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ColoState Spring 2016 Math 561 Homework 4 Due Mon. 03/28/2016 Name: CSUID: (20 points) Problem 1. Let A ∈ Rm×n , r = rank(A), σ1 ≥ σ2 ≥ · · · ≥ σr > 0 be the singular values of A, and r ∑ A= σj uj vjT , j=1 where uj , vj are respectively the j-th column vectors of U, V in the singular value decomposition. For 1 ≤ k ≤ r, deﬁne k ∑ Ak = σj uj vjT . j=1 Show that ∥A − Ak ∥2 = min ∥A − B∥2 . B ∈ Rm×n rank(B) = k (20 points) Problem 2. Let√A be an n × n SPD matrix, b ∈ Rn , Ax∗ = b, ϕ(x) = ⟨Ax, x⟩ − 2⟨b, x⟩, and ∥x∥A = ⟨Ax, x⟩ for x ∈ Rn . (1) Show that ⟨Ax, y⟩ deﬁnes an inner product on Rn . (2) Show that ϕ(x) − ϕ(x∗ ) = ∥x − x∗ ∥2A . (20 points) Problem 3. Apply SVD to image compression as shown in Textbook p113 Example 3.4. Try any of these three images: clown.mat detail.mat mandrill.mat Test at least three diﬀerent values for k and report your relative errors and compression ratios. (20 points) Problem 4. Implement the Jacobi and Gauss-Seidel iterative methods in Matlab and test them with the 2-dim Poisson’s equation with the following data: the unit square [0, 1]×[0, 1], the exact solution u(x, y) = sin(πx) sin(πy), and n = 16, 32, 64, 128, respectively. Recall that for the Gauss-Seidel method, the Red-Black ordering works better than the natural ordering. Present the errors of the numerical solutions in the ∞− norm for both methods. (20 points) Problem 5. Implement the Steepest Descent method in Matlab and test it with the 2-dim Poisson’s equation with the following data: the unit square [0, 1] × [0, 1], the exact solution u(x, y) = sin(πx) sin(πy), and n = 16, 32, 64, 128, respectively. Present the errors of the numerical solutions in the ∞− norm.