To find a simple function that fits the points you provided, we can analyze the relationship between $n$ and the corresponding function values.

The points you provided are:

$(128, 12.18), (256, 20.04), (512, 35.59), (1024, 66.09), (2048, 119.27), (4096, 214.26), (4096, 214.26), (8192, 397.62), (8192, 397.62)$

Step 1: Try Common Functional Forms

We can start by testing some common functional forms like:

1. Linear: $f(n) = a n + b$
2. Power Law: $f(n) = a n^k$
3. Logarithmic: $f(n) = a \log_b(n)$
4. Root function: $f(n) = a \sqrt{n}$
5. Log-Linear: $f(n) = a n \log_b(n)$

Step 2: Power Law Hypothesis

We will start by testing a power law hypothesis of the form:

$f(n) = a n^k$

To determine the values of $a$ and $k$, we can take the logarithm of both sides:

$\log(f(n)) = \log(a n^k) = \log(a) + k \log(n)$

This is now in the form of a linear equation $y = mx + b$, where $y = \log(f(n))$ and $x = \log(n)$. We can then perform linear regression on $\log(f(n))$ versus $\log(n)$ to find $k$ and $\log(a)$.

Step 3: Logarithmic Plot and Linear Regression

I'll calculate the logarithmic values for $n$ and $f(n)$, then use linear regression to estimate $k$ and $a$. numpy as np import matplotlib.pyplot as plt from sklearn.linear_model import LinearRegression

Given points

n_values = np.array([128, 256, 512, 1024, 2048, 4096, 4096, 8192, 8192]) f_values = np.array([12.18, 20.04, 35.59, 66.09, 119.27, 214.26, 214.26, 397.62, 397.62])

Taking log of n and f(n)

log_n = np.log(n_values) log_f = np.log(f_values)

Linear regression to find k and log(a)

model = LinearRegression() model.fit(log_n.reshape(-1, 1), log_f)

The slope is k and the intercept is log(a)

k = model.coef_[0] log_a = model.intercept_

Calculate a from log(a)

a = np.exp(log_a)

k, aThe results of the regression are:

• $k \approx 0.848$
• $a \approx 0.187$

Thus, the power law function that best describes the data is approximately:

$f(n) = 0.187 n^{0.848}$

Step 4: Plot the Data and the Fitted Function

Let's plot the given points along with the fitted power law function to visualize how well it fits the data. Generate values from the fitted function for plotting n_range = np.linspace(min(n_values), max(n_values), 500) fitted_values = a * n_range**k

Plot the data points and the fitted curve

plt.figure(figsize=(8,6)) plt.scatter(n_values, f_values, color='blue', label='Data Points') plt.plot(n_range, fitted_values, color='red', label=f'Fitted Function: $f(n) = 0.187 n^{0.848}$') plt.xlabel('n') plt.ylabel('f(n)') plt.xscale('log') plt.yscale('log') plt.legend() plt.title('Fitting a Power Law to the Data') plt.grid(True) plt.show()Here is the plot showing the data points along with the fitted power law function $f(n) = 0.187 n^{0.848}$. As we can see, the function fits the data reasonably well.

Would you like more details on how this function was derived, or do you have any other questions? Here are some related questions you may find interesting:

1. How would this function behave for large values of $n$?
2. What other types of functions could potentially fit this data?
3. How can we determine the accuracy of this model?
4. How would you fit a logarithmic or exponential function to this data?
5. What would happen if we used a different regression method?

Tip: When fitting functions, it's important to consider the scale of the data. Using log-log plots (as in this case) is often helpful for fitting power law functions.