title":"Week 28","authors":"N. Newton","doi":"10.4324/9780367821968-28","DOIUrl":"https://doi.org/10.4324/9780367821968-28","url":null,"abstract":"","PeriodicalId":363854,"journal":{"name":"Day-by-Day Math Thinking Routines in Kindergarten","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123295968","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-02-20DOI: 10.1142/9781938134234_0011
N. Newton
{"title":"Week 11","authors":"N. Newton","doi":"10.1142/9781938134234_0011","DOIUrl":"https://doi.org/10.1142/9781938134234_0011","url":null,"abstract":"","PeriodicalId":363854,"journal":{"name":"Day-by-Day Math Thinking Routines in Kindergarten","volume":"266 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131860933","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-02-20DOI: 10.1142/9781938134234_0006
N. Newton
{"title":"Week 6","authors":"N. Newton","doi":"10.1142/9781938134234_0006","DOIUrl":"https://doi.org/10.1142/9781938134234_0006","url":null,"abstract":"","PeriodicalId":363854,"journal":{"name":"Day-by-Day Math Thinking Routines in Kindergarten","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127667889","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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