When using mathematical methods to analyze and solve practical problems, it is required to find out the inherent laws from the actual complex relations, and then describe and explain them with mathematical languages-numbers, formulas, charts, symbols, etc. , and then through mathematical and computer processing-calculation, iteration, etc., to get quantitative results. For people to analyze, predict, make decisions and control. The process of reasonably simplifying practical problems into mathematical problems and solving them is to establish mathematical models. This successful method and technology are embodied in the university teaching activities of cultivating specialized talents, that is, mathematical modeling teaching and competition. In short, mathematical modeling is the process of applying mathematical models to solve various practical problems, that is, abstracting, simplifying and determining variables and parameters, applying certain laws to establish the relationship between variables and parameters, and then solving mathematical problems with computers, and explaining, testing and evaluating the obtained solutions to determine whether they can be used to solve mathematical problems (or mathematical models) of practical problems for many times.
Second, the characteristics of mathematical modeling
Since 1985 held the annual Mathematical Modeling Competition (MCM) in the United States, and since 1992 held the first national mathematical modeling competition for college students in China, it has become one of the important items of the national college students' science and technology competition, and it is a mass scientific and technological activity for college students all over the country. The competition requires students (of any major) to participate in the competition in groups of three, and they can freely collect information, investigate and study, including using computers and any software, and even inquire online, but they are not allowed to discuss with anyone outside the team. In three days, they can complete a paper including the hypothesis, establishment and solution of the model, the design of calculation method and the computer realization of the solution, as well as the analysis and test of the results and the improvement of the model. This activity has played a positive role in improving the quality of college students and promoting the reform of mathematics and computer teaching in colleges and universities.
Over the years, the annual national and international mathematical modeling competition for college students has brought new ideas and evaluation standards to the traditional higher mathematics education reform. Mathematical modeling course has also expanded from training only for contestants to more popular elective courses, and mathematical experiment has emerged as a new course. The focus of mathematical modeling and mathematical experiment teaching is the deep application and problem-oriented design of advanced mathematics and modern mathematics, rather than the in-depth discussion and systematic demonstration of classical theories. Mathematical modeling problems mostly come from some specific scientific research topics or practical engineering problems, which are different from ordinary mathematical exercises or competition questions. The characteristics of mathematical modeling problems are: application oriented to real life, relevant scientific research background, strong comprehensiveness, wide coverage, complex relationship between factors, lack of sufficient standardization, difficulty in applying traditional and mature solutions, huge data, complex algorithms, flexible results and certain accompanying conditions. Many problems can only get approximate solutions.
On the other hand, modeling is different from theoretical research, which focuses on dealing with practical problems rather than deep pure mathematical theory or world problems. Therefore, solving modeling problems mostly depends on various auxiliary tools or means, especially the application of computer software, which greatly improves the efficiency and quality of solving problems. In a word, mathematical modeling is a technology application course, not a basic education course. It emphasizes how to solve problems better and faster, and how to make full use of various scientific and technological means as technical support, so the application of computers has become an indispensable basic component. There are two main parts of computer technology related to this: one is how to transform or express actual problems or models into algorithms that can be realized by computer software or programming; The second is which application software or programming technology can be used to solve these problems. Obviously, the latter is the basis of the former, and only when the tool scheme is determined can there be a corresponding solution.
Due to the above characteristics of mathematical modeling, it is determined that mathematical modeling is closely related to computers. Computer plays an important role in the cultivation of mathematical modeling ideas, mainly providing powerful tools and technical support, which is the basis for better and faster modeling. The computer level can be said to determine the overall modeling level of a team.
Third, the relationship between mathematical modeling and computer
The appearance of computer is the product of mathematical modeling. In the 1920s and 1940s, in order to study the trajectory of ballistic missiles, the United States urgently needed a calculation tool to replace manual calculation, and computers came into being under this background. The appearance and development of computers have greatly promoted mathematical modeling activities, and the high-speed computing ability of computers is very suitable for numerical calculation in the process of mathematical modeling. Its large storage capacity and network communication function make it convenient and effective to store and retrieve data in the process of mathematical modeling. Its multimedia makes it possible to simulate some problems in mathematical modeling more realistically on the computer; Its intelligence can remind and help us to solve the mathematical model at any time. In addition, the emergence of a number of excellent mathematical software, such as Mathlab, Maple, SAS, SPSS and so on, makes mathematical modeling even more powerful. Furthermore, mathematical modeling is closely related to real life, and the amount of data collected is large and complex, such as DVD online rental, Yangtze River water quality evaluation and prediction, bank loans and installment payment. This often requires a lot of calculation and can be done quickly and simply with the help of a computer. Mathematical modeling contest is different from the mathematical contest (pure mathematical contest) mentioned before. It needs a computer, even without it, but it is not a pure computer competition. It involves physics, chemistry, biology, medicine, electronics, agriculture, military affairs, management and other disciplines and fields, but it is not limited by any specific discipline or field. The process of mathematical modeling needs to go through several steps, such as model hypothesis, model establishment, model solution, model analysis and test, model application and so on. All these are accompanied by the use of computers. For example, when solving the model, it is necessary to calculate, compile software and draw graphics on the computer. And the printer can be used at any time in mathematical modeling competition. At the same time, the study of mathematical modeling also greatly promotes the cultivation of computer ability. For example, applying for graduate students majoring in computer science requires high mathematics; In the research of computer science, a strong mathematical foundation is also required to write a paper with considerable depth. The development of computer science is also based on mathematics. Many scientists who have made outstanding contributions to the development of computers come from mathematics majors. Obviously, an important part of the competition is to use computers to solve problems, which is very obvious for improving the ability of using computers.
The purpose of mathematical modeling is to establish mathematical modeling consciousness and cultivate students' creative thinking ability. Among many thinking activities, innovative thinking is the highest level of thinking activities and the necessary ability to develop innovative talents. To cultivate creative thinking ability, it is necessary to train students to use basic theories flexibly to solve practical problems. Cultivating students' modeling consciousness in mathematics teaching is essentially to cultivate and develop students' creative thinking ability. Because the modeling activity itself is a creative thinking activity, which is both theoretical and practical, and also requires the profundity and flexibility of thinking. Moreover, in the process of modeling activities, students can be trained to use the conditions of given problems independently and consciously, seek the best methods and ways to solve problems, and cultivate their abilities of imagination, intuitive thinking, guessing, transformation and construction. These mathematical abilities are the most basic characteristics of creative thinking. In the process of cultivating innovative thinking, a certain computer foundation is required. Only with certain computer knowledge can we better process data, discover the internal relations between things, better transform knowledge and better construct the optimal model. In a word, having the necessary computer knowledge is the key to cultivate the consciousness of modeling and the premise to cultivate the innovative ability of mathematical models. Computers also provide powerful tools for mathematical modeling competitions.
Fourth, the application of computer in mathematical modeling
The use of computers not only makes it convenient for us to search the knowledge and related documents involved in modeling problems on the Internet, but also makes it convenient for us to process data, solve models and test models.
Computer software related to modeling is the software that we must master when building and processing models. They each have their own characteristics. When using them, we should pay attention to distinguish their advantages and disadvantages and choose more suitable software to deal with the problem. Commonly used software includes the following types:
1, general mathematical software. Mainly including Matlab, Mathematica, Maple, Mathcad, the ability and usage are similar. They are mainly used to draw graphs and calculate known functions, and support complete symbolic operation, accurate calculation and approximate calculation with arbitrary precision. They can solve common problems in many fields, such as calculus, analytic geometry, linear algebra, differential equations, calculation methods, probability statistics and so on. However, they also have their own characteristics: Mathematica, for example, has strong symbolic calculation ability, while Matlab has more advantages in numerical calculation, matrix calculation and graphic drawing, and can be used together.
2.Lingo/Lindo's special mathematical software for calculating optimization problems. Lindo is used to solve linear programming and quadratic programming. Lingo not only has all the functions of Lindo, but also can be used to solve nonlinear programming, some linear and nonlinear equations and algebraic equations. Both can solve integer programming. .
3. Statistical analysis software, SPSS is called sociological statistical software package, and its main functions are: basic statistical analysis, definition table and comparison average; General linear mode; Correlation analysis; Regression analysis, logistic linear analysis, cluster and discriminant analysis, factor analysis, nonparametric test, time series, proportion, multivariate response, etc. SAS provides many database query and statistics functions, and provides rich functional support in the classical processing and calculation of probability statistics. It is statistical professional software.
4. There are many high-level programming languages, such as C, C++, C#, Basic, Delphi and Java.
5. Drawing software. Adding some charts to the attachment can add color to the article. Mathematical software can only draw graphs of known functions. If you want to draw a rough figure, you must use drawing software. You can use geometric sketchpad, Photoshop, Flash, etc. Therefore, the future trend of mathematical modeling competition is to require students to know all aspects of knowledge and have higher requirements for students' computer knowledge. In recent years, almost all the competition topics in mathematical modeling competitions involve a lot of calculations or logical operations, and it is difficult to achieve good results without mastering the use of computers and related mathematical software. Because the contest topics come from different fields, and we don't know them in advance, we can find relevant information quickly by using the internet, which is also helpful to get good results in the contest. Therefore, there is an inseparable relationship between computer and mathematical modeling. The organic combination of the two has effectively improved college students' flexible application of theoretical knowledge, knowledge transfer, practical application ability and ability to analyze and solve problems.
Verb (abbreviation of verb) conclusion
I participated in two mathematical modeling competitions during my college years, and I also participated in the counseling of mathematical modeling competitions in recent years, from which I can deeply understand the ups and downs and the essence of mathematical modeling competitions. It is not only beneficial for students to master and use knowledge better, but also beneficial to scientific research and teaching in colleges and universities, so that students and teachers can automatically form a good habit of being diligent in thinking in their usual study and work. Mathematical modeling competition is very similar to the working conditions of students after graduation, and it is an all-round cultivation of students' business, ability and quality, especially their open thinking and innovative consciousness. The development of this activity is conducive to the cultivation of students' all-round quality, which not only enriches and enlivens the extracurricular life of students. Many students who have participated in the training have the same experience, and once they participate in the competition, they will benefit for life. Mathematical modeling is the road to success in the future. Regardless of the ranking, every contestant is a winner. In a word, using computer technology for mathematical modeling will be beneficial to the establishment, solution, calculation and expression of mathematical models, create an ideal background for explorers, and make our computers better and more vivid. The application of computer in mathematical modeling will make the process of mathematical modeling even more powerful. The application of mathematical modeling method in computer makes the computer develop rapidly, and the combination of computer technology and mathematical modeling will certainly promote its rapid development.