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project1/Project1_Q1.py

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