Objective: This activity is designed to help you understand the concept of Lagrange interpolation, a polynomial interpolation method used in numerical analysis and applied mathematics.

Material:

• Internet access
• Pen and paper
• A computer for running Python code

Instructions:

Part 1: Understanding Lagrange Interpolation

1. Research: Start by researching the definition and basics of Lagrange interpolation. Some key points to understand are:
   • What is Lagrange interpolation?
   • How does it work?
   • What is its mathematical formula?
   • When is it used?
2. Conceptual Understanding: Once you’ve read about Lagrange interpolation, try to explain the concept to someone else or write down your understanding. This helps consolidate your learning.
3. Examples: Look up examples of Lagrange interpolation. This could be mathematical examples where you’re given a set of points and you need to find the interpolating polynomial using the Lagrange formula.

Part 2

1. Setup your development environment. Open your web browser and go to https://replit.com/. Sign up for a new account if you don’t already have one, or log in.
2. Start a new Python project. Click on the “+ New Repl” button on the top right, select “Python” as the language, give your project a name, and click on “Create Repl”.
3. Write the Lagrange interpolation function. In the code editor, write a function named lagrange_interpolation. This function should take in three parameters: a list of x values (x_values), a list of y values (y_values), and the x value to interpolate (x). Use the provided code snippet for this step:

def lagrange_interpolation(x_values, y_values, x):
n = len(x_values)
y = 0
for i in range(n):
p = 1
for j in range(n):
if i != j:
p *= (x - x_values[j]) / (x_values[i] - x_values[j])
y += y_values[i] * p
return y

4. Test the Lagrange interpolation function. Still in the same Python file, create a list of x values (x_values) and a list of y values (y_values). Call the lagrange_interpolation function with these lists and the x value to interpolate. Print the result. Use the provided code snippet for this step:

x_values = [0, 1, 2, 3]
y_values = [1, 2, 1, 10]
print(lagrange_interpolation(x_values, y_values, 2.5)) # Outputs: 3.75

5. Run the code. Click the ‘Run’ button at the top of the page. The output of the code will be displayed on the right side of the page.
6. Analyze the results.
   • Try changing the x value in the lagrange_interpolation function call. What happens when you interpolate at a value of x that is within your list of x values? What happens when you interpolate at a value of x that is outside your list of x values?
   • Try changing the y_values. How do the results change when you alter these values?
7. Write a short report. After performing the above steps, write a short report explaining your findings. Include the answers to the questions asked in the analysis step. Also, discuss any challenges you faced and how you overcame them.

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