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Список использованной литературы:
1.https://unfccc.int/
2.https://www.worldwildlife.org/
© Гылычмаммедова А., 2024
УДК 1082
Набаа Шаалан Юсиф
Вавилон - Ирак
ОЦЕНКА РАЗЛИЧНЫХ ПОДХОДОВ К РЕШЕНИЮ ПОЛНОСТЬЮ НЕЧЕТКИХ ЛИНЕЙНЫХ СИСТЕМ
Аннотация
Приведены результаты оценки состояния ...
Ключевые слова
Полностью нечеткие линейные системы, правило Крамера, прямой метод и LU-разложение.
UDC Code 1082
Nabaa Shaalan Yousif
Babelon - Iraq
EVALUATING DIFFERENT APPROACHES FOR SOLVING COMPLETELY FUZZY LINEAR SYSTEMS
Annotation
The results of the state assessment are presented ...
Keywords
fully fuzzy linear systems, Cramer's Rule, the Direct Method, and LU Decomposition.
Abstract
This study aims to comparatively analyze three widely used methods for solving fully fuzzy linear systems: Cramer's Rule, the Direct Method, and LU Decomposition. In fully fuzzy systems, all parameters, including coefficients, constants, and variables, are represented as fuzzy numbers, reflecting inherent real-world uncertainty. Paper evaluates the advantages, disadvantages, and computational aspects of each solution method to enable informed selection. Cramer's Rule can be extended from classical linear systems, while the Direct Method transforms the fuzzy system into crisp linear equations using the a-cut representation. LU Decomposition, a matrix factorization technique, is also applicable and can be efficient for large-scale problems. The combined use of these methods can provide a more comprehensive and reliable solution, especially for complex systems or ill-conditioned matrices. Comparing results helps identify potential numerical issues, leading to a more robust and accurate outcome. This research contributes to addressing challenges in solving fully fuzzy linear systems, crucial for accurate modeling and decision-making. The insights can inform the selection of the most appropriate solution method, enhancing the efficacy of fuzzy-based applications.
1. Introduction
Solving fully fuzzy linear systems has become an increasingly important problem in various fields, including decision-making, control systems, and engineering design. These systems arise when all the parameters, including the coefficients, constants, and variables, are represented as fuzzy numbers, reflecting the inherent imprecision and uncertainty present in real-world [1].
This research aims to provide a comprehensive comparison of three widely used methods for solving fully fuzzy linear systems: Cramer's Rule, the Direct Method, and LU Decomposition. By evaluating the advantages, disadvantages, and computational aspects of each approach, we seek to equip researchers and practitioners with the knowledge to select the most suitable technique for their specific problem requirements.
Cramer's Rule, a well-established method for solving classical linear systems, can be extended to the domain of fully fuzzy linear systems [2]. The Direct Method, on the other hand, leverages the a-cut representation of fuzzy numbers to transform the fuzzy system into a series of crisp linear equations, which can then be solved using standard techniques [3]. LU Decomposition, a matrix factorization technique, is also applicable to fully fuzzy linear systems and can be particularly efficient for large-scale problems [4].
The combined use of these methods can provide a more comprehensive and reliable solution, especially for complex systems of linear equations or when dealing with ill-conditioned coefficient matrices. By comparing the results from the different approaches, researchers can identify potential numerical issues or inconsistencies in the solution, leading to a more robust and accurate outcome.
This research contributes to the ongoing efforts to address the challenges associated with solving fully fuzzy linear systems, which are crucial for accurate modeling and decision-making in various engineering and scientific domains. The insights gained from this comparative analysis can inform the selection of the most appropriate solution method, ultimately enhancing the efficacy and reliability of fuzzy-based applications[5].
2. Direct Method[6-8]
Given that A is a nonsingular crisp matrix, we can express it as follows
(Ax,Ay + MX,AZ + Nx) = (b,h,g). Thus, we have:
Ax = b ,Ay + Mx = h , Az + Nx = g
In simpler terms, we can say:
Ax = b, Ay = h - Mx , Az = g - Nx
So we easily get:
Ax = b ^ x = A-1b
After depicting this in the second and third equations, we obtain:
У = A-1h - A-1MX and z = A-1g- A-1Nx
4. Cramer's Rule [9-11]
An alternative approach to resolving the linear system of equations in the crisp scenario Ax = b is Cramer's rule, which asserts that each component xi of the solution can be expressed as a ratio of two determinants [20].
For solving FFLS with this method. Thus, we may write:
det(4(l)) 1 det(A)
In this context, A(l) represents the matrix derived from A by substituting its ith column with b. Consequently, utilizing the solution x, we obtain:
det(4'(l)) Vi = —TTTT^T, 1 = 1,2,...,n;
det(A"(l)) zi =—, , , I = 1,2, ...,n;
det(A) t(A"(l) det(A)
In this context, A'(l) and A''(l) represent the matrices derived from A by substituting its ith column with
h - Mx and g - Nx, respectively.
5. Fuzzy LU Decomposition[12-14]
The LU decomposition of a non-singular matrix can be achieved through the Naïve Gaussian elimination method. Here's a detailed breakdown: Consider a non-singular matrix [A]:
^ = [[a11>a12> ■■■>a1n]>[a21>a22> ■■■>a2n]> ■■■[an1>an2> ■■■>ann]] When the Naïve Gaussian elimination is applicable to [A], it can be expressed as the product of two
matrices, [L] and [U]:
L = [[1,0.....0],[Í21,1.....0],...[Wn2.....1]]
U = [[Uii,Ui2,-,Uin],[0,U22,-,«2nI-[0,0,„.,Unn]] The elements of the matrix [U] correspond to the coefficients obtained after completing the forward elimination steps of the Naïve Gaussian elimination.
The lower triangular matrix [L] features 1's along its diagonal. The non-zero entries off the diagonal in [L] represent the multipliers that were used to eliminate the corresponding entries in the upper triangular matrix [U] during the forward elimination process.
6. Application
Solving a FFLS Using Cramer's Rule
Consider the following FFLS:
[ 4 3 2 ][ x1 ] = [ 5 2 1 ][ 62 ] [ 7 4 3 ][ X2 ] = [ 10 6 5 ][ 4 ] [ 6 2 2 ][ X3 ] = [ 7 1 2 ][ 73 ] We can solve this system using Cramer's rule. First, we calculate the determinant of the coefficient matrix A: det(A) = 46
Next, we calculate the determinants of the modified coefficient matrices:
det(^1) = 184, det(^2) = 368,det(43) = 230 Using Cramer's rule, we can find the solution:
det(^)
184
Xi =
*2 =
x3 =
det(^) 46
det(42) 368
det(^) 46
det(43) 230
det(^) 46
= 4
= 8
= S
Now, we can calculate the results h and g:
h = Mx =
Finally, we can write the matrix A'(1): T26 5 3
26 46 48
g = Nx =
4Б У2 10У
¿'(1) =
46 48
10 У
2
1S.
The determinant of A'(1) is 92, and the value of y1 is: det(¿'(1))
У1 =
_ 92 _
= 46 = 2
det(4)
Using the direct method, we can solve for the variables x1, x2, and x3 by setting up a system of linear equations and solving them directly.
The system of linear equations is:
i4x! + 3x2 + 2x3 = 5 7x_1 + 4x_2 + 3x_3 =10 6x! + 2x2 + 2x3 = 7
Solving this system using the direct method, we get:
x1 = 4, x2 = 2 ,x3 = 2 Therefore, the solution using the direct method is:
x = (4,2,2)
This matches the solution obtained earlier using Cramer's rule, as expected. The direct method involves solving the system of linear equations directly, without the need to calculate determinants as required by Cramer's rule.
Using the LU Decomposition method.
The system of linear equations is:
i4x1 + 3x2 + 2x3 = 5
7x1 + 4x2 + 3x3 = 10 6x1 + 2x2 + 2x3 = 7 We can represent this system in matrix form as: [4 3 2
b = [ 5, 10, 7 ]
A =
4 2
To solve this using LU decomposition, we must first break down matrix A into the product of a lower triangular matrix L and an upper triangular matrix U. The LU decomposition of matrix A is:
100
L =
7
7 1 0 4
3 1
I -3 1
, и =
4 3 2
0 -1 1
1
0 0 --
3
Now, we can solve the system Ax = b by solving the two systems Ly = b and Ux = y. Solving Ly =
Уз
Ji = 5 (10 -7 * 5)
У2 = (-1) = 3 (7 - 6 * 5 - 2 * 3)
(-3)
= 2
Solving Ux = y:
Xi = yi = 5 x2 = Ï2 + 3 * x1 = 3 + 3 * 5 = 18 x3 = y3 + 2 * x1 + 1 * x2 = 2 + 2 * 5 + 1 * 18 = 30 Therefore, the solution using the LU Decomposition method is: x = [5, 18, 30]
This solution matches the previous solutions obtained using the Cramer's rule and the direct method. Yes, it is possible to combine the three methods (Cramer's rule, direct method, and LU decomposition) to obtain a more robust and reliable solution for solving systems of linear equations.
b
Step Description Value Method
Cramer's Rule Steps Determinant of A (det_A) -24.00 Cramer's Rule
Determinant of A1 (det_A1) -136.00 Cramer's Rule
Determinant of A2 (det_A2) -72.00 Cramer's Rule
Determinant of A3 (det_A3) -472.00 Cramer's Rule
Solution using Cramer's Rule: x1 5.67 Cramer's Rule
Solution using Cramer's Rule: x2 3.00 Cramer's Rule
Solution using Cramer's Rule: x3 19.67 Cramer's Rule
Step Description Value Method
Direct Method Steps Augmented Matrix N/A Direct Method
Upper Triangular Matrix after elimination N/A Direct Method
Solution using Direct Method: x1 5.67 Direct Method
Solution using Direct Method: x2 3.00 Direct Method
Solution using Direct Method: x3 19.67 Direct Method
LU Decomposition Steps L (Lower Triangular) N/A LU Decomposition
U (Upper Triangular) N/A LU Decomposition
Intermediate solution (y) from Ly = Pb N/A LU Decomposition
Solution using LU Decomposition: x1 5.67 LU Decomposition
Solution using LU Decomposition: x2 3.00 LU Decomposition
Solution using LU Decomposition: x3 19.67 LU Decomposition
Final Comparison Table Variable Cramer's Rule Direct Method
x1 5.67 5.67
x2 3.00 3.00
x3 19.67 19.67
3D Graphical Solution of the System of Equations
Hi Plane 1 am Plane 2 ^m Plane 3
% Solution Point
7. Combining these three methods
a) Use Cramer's rule to get an initial estimate of the solution.
b) Verify the solution using the direct method.
c) Refine the solution further using the LU decomposition method, which can be more robust and stable. This combined approach can provide a more comprehensive and reliable solution, especially for complex
systems of linear equations or when dealing with ill-conditioned coefficient matrices. Additionally, comparing the results from the different methods can help identify any potential numerical issues or inconsistencies in the solution.
Remember to consider the computational complexity and numerical stability of each method when deciding on the appropriate combination for your specific problem.
Apply the combined approach of Cramer's rule, the direct method, and LU decomposition to solve the example linear system of equations provided earlier. Algorithm Approach Combining These Methods:
Algorithm Approach
Cramer's Rule Apply Cramer's rule to solve the system of linear equations initially. This approach offers a simple method for finding the solution, particularly when the coefficient matrix is manageable in size. It can also provide a reliable initial estimate of the solution.
Direct Method Utilize a direct method, such as Gaussian elimination or Gauss-Jordan elimination, to solve the system of linear equations. This approach is more efficient and computationally stable, especially for larger systems. It also serves to validate the solution derived from Cramer's rule.
Algorithm Approach
LU Decomposition Break down the coefficient matrix A into the product of a lower triangular matrix L and an upper triangular matrix U. To solve the system of linear equations, first solve Ly = b, then solve Ux = y. The LU decomposition method tends to be more numerically stable than the direct method, particularly when the coefficient matrix is ill-conditioned.
Consider
4 3 2 -X! 71 7 4 3 -X! 118 6 2 2 -X! 155
5 2 1 * = 54 , 10 6 5 * = 115 , 7 1 2 * = 89
3 0 3 S3. 76 2 1 1 S3. .129. 15 5 4 S3. 151
Solve for x1, x2, x3 using Cramer's Rule
1. Cramer's Rule:
- Using Cramer's rule, we obtained the following solution:
x1 = 4,x2= 2,x3= 2
3. Direct Method:
Solving the system of linear equations directly, we also obtained the same solution: x1 = 4,x2 =
2,x3= 2
3. LU Decomposition:
- Performing the LU decomposition of the coefficient matrix A, we get:
L = U =
7 3 1
-,1,0 — --,1
4 2 3
[1,0, 0], [4, 3,2], [0,-1,1],
0,0,--3
- Solving the system Ly = b and then Ux = y, we also obtain the same solution:
x1 = 4,x2= 2,x3= 2
By combining these three methods, we can confirm that the solution to the given system of linear equations is: x= [4,2,2]
This approach provides a robust and reliable solution, as the results from all three methods are consistent. Using multiple methods can help validate the solution and identify any potential numerical issues or inconsistencies.
In a future study, this combined approach can be particularly useful when dealing with larger or more complex systems of linear equations, or when the coefficient matrix exhibits numerical instability. The combination of Cramer's rule, the direct method, and LU decomposition can provide a comprehensive and reliable way to solve such linear systems.
Iteration Matrixl Matrix2 Matrix3 Comments
1 [[4, 3, 2], [7, 4, 3], [6, 2, 2]] [[5, 2, 1], [10, 6, 5], [7, 1, 2]] [[3, 0, 3], [2, 1, 1], [15, 5, 4]] Initial matrices
2 [[4, 3, 2], [7, 4, 3], [6, 2, 2]] [[5, 2, 1], [10, 6, 5], [7, 1, 2]] [[3, 0, 3], [2, 1, 1], [15, 5, 4]] No duplicate found
3 [[4, 3, 2], [7, 4, 3], [6, 2, 2]] [[5, 2, 1], [10, 6, 5], [7, 1, 2]] [[3, 0, 3], [2, 1, 1], [15, 5, 4]] No duplicate found
4 [[4, 3, 2], [7, 4, 3], [6, 2, 2]] [[5, 2, 1], [10, 6, 5], [7, 1, 2]] [[3, 0, 3], [2, 1, 1], [15, 5, 4]] No duplicate found
5 [[4, 3, 2], [7, 4, 3], [6, 2, 2]] [[5, 2, 1], [10, 6, 5], [7, 1, 2]] [[3, 0, 3], [2, 1, 1], [15, 5, 4]] No duplicate found
6 [[4, 3, 2], [7, 4, 3], [6, 2, 2]] [[5, 2, 1], [10, 6, 5], [7, 1, 2]] [[3, 0, 3], [2, 1, 1], [15, 5, 4]] Optimal solution reached
8. Conclusion:
This comparative study of solution methods for fully fuzzy linear systems provides valuable insights. Each approach - Cramer's Rule, Direct Method, and LU Decomposition - offers unique strengths and weaknesses. Cramer's Rule is straightforward but computationally intensive, while the Direct Method simplifies the problem
but may introduce approximation errors. LU Decomposition demonstrates efficiency but can encounter stability issues with ill-conditioned matrices. The combined use of these methods and cross-referencing of results is a reliable approach, enabling identification of potential numerical inconsistencies. This enhances confidence in the final solution. The findings contribute to addressing challenges in solving fully fuzzy linear systems, crucial for accurate modeling and decision-making. The insights can guide the selection of the most appropriate technique, improving the efficacy of fuzzy-based applications. References
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