Question (CMCM-93B) gives the results of the national 12 football teams in the competition, and requires:
( 1)? Design an algorithm, rank teams according to these results, and give the results.
(2)? The algorithm can be extended to any n teams.
(3)? Discuss what it takes to arrange these data in your way.
Judging from the game results given in the table, the data is irregular, and there may be three, two, one game or even no game between the two teams.
One? A reasonable assumption
1? The ranking is only based on the existing competition results, without considering other factors.
Every game has the same importance and credibility, and different games are independent.
3? The irregularity of competition data is caused by the arrangement of the competition, not the victory or defeat in the competition.
4? The game is played according to the three-point system.
Two? analyse
What is the ranking: winning or losing? Strength? League, total points. The data is irregular and the total score is weak. And consider the difference between the winning team and the strong team.
Objective: To propose an algorithm of different rules of game data, which can reflect the real level of each team as reasonably as possible.
Three? model
1? Complete integration method
2? Average integral method
3? Consider the opponent's strength:
The winning team scored more points and the winning team scored less points. Ti's average score for Tj? , Tj's strength coefficient? What about Ti's score on Tj? , the total score of Ti
The matrix is expressed as
Y=AXX: strength coefficient y: ranking a: score matrix.
X and y are unknowns and reflect the strength of each team, so they should be proportional, that is, AX=? X and a are nonnegative irreducible matrices.
Four? Analysis results
Give a ranking:
Model test: given the strength coefficient x, simulate the game with a computer, generate the game results, get the score matrix and sort it. Compare the result with x and calculate the deviation.
mathematical modeling
Real problems? ——? Mathematical model-seeking mathematical solution-practical solution
Complete model
1? Modeling (from reality to mathematics);
Understand the background (research), analyze the problems and put forward the modeling basis.
Reasonable assumption: simplify the problem; The necessary premise of mathematical methods used in the model.
Use appropriate methods to build the model.
2? Solution of model (from mathematics to mathematics)
3? Analysis and test of the model;
result analysis
model testing
Stability and sensitivity analysis
Comparison between old and new models
error analysis
One? From practice to mathematics
1? Understand the background and previous work.
2? Consider all factors:
List all the factors.
Select the main factors to be included in the model.
Revise the model considering secondary factors.
3? The essence of analytical mathematics
System optimization design
Differential equation model
Statistical model
Interpolation and fitting model
Computer simulation
4? A reasonable assumption
Grasp the main factors and highlight the essence of the problem.
Practical problems are idealized and approximated to meet the requirements of the model.
Second, from mathematics to practice
1? Analyze the results from a practical point of view
2? error analysis
3? Stability analysis and sensitivity analysis
4? Models? Comparison of
5? Model checking, computer simulation