I.The Basic Information of the Course
Course Number: 202032020003
The English Name of the Course：Numerical Computation
The Chinese Name of the Course：数值计算
In-classHours andAllocationTotal class hours: 32, 28 class hours in class, 4 class hours in class discussion
Credit(s):2
Semester:1
AppliedDiscipline(Professional Degree Category)：Engineering
CourseObjectOriented：Master
EvaluationMode:Closed-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
Higher Mathematics, Linear Algebra, Probability Statistics, etc.
III. The Objectives and Requirements of the Course
(1) We should cultivate postgraduates with a noble scientific outlook, firm ideals and beliefs, and consciously carry forward and practice the core values of socialism.
(2) Possesses considerable mathematical theoretical derivation and application ability of numerical algorithms, and is familiar with the basic ideas and classic algorithms of numerical computes for scientific computations. For practical problems, we can use the idea of numerical analysis to establish mathematical models and give effective numerical solutions to train students to be self-active in solving practical problems in engineering and to use computers to perform scientific computations.
(3) Final scores are assessed on the three levels of closed-book exam (70%), class discussion (10%) and major assignments (20%), and finally evaluated on a 100-point scale.
(4) Classroom discussions (accounting for 10%) mainly contain the content of curriculum ideology and politics. Teachers use the relevant content from ancient times to the present as case guides, allowing students to fully consult the materials and actively speak, and use this part of the results to praise students' Patriotic feelings, while stimulating students' interest in learning.
IV. The Content of the Course
Numerical Computation is a course that studies the numerical computation methods and theories of solving various mathematical problems. It is rich in content and very practical. The research method is profound and has its own theoretical system; the main content of the 32-class-hour Numerical Computation course is as follows:
1. Introduction to Computation Methods 4 class hours
The significance of scientific computation, basic knowledge of error, vector norm, matrix norm, operator norm, MATLAB software;
2. Polynomial Interpolation and Spline Interpolation 6 class hours
The existence and uniqueness of the solution to the interpolation problem, Lagrange interpolation, Newton interpolation and divided differences, difference, Hermit interpolation, cubic spline interpolation;
3. Function Approximation and Computation 4 class hours
Inner product and orthogonal polynomials, best-square approximation of continuous function space, least-squares method of curve fitting problem;
4. Numerical Integration and Numerical Differentiation 4 class hours
Algebraic accuracy, interpolation type quadrature formula, Romberg integration method, Gauss type quadrature formula;
5. Numerical Solution of Linear Equations 8 class hours
Gauss elimination method, LU factorization, Cholesky factorization of symmetric positive definite matrices, chasing method for solving tridiagonal equations, concepts and properties of condition number, spectral radius, pathological description of matrices and equations, construction of iterative method, iterative matrix, Jacobi iterative method, Gauss-Seidel iterative method, super-slack iterative method, iterative method of convergence criterion method;
6. Finding Roots of Nonlinear Equations 2 class hours
The inter-partition method for finding the real roots of equations, the iterative method for a single equation, the Newton method for a single equation, polynomial root finding, nonlinear equations；
7. Numerical solution of initial value problem of ordinary differential equations 4 class hours
Euler formula and its improvement, Runge-Kutta method, convergence and stability of single-step method, finite difference method.
The study of mathematics is mainly divided into two categories: theoretical analysis and numerical computation. The course of numerical computation is different from the theoretical analysis emphasized in higher mathematics. Numerical computation pays more attention to how to use the results of these theoretical analyses for scientific computation. For example, the solution of linear equations uses the matrix factorization theory in linear algebra. The function interpolation part is based on the theory of function space in linear algebra and the differential median theorem in analysis. The differential quotient numerical differential formula applies higher mathematics. In the Taylor formula, in the teaching process, it should be highlighted that the course is computer-oriented, focusing on the characteristics of computer-executable computation algorithms. At the same time, this course introduces the most basic principles and methods in numerical computation. To achieve programming, we must emphasize algorithm design and efficiency analysis. Therefore, in addition to teaching the basic methods and principles in the teaching process, more attention should be paid to the algorithm design and analysis of commonly used numerical computations in order to train students' ability and skills in algorithm design and analysis.
Course thinking case: The numerical computation course is rich in content and profound in research methods. It is a very practical mathematics course, which can play a major role in cultivating students' ability to apply mathematics and the development of innovative thinking. This course introduces the following two cases in the course of teaching:
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 = be called "Zu rate".
Case 2: Western mathematics research uses mathematics as a way to understand nature and the laws of the universe. They tend to think philosophically and use deductive reasoning to prove conclusions. For example, the ancient Greek mathematician Euclid's masterpiece "Original Geometry". He completed an epoch-making work—transforming empirical science established through experiments and observations into deductive science, systematically introducing logical proofs into mathematics, and the axioms and theorems adopted by Euclidean in the "Original Geometry" all are carefully considered and selected, and the content is organized according to a rigorous scientific system to make it systematic and theoretical.
Comparing the history of the development of mathematics at home and abroad, I realized the gap between my country's mathematics research and foreign advanced level, and strived for the great rejuvenation of the Chinese nation.
V. Reference Books, Reference Literatures, and Reference Materials
A. Text Books, Monographs and References
1. Shen Yan, Yang Lihong, etc. Higher 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. Xu Feng. Numerical analysis. Press of China University of Science and Technology. 2017.
4. Walter Gautschi. Numerical analysis. World Book Publishing Company, 2015.
5. Li Qingyang, Yi Dayi, etc. Numerical Analysis (Fifth Edition). Tsinghua University Press, 2018.
6.David Kincaid, Ward Cheney. Numerical Analysis: Mathematics of Scientific Compution (Third Edition). China Machine Press,2003.
B. Learning Resources
1. http://math.hrbeu.edu.cn
2. Address of the MOOC learning platform:
https://mooc1.chaoxing.com/course/201826509.html（数值分析MOOC）.
3. Chinese University MOOC platform: calculation method. Dalian University of Technology.
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Leader in charge of teaching at the College (Signature)：
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