"Numerical Analysis" Course Design
Require:
(1) A group of 3 people will do a design topic, and do the following designs in turn;
(2) Each group selects one student to participate in the defense. Those who fail to pass the defense will be regarded as failing grades;
(3) During the course design period, attendance records are strictly implemented, and students are required to go to designated classrooms;
(4) Submit course design papers in strict accordance with the requirements of course design;
(5) Papers are collected by class before 10:00 a.m. on the Friday of the twelfth week and handed over to 631 of the Basic Laboratory Building. The defense will be held at 10:05 a.m. on Friday the twelfth week.
(Complete course design and ppt are not shared, private chat is required)
Topic (1)
1. Consider the two-point boundary value problem
It is easy to know that its exact solution is
In order to discretize the differential equation, divide the interval equally, let , get the differential equation
Simplified to
Therefore, the coefficient matrix of the linear equation system obtained after discretization is
Yes,,, respectively use the hyper-relaxation iterative method of , , and , to solve the linear equation system, requiring 4 significant figures, then compare the error with the exact solution, explore the value that makes the hyper-relaxation iterative method converge faster, and analyze the results . Change, discuss the same issue.
Topic (2)
2. First use the computer language you are familiar with to write the non-selected pivot, column pivot and complete pivot Gauss elimination method into a general subroutine, and then use the program you wrote to solve the following equations (considering from 120 to 130 )
=；
What other methods can be used to solve the above equations? Choose some of these methods to program the computer to solve the above equations, explain which method is the most suitable; compare and analyze the calculation results, and talk about your views on these methods.
Topic (3)
3. Linear model in real estate valuation
Among them, respectively represent the tax, the number of bathrooms, the floor area, the living area, the number of garages, the number of houses, the number of bedrooms, the age of the house, the type of building, the size of the apartment and the number of fireplaces, representing the price of the house. Now according to the following 28 sets of data, the least square results of the parameters in the model are obtained.
| 25.9 |
29.5 |
27.9 |
25.9 |
29.9 |
29.9 |
30.9 |
| 28.9 |
84.9 |
82.9 |
35.9 |
31.5 |
31.0 |
30.9 |
| 30.0 |
28.9 |
36.9 |
41.9 |
40.5 |
43.9 |
37.5 |
| 37.9 |
44.5 |
37.9 |
38.9 |
36.9 |
45.8 |
41.0 |
| 4.9176 |
1.0 |
3.4720 |
0.9980 |
1.0 |
7 |
4 |
42 |
3 |
1 |
0 |
| 5.0208 |
1.0 |
3.5310 |
1.5000 |
2.0 |
7 |
4 |
62 |
1 |
1 |
0 |
| 4.5429 |
1.0 |
2.2750 |
1.1750 |
1.0 |
6 |
3 |
40 |
2 |
1 |
0 |
| 4.5573 |
1.0 |
4.0500 |
1.2320 |
1.0 |
6 |
3 |
54 |
4 |
1 |
0 |
| 5.0597 |
1.0 |
4.4550 |
1.1210 |
1.0 |
6 |
3 |
42 |
3 |
1 |
0 |
| 3.8910 |
1.0 |
4.4550 |
0.9880 |
1.0 |
6 |
3 |
56 |
2 |
1 |
0 |
| 5.8980 |
1.0 |
5.8500 |
1.2400 |
1.0 |
7 |
3 |
51 |
2 |
1 |
1 |
| 5.6039 |
1.0 |
9.5200 |
1.5010 |
0.0 |
6 |
3 |
32 |
1 |
1 |
0 |
| 15.4202 |
2.5 |
9.800 |
3.4200 |
2.0 |
10 |
5 |
42 |
2 |
1 |
1 |
| 14.4598 |
2.5 |
12.8000 |
3.000 |
2.0 |
9 |
5 |
14 |
4 |
1 |
1 |
| 5.8282 |
1.0 |
6.4350 |
1.2250 |
2.0 |
6 |
3 |
32 |
1 |
1 |
0 |
| 5.3003 |
1.0 |
4.9883 |
1.5520 |
1.0 |
6 |
3 |
30 |
1 |
2 |
0 |
| 6.2712 |
1.0 |
5.5200 |
0.9750 |
1.0 |
5 |
2 |
30 |
1 |
2 |
0 |
| 5.9592 |
1.0 |
6.6660 |
1.1210 |
2.0 |
6 |
3 |
32 |
2 |
1 |
0 |
| 5.0500 |
1.0 |
5.0000 |
1.0200 |
0.0 |
5 |
2 |
46 |
4 |
1 |
1 |
| 5.6039 |
1.0 |
9.5200 |
1.5010 |
0.0 |
6 |
3 |
32 |
1 |
1 |
0 |
| 8.2464 |
1.5 |
5.1500 |
1.6640 |
2.0 |
8 |
4 |
50 |
4 |
1 |
0 |
| 6.6969 |
1.5 |
6.0920 |
1.4880 |
1.5 |
7 |
3 |
22 |
1 |
1 |
1 |
| 7.7841 |
1.5 |
7.1020 |
1.3760 |
1.0 |
6 |
3 |
17 |
2 |
1 |
0 |
| 9.0384 |
1.0 |
7.8000 |
1.5000 |
1.5 |
7 |
3 |
23 |
3 |
3 |
0 |
| 5.9894 |
1.0 |
5.5200 |
1.2560 |
2.0 |
6 |
3 |
40 |
4 |
1 |
1 |
| 7.5422 |
1.5 |
4.0000 |
1.6900 |
1.0 |
6 |
3 |
22 |
1 |
1 |
0 |
| 8.7951 |
1.5 |
9.8900 |
1.8200 |
2.0 |
8 |
4 |
50 |
1 |
1 |
1 |
| 6.0931 |
1.5 |
6.7265 |
1.6520 |
1.0 |
6 |
3 |
44 |
4 |
1 |
0 |
| 8.3607 |
1.5 |
9.1500 |
1.7770 |
2.0 |
8 |
4 |
48 |
1 |
1 |
1 |
| 8.1400 |
1.0 |
8.0000 |
1.5040 |
2.0 |
7 |
3 |
3 |
1 |
3 |
0 |
| 9.1416 |
1.5 |
7.3262 |
1.8310 |
1.5 |
8 |
4 |
31 |
4 |
1 |
0 |
| 12.0000 |
1.5 |
5.0000 |
1.2000 |
2.0 |
6 |
3 |
30 |
3 |
1 |
1 |
Topic (4)
4. In the design of sewage pipelines, engineers are concerned with the relationship between the slope of the pipeline, the diameter of the pipe, and the flow of sewage. For circular cross-section pipes, there is the following empirical formula between these quantities:
Among them, Q represents the flow rate (), S represents the slope of the pipeline (m/m), and D represents the diameter of the round pipe (m).
An empirical parameter determined through experiments. There is a set of experimental data as follows:
| Experiment number |
S |
||
| 1 2 3 4 5 6 7 8 9 |
0.302 0.604 0.906 0.302 0.604 0.902 0.302 0.604 0.906 |
0.001 0.001 0.001 0.01 0.01 0.01 0.05 0.05 0.05 |
0.0385 0.2283 0.6655 0.1293 0.7948 2.3100 0.3053 1.8975 5.5000 |
Use appropriate numerical methods to find
Knowledge involved - least squares fitting.
Topic (5)
5. First use the computer language you are familiar with to write the square root method and the catch-up method into a general subroutine, and then use the program you wrote to solve the linear equation system, among which the coefficient matrix is randomly selected, and the coefficient matrix is a matrix of order 120 (symmetrical positive definite)
What other methods can be used to solve the above equations? Then select some of the methods to solve the above equations; compare and analyze the calculation results.
Topic (6)
6. Use a familiar computer language to program on the computer to complete
（1）用复化梯形公式、复化Simpson公式和复化Cotes公式计算积分，自己设置不同精度要求，对结果进行比较分析。
（2）用Romberg积分法计算积分，自己设置不同精度要求，对结果进行比较分析。
（3）记，在上面的计算中只取4位有效数字或7位有效数字，计算结果又有什么不同。
（4）上面计算精度可达8-20位有效数字吗？若可以，请说明实现过程，并举例。
题目（七）
7、给定单摆方程初值问题
其中g=9.8,l=25.
- 取初始偏离角度
- 取初始偏离角度
其精确解为。分别对上述两种情况按照下列方法求出其数值解，比较各方法的优缺点，并将计算结果与精确解做比较（列表、画图）。
（方案I）欧拉法，步长h = 0.025, h = 0.1；
（方案II）改进的欧拉法，步长h = 0.05, h = 0.1；
（方案III）四阶经典龙格—库塔法，步长h = 0.1。
题目（八）
8、用熟悉的计算机语言编程上机完成
（1）用Newton-Cotes公式计算积分的近似值，自己设置不同精度要求，对结果进行比较分析。
（2）用Romberg积分法计算积分的近似值，自己设置不同精度要求，对结果进行比较分析；与（1）的结果进行比较分析，谈谈你的体会。
（3）记，在上面的计算中只取4位有效数字或7位有效数字，计算结果有什么不同。
（4）上面计算精度可达8-20位有效数字吗？若可以请说明实现过程，并举例。
题目（九）
9、土木工程和环境工程师在设计一条排水渠道时必须考虑渠道的各种参数（如宽度，深度，渠道内壁光滑度）及水流速度、流量、水深等物理量之间的关系。
假设修一条横断面为矩形的水渠，其宽度为B，假定水流是定常的，也就是说水流速度不随时间而变化。
根据质量守恒定律可以得到
Q=UBH （1.1）
其中Q 是水的流量（），U是流速（），H是水的深度（）。
在水工学中应用的有关流速的公式是
（1.2）
这里n是Manning粗糙系数，它是一个与水渠内壁材料的光滑性有关的无量纲量；S是水渠的斜度系数，也是一个无量纲量，它代表水渠底每米内的落差。
把（1.2）代入（1.1）就得到
（1.3）
为了不同的工业目的（比如说要把污染物稀释到一定的浓度以下，或者为某工厂输入一定量的水），需要指定流量Q和B，求出水的深度。这样，就需要求解
（1.4）
一个具体的案例是
求出渠道中水的深度H。
所涉及的知识——非线性方程解法。
题目（十）
10、用熟悉的计算机语言编程上机完成
（1）用步长自动减半的复化梯形公式、复化Simpson公式和复化Cotes公式计算积分，自己设置不同精度要求，对结果进行比较分析。
（2）用步长不自动减半的（即一般的）复化梯形公式、复化Simpson公式和复化Cotes公式计算积分，自己设置不同精度要求，对结果进行比较分析；与（1）的结果进行比较。
（3）记，在上面的计算中只取4位有效数字或7位有效数字，计算结果有什么不同。
（4）上面计算精度可达8-20位有效数字吗？若可以请说明实现过程，并举例。
题目（十一）
11、某城市在1900-1990年间，每隔10年统计一次该城市的人口数量，得到结果如下（单位为：百万）：
| x（年） |
1900 |
1910 |
1920 |
1930 |
1940 |
1950 |
1960 |
1970 |
1980 |
1990 |
| Y（人数） |
75.995 |
91.972 |
105.711 |
123.203 |
131.669 |
150.697 |
179.323 |
203.212 |
226.505 |
249.633 |
分别使用不同的插值方法，对没有进行人口统计的年限的人口数量进行预测，并画出效果图。
题目（十二）
12、船舶邦戎曲线是由一组船舶横剖面的面积曲线组成的，其中每条曲线表示该处横剖面在不同水线以下浸入水中的面积。邦戎曲线是船体纵向积分的基础，利用它可以计算船舶在不同吃水下的排水体积和浮心位置，进而为船舶的稳性与强度计算提供基本数据。因而邦戎曲线的精确性直接影响到船舶的安全性。传统邦戎曲线的计算以型值表为基础，利用梯形积分法，把船体某一横截面上各水线之间的面积近似成梯形，然后把这些小梯形的面积求和得到，但梯形法只有一阶代数精度，对稳性计算要求较高的液货船来说，似乎不够精确。利用表1中的数据，采用三次样条插值法构造精度更高的计算船舶邦戎曲线的数值方法，并绘制船舶的邦戎曲线。
题目（十三）
13、选用Jacobi迭代法、Gauss-Seidel迭代法和超松弛迭代法求解下面的方程组（考虑从200到220）
=；
考虑初值的变化和松弛因子的变化收敛效果的影响；对上述方程组还可以采用哪些方法求解？选择其中一些方法编程上机求解上述方程组，说明最适合的是什么方法；将计算结果进行比较分析，谈谈你对这些方法的看法。
题目（十四）
14、先用你所熟悉的计算机语言将平方根法和改进的平方根法编写成通用的子程序，然后用你编写的程序求解线性方程组，其中
对上述方程组还可以采用哪些方法求解？再选用其中一些方法求解上述方程组；对计算结果进行比较分析。