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Asked: 2024-09-27T15:13:22+05:30 In:

How to Implement the Blancmange Function Calculation with Iteration Optimization in Python?

I recently stumbled upon this interesting concept called the Blancmange function while diving into some mathematical curiosities, and I thought it might spark some fun discussions or coding challenges here. So, I have a little problem I’d love for you all to help brainstorm or solve!

Here’s the lowdown: The Blancmange function is this fascinating continuous curve that’s defined over the interval [0, 1] and showcases some really curious properties. It’s constructed using the height of triangular peaks that are infinitely repeated, with each peak getting smaller and smaller as they extend outwards. The visual of it is quite mesmerizing, and it has some very remarkable traits that make it an intriguing topic for coding!

So, here’s the challenge—can you write a function that calculates the value of the Blancmange function at a given point x in the range [0, 1]?

To make it a bit more interactive, let’s say you need to accommodate a parameter `n`, which refers to how many iterations of the peaks you want to calculate. The more iterations you include, the more accurate your result will be. But be warned! Too many iterations can lead to slower performance and potentially long runtime.

I can already imagine different approaches people might take. Some could try brute-forcing the calculation by explicitly summing the heights of triangles, while others might find clever mathematical shortcuts or optimizations. It’d be really cool to see the different coding styles and how you tackle the problem!

Also, if you want to get creative, it would be great to add some features, like plotting the function for a range of points or visualizing the effect of increasing `n`. Wouldn’t that be a fun addition?

Looking forward to seeing all your coding wizardry!

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    2 Answers

    1. Blancmange Function Calculator

      Here’s a simple Python function to calculate the value of the Blancmange function at a given point x in the range [0, 1] with a parameter n for iterations!

      
def blancmange(x, n):
    total_height = 0
    for i in range(n):
        # The width of each triangle peak decreases as 1/2^i
        width = 1 / (2 ** i)
        # The peak height decreases too, specifically it's 1 - 2 * |x - (0.5 * (2*k + 1))| for each peak k
        for k in range(2**i):
            peak_center = k * width
            peak_height = 1 - 2 * abs(x - (peak_center + width / 2))
            if peak_height > 0:
                total_height += peak_height
    return total_height

      Usage Example

      To use this function, you can call it like this:

      
result = blancmange(0.3, 10)  # Calculate the value at x = 0.3 with 10 iterations
print(result)  # This will print the calculated value!

      Plotting the Function

      For a cool visual, you might want to plot the function using matplotlib!

      
import matplotlib.pyplot as plt
import numpy as np

x_values = np.linspace(0, 1, 100)  # 100 points between 0 and 1
y_values = [blancmange(x, 10) for x in x_values]

plt.plot(x_values, y_values)
plt.title('Blancmange Function Plot')
plt.xlabel('x')
plt.ylabel('Blancmange(x)')
plt.grid(True)
plt.show()

      Feel free to tweak the number of iterations and see how it changes the results!

    2. Blancmange Function Calculation

      The Blancmange function can be approximated using a summation of triangular peaks defined over the interval [0, 1]. To compute the value of the function at a given point x with respect to a number of iterations n, we can utilize the following Python function:

              
def blancmange(x, n):
    if x < 0 or x > 1:
        raise ValueError("x must be in the range [0, 1]")
    result = 0.0
    for k in range(n):
        peak_height = max(0, 1 - abs(x * 2 - (k + 1)))
        result += peak_height * (1 / (2 ** k))
    return result

# Example of usage:
x_value = 0.5  # Point at which we want to evaluate the function
n_iterations = 10  # Number of iterations for peak calculation
print(f"Blancmange Function at x={x_value} with n={n_iterations}: {blancmange(x_value, n_iterations)}")

      In the above code, the function blancmange calculates the height of the Blancmange function at a specified point x after n iterations. It sums the contributions of each triangular peak where heights decrease by a factor of 2 for every iteration. Additionally, to visualize the function over a range of points, you can plot it using libraries like Matplotlib in Python:

              
import numpy as np
import matplotlib.pyplot as plt

x_values = np.linspace(0, 1, 100)
y_values = [blancmange(x, n_iterations) for x in x_values]

plt.plot(x_values, y_values)
plt.title("Blancmange Function Visualization")
plt.xlabel("x")
plt.ylabel("Blancmange(x)")
plt.grid()
plt.show()

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