I.The Basic Information of the Course
Course Number:202032023009
The EnglishName of theCourse：Scientific andEngineeringComputing
The ChineseName of theCourse：科学与工程计算
In-classHours andAllocationTotal class hours: 48, 40 in class, 8 in class discussion and on computer
Credit(s):3
Semester:1
AppliedDiscipline(Professional Degree Category)：Engineering Discipline
CourseObjectOriented：Doctor
EvaluationMode:Sample:Open-book Examination + Major Assignments
TeachingMethod:Blended Teaching
CourseOpening Department:School of Mathematical Sciences
Notes:Filling explanation to applied discipline: the general courses like foreign languages, ideological and political theory course may fill in “AllDisciplines”,Mathematics general course may fill inScience andEngineeringDiscipline, and the other courses fill in “Name ofApplicableDiscipline” according to actual teaching objects, and the number of applicable discipline may be more than one.
II.Prerequisite Course
NumericalComputation, Linear Algebra, Differential Equation, Fundamentals of Computer.
III. The Objectives and Requirements of the Course
The Objectives of the Course：
"Scientific and engineering computing" is the numerical approximate solution method of the differential equation mathematical model or statistical model when it appears in engineering and scientific research. Through the study of this course, you can understand the basic principles of scientific computing, master the scientific computing methods and theories of some typical problems, have the ability to solve general scientific computing problems on the computer, and lay the foundation for subsequent scientific research.
Through the course study, you can understand the latest scientific research results related to the course at home and abroad, be familiar with the relevant theories and applications of numerical analysis, scientificcomputation, and matrixcomputation. At the end of the course, we will focus on the numerical solution of differential equation mathematical models or statistical models that appear in engineering and scientific research.
The Requirements of the Course：
(1) Be familiar with and master the basic concepts and theories of numerical approximation, numerical algebra and numerical solution of ordinary differential equation in numerical analysis theory; focus on some classical numericalcomputationmethods, such as interpolation, function approximation, numerical integration,linear equations solution and non-linear equations solution.
(2) Master several standard methods of matrixcomputation, such as matrix sequence, matrix series and matrix computes, matrix factorization methods and applications, such as LU factorization, QR factorization, singular value factorization, etc.; understand the matrix eigenvalues estimation and characterization, such as eigenvalue bound estimation and included areas, etc.; understand the Krylov subspace method, familiar with the steepest descent method, conjugate gradient method, least squares method, etc.
(3) Understand the finite difference method, including understanding the general concept of partial differential equations, the difference format of the parabolic equation, the difference format of the hyperbolic equation, the difference format of the convection diffusion equation, the difference format of the elliptic equation; understand and be familiar with the finite element method, including understanding from function expansion to variational principle, one-dimensional finite element method, two-dimensional finite element method, Galerkin method and its extension, error estimation, etc.
(4) Master modern compression technology, including fast Fourier transform, triangle interpolation, discrete cosine transform.
(5) Final scores are assessed on the three levels of closed-book examination (50%), class discussion (20%) and major assignments (30%), and finally evaluated on a 100-point scale.
(6) Classroom discussions (accounting for 20%) mainly include the content of course ideology and politics. Teachers use the relevant content in our country since ancient times as a case guide to allow students to fully consult the materials and actively speak, using this part of the results to praise students' Patriotic feelings, while stimulating students' interest in learning.
IV. The Content of the Course
Part I: Related Theory of Numerical Analysis (10 class hours)
This part mainly includes the review and extension of the previous "numerical computation" course content, focusing on spline interpolation and B-spline interpolation, introducing functional and error analysis; adaptive integration method; computation of eigenvalue eigenvectors; cases 1.
Part 2: Related Theory of Matrix Computation (12 class hours)
The main content includes the basic problems and sources of matrix computation, ill-posed problems and numerical stability; several standard methods of basic tools for matrix computation, including multiple factorization forms of matrix and their applications, such as Schur factorization and singular value factorization, QR, QL factorization and standard orthonormalization process, Krylov subspace method and matrix eigenvalue problem, etc. Case 2.
Part 3: Numerical solution method of differential equation mathematical model (16 class hours)
The main contents of this part include the finite difference method and the finite element method, such as the general concept of partial differential equations, the difference format of the parabolic equation, the difference format of the hyperbolic equation, the difference format of the convection diffusion equation, the difference format of the elliptic equation; To the variational principle, one-dimensional finite element method, two-dimensional finite element method, Galerkin method and its extension, error estimation.
Part IV: Modern Engineering Numerical Computing Technology (10 class hours)
Compression is the core of numerical analysis, although it is often hidden in interpolation, least squares, and Fourier analysis. The main contents of this part include fast Fourier transform, triangular interpolation, discrete cosine transform and their applications, case 3.
Course Ideological Case Teaching:
Case 1: What is computational mathematics
Modern science and technology are developing very quickly. They have a common feature, that is, they all have a lot of data problems. For example, to launch a satellite to detect the mysteries of the universe, from the beginning of the satellite century to the launch and recovery, scientists, engineering technicians, and workers will have to comprehensively design and produce the overall satellite and components, and design the selected rocket. And production, there are a lot of data to be calculated accurately. When launching and recovering, there are precise calculations regarding launch angle, orbit, remote control, recovery fall angle, etc. As another example, high-energy physics experiments are conducted in high-energy accelerators to study the properties of elementary particles with high energy, their interactions and conversion laws. There are also a lot of data calculation problems. The calculation problem can be counted as a common problem in all fields of modern society. Industry, agriculture, transportation, medical care, culture and education, etc., which line and industry have a lot of data to be calculated, through data analysis, in order to grasp the development of things about the law. A discipline that studies the solution of computational problems and related mathematical theoretical problems is called computational mathematics. Computational mathematics belongs to the category of applied mathematics. It mainly studies how to solve related mathematical and logical problems effectively by computers.
Computational mathematics is also called numerical computation method or numerical analysis. The main content includes the numerical solution of algebraic equations, linear algebraic equations, differential equations, numerical approximation of functions, the solution of matrix eigenvalues, optimization computation problems, probability and statistical computation problems, etc., it also includes the existence, uniqueness, convergence and error analysis of the solution.
We know that there are no root-finding formulas for algebraic equations of degree five and above. Therefore, the solution to a higher-order algebraic equation of degree five or more is generally only an approximate solution to it. The method of seeking an approximate solution is numerical analysis. For general transcendental equations, such as logarithmic equations, triangular equations, etc., only numerical analysis can be used. How to find out the computation method which is more concise, less error and less time is the main subject of numerical analysis.
In the method of solving equations, one of the commonly used methods is the iterative method, also known as the successive approximation method. The computation of the iterative method is relatively simple and easy to carry out. Iterative methods can also be used to solve linear equations. To find the approximate solution of the system of equations, an appropriate iterative formula should also be selected, so that the convergence speed is fast and the approximation error is small.In the solution of linear algebraic equations, the Seidel iteration method, the conjugate slope method, the ultra-relaxation iteration method, etc. are commonly used. In addition, some relatively old common elimination methods, such as the Gaussian method and the chasing method, can also be widely used under the condition of using computers.
In the computation method, numerical approximation is also a commonly used basic method. Numerical approximation is also called approximate replacement, that is, a simple function is used to replace a more complex function, or a function that cannot be represented by an analytical expression. The basic method of numerical approximation is interpolation. The trigonometric function table in elementary mathematics and the correction value in the logarithm table are made according to the interpolation method.
When encountering differentiation and integration, how to use a simple function to approximate replace the given function in order to easily find and integrate is also a main content of the computation method. The numerical solution of differential equations is also an approximate solution. The numerical solutions of ordinary differential equations are Euler method, prediction correction method and so on. For the initial value problem or boundary value problem of partial differential equations, the finite difference method and the finite element method are commonly used at present.
The basic idea ofthe finite difference method is to replace the differential equations and definite solution conditions of continuous variables with discrete difference equations containing only a limited number of unknowns. Find the solution of the difference equation as the approximate solution of the partial differential equation.
The finite element method was developed in modern times, and it is based on the variational principle and the difference between the divisions. It has been widely used in solving elliptic equation boundary value problems. Many people are studying the finite element method to solve hyperbolic and parabolic equations.
Case 2: The contribution of ancient Chinese mathematicians to computational mathematics (Zu Chongzhi, Liu Hui)
Zu Chongzhi's outstanding achievement in mathematics is about the calculation of pi. Before the Qin and Han dynasties, people used "the diameter of a week" as the pi, which is the "old rate". Later, it was found that the error of the ancient rate is too large, and the pi should be "one round diameter and more than Wednesday", but there are different opinions on how much is left. Until the Three Kingdoms period, Liu Hui proposed a scientific method for calculating the pi-"cutting circle", which approximates the circumference by using the circumference of the regular polygon inscribed in the circle. Liu Hui calculated that 96 polygons were inscribed in the circle, andπ=3.14 was obtained, and pointed out that the more the number of sides inscribed in the regular polygon, the more accurate the value ofπobtained. Based on the achievements of his predecessors, Zu Chongzhi worked hard and repeated calculations to findπbetween 7.1415926 and 3.1415927. And the approximate value in the form ofπfraction is obtained, which is taken as the approximate rate and as the density rate, where the six decimal places is 3.141929, which is the fraction of the numerator denominator within 1000 that is closest to the value ofπ. What method Zu Chongzhi used to obtain this result cannot be investigated at present. If we envisage him to follow Liu Hui's "circumcision technique", he must calculate the 16.384 polygon inscribed in the circle. How much time and labor will be required! It can be seen that his tenacious perseverance and intelligence in academics are admirable. Zu Chongzhi's calculation of the density rate has been obtained by foreign mathematicians for more than a thousand years. To commemorate Zu Chongzhi's outstanding contributions, some foreign mathematics historians suggested thatπ= be called "Zu rate".
Case 3: The book of changes and the origin of Mathematics
Ancient Chinese mathematics developed from the formation of the "Nine Chapters Arithmetic" in the Han Dynasty to its peak in the Song and Yuan Dynasties, during which many important mathematical works appeared. During the Han and Tang Dynasties, there were "Ten Books of Mathematical Classics". In addition to "Nine Chapters of Arithmetic", there were also "Islandic Mathematical Classics", "Five Cao Calculation Classic ", "Sunzi Calculation Classic ", "Xia Houyang Calculation Classic", "Zhang QiuJian Calculation Classic ", "Five Classics" , "Conjure", "Ancient calculation classic", and "Legend of Mathematics"; there are four major mathematicians in the Song and Yuan Dynasties, Including Qin Jiushao's "Nine Chapters of Several Books", Li Ye wrote "The Round Mirror", Yang Hui wrote "Detailed Algorithms of Nine Chapters", "Yang Hui Algorithm", etc., and Zhu Shijie wrote "Si Yuan Yu Jian". After the Song and Yuan Dynasties, a famous mathematician Cheng Dawei wrote the "Algorithm of Algorithms" in the Ming Dynasty, etc. Although Zhouyi is not a special mathematics book, it is an important ancient book, which has been passed down to the world and has an important influence on ancient Chinese mathematics. The study from the "Nine Chapters of Arithmetic" in the Han Dynasty to the Song and Yuan Dynasties and the development of mathematics in the Ming Dynasty can be seen that most of the important mathematics works that have been around for a long time still have traces of the influence of the "Book of Changes".Qin Jiushao, the word is ancient. He was born in the first year of Jiading in the Southern Song Dynasty (1208); he died in Meizhou in the second year of 1261, an Ancient Chinese mathematician.His representative works include "Nine Chapters of Mathematics", and created important mathematical methods such as Dayan seeking a skill, three oblique quadrature, Qin Jiu Shao algorithm and so on.
Dayan seeking a skill: Dayan's problem stems from the problem of "the number of things is unknown" in "The Grandson": "There are things today, the number is unknown, two of the three and three are left, three of the five and five are left, seven and seven What is the remaining number two, and ask the geometry of things?" This is a problem of solving a system of congruence equations in modern number theory. Song mathematician Qin Jiushao made a systematic discussion on the solution of such problems in "Several Books and Nine Chapters" (completed in 1247), and called it "Dayan seeking a skill".
Three oblique quadrature technique: Qin Jiushao, a famous mathematician in my country, proposed the "three oblique quadrature technique" in "Nine Chapters of Several Books". He called the three sides of the triangle respectively small oblique, medium oblique and large oblique. "Surgery" is the method. Tri-slope quadrature is to add the small slanted square to the large slanted square and send it to the slanted square, take half of the remainder after the subtraction, and multiply it to get a number. After subtracting the remainder, the number obtained by dividing the remainder by 4 is "real", and 1 is regarded as "corner", and the area is obtained after square rooting.
Zhu Shijie and Quaternary Technique, Zhu Shijie (1249-1314), Han Qing, Han nationality, Yanshan (now Beijing) surname, mathematician and educator of the Yuan Dynasty, engaged in mathematics education all his life. He has the reputation of "the greatest mathematician in the medieval world". He developed the "Quaternary Technique" on the basis of the Tianyuan technique at that time, that is, listing the quaternary high-degree polynomial equations and the method of solving the elimination. In addition, he also created the "stacking method", which is the summation method of the higher-order arithmetic sequence, and the "stroke difference technique", that is, the higher-order interpolation method. The main works are "Enlightenment of Mathematics" and "Si Yuan Yu Jian".
In the Yuan Dynasty, the famous mathematician Zhu Shijie's "Four Yuan Yu Jian" discussed the problems of solving multivariate higher-order equations, and was called by the American science historian George Sarton (G. Sarton). The most important Chinese mathematics book, and one of the most outstanding mathematics books in the Middle Ages. The "Preface" of the book was written by his friend Mo Ruo, which said: "It's just one, all the things are from the beginning. One and two, two and four, four and eight. Is the poor the number that is natural? The book reveals its secrets, the book of the "Nine Chapters of the Yellow Emperor" has nine chapters, and its technique is six hundred and forty-four. Including three talents, there are many bypasses." It is also believed that the earliest mathematics originated from the book.
The "Algorithm of Algorithm", written by the mathematician Cheng Dawei at the end of the Ming Dynasty, is an abacus book and has a long history. Nine Palaces and Eight Diagrams.Among them, "General Talk" said: "What counts? It's from pictures and books! Fu Xi gets the hexagrams, Da Yu gets the order, and the Saints get things by doing things. There are countless members, law calendars, soldiers, and slacks, and they are innumerable, but they should be based on "Yi" and "Fan".He also said in the book "After the Directed Algorithms" that "the number is one of the six arts, and it is still alive, it is a master of the drama, the dragon and the horse are negative, and the number is the beginning. Therefore, the sage inherits the poles of the heavens, so the people whobelieve in the same measure and establish the people's faith, but the yellow bell nine inches."
It should be further pointed out that ancient mathematicians attributed the origin of mathematics to the Book of Changes, not only to determine the source of mathematics from the perspective of the history of mathematics development. What's more, it shows that there is a close relationship between the principle of Zhouyi and mathematical research.
V. Reference Books, Reference Literatures, and Reference Materials
A. Text Books, Monographs and References
1. E. Cresseg.Higher Engineering Mathematics (Sixth Edition in 1988). World Book Publishing Company, 1992.
2. Zhang Pingwen, Li Tiejun. Numerical Analysis. Peking University Press, 2007.
3. Xiong Chunguang, Li Yuan. Scientific and Engineering Computing (Second Edition). Hua University Press, 2015.
4. Shen Yan, Yang Lihong, etc. Higher numerical computation. Tsinghua University Press, 2016.
5.Liu Jijun. Algebraic numerical computation method. Science Press, 2016.
6. Zheng Huiluo, Chen Shaolin, etc. Numerical computation method. Wuhan University Press, 2002.
7.Li Qingyang, Yi Dayi and others. Numerical algorithm analysis and efficient algorithm design. Huazhong University of Science and Technology Press, 2018.
8.Li Qingyang, Yi Dayi and others. Numerical algorithm analysis and efficient algorithm design. Huazhong University of Science and Technology Press, 2018.
9. David Kincaid, Ward Cheney. Numerical Analysis: Mathematics of Scientific Computation (Third Edition). China Machine Press,2003.
10. Walter Gautschi. Numerical analysis. World Book Publishing Company, 2015.
11.Gene H. Golub, Charles F. Van Loan. Matrix computations. People's Posts and Telecommunications Press, 2014.
12.Wu Qun, Zhou Lingjun, Yin Junfeng. Matrix analysis. Tongji University Press, 2017.
B. Learning Resources
1. http://math.hrbeu.edu.cn
2. Address of the MOOC learning platform:
https://mooc1.chaoxing.com/course/201826509.html(Numerical analysis MOOC).
3. Chinese University MOOC platform: computation method. Dalian University of Technology.
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Leader in charge of teaching at the College (Signature)：
Date：
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