Ledoit-Wolf Shrinkage

Time Limit: 2sMemory: 256MB

Problem

In high-dimensional portfolio management, the sample covariance matrix is noisy and can be singular. Ledoit-Wolf shrinkage stabilizes it by blending with a structured target:

$\hat{\Sigma}_{\text{shrunk}} = \alpha F + (1 - \alpha) \hat{\Sigma}$

Given a 2x2 sample covariance matrix $\hat{\Sigma}$ , a shrinkage target $F$ , and the optimal shrinkage intensity $\alpha$ , compute the shrunk covariance matrix.

Input Format

Three lines:

Line 1: four floats s11 s12 s21 s22 (sample covariance, row-major)
Line 2: four floats f11 f12 f21 f22 (shrinkage target, row-major)
Line 3: one float alpha (shrinkage intensity)

Output Format

Three space-separated values: c11 c12 c22 (the 3 unique elements of the symmetric shrunk matrix), each to 4 decimal places.

Examples

Example 1

Input(Three lines:)

0.04 0.01 0.01 0.09 0.04 0.0 0.0 0.09 0.3

Output

0.0400 0.0070 0.0900

c11 = 0.3*0.04 + 0.7*0.04 = 0.04. c12 = 0.3*0 + 0.7*0.01 = 0.007. c22 = 0.3*0.09 + 0.7*0.09 = 0.09.

Example 2

Input(Three lines:)

0.10 0.03 0.03 0.08 0.05 0.0 0.0 0.05 0.5

Output

0.0750 0.0150 0.0650

c11 = 0.5*0.05 + 0.5*0.10 = 0.075. c12 = 0.5*0 + 0.5*0.03 = 0.015. c22 = 0.5*0.05 + 0.5*0.08 = 0.065.

Constraints

•4 space-separated floats for sample covariance (row-major 2x2): s11 s12 s21 s22
•4 space-separated floats for shrinkage target (row-major 2x2): f11 f12 f21 f22
•1 float for shrinkage intensity alpha in [0, 1]
•Output 3 unique elements (c11 c12 c22) to 4 decimal places

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