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above is a formula for a recurrence relation with U0=1 and U1=9, but there are some error so Uerror0=1.01 and Uerror1=9.01. how do i make a recurrence relation formula for the abs error for each n and proof if it the same with the first recurrence relation formula or not.
Explain it step by step. please help this is so important for my school life

Given the recurrence relation:

\[
U_n = \frac{9}{35} U_{n-1} - \frac{4}{35} U_{n+1}, \quad n \in \mathbb{N}
\]

with initial conditions \( U_0 = 1 \) and \( U_1 = 9 \), and the perturbed values \( U_{\text{error}0} = 1.01 \) and \( U_{\text{error}1} = 9.01 \), we want to determine the recurrence relation for the absolute error at each \( n \) and prove whether this new recurrence relation matches the initial one.

### Step-by-Step Solution:

1. **Define the Absolute Error**:

   Let the absolute error for each \( n \) be defined as:

   \[
   E_n = |U_{\text{error}n} - U_n|
   \]

   where \( U_{\text{error}n} \) is the perturbed value and \( U_n \) is the exact value from the original recurrence.

2. **Substitute Perturbed Values into the Recurrence Relation**:

   The perturbed recurrence relation can be written using the error values:

   \[
   U_{\text{error}n} = \frac{9}{35} U_{\text{error}(n-1)} - \frac{4}{35} U_{\text{error}(n+1)}
   \]

   From this, subtract the original recurrence relation:

   \[
   U_n = \frac{9}{35} U_{n-1} - \frac{4}{35} U_{n+1}
   \]

   to obtain the recurrence relation for the difference:

   \[
   U_{\text{error}n} - U_n = \frac{9}{35} (U_{\text{error}(n-1)} - U_{n-1}) - \frac{4}{35} (U_{\text{error}(n+1)} - U_{n+1}).
   \]

3. **Define the Error Recurrence**:

   Letting \( E_n = U_{\text{error}n} - U_n \), the above relation simplifies to:

   \[
   E_n = \frac{9}{35} E_{n-1} - \frac{4}{35} E_{n+1}.
   \]

4. **Absolute Value of Error Relation**:

   Since \( E_n \) is already the difference, to find the recurrence for the absolute error, we take:

   \[
   |E_n| = \left|\frac{9}{35} E_{n-1} - \frac{4}{35} E_{n+1}\right|.
   \]

   However, because \( |E_n| \neq E_n \) when the error terms can be negative, this expression simplifies differently depending on sign analysis.

5. **Comparison with the Original Recurrence Relation**:

   The form of the recurrence relation of the error matches that of the original recurrence. However, the values differ based on the initial conditions of \( E_0 \) and \( E_1 \). Specifically, the growth or decay of the error depends on whether the recurrence leads to an amplification or reduction of errors in subsequent steps, which depends on the coefficients in the recurrence.

### Conclusion:

- **The recurrence relation of the absolute error** \( |E_n| \) indeed has the same form as the original recurrence relation.
- **However, the propagation of errors is highly dependent on initial error values and might behave differently** because initial perturbations might amplify or dampen, depending on the specific numerical characteristics.

Would you like more details on how the absolute error evolves over time, or on analyzing the stability of this error recurrence relation?

### Related Questions:

1. How does the initial perturbation affect the long-term behavior of the error in the recurrence?
2. What is the role of the coefficients \( \frac{9}{35} \) and \( \frac{4}{35} \) in the stability of the recurrence?
3. Can we derive a closed-form expression for \( U_n \) or \( E_n \)?
4. How does the nature of the recurrence (linear, non-linear) affect error propagation?
5. What happens if we introduce a different type of perturbation, such as multiplicative noise?

### Tip:

For recurrence relations, always examine initial conditions carefully, as they significantly influence the evolution of the sequence, especially when analyzing error propagation.

The query submitted by the user: above is a formula for a recurrence relation with U0=1 and U1=9, but there are some errors so Uerror0=1.01 and Uerror1=9.01. How do I make a recurrence relation formula for the absolute error for each n and prove if it is the same as the first recurrence relation formula or not? Explain it step by step. Please help, this is so important for my school life.

Understand how to derive the recurrence relation for the absolute error in a sequence with given initial conditions. The problem involves key mathematical concepts such as recurrence relations, error analysis, and algebra. Suitable for high school students in Grades 10-12.

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