I.The Basic Information of the Course
Course Number:202012420027
The English Name of the Course：Numerical Analysis
The Chinese Name of the Course：数值分析
In-classHours andAllocation:total class hours:32, classroom teaching: 26classhours, classroom discussion: 6 classhours, 6 classhours on computer
Credit(s):2
Semester:2
AppliedDiscipline(Professional Degree Category)：Mathematics
CourseObjectOriented：AcademicMaster
EvaluationMode:Process Evaluation+ MajorAssignment
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
Mathematical Analysis, Advanced Algebra, Real Variable Functions, Probability Theory, Ordinary Differential Equations, a Computer Programming Language
III. The Objectives and Requirements of the Course
The Objectives of the Course:
(1) The course will be based on scientific computing theory and methods, computer technology as a means, and mathematical software as a tool, focusing on guiding and inspiring students to think.
(2)Focus on finding ways to solve problems; focus on the cultivation of innovative awareness and innovative thinking; focus on introducing the latest achievements in teaching reform into the classroom, introduce cutting-edge issues of subject development and future trends to students in a timely manner and integrate knowledge, ability and quality education into The integration makes teaching more characteristic of the times and promotes the connotation development of curriculum reform.
(3) Through the study of this course, graduate students can master the modern methods of interpolation, curve fitting, numerical integration and numerical differentiation, rooting of nonlinear equations, numerical solutions of linear and nonlinear equations, numerical solutions of initial value problems of ordinary differential equations, etc. The commonly used numerical computation methods and basic theories of computers, understand the latest theories and hot issues in this discipline in recent years, master their basic processing ideas and solutions, and lay a good foundation for solving scientific computing problems on computers.
The Requirements of the Course:
(1) Master the numerical methods and related theories commonly used in scientific computing, master the methods of engaging in large-scale scientific computing, train students to have the ability to engage in large-scale scientific computing and scientific computing, understand the development direction of computational mathematics, for further Scientific research and learning lay the foundation.
(2) The final score is based on a combination of process assessment (50%) and major work (50%), and is evaluated on a 100-point scale.
(3) 50% of major assignments include classroom seminars participated by students, understanding and publicizing of ideological and political cases, and reading of related documents.
IV. The Content of the Course
The main teaching contents of the numerical analysis course are as follows:
(1) Error Theory (2 class hours)
The content of this part mainly includes: the objects, functions and characteristics of numerical analysis; the basic concepts of error; sources and classification; the technology of algorithm design in numerical calculation;Course Ideological Case 1.
(2) Interpolation and Approximation Theory (8 class hours)
This course mainly teaches several classic interpolation techniques research, the best square approximation theory of functions, orthogonal polynomials, least squares algorithm of curve fitting; triangular polynomial approximation and fast Fourier transform.
(3) Numerical Integration and Numerical Differentiation (6 class hours)
This course mainly teaches the basic idea of numerical integration, Newton-Cotes formula and Romberg quadrature formula and the basic algorithm of numerical differentiation.
(4)Searching Roots of Nonlinear Equations (4 class hours)
Focus on introducing the basic idea of searching roots of nonlinear equations and Newton iterative method; numerical solution of nonlinear equations.
(5) Numerical solution of linear equations (8 class hours)
This course mainly teaches the direct method of solving linear equations based on several matrix factorization, the matrix and vector norm, and the condition number of the matrix; the basic idea of the iterative method based on matrix splitting and the theory of Convergence Analysis, Conjugate Gradient Method; course thinking and politics Case 2.
(6) Numerical Solution of Initial Value Problems of Ordinary Differential Equations (4 hours)
This course mainly teaches the convergence and stability of Euler method, Runge-Kutta method, and single-step method.
Course thinking case:
The task of this course is to give students a systematic understanding of numerical analysis and the numerical computation methods commonly used by computer students through lectures. Skillfully use computer design algorithms to solve practical problems in engineering.
Case 1: 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, and π=3.14 was obtained, and pointed out that the more the number of sides inscribed in the regular polygon, the more accurate the value of π obtained. Based on the achievements of his predecessors, Zu Chongzhi worked hard and repeated calculations to find π between 7.1415926 and 3.1415927. And the approximate value in the form of π fraction 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 of π. 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 Chong'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 that π is called "Zu rate".
Case 2: Partial Differential Equations of Mathematics
(1) The origin of partial differential equations:
If an unknown function appearing in a differential equation contains only one independent variable, this equation is called an ordinary differential equation, also referred to as a differential equation; if a partial derivative of a multivariate function appears in a differential equation, or if the unknown function is related to several variables, and the derivative of the unknown function to several variables appears in the equation, then this differential equation is a partial differential equation.
In the ever-changing development of science and technology, it is not enough to describe many problems studied by a function of independent variables, and many problems are described by functions of multiple variables. For example, from a physical point of view, physical quantities have different properties. Temperature, density, etc. are described by numerical values called scalar quantities; velocities, gravitation of electric fields, etc. not only differ in numerical values, but also have directions. These quantities are called Vector; the quantity described by the tension state of an object at a point is called a tensor, and so on. These quantities are not only related to time, but also to spatial coordinates, which must be represented by functions of multiple variables.
It should be pointed out that for all possible physical phenomena, it can only be idealized as a function of some multiple variables, such as the density of the medium. In fact, the density "at one point" does not exist. And we regard the density at a point as the ratio of the mass and volume of matter when the volume shrinks indefinitely, which is idealized. The same is true for the temperature of the medium. In this way, functional equations of multiple variables ideal for studying certain physical phenomena are produced. Such equations are partial differential equations.
The discipline of calculus equations originated in the eighteenth century. Euler first proposed the second-order equations of string vibration in his writings. Soon afterwards, French mathematician D'Alembert also proposed special partial differential equations in his book "On Dynamics". These works did not attract much attention at the time. In 1746, in his paper "Study on the Curves Formed when Tensioned Strings Vibrate", D'Alembert proposed to prove that an infinite variety of curves that differ from sinusoids are modes of vibration. In this way, the discipline of partial differential equations was initiated by the study of string vibration.
The Swiss mathematician Daniel Bernoulli, who was the same as Euler, also studied the problems of mathematical physics, and proposed a general method to understand the vibration problem of the elastic system, which has a relatively large influence on the development of partial differential equations. Lagrange also discussed first-order partial differential equations, which enriched the subject.
Partial differential equations developed rapidly in the 19th century. At that time, the study of mathematical physics problems flourished, and many mathematicians contributed to the solution of mathematical physics problems. Here we should mention French mathematician Fourier, who was an outstanding mathematician when he was young. In the study of heat flow, he wrote "Analysis Theory of Heat". In the article, he proposed the thermal equation in three-dimensional space, which is a kind of partial differential equation. His research has a great influence on the development of partial differential equations.
(2) Contents of partial differential equations:
What is the partial differential equation? What does it include? Here we can introduce from the study of an example.
String vibration is a kind of mechanical motion. Of course, the basic law of mechanical motion is F=ma of particle mechanics, but the string is not a particle, so the law of particle mechanics does not apply to the study of string vibration. However, if we divide the string finely into a number of extremely small segments, each segment is abstractly regarded as a particle, so that we can apply the basic laws of particle mechanics.
Strings refer to thin and long elastic materials. For example, the strings used in stringed instruments are long, thin, and elastic. When playing, the string is always tight with a tension, which is tens of thousands of times greater than the weight of the string. When the performer uses a thin piece to toggle or pull on the string with a bow, although only the string of a piece of contact vibrates, due to the action of tension, it spreads to vibrate the entire string.
(3) Classification of partial differential equations:
By differential analysis, the displacement of a point on the string can be obtained as a partial differential equation whose position and time are independent variables. There are many types of partial equations, generally including elliptic partial differential equations, parabolic partial differential equations, and hyperbolic partial differential equations. The above example is the string vibration equation, which belongs to the wave equation in the mathematical physics equation, that is, the hyperbolic partial differential equation.
(4) About the definite solution conditions of partial differential equations:
Partial differential equations generally have an infinite number of solutions, but when solving specific physical problems, you must choose the solution you need from them, so you must also know the additional conditions. Partial differential equations are expressions of the common law of the same type of phenomenon, and just knowing this common law is not enough to grasp and understand the specificity of specific problems, so as far as physical phenomena are concerned, the specificity of each specific problem lies in the research object. The specific conditions are the initial conditions and boundary conditions.
Take the example of string vibration mentioned above. For a stringed instrument with the same string, if one kind pulls the string with a thin piece and the other kind pulls the string with a bow, then the sounds they make are different. The reason is that the vibration at the "initial" moment of "pulling" or "pulling" is different, so the subsequent vibrations are also different.
Case 3:Zhao Shuang and Pythagorean Theorem;Zhao Shuang, also known as Ying, Zi Junqing, a Chinese mathematician. Wu people from the late Eastern Han Dynasty to the Three Kingdoms period. He is a famous mathematician and astronomer in the history of my country. Unknown life, about 182-250 years.He explained in detail the Pythagorean Theorem in the "Zhou Bi Suan Jing", and expressed the Pythagorean Theorem as: "The Pythagorean Pythagoreans are multiplied by each other, and are equal to the string. The square root is divided by the string." A new proof is also given: "According to the string diagram, the Pythagorean can be multiplied by Zhu Shi 2 and doubled by Zhu Shi 4, and the difference of Pythagorean can be multiplied to be middle yellow, plus difference real, and italso becomes a real string."
There is a similar situation in astronomy. If we want to predict the motion of celestial bodies through calculation, we must know the mass of these celestial bodies. In addition to the general formula of Newton’s law, we must also know the initial state of the celestial system we are studying. At the beginning, the distribution of these celestial bodies and their speed. When solving any mathematical physics equation, there will always be similar additional conditions.
As far as string vibration is concerned, the string vibration equation only represents the mechanical law of the inner point of the string, and it does not hold for the end point of the string, so the boundary conditions must be given at both ends of the string, that is, considering the physics on the boundary where the research object is located situation. Boundary conditions are also called boundary value problems.
Of course, there are still "problems without initial conditions" in objective reality, such as fixed-field problems (electrostatic field, stable concentration distribution, stable temperature distribution, etc.), and "problems without boundary conditions", such as focusing on research not close to both ends ,that string becomes an abstract string without boundaries.
Mathematically, the initial conditions and boundary conditions are called definite solution conditions. Partial differential equations themselves express the commonality of the same type of physical phenomenon, and serve as the basis for solving problems; the conditions of the solution reflect the individuality of the specific problem, and it proposes the specific situation of the problem. The combination of equations and definite solution conditions is called a definite solution problem.
(5) About the solution of partial differential equations:
To find a definite solution to a partial differential equation, you can first find its general solution, and then use the definite solution conditions to determine the function. But generally speaking, it is not easy to find the general solution in practice, and it is more difficult to determine the function with the definite solution condition.
Partial differential equations can also be solved using the separation coefficient method, also known as Fourier series; the separation variable method, also known as Fourier transform or Fourier integral. The separation coefficient method can solve the definite solution problem in the bounded space, the separation variable method can solve the definite solution problem in the unbounded space; the Laplace transform method can also be used to solve the definite solution of the mathematical and physical equations in one-dimensional space. The Laplace transform of the equation can be transformed into an ordinary differential equation, and the initial conditions are also taken into account, and the inversion can be performed after solving the ordinary differential equation. It should be pointed out that although there are various solutions to the definite solutions of partial differential equations, we cannot ignore that there are many definite solution problems that cannot be solved strictly for some reasons. We can only use the approximation method to find an approximate solution that meets the actual degree of approximation.
Common methods include variational method and finite difference method. The variational method is to convert the definite solution problem into a variational problem, and then find the approximate solution of the variational problem; the finite difference method is to convert the definite solution problem into an algebraic equation, and then use a computer to calculate; there is also a more meaningful simulation method, it replaces the definitive solution of a physical problem under study with another physical problem experimental study. Although the physical phenomena are different in nature, they are abstractly expressed to be the same definite solution problem in mathematics. For example, studying the stable temperature distribution in an irregularly shaped object is mathematically the boundary value problem of the Laplace equation. Because it is difficult to solve, the corresponding electrostatic field or steady current field experimental research can be done to measure the electric potential everywhere in the field, thus also solving the temperature distribution problem in the stable temperature field under study.
With the expansion of the phenomena studied in physical science in both breadth and depth, the application of partial differential equations is more extensive. From the perspective of mathematics itself, the solution of partial differential equations has prompted the development of mathematics in various aspects such as function theory, variational methods, series expansion, ordinary differential equations, algebra, and differential geometry. From this perspective, partial differential equations become the center of mathematics.
V. Reference Books, Reference Literatures, and Reference Materials
A. Text Books, Monographs and References
1. Shen Yan, Yang Lihong, etc. Advanced Numerical Computing. Tsinghua University Press, 2016.
2. Li Qingyang, Yi Dayi, etc. Numerical Algorithm Analysis and Efficient Algorithm Design. Huazhong University of Science and Technology Press, 2018.
3. Li Qingyang, Yi Dayi, etc. Numerical Analysis (Fifth Edition). Tsinghua University Press, 2018.
4. David Kincaid, Ward Cheney. Numerical Analysis: Mathematics of Scientific Computation (Third Edition). China Machine Press, 2003.
5.Walter Gautschi. Numerical analysis. World Book Publishing Company, 2015。
B. Learning Resources
1. http://math.hrbeu.edu.cn
2.China University MOOC Platform (China University MOOC) Address:
https://mooc1.chaoxing.com/course/201826509.html.
3. China University MOOC Platform (China University MOOC): Computation Method. Numerical Analysis (Dalian University of Technology: Guan Bo Team).
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