From 4b5b5eeb4c3800e99e6c0628fcf8062fed12660e Mon Sep 17 00:00:00 2001 From: Zhe Yuan Date: Fri, 26 Sep 2025 22:16:05 -0400 Subject: [PATCH] first commit --- README.md | 0 project1/Project1.md | 85 +++++++ project1/Project1_A1.txt | 67 ++++++ project1/Project1_A2.txt | 15 ++ project1/Project1_A2d_orthogonality_loss.png | Bin 0 -> 75982 bytes project1/Project1_A2d_time_taken.png | Bin 0 -> 89336 bytes ...oject1_A3c_least_squares_approximation.png | Bin 0 -> 72403 bytes project1/Project1_A3e_coefficient_error.png | Bin 0 -> 38823 bytes project1/Project1_Q1.py | 107 +++++++++ project1/Project1_Q2.py | 209 ++++++++++++++++++ project1/Project1_Q3.py | 166 ++++++++++++++ project1/README.md | 13 ++ project1/requirements.txt | 2 + 13 files changed, 664 insertions(+) create mode 100644 README.md create mode 100644 project1/Project1.md create mode 100644 project1/Project1_A1.txt create mode 100644 project1/Project1_A2.txt create mode 100644 project1/Project1_A2d_orthogonality_loss.png create mode 100644 project1/Project1_A2d_time_taken.png create mode 100644 project1/Project1_A3c_least_squares_approximation.png create mode 100644 project1/Project1_A3e_coefficient_error.png create mode 100644 project1/Project1_Q1.py create mode 100644 project1/Project1_Q2.py create mode 100644 project1/Project1_Q3.py create mode 100644 project1/README.md create mode 100644 project1/requirements.txt diff --git a/README.md b/README.md new file mode 100644 index 0000000..e69de29 diff --git a/project1/Project1.md b/project1/Project1.md new file mode 100644 index 0000000..66e352f --- /dev/null +++ b/project1/Project1.md @@ -0,0 +1,85 @@ +# MATH 6601 - Project 1 Report + +Author: Zhe Yuan (yuan.1435) + +Date: Sep 2025 + +### 1. Projections + +**(a)** +The projection of vector $b$ onto the column space of a full-rank matrix $A$ is given by $p=A(A^*A)^{-1}A^*b$. A function `proj(A, b)` was implemented based on this formula (see `Project1_Q1.py`). For $\epsilon = 1$, the projection is: +`p ≈ [1.85714286, 1.0, 3.14285714, 1.28571429, 3.85714286]` + +**(b)** +As $\epsilon \rightarrow 0$, the normal equation method numerically unstable. A more robust SVD-based method, `proj_SVD(A, b)`, was implemented to handle this (see `Project1_Q1.py`). + +Results: +| $\epsilon$ | Normal Equation Method `proj(A, b)` | SVD Method `proj_SVD(A, b)` | Difference (Normal Eq - SVD) | +|:---:|:---|:---|:---| +| **1.0** | `[1.85714286 1. 3.14285714 1.28571429 3.85714286]` | `[1.85714286 1. 3.14285714 1.28571429 3.85714286]` | `[ 1.78e-15 -2.22e-15 -8.88e-16 8.88e-16 0.00e+00]` | +| **0.1** | `[1.85714286 1. 3.14285714 1.28571429 3.85714286]` | `[1.85714286 1. 3.14285714 1.28571429 3.85714286]` | `[ 7.28e-14 -4.44e-16 -2.66e-14 -1.62e-14 -5.28e-14]` | +| **0.01** | `[1.85714286 1. 3.14285714 1.28571429 3.85714286]` | `[1.85714286 1. 3.14285714 1.28571429 3.85714286]` | `[-5.45e-12 -7.28e-12 -3.45e-13 -4.24e-12 -4.86e-12]` | +| **1e-4** | `[1.85714297 1.00000012 3.14285716 1.28571442 3.85714302]` | `[1.85714286 1. 3.14285714 1.28571429 3.85714286]` | `[1.11e-07 1.19e-07 1.92e-08 1.36e-07 1.67e-07]` | +| **1e-8** | **Error:** `LinAlgError: Singular matrix` | `[1.85714286 1. 3.14285714 1.28571428 3.85714286]` | `Could not compute difference due to previous error.` | +| **1e-16**| **Error:** `ValueError: Matrix A must be full rank` | `[1.81820151 1. 3.18179849 2.29149804 2.89030045]` | `Could not compute difference due to previous error.` | +| **1e-32**| **Error:** `ValueError: Matrix A must be full rank` | `[2. 1. 3. 2.5 2.5]` | `Could not compute difference due to previous error.` | + +Numerical experiments show that as $\epsilon$ becomes small, the difference between the two methods increases. When $\epsilon$ becomes very small (e.g., 1e-8), the normal equation method fails while the SVD method continues to provide a stable solution. + +*** + + +### 2. QR Factorisation + +**(a, b, c)** +Four QR factorization algorithms were implemented to find an orthonormal basis $Q$ for the column space of the $n \times n$ Hilbert matrix (for $n=2$ to $20$): +1. Classical Gram-Schmidt (CGS, `cgs_q`) +2. Modified Gram-Schmidt (MGS, `mgs_q`) +3. Householder QR (`householder_q`) +4. Classical Gram-Schmidt Twice (CGS-Twice, `cgs_twice_q`) + +For implementation details, please see the `Project1_Q2.py` file. + +**(d)** +The quality of the computed $Q$ from each method was assessed by plotting the orthogonality loss, $log_{10}(||I - Q^T Q||_F)$, versus the matrix size $n$. + +![Orthogonality Loss Plot](Project1_A2d_orthogonality_loss.png) + +A summary of the loss and computation time for n = 4, 9, 11 is provided below. + + + +**(e)** +The speed and accuracy of the four methods were compared based on the plots. + +![Time Taken Plot](Project1_A2d_time_taken.png) + +- **Accuracy:** Householder is the most accurate and stable. MGS is significantly better than CGS. CGS-Twice improves on CGS, achieving accuracy comparable to MGS. CGS is highly unstable and loses orthogonality quickly. +- **Speed:** CGS and MGS are the fastest. Householder is slightly slower, and CGS-Twice is the slowest. + +*** + +### 3. Least Square Problem + +**(a)** +The problem is formulated as $Ax = b$, where $A$ is an $(N+1) \times (n+1)$ vandermonde matrix, $x$ is the $(n+1) \times 1$ vector of unknown polynomial coefficients $[a_0, ..., a_n]^T$, and $b$ is the $(N+1) \times 1$ vector of known function values $f(t_j)$. + +**(b)** +The least squares problem was solved using the implemented Householder QR factorization (`householder_lstsq` in `Project1_Q3.py`, the same function in Question 2) to find the coefficient vector $x$. + +**(c)** +The function $f(t) = 1 / (1 + t^2)$ and its polynomial approximation $p(t)$ were plotted for $N=30$ and $n=5, 15, 30$. + +![Least Squares Plot](Project1_A3c_least_squares_approximation.png) + +When $n=15$, the approximation is excellent. When $n=30$, the polynomial interpolates the points but exhibits wild oscillations between them. + +**(d)** +No, $p(t)$ will not converge to $f(t)$ if $N = n$ tends to infinity. Polynomial interpolation on equally spaced nodes diverges for this function, as demonstrated by the severe oscillations in the $n=30$ case. + +**(e)** +The error between the computed and true coefficients was plotted against the polynomial degree $n$. + +![Coefficient Error Plot](Project1_A3e_coefficient_error.png) + +The error in the computed coefficients grows significantly as $n$ increases. \ No newline at end of file diff --git a/project1/Project1_A1.txt b/project1/Project1_A1.txt new file mode 100644 index 0000000..fb95dbe --- /dev/null +++ b/project1/Project1_A1.txt @@ -0,0 +1,67 @@ +Question 1(a): +Projection of b onto the column space of A(1): (Using proj function) +[1.85714286 1. 3.14285714 1.28571429 3.85714286] +Projection of b onto the column space of A(1): (Using proj_SVD function) +[1.85714286 1. 3.14285714 1.28571429 3.85714286] +Difference between the two methods: +[ 1.77635684e-15 -2.22044605e-15 -8.88178420e-16 8.88178420e-16 + 0.00000000e+00] + +Question 1(b): + +For ε = 1.0: +Projection of b onto the column space of A(1.0): +[1.85714286 1. 3.14285714 1.28571429 3.85714286] +Projection of b onto the column space of A(1.0) using SVD: +[1.85714286 1. 3.14285714 1.28571429 3.85714286] +Difference between the two methods: +[ 1.77635684e-15 -2.22044605e-15 -8.88178420e-16 8.88178420e-16 + 0.00000000e+00] + +For ε = 0.1: +Projection of b onto the column space of A(0.1): +[1.85714286 1. 3.14285714 1.28571429 3.85714286] +Projection of b onto the column space of A(0.1) using SVD: +[1.85714286 1. 3.14285714 1.28571429 3.85714286] +Difference between the two methods: +[ 7.28306304e-14 -4.44089210e-16 -2.66453526e-14 -1.62092562e-14 + -5.28466160e-14] + +For ε = 0.01: +Projection of b onto the column space of A(0.01): +[1.85714286 1. 3.14285714 1.28571429 3.85714286] +Projection of b onto the column space of A(0.01) using SVD: +[1.85714286 1. 3.14285714 1.28571429 3.85714286] +Difference between the two methods: +[-5.45297141e-12 -7.28239691e-12 -3.44613227e-13 -4.24371649e-12 + -4.86499729e-12] + +For ε = 0.0001: +Projection of b onto the column space of A(0.0001): +[1.85714297 1.00000012 3.14285716 1.28571442 3.85714302] +Projection of b onto the column space of A(0.0001) using SVD: +[1.85714286 1. 3.14285714 1.28571429 3.85714286] +Difference between the two methods: +[1.11406275e-07 1.19209290e-07 1.91642426e-08 1.35516231e-07 + 1.67459071e-07] + +For ε = 1e-08: +LinAlgError for eps=1e-08: Singular matrix +Projection of b onto the column space of A(1e-08) using SVD: +[1.85714286 1. 3.14285714 1.28571428 3.85714286] +Difference between the two methods: +Could not compute difference due to previous error. + +For ε = 1e-16: +ValueError for eps=1e-16: Matrix A must be full rank. +Projection of b onto the column space of A(1e-16) using SVD: +[1.81820151 1. 3.18179849 2.29149804 2.89030045] +Difference between the two methods: +Could not compute difference due to previous error. + +For ε = 1e-32: +ValueError for eps=1e-32: Matrix A must be full rank. +Projection of b onto the column space of A(1e-32) using SVD: +[2. 1. 3. 2.5 2.5] +Difference between the two methods: +Could not compute difference due to previous error. diff --git a/project1/Project1_A2.txt b/project1/Project1_A2.txt new file mode 100644 index 0000000..2c05228 --- /dev/null +++ b/project1/Project1_A2.txt @@ -0,0 +1,15 @@ + +Table of Orthogonality Loss and Time for n=4, 9, 11: +n Method Loss Time (s) +4 Classical Gram-Schmidt -10.31105413525148 4.2772293090820314e-05 +4 Modified Gram-Schmidt -12.374458030887125 5.888938903808594e-05 +4 Householder -15.351739155108316 0.00014197826385498047 +4 Classical Gram-Schmidt Twice -15.570756316229907 8.988380432128906e-05 +9 Classical Gram-Schmidt 0.3894252720607535 0.00028896331787109375 +9 Modified Gram-Schmidt -5.257678985289507 0.00034956932067871095 +9 Householder -14.715202433670639 0.0006194353103637695 +9 Classical Gram-Schmidt Twice -8.398044187428138 0.0006073951721191406 +11 Classical 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SJ>f$4z>?p;J89S9v;QBIji?p? literal 0 HcmV?d00001 diff --git a/project1/Project1_Q1.py b/project1/Project1_Q1.py new file mode 100644 index 0000000..653dea2 --- /dev/null +++ b/project1/Project1_Q1.py @@ -0,0 +1,107 @@ +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) + diff --git a/project1/Project1_Q2.py b/project1/Project1_Q2.py new file mode 100644 index 0000000..378362f --- /dev/null +++ b/project1/Project1_Q2.py @@ -0,0 +1,209 @@ +import numpy as np +import time +import matplotlib.pyplot as plt + +def cgs_q(A: np.ndarray) -> np.ndarray: + """ + Perform Classical Gram-Schmidt orthogonalization on matrix A. + + Parameters: + A (np.ndarray): The input matrix to be orthogonalized. + + Returns: + np.ndarray: The orthogonalized matrix Q. + """ + m, n = A.shape + Q = np.zeros((m, n), dtype=A.dtype) + + for j in range(n): + v = A[:, j] + for i in range(j): + # r_ij = q_i^* a_j + r_ij = np.dot(Q[:, i].conj(), A[:, j]) + # v = a_j - r_ij * q_i + v = v - r_ij * Q[:, i] + # v = a_j - sum_{i=1}^{j-1} r_ij * q_i + # r_jj = ||v|| + r_jj = np.linalg.norm(v) + #if r_jj < 1e-14: + # print(f"Warning: Vector {j} is close to zero after orthogonalization.") + Q[:, j] = v / r_jj + + return Q + +def mgs_q(A: np.ndarray) -> np.ndarray: + """ + Perform Modified Gram-Schmidt orthogonalization on matrix A. + + Parameters: + A (np.ndarray): The input matrix to be orthogonalized. + + Returns: + np.ndarray: The orthogonalized matrix Q. + """ + m, n = A.shape + Q = np.zeros((m, n), dtype=A.dtype) + V = A.copy() + + for j in range(n): + for i in range(j): + # r_ij = q_i^* a_j + r_ij = np.dot(Q[:, i].conj(), V[:, j]) + # a_j = a_j - r_ij * q_i + V[:, j] = V[:, j] - r_ij * Q[:, i] + # r_jj = ||a_j|| + r_jj = np.linalg.norm(V[:, j]) + #if r_jj < 1e-14: + # print(f"Warning: Vector {j} is close to zero after orthogonalization.") + Q[:, j] = V[:, j] / r_jj + return Q + +def householder_q(A: np.ndarray) -> np.ndarray: + """ + Perform Householder QR factorization on matrix A to obtain orthogonal matrix Q. + + Parameters: + A (np.ndarray): The input matrix to be orthogonalized. + + Returns: + np.ndarray: The orthogonalized matrix Q. + """ + m, n = A.shape + Q = np.eye(m, dtype=A.dtype) + R = A.copy() + + for k in range(n): + # Extract the vector x + # Target: x -> [||x||, 0, 0, ..., 0]^T + x = R[k:, k] # x = [a_k, a_(k+1), ..., a_m]^T + s = np.sign(x[0]) if x[0] != 0 else 1 + e = np.zeros_like(x) # e = [||x||, 0, 0, ..., 0]^T + e[0] = s * np.linalg.norm(x) + # Compute the Householder vector + # x - e is the reflection plane normal + u = x - e + if np.linalg.norm(u) < 1e-14: + continue # Skip the transformation if u is too small + v = u / np.linalg.norm(u) + # Apply the Householder transformation to R[k:, k:] + R[k:, k:] -= 2 * np.outer(v, v.conj().T @ R[k:, k:]) + # Update Q + # Q_k = I - 2 v v^* + Q_k = np.eye(m, dtype=A.dtype) + Q_k[k:, k:] -= 2 * np.outer(v, v.conj().T) + Q = Q @ Q_k + return Q + +def cgs_twice_q(A: np.ndarray) -> np.ndarray: + """ + Perform Classical Gram-Schmidt orthogonalization twice on matrix A. + + Parameters: + A (np.ndarray): The input matrix to be orthogonalized. + + Returns: + np.ndarray: The orthogonalized matrix Q. + """ + return cgs_q(cgs_q(A)) + +def ortho_loss(Q: np.ndarray) -> float: + """ + Compute the orthogonality loss of matrix Q. + + Parameters: + Q (np.ndarray): The orthogonal matrix to be evaluated. + + Returns: + float: The orthogonality loss defined as log10(||I - Q^* Q||_F). + """ + m, n = Q.shape + I = np.eye(n, dtype=Q.dtype) + E = I - Q.conj().T @ Q + loss = np.linalg.norm(E, ord='fro') + return np.log10(loss) + +def build_hilbert(m: int, n: int) -> np.ndarray: + """ + Build an m x n Hilbert matrix. + + Parameters: + m (int): Number of rows. + n (int): Number of columns. + + Returns: + np.ndarray: The m x n Hilbert matrix. + """ + H = np.zeros((m, n), dtype=float) + for i in range(m): + for j in range(n): + H[i, j] = 1.0 / (i + j + 1) + return H + + +def question_2bcd(n_min=2, n_max=20): + # Question 2(b) + ns = list(range(n_min, n_max + 1)) + methods = { + "Classical Gram-Schmidt": cgs_q, + "Modified Gram-Schmidt": mgs_q, + "Householder": householder_q, + "Classical Gram-Schmidt Twice": cgs_twice_q + } + losses = {name: [] for name in methods.keys()} + times = {name: [] for name in methods.keys()} + + for n in ns: + H = build_hilbert(n, n) + # run ten times and take the average + times_temp = {name: [] for name in methods.keys()} + losses_temp = {name: [] for name in methods.keys()} + for _ in range(10): + for name, method in methods.items(): + start_time = time.time() + Q = method(H) + end_time = time.time() + loss = ortho_loss(Q) + times_temp[name].append(end_time - start_time) + losses_temp[name].append(loss) + for name in methods.keys(): + times[name].append(np.mean(times_temp[name])) + losses[name].append(np.log(np.mean(np.exp(losses_temp[name])))) + #print(f"n={n}, Method={name}, Loss={loss}, Time={end_time - start_time}s") + + # Plot the orthogonality loss + plt.figure(figsize=(10, 6)) + for name, loss in losses.items(): + plt.plot(ns, loss, marker='o', label=name) + plt.xlabel('Matrix Size n') + plt.ylabel('Orthogonality Loss (log10 scale)') + plt.title('Orthogonality Loss vs Matrix Size for Different QR Methods') + # set x ticks to be integers + plt.xticks(ns) + plt.legend() + plt.grid(True, alpha=0.3) + plt.savefig('Project1_A2d_orthogonality_loss.png') + #plt.show() + # Plot the time taken + plt.figure(figsize=(10, 6)) + for name, time_list in times.items(): + plt.plot(ns, time_list, marker='o', label=name) + plt.xlabel('Matrix Size n') + plt.ylabel('Time Taken (seconds)') + plt.title('Time Taken vs Matrix Size for Different QR Methods') + # set x ticks to be integers + plt.xticks(ns) + plt.legend() + plt.grid(True, alpha=0.3) + plt.savefig('Project1_A2d_time_taken.png') + #plt.show() + + # Table of results (n=4, 9, 11) + print("\nTable of Orthogonality Loss and Time for n=4, 9, 11:") + print(f"{'n':<5} {'Method':<30} {'Loss':<20} {'Time (s)':<10}") + for n in [4, 9, 11]: + for name in methods.keys(): + idx = ns.index(n) + print(f"{n:<5} {name:<30} {losses[name][idx]:<20} {times[name][idx]:<10}") + +if __name__ == "__main__": + question_2bcd() \ No newline at end of file diff --git a/project1/Project1_Q3.py b/project1/Project1_Q3.py new file mode 100644 index 0000000..77e0e4f --- /dev/null +++ b/project1/Project1_Q3.py @@ -0,0 +1,166 @@ +import numpy as np +import matplotlib.pyplot as plt + +# householder function from question 2 +def householder_qr(A: np.ndarray) -> tuple[np.ndarray, np.ndarray]: + """ + Perform Householder QR factorization on matrix A to obtain orthogonal matrix Q. + + Parameters: + A (np.ndarray): The input matrix to be orthogonalized. + + Returns: + tuple[np.ndarray, np.ndarray]: The orthogonal matrix Q and upper triangular matrix R such that A = QR. + """ + m, n = A.shape + Q = np.eye(m, dtype=A.dtype) + R = A.copy() + + for k in range(n): + # Extract the vector x + # Target: x -> [||x||, 0, 0, ..., 0]^T + x = R[k:, k] # x = [a_k, a_(k+1), ..., a_m]^T + s = np.sign(x[0]) if x[0] != 0 else 1 + e = np.zeros_like(x) # e = [||x||, 0, 0, ..., 0]^T + e[0] = s * np.linalg.norm(x) + # Compute the Householder vector + # x - e is the reflection plane normal + u = x - e + if np.linalg.norm(u) < 1e-14: + continue # Skip the transformation if u is too small + v = u / np.linalg.norm(u) + # Apply the Householder transformation to R[k:, k:] + R[k:, k:] -= 2 * np.outer(v, v.conj().T @ R[k:, k:]) + # Update Q + # Q_k = I - 2 v v^* + Q_k = np.eye(m, dtype=A.dtype) + Q_k[k:, k:] -= 2 * np.outer(v, v.conj().T) + Q = Q @ Q_k + + return Q, R + +def householder_lstsq(A: np.ndarray, b: np.ndarray) -> np.ndarray: + """ + Solve the least squares problem Ax = b using Householder reflections. + + Parameters: + A (np.ndarray): The input matrix A. + b (np.ndarray): The right-hand side vector b. + + Returns: + np.ndarray: The solution vector x. + """ + """ + Ax = b, A = QR + Rx = Q^*b + x = R^{-1}Q^*b + R is upper triangular + Q is orthogonal + """ + m, n = A.shape # m is N + 1, n is n + 1 + Q, R = householder_qr(A) + # Compute Q^*b + Qt_b = Q.conj().T @ b + # Solve Rx = Q^*b using back substitution + x = np.zeros(n, dtype=A.dtype) + for i in range(n-1, -1, -1): + x[i] = (Qt_b[i] - R[i, i+1:] @ x[i+1:]) / R[i, i] + return x + +def f1(t: float) -> float: + """ + f1(t) = 1 / (1 + t^2) + """ + return 1.0 / (1.0 + t**2) + +def build_A(N: int, n: int, f: callable) -> tuple[np.ndarray, np.ndarray]: + """ + Build the matrix A for the least squares problem. + + Parameters: + n + 1 (int): The number of columns in the matrix A. + + Returns: + np.ndarray: The constructed matrix A. + """ + """ + {t_j}j=0->N, t_0=-5, t_N=5, t_j divides [-5, 5] equally + A = [[1, t_0, t_0^2, t_0^3, ..., t_0^n], + [1, t_1, t_1^2, t_1^3, ..., t_1^n], + ... + [1, t_N, t_N^2, t_N^3, ..., t_N^n]] (N + 1) x (n + 1) + + b = [f(t_0), f(t_1), ..., f(t_N)]^T (N + 1) x 1 + x = [a_0, a_1, a_2, ..., a_n]^T (n + 1) x 1 + Ax = b, x is the coeffs of the polynomial + """ + A = np.zeros((N + 1, n + 1), dtype=float) + t = np.linspace(-5, 5, N + 1) + for i in range(N + 1): + A[i, :] = [t[i]**j for j in range(n + 1)] + b = np.array([f(t_i) for t_i in t]) + return A, b + +def question_3c(): + Ns = [30, 30, 30] + ns = [5, 15, 30] + # p(t) = a_n t^n + a_(n-1) t^(n-1) + ... + a_1 t + a_0 + # plot f(t) and p(t) on the same figure for each (N, n) + t = np.linspace(-5, 5, 1000) + f_t = f1(t) + plt.figure(figsize=(15, 5)) + for i in range(len(Ns)): + N = Ns[i] + n = ns[i] + A = np.zeros((N + 1, n + 1), dtype=float) + A, b = build_A(N, n, f1) + x = householder_lstsq(A, b) + p_t = sum([x[j] * t**j for j in range(n + 1)]) + plt.subplot(1, 3, i + 1) + plt.plot(t, f_t, label='f(t)', color='blue') + plt.plot(t, p_t, label='p(t)', color='red', linestyle='--') + plt.title(f'N={N}, n={n}') + plt.xlabel('t') + plt.ylabel('y') + plt.legend() + plt.grid(True, alpha=0.3) + plt.suptitle('Least Squares Polynomial Approximation using Householder QR') + plt.tight_layout(rect=[0, 0.03, 1, 0.95]) + plt.savefig('Project1_A3c_least_squares_approximation.png') + #plt.show() + +def f2(n:int, t: float) -> float: + """ + f2(t) = t^n + t^(n-1) + ... + t^2 + t + 1 + """ + return sum([t**i for i in range(n + 1)]) + +def question_3e(): + # n = 4, 6, 8, 10, ..., 20 + ns = list(range(4, 21, 2)) + # N = 2n + # quantity is log10||(a_0-1,...,a_n-1)||_2) + losses = [] + for n in ns: + N = 2 * n + A, b = build_A(N, n, lambda t: f2(n, t)) + x = householder_lstsq(A, b) + true_coeffs = np.array([1.0 for _ in range(n + 1)]) + loss = np.linalg.norm(x - true_coeffs) + losses.append(np.log10(loss)) + plt.figure(figsize=(10, 6)) + plt.plot(ns, losses, marker='o') + plt.xlabel('Polynomial Degree n') + plt.ylabel('log10(||a - true_coeffs||_2)') + plt.title('Coefficient Error vs Polynomial Degree using Householder QR') + plt.xticks(ns) + plt.grid(True, alpha=0.3) + plt.savefig('Project1_A3e_coefficient_error.png') + #plt.show() + + +if __name__ == "__main__": + question_3c() + question_3e() + + diff --git a/project1/README.md b/project1/README.md new file mode 100644 index 0000000..e32ca4f --- /dev/null +++ b/project1/README.md @@ -0,0 +1,13 @@ +### Create a virtual environment and install dependencies + +```bash +conda create -n math6601_env python=3.10 +conda activate math6601_env +pip install -r requirements.txt +``` + +### Question 1 + +```bash +python Project1_Q1.py > Project1_A1.txt +``` diff --git a/project1/requirements.txt b/project1/requirements.txt new file mode 100644 index 0000000..806f221 --- /dev/null +++ b/project1/requirements.txt @@ -0,0 +1,2 @@ +numpy +matplotlib \ No newline at end of file