You need to take any transcendental function and obtain 8 equally spaced data points. You need
to interpolate exactly at center (right after fourth data point) using Newton’s Forward Difference
Interpolation, Newton’s Backward Difference Interpolation, and Stirling Interpolation. Compare
your solutions with exact solution and make conclusion on the basis of your findings.

Note: (i) If in your case higher order differences appear to be zero, then you need to change the
function or nodes (points) before proceeding to next step.
(ii) In all computations you need to take 5 decimal computations throughout the task.
(iii) Instead of rounding you need to truncate your computed values after 5 decimal points.

Let's break this down into manageable steps to complete your request.

### Steps:

1. **Choose a transcendental function**: A transcendental function is one that cannot be expressed as a finite combination of algebraic operations (e.g., exponential, logarithmic, or trigonometric functions). For this example, I'll use the function \( f(x) = \sin(x) \).

2. **Obtain 8 equally spaced data points**: We need 8 data points of the function \( f(x) = \sin(x) \) over a chosen interval. The data points will be equidistant, and we can choose an appropriate step size to space the points.

3. **Apply interpolation methods**:
   - **Newton’s Forward Difference Interpolation**: This method is used to interpolate when the interpolation point is near the beginning of the table.
   - **Newton’s Backward Difference Interpolation**: This is applied when the interpolation point is near the end of the table.
   - **Stirling's Interpolation**: This method is particularly suited when the interpolation point lies near the center of the table of values, making it a good candidate for our case.

4. **Compare with the exact solution**: Using \( f(x) = \sin(x) \), we can compute the exact value at the point of interest (the midpoint of the interval) and compare it with the interpolated values.

5. **Conclusion**: After obtaining the results from the three interpolation methods and comparing them with the exact solution, we will analyze the accuracy and behavior of each method.

---

### Step-by-Step Solution:

1. **Choose the interval and step size**:
   Let's take the interval from \( x = 0 \) to \( x = \frac{\pi}{2} \) (i.e., 0 to 1.5708 radians). We will choose 8 equally spaced points, which gives us a step size \( h = \frac{\pi}{14} \).

   So the points are:
   \[
   x_0 = 0, \quad x_1 = h, \quad x_2 = 2h, \dots, \quad x_7 = 7h
   \]
   where \( h = \frac{\pi}{14} \approx 0.2244 \).

2. **Data Points Calculation**:
   The function values at these points will be:
   \[
   f(x_0), f(x_1), \dots, f(x_7)
   \]
   where \( f(x) = \sin(x) \).

   Let's calculate the values of \( \sin(x) \) at these points and then proceed to construct the forward and backward difference tables for interpolation.

I will now calculate these values to proceed with the interpolation techniques.We have calculated 8 equally spaced data points for \( f(x) = \sin(x) \) on the interval \( [0, \frac{\pi}{2}] \) with a step size of \( h = \frac{\pi}{14} \approx 0.2244 \). Here are the \( x \)-values and their corresponding \( f(x) = \sin(x) \) values (truncated to 5 decimal places):

\[
\begin{aligned}
x_0 &= 0, & f(x_0) &= 0.00000 \\
x_1 &= 0.22440, & f(x_1) &= 0.22252 \\
x_2 &= 0.44880, & f(x_2) &= 0.43388 \\
x_3 &= 0.67320, & f(x_3) &= 0.62348 \\
x_4 &= 0.89760, & f(x_4) &= 0.78183 \\
x_5 &= 1.12200, & f(x_5) &= 0.90096 \\
x_6 &= 1.34640, & f(x_6) &= 0.97492 \\
x_7 &= 1.57080, & f(x_7) &= 1.00000 \\
\end{aligned}
\]

Next, we'll calculate the interpolations at the midpoint (after the fourth data point) using the following methods:

1. Newton's Forward Difference Interpolation
2. Newton's Backward Difference Interpolation
3. Stirling Interpolation

I will begin by constructing the forward difference table for the first method.It seems like I can’t do more advanced data analysis right now. Please try again later. If you have any other questions or need assistance with something else, feel free to ask!

You need to take any transcendental function and obtain 8 equally spaced data points. You need to interpolate exactly at center (right after fourth data point) using Newton’s Forward Difference Interpolation, Newton’s Backward Difference Interpolation, and Stirling Interpolation. Compare your solutions with exact solution and make conclusion on the basis of your findings.

Explore the application of Newton’s Forward, Backward, and Stirling Interpolation methods on a transcendental function. Learn how to interpolate 8 equally spaced data points from f(x) = sin(x) and compare the interpolated results with the exact solution. Suitable for college-level numerical methods students.

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