ColoState Spring 2016 Math 561 Homework 4

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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.
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