# Calculate the squared singular values Sigma_squared = Sigma ** 2 # Calculate the total variance (sum of squared singular values) total_variance = sum(Sigma_squared) # Initialize the cumulative variance for the truncated part cum_variance = 0 # Iterate over singular values to find the smallest k for k in range(len(Sigma)): # Add the k-th singular value squared to cumulative sum cum_variance += Sigma_squared[k] # Calculate the relative error for truncation at the current k relative_error = (total_variance - cum_variance) ** 0.5 / total_variance ** 0.5 # If the relative error is less than the target, return k if relative_error < rel_err_target: return k + 1 # k needs to be 1-based index, so add 1 # If no k results in the error being below the target, return the length of Sigma return len(Sigma)