Problem Statement

Given the eigenvalues (or squared singular values) from a PCA decomposition, compute the explained variance ratio for each component and the cumulative explained variance.

Algorithm:

1. Let the total variance = sum of all eigenvalues.
2. The ratio for eigenvalue λ_i is λ_i / total.
3. The cumulative ratios are the running sum of the individual ratios.

Function signature:

def explained_variance(evals: list) -> tuple:
    # returns (ratios, cumulative)
    # both are lists of floats in the same order as input

Input Format

Line 1: k — number of eigenvalues
Lines 2..k+1: one float per line — eigenvalues in descending order

Output Format

Line 1: k space-separated ratios (10 significant figures each)
Line 2: k space-separated cumulative ratios (10 significant figures each)

Example

Input:

4
10.0
4.0
3.0
3.0

Output:

0.5 0.2 0.15 0.15
0.5 0.7 0.85 1

Notes

• The ratios always sum to 1.0.
• The cumulative array's last value is always 1.0.
• In PCA, eigenvalues are typically sorted in descending order, so the first component explains the most variance.