To verify the data using the discrepancy value ($\Delta x^*$) as described in the text, follow these steps:

1. Understanding the concept

The text explains that a well-posed system ($\text{корректная задача}$) is sensitive to small perturbations in the right-hand side vector ($\Delta b$) and/or the coefficients ($\Delta a_{ij}$) of the matrix $A$. A system is considered poorly conditioned if these perturbations cause large errors in the solution ($\Delta x^*$).

The sensitivity is quantified using the condition number of the matrix $A$, denoted as $\text{cond}(A) = \|A\| \cdot \|A^{-1}\|$. A high condition number indicates a poorly conditioned system.

2. Steps to calculate and verify the solution

To validate your solution:

1. Condition number:

   • Compute $\text{cond}(A) = \|A\| \cdot \|A^{-1}\|$. This tells you how sensitive the system is to errors.
   • If $\text{cond}(A)$ is very large, the system may be unstable.

2. Compute residual ($r$):

   • Use the formula $r = Ax - b$, where $A$ is the coefficient matrix, $x$ is your solution vector, and $b$ is the right-hand side vector.
   • A small residual (close to 0) indicates the solution is consistent with the system.

3. Relative error:

   • Estimate the relative error in the solution due to perturbations ($\Delta x^*$) using: $\frac{\|\Delta x^*\|}{\|x\|} \approx \text{cond}(A) \cdot \frac{\|\Delta b\|}{\|b\|}$
   • Here, $\Delta b$ represents any small changes or errors in $b$.

4. Validation:

   • Compare the calculated relative error and condition number to thresholds that indicate whether the system is well-posed.

3. Applying this to your solution

If the solution vector is: $x = [0.4615, 1.9615, 2.4615, 1.9615, 0.4615]$ and you have the system $Ax = b$, you need:

• Matrix $A$
• Right-hand side vector $b$

Example:

• If $A$ and $b$ are known, calculate $r = Ax - b$.
• Compute the condition number $\text{cond}(A)$.
• Verify that small changes in $b$ (e.g., $\Delta b$) result in proportional changes in $x$ according to the formula above.

Let me know if you need assistance with the calculations!