108 lines
3.4 KiB
Python
108 lines
3.4 KiB
Python
import numpy as np
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def proj(A:np.ndarray, b:np.ndarray) -> np.ndarray:
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"""
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Project vector b onto the column space of matrix A using the formula:
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proj_A(b) = A(A^* A)^(-1) A^* b
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Parameters:
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A (np.ndarray): The matrix whose column space we are projecting onto.
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b (np.ndarray): The vector to be projected.
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Returns:
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np.ndarray: The projection of b onto the column space of A.
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"""
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# Check if A is full rank
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if np.linalg.matrix_rank(A) < min(A.shape):
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raise ValueError("Matrix A must be full rank.")
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# Compute the projection
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AhA_inv = np.linalg.inv(A.conj().T @ A)
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projection = A @ AhA_inv @ A.conj().T @ b
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return projection
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def proj_SVD(A:np.ndarray, b:np.ndarray) -> np.ndarray:
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"""
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Project vector b onto the column space of matrix A using SVD.
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Parameters:
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A (np.ndarray): The matrix whose column space we are projecting onto.
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b (np.ndarray): The vector to be projected.
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Returns:
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np.ndarray: The projection of b onto the column space of A.
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"""
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# A = U S V^*
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U, S, Vh = np.linalg.svd(A, full_matrices=False)
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"""
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proj_A(b) = A(A^* A)^(-1) A^* b
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= U S V^* (V S^2 V^*)^(-1) V S U^* b
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= U S V^* V S^(-2) V^* V S U^* b
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= U U^* b
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"""
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# Compute the projection using the SVD components
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projection = U @ U.conj().T @ b
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return projection
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def build_A(eps: float) -> np.ndarray:
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"""
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Build the matrix A(ε) as specified in the problem statement.
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"""
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A = np.array([
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[1.0, 1.0, 0.0, (1.0 - eps)],
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[1.0, 0.0, 1.0, (eps + (1.0 - eps))],
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[0.0, 1.0, 0.0, eps],
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[1.0, 0.0, 0.0, (1.0 - eps)],
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[1.0, 0.0, 0.0, (eps + (1.0 - eps))]
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], dtype=float)
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return A
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def build_b() -> np.ndarray:
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"""
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Build the vector b as specified in the problem statement.
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"""
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b = np.array([2.0, 1.0, 3.0, 1.0, 4.0])
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return b
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def question_1a(A: np.ndarray, b: np.ndarray):
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# Question 1(a)
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print("Question 1(a):")
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projection = proj(A, b)
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print("Projection of b onto the column space of A(1): (Using proj function)")
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print(projection)
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projection_svd = proj_SVD(A, b)
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print("Projection of b onto the column space of A(1): (Using proj_SVD function)")
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print(projection_svd)
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print("Difference between the two methods:")
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print(projection - projection_svd)
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def question_1b(A: np.ndarray, b: np.ndarray):
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# Question 1(b)
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print("\nQuestion 1(b):")
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for eps in [1.0, 1e-1, 1e-2, 1e-4, 1e-8, 1e-16, 1e-32]:
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print(f"\nFor ε = {eps}:")
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A_eps = build_A(eps)
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try:
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projection_eps = proj(A_eps, b)
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print(f"Projection of b onto the column space of A({eps}):")
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print(projection_eps)
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except np.linalg.LinAlgError as e:
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print(f"LinAlgError for eps={eps}: {e}")
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except ValueError as ve:
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print(f"ValueError for eps={eps}: {ve}")
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projection_svd_eps = proj_SVD(A_eps, b)
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print(f"Projection of b onto the column space of A({eps}) using SVD:")
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print(projection_svd_eps)
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print("Difference between the two methods:")
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try:
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print(projection_eps - projection_svd_eps)
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except:
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print("Could not compute difference due to previous error.")
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projection_eps = None # Reset for next iteration
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if __name__ == "__main__":
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A = build_A(eps=1.0)
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b = build_b()
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question_1a(A, b)
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question_1b(A, b)
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