Math 641-600 — Fall 2015

Assignments

Assignment 1 - Due Wednesday, September 9, 2015.


Assignment 2 - Due Wednesday, September 16, 2015.


Assignment 3 - Due Wednesday, September 23, 2015.


Assignment 4 - Due Friday, October 2, 2015.


Assignment 5 - Due Friday, October 9.


Assignment 6 - Due Friday, October 30.


Assignment 7 - Due Friday, November 6.


Assignment 8 - Due Friday, November 13.


Assignment 9 - Due Friday, November 20.


Assignment 10 - Due Monday, November 30.


Extra Problems - These are not to be handed in.

  1. Let $L$ be in $\mathcal B (\mathcal H)$.
    1. Show that $\|L^k\| \le \|L\|^k$, $k=2,3,\ldots$.
    2. (We did this in class.) Let $|\lambda| \|L\|<1$. Show that \[ \big\|(I - \lambda L)^{-1} - \sum_{k=0}^{n-1}\lambda^k L^k\big\| \le \frac{|\lambda|^k \|L\|^k}{1 - |\lambda| \|L\|}. \]
    3. Let $L$ be as in problem 6, HW8. Estimate how many terms it would require to approximate $(I - \lambda L)^{-1}$ to within $10^{-8}$, if $|\lambda|\le 0.1$.
  2. Use Newton's method (see text, problem 3.6.3) to approximate the cube root of 2. Show that the method converges.
  3. Section 4.1: 6
  4. Section 4.2: 1, 4, 8
  5. Let $Lu=-u''$, $u(0)=0$, $u'(1)=2u(1)$.
    1. Show that the Green's function for this problem is \[ G(x,y)=\left\{ \begin{array}{rl} -(2y-1)x, & 0 \le x < y \le 1\\ -(2x-1)y, & 0 \le y< x \le 1. \end{array} \right. \]
    2. Verify that $0$ is not an eigenvalue for $Kf(x) := \int_0^1G(x,y)f(y)dy$.
    3. Show the orthonormal set of eigenfunctions for $L$ form a complete set in $L^2[0,1]$. (Hint: use tthe results from problem 4, HW10.

Updated 12/8/2015.