Math 641-600 Assignments
Assignment 1 - Due Wednesday, September 6.
- Read sections 1.1-1.5.
- Do the following problems.
- Section 1.1, problem 5, page 49.
- Section 1.1 problem 9(d), page 50. (Do up to degree 3.)
- Let x be in R2 and set
||x||p =
(|x|p+|y|p|1/p. For p=1, 3/2, 2, 4,
use your favorite software (MATLAB is mine) to plot, on the same set of
axes, ||x||p = 1.
- Let U be a subspace of an inner product space V, with the inner
product and norm being < ·,· > and ||·||
Also, let v ∈ V, but suppose v is not in U. Show that if there is a
u0 for which
minu ∈ U ||v-u|| = ||v-u0||,
then v-u0 is orthogonal to every u ∈ U. Also,
show that u0 is unique, if it exists.
- Section 1.2, problem 1, page 50.
- Section 1.2, problem 6(d), page 51.
- Section 1.2, problem 9, page 52.
Assignment 2 - Due Friday, September 15.
- Read sections 1.1-1.5.
- Do the following problems.
- Section 1.2, problem 10(a), page 52.
- Let A and B be n×n matrices. Suppose that the range of B, M
= R(B), is an invariant subspace for A. Show that there is an
n×n matrix X such that AB = BX.
- Let U be a unitary, n×n matrix. Show that the following hold.
- < Ux, Uy > = < x, y >
- The eigenvalues of U all lie on the unit circle, |λ|=1.
- Suppose that U is real as well as unitary. (Such matrices are
orthogonal.) In an odd dimensional space, show that either 1
or − 1 is an eigenvalue of U. (It's possible for both 1
and − 1 to be eigenvalues.)
- Do an internet search for Gershgorin's theorem. Write
out both a statement and a proof of the theorem, even if all you do is
copy by hand what's on the web site. (Please reference the site used.)
Use Gershgorin's theorem to answer the questions below.
- Show that the eigenvalues of the matrix A (pg. 23) satisfy
λj ≤ 0. (The book directly shows that they are
actually negative.)
- Consider the matrix B =
4 1 1
0 3 -2
1 2 4
Show that the eigenvalues of B are all in the right-half plane. (I.e.,
Re(λ) > 0.)
- Section 1.3, problem 2, page 53.
- Section 1.4, problem 2, page 54.
- Section 1.4, problem 3, page 54.
Assignment 3 - Due Wednesday, September 27.
- Read sections 2.1-2.2.2.
- Do the following problems.
- Let V be a finite dimensional inner product space and let M be a
subspace of V. The orthogonal complement of M is
M⊥ = {v ∈ V | < v,w> = 0 for all w ∈ M}
Show that V = M⊕M⊥. (You may need to look up the
definition of ⊕, which symbolizes the direct sum of
vector spaces.)
- Consider the matrix A =
1 1 2
-1 -1 -2
2 2 4
Let b = (b1 b2
b3)T. Form the augmented matrix [A |
b]. Put this matrix in reduced row echelon form. This gives two
conditions of the form wTb = 0 for Ax
= b to have a solution. Show that in both cases
ATw = 0. Explain.
- Suppose data samples y-2, y-1,
y0, y1, y2 are taken at the times t =
t0 − 2h, t0 − h, t0,
t0 + h, t0 + 2h.
- Find the linear least squares fit to the data, Y(t) = a + bt.
- Show that Y(t0) = (y−2 +
y−1 + y0 + y1 +
y2)/5.
- Smoothing filter. A strategy for removing noise from
data is use the average at t0 to replace the actual data at
t0. That is, we replace y(t0) by
Y(t0). Perform the following numerical experiment. Let y(t)
= sin(πt), where −1 ≤ t ≤1. Sample y at 101 equally
spaced points, starting at t = −1 and ending at t = 1. (This
means that h = 0.02). To each y(t), add random noise between
−0.05 and +0.05. This gives a vector yn(t). Use the strategy
above to filter yn. How well does it compare to y? (In Matlab, you can
use the
filter
command. But be careful! This gives
something slightly different from what you want.) Make separate plots
of y-yn and y-Y vs. t.
- Consider the matrix:
- Find the SVD for B.
- For a square matrix, the ratio of the largest to smallest
singular values is called the condition number of the matrix. Find the
condition number for B.
- Suppose that û is the measured or calculated estimate of a
vector u. The relative error is then
rerr(u) = || û - u || · || u ||-1.
Solve the systems Bu = [1 1]T and Bû = [0.999
1.001]T exactly. Calculate rerr(u). How does
this compare with the relative error in the right hand side, which is
0.001? How does the ratio of rerr(u) to 0.001 compare with the
condition number for this problem? Briefly explain your results.
- Section 1.5, problem 4, page 55.
- Section 1.5, problem 10, page 56.
- Section 2.1, problem 4, page 94.
Assignment 4 - Due Wednesday, October 11.
- Read sections 2.1-2.2.3.
- Do the following problems.
- Recall that a function f:[0,1] → R is
measurable if and only if for all c ∈ R the set
f −1(c,∞) is measurable. Let A be a Lebesgue
measurable subset of the interval [0,1], with measure m(A).
- Show that the characteristic function of A, which is defined by
χ(t) = 1 if t ∈ A, and χ(t) = 0 if t ∉ A,
is measurable.
- Show that ∫ 01χ(t) dt = m(A).
- Note that χ has a range consisting of two values, 0 and
1. Characteristic functions are special cases of simple
functions. A measurable function g:[0,1] → R is
simple if and only if its range consists of a finite number
of distinct values, a1, ..., an. Let
Ej = g −1{aj}. Show that
∫ 01g(t)dt = a1m(E1) +
... + anm(En). Hint: Verify that g(t) =
a1χ1(t) + ... +
anχn(t), where χj is the
characteristic function of Ej, and then use the linearity
of the integral and part (b) above.
- Let F(s) = ∫ 0∞ e − s
t f(t)dt be the Laplace transform of f ∈
L1([0,∞)). Use the Lebesgue dominated convergence
theorem to show that F is continuous from the right at s = 0. That is,
show that
lims ↓ 0 F(s) = F(0)= ∫
0∞f(t)dt
- Section 2.1, problem 10, page 94.
- Calculus problem: Let g be C2 on an interval
[a,b]. Let h = b − a. Show that if g(a) = g(b) = 0, then
||g||C[a,b] ≤ (h2/8)
||g′′||C[a,b].
Give an example that shows
that 1/8 is the best possible constant.
- Use the previous problem to show that if f ∈
C2[0,1], then the equally spaced linear spline interpolant
fn satisfies
||f −
fn||C[a,b] ≤ (8n2) −
1 ||f′′||C[0,1]
- Section 2.2, problem 2(a), page 95. (Use the real form of the
Fourier series; see p. 75 for formulas.)
- Section 2.2, problem 2(c), page 95. (Use the real form of the
Fourier series; see p. 75 for formulas.)
Assignment 5 - Due Monday, November 6.
- Read sections 2.2.7, 3.2.
- Do the following problems.
- Prove the Convolution Theorem for the DFT. (See Notes on the
Discrete Fourier Transform, pg. 3.)
- Let a,x,y be n-periodic sequences, and let A, X, Y be column
vectors with entries a0, ..., an-1, etc. Show
that the convolution y = a∗x is equivalent to the matrix
equation Y = αX, where α is an n×n matrix whose
first column is A, and whose remaining columns are A with the entries
cyclically permuted. For example, if n = 4, and A = (a b c
d)T, then α =
a d c b
b a d c
c b a d
d c b a
Such matrices are called cyclic. Use the DFT and the convolution
theorem to find the eigenvalues of α.
- Section 2.2 problem 18(a,b,c), page 50.
- Let ψ(x) be the Haar wavelet. Show that if f ∈
L2(R) is a uniformly continuous function on
R, then the wavelet coefficient
djk = 2j < f(x),
ψ(2jx - k) >
satisfies the bound |
djk | ≤ 2-1
ω(f,2-j-1), where ω is the modulus of
continuity.
- Working in the Haar MRA, where the scaling function is
φ(x)=N1(x), let f1 ∈ S1 be
defined by
Express f1 in terms of the φ(2x-k)'s, then decompose
f1 into f0 ∈ S0 plus
w0 ∈ W0. Sketch all three functions.
Assignment 6 - Due Monday, November 13.
- Read sections 3.2, 3.3
- Do the following problems.
- Finite element problem. In this problem, let
< f,g >H = ∫01 f ′(x) g
′(x) dx,
where f,g are continuous functions vanishing at x = 0 and x = 1, and
have L2 derivatives. We want to solve the boundary value
problem (BVP) -y'' = f(x), y(0) = y(1) = 0.
- Write the BVP in weak form.
- Let f(x) = x2 in the rest of the problem. Find the
exact solution to the BVP for this choice of f.
- Let φj(x) := N2(nx-j+1), where
N2 is the linear B-spline. Find the βj's,
where
βj = < y,φj >H
= ∫01 f(x)
φj(x) dx, j=1 ... n-1.
- Show that Φkj = < φj,
φk >H, the k-j entry in the Gram matrix for the
problem, satisfies
Φj,j = 2n, j = 1 ... n-1
Φj,j-1 = - n, j = 2 ... n-1
Φj,j+1 = - n, j = 1 ... n-2
Φj,k = 0, all other possible k.
For example, if n=5, then Φ is
10 -5 0 0
-5 10 -5 0
0 -5 10 -5
0 0 -5 10
- Numerically solve Φα = β for n = 10, 25, 50. Use
your favorite software (mine is MATLAB) to plot the exact solution y
and, for each n, the linear finite element approximation to y,
v(x) = ∑j αj φj(x),
which is also the least squares approximation to y in the inner
product < · , · >H defined above.
- Let V be a Banach space. Show that a linear operator L:V → V
is bounded if and only if L is continuous.
- Let M be a closed subspace of a Hilbert space H. Let h be in H,
and let p be the unique minimizer of || h - u|| over all u in
M. Define the operator P:H → H by Ph = p. The operator P is
called the projection of H onto M. Show that the following
are true.
- P is a bounded linear operator, with ||P||=1.
- P2 = P.
- Range(P) = M and Null(P) = M⊥.
- P is self-adjoint, i.e. P* = P.
- Section 3.2 problem 3(b), page 128. (Assume the appropriate
operators are closed and that λ is real.)
Assignment 7 - Due Wednesday, November 22.
- Read sections 3.4, 3.5
- Do the following problems.
- Weak solutions. In this problem, let
< f,g >H = ∫01(f
′(x) g ′(x) +P(x)f(x)g(x))dx,
where f,g are continuous functions vanishing at x = 0 and x = 1, and
having L2 derivatives. In addition, we assume that P is
continuous and strictly positive on [0,1]. Show that the boundary
value problem (BVP) below has a unique weak solution in H, given that
H is known to be a Hilbert space. (Hint: use the Riesz Representation
Theorem.)
-y''+P(x)y = h(x), y(0) = y(1) = 0, h ∈
L2[0,1].
- Section 3.2 problem 3(d), page 128. (Assume the appropriate
operators are closed and that λ is real.)
- Section 3.3 problem 2, page 129. (Assume the appropriate
operators are closed and that λ is real.)
- A sequence {fn} in H is said to be weakly
convergent to f∈H if and only if lim n →
∞ < fn,g> = < f,g> for every
g∈H. When this happens, we write f = w-lim fn. For
example, in class we showed that if {φn} is any
orthonormal sequence, then φn converges weakly to
0. One can show that every weakly convergent sequence is a bounded
sequence; that is, there is a constant C such that ||fn||
≤ C for all n. Prove the following:
Let K be a compact linear operator on a Hilbert space H. Show
that if fn weakly converges to f, then Kfn
converges strongly to Kf -- that is, lim n →
∞ || Kfn - Kf || =0.
Hint: Suppose this doesn't happen, then there will be a subsequence of
{fn}, say {gk}, such that
|| Kgk - Kf || ≥ ε
for all k. Use this and the compactness of K to
arrive at a contradiction.
Assignment 8 - Due Wednesday, November 29.
- Read sections 3.6, 4.1
- Do the following problems. For problems 1-3, the operator K is given by
Ku(x) = ∫0π k(x,t)u(t)dt, where k(x,t) =
2cos(x)cos(t) + sin(x)sin(t)
- Verify that K is a compact, self-adjoint operator on
L2[0, π]. Find the nonzero eigenvalues and corresponding
eigenvectors for K. (There are two nonzero eigenvalues,
μ1 > μ2 > 0.) Also, show that
||K||op = π, and that ||k||L2 =
5½π/2.
- Use problem 1 and the appropriate formula from p. 119 of the text
to find the resolvent for L = I - λK. In addition, show that
Knu(x) = ∫0π
kn(x,t)u(t)dt, where
kn(x,t) =
2μ1n-1 cos(x)cos(t) +
μ2n-1sin(x)sin(t). Use Kn
and the Neumann expansion from section 3.6 to calculate the resolvent
when λπ < 1. Sum the series that you get and compare it
with your previous expression for the resolvent.
- Galerkin's method. Use your favorite software to do the following
problem. Solve u - ¼Ku = f, where f(x) = exp(x), using the
resolvent from problem 2. Then, for n = 5, 10, 30, find the
approximate solution to the same problem using the finite element
Galerkin approach outlined in section 3.6. Plot the results.
- Section 3.6, problem 3, page 131.
Updated 11/22/06 (fjn).