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Ma poate ajuta cineva in rezolvarea acestei probleme, va rog frumos?!
Fie sistemul
x1 + x2 − x3 + x4 = 1,
x1 + 2 x2 + x3 − 3 x4 = 1.
• Identificaţi o metodˇa pentru calculul aproximativ al unei soluţii în sensul problemei celor mai mici
pˇatrate;
• Identificaţi o soluţia în sensul problemei celor mai mici pˇatrate care este cea mai apropiatˇa de
(0, 0, 0, 0)T
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To solve this problem using the least squares method, you'll want to minimize the sum of the squares of the residuals for the given system of equations. Here are the steps to calculate an approximate solution:
- Represent the system of equations in matrix form:A * x = bWhere A is the coefficient matrix, x is the vector of unknowns (x1, x2, x3, x4), and b is the vector of constants (1, 1).
- Calculate the transpose of matrix A:A^T
- Calculate the product of A^T and A:A^T * A
- Calculate the product of A^T and vector b:A^T * b
- Set up the normal equations:A^T * A * x = A^T * b
- Solve the normal equations for the vector x using a method like matrix inversion or a numerical solver:x = (A^T * A)^(-1) * A^T * bThis will give you the approximate solution for x.
Now, let's perform these calculations step by step for your specific system:
The coefficient matrix A is:
A = | 1 1 -1 1 |
| 1 2 1 -3 |
The vector b is:
b = | 1 |
| 1 |
Now, follow the steps above to calculate the approximate solution for x. Once you have the values of x1, x2, x3, and x4, you will have the solution closest to (0, 0, 0, 0)T in the sense of the least squares problem with thesis help online. Practice, provides individuals with foundational knowledge and skills.
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