Numerical Mathematics, Spring '13 (131)

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mankind so long bled and suffered, we have yet gained little if we countenance a political
intolerance as despotic, as wicked, and capable of as bitter and bloody persecutions.
— Thomas Jefferson (1743-1826) in his First Inaugural Address

Homework List

✈ Jump down to 'This Week.'

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Week 1

Monday, Jan 14 — First day of class
Read §1.1.
Pg 12: #3, 9, 11a, 12b, 17.
Wednesday, Jan 16
1. Convert the pseudocode to Python in: Pg 14: #15.
2. Use a Python loop to answer: Pg 15: #4.
3. Write Python code to calculate: \[ \textbf{Mean:}\quad \mu = \frac1n \sum_{k=1}^n a_k \] \[ \textbf{Variance:}\quad \sigma^2 = \frac1{n-1} \sum_{k=1}^n (a_k - m)^2 \]
Friday, Jan 18 - Last day to drop/add a class
❄ No class — classes before 10:00 am are cancelled due to snow ➀
¤ Read through the Python_Intro class notes.
1. Investigate Python's lists.
2. Write variance as a Python function.

Week 2

Monday, Jan 21 —
◊ No class — Martin Luther King, Jr, Day
Wednesday, Jan 23
¤ Defining functions with Python:
1. Write a function to compute:
  1. The area of a square.
  2. The area of a rectangle.
  3. The area of a circle.
2. The length of a string is len(S), so the value of len('allen') is 5.
  Write a function named right_justify that takes a string named S as a parameter and prints the string with enough leading spaces so that the last letter of the string is in column 70 of the display.
Friday, Jan 25
¤ Define the following functions in Python:
1. s(a) that returns the sum of the values in the list a.
2. gauss5(f) that returns the Gauss-5 quadrature of the integral $\displaystyle\int_{-1}^1 f(t)\,dt$ for the function f defined outside of gauss5. Use the lists x and w for the nodes and weights, respectively.
  - Test your function with $f(t) = \dfrac{1}{\sqrt{2\pi}}\,e^{-t^2/2}$. The integral is $\approx 0.6826894920$.
  - Recall that the Gauss-5 nodes and weights are \[ \begin{bmatrix} x_k & w_k \\ \hline -0.9061798457 & 0.2369268851 \\ -0.5384693100 & 0.4786286705 \\ 0.0 & 0.5688888889 \\ 0.5384693100 & 0.4786286705 \\ -0.9061798457 & 0.2369268851 \end{bmatrix} \] and the Gauss-5 quadrature rule is $\operatorname{Gauss}_5(f) = \displaystyle\sum_{k=1}^5 f(x_k)\cdot w_k$.
  - Write a function gauss(x,w,f) that does the quadrature with the nodes and weights of an arbitrary size, and function f as arguments. Test with the function $f$ above and $n=20$: \[ \begin{bmatrix} i & x_i & w_i \\ \hline 1 & -0.9931285991850949247861 & 0.017614007139152118312 \\ 2 & -0.9639719272779137912677 & 0.040601429800386941331 \\ 3 & -0.9122344282513259058678 & 0.0626720483341090635695 \\ 4 & -0.8391169718222188233945 & 0.083276741576704748725 \\ 5 & -0.7463319064601507926143 & 0.1019301198172404350368 \\ 6 & -0.6360536807265150254528 & 0.1181945319615184173124 \\ 7 & -0.5108670019508270980044 & 0.1316886384491766268985 \\ 8 & -0.3737060887154195606725 & 0.1420961093183820513293 \\ 9 & -0.2277858511416450780805 & 0.149172986472603746788 \\ 10 & -0.0765265211334973337546 & 0.1527533871307258506981 \\ 11 & 0.0765265211334973337546 & 0.152753387130725850698 \\ 12 & 0.2277858511416450780805 & 0.1491729864726037467878 \\ 13 & 0.3737060887154195606725 & 0.142096109318382051329 \\ 14 & 0.5108670019508270980044 & 0.131688638449176626899 \\ 15 & 0.6360536807265150254528 & 0.11819453196151841731 \\ 16 & 0.7463319064601507926143 & 0.101930119817240435037 \\ 17 & 0.8391169718222188233945 & 0.08327674157670474872 \\ 18 & 0.9122344282513259058678 & 0.062672048334109063569 \\ 19 & 0.9639719272779137912677 & 0.040601429800386941331 \\ 20 & 0.9931285991850949247861 & 0.017614007139152118312 \end{bmatrix} \] Gaussian-20 Nodes and Weights

Week 3

Monday, Jan 28
◊ Write Python functions to:
1. Calculate the $\sin$ of a list of values using map
2. Calculate the $\sin$ of a list of values using a for loop
3. Reverse a list keeping only every third entry.
Wednesday, Jan 30
◊ Convert these Maple functions to Python:
1. $\Delta$ := (n,k) $\to$ piecewise(k<n, $\displaystyle\sum_{j=0}^k$(-1)$^j$*binomial(k,j)*a[n-j], 0):
2. M := N $\rightarrow$ Matrix(N,N, (i,j)$\rightarrow$ $\Delta$(i, j-1))
3. DT := (p,h) $\rightarrow$ subs([seq(a[i]=p(i*h), i=1..degree(p(x))+2)], M(degree(p(x))+2)))
Friday, Feb 1
❄ No class today — wind and snow again… ➁
☞ Homework: Due next ~~Wednesday, 2/6~~ Friday, 2/8. Either send your function to me as a text file or in an email.
- Write a Python function gauss_kronrod that uses the Gauss-Kronrod 7-15 quadrature to approximate the integral $\displaystyle\int_{-1}^1 f(t)\,dt$ and estimates the error.
  def gauss_kronrod(f): ⋮ return (approxInt, err)
- A reference to refresh your memory: Gauss–Kronrod quadrature formula.

Week 4

Monday, Feb 4
❄ No class today — classes before 10:00 am are cancelled ➂
Wednesday, Feb 6
§1.2, pg 31. #1, 2, 3, 4a-d.
Friday, Feb 8
☞ Remember: The Gauss–Kronrod Python program homework is due today
§2.2, pg 68. #17 (also add "b. with using a series."), 20, 26cd.
◊ Look at Computer Exercises #17, pg 73.

Week 5

Monday, Feb 11
§3.2, pg 101. #3, 4, 9, 17, Look at 39.
Wednesday, Feb 13
§3.3, pg 119. #4, 10.
◊ Look at Computer Exercises #4, pg 121.
Friday, Feb 15
§5.7, pg 200. #1, 2, 3, 7.
◊ Computer Exercises #5a, pg 204.

Week 6

Monday, Feb 11
¤ Due: Wed 2/20:
Set $ N(x) = \displaystyle\int_0^x e^{-t^2/2}\,dt $. Compare the values from Maple's int to the Python program romberg:
1. for $ N(1.0) $
2. for $ N(1.0) $ after substituting $ -t^2/2=\ln(u) $. Note: revised substitution (2/19/13) — this one's much easier; it doesn't require complex integration.
Wednesday, Feb 13
§6.1, pg 227. #1, 3, Look at 8.
¤ Prepare your one-page (8.5"$\times$11" max) note sheet for ~~Friday's~~ Monday's test.
Friday, Feb 15
☞ ¡The test has moved to Monday! (Avoiding the potential ice storm and by a slim majority vote.)
¤ Study the Adaptive Simpson's Notes and Programs for next Wednesday.

Week 7

Monday, Feb 25
◊ ¡Test today!
Wednesday, Feb 27
§7.1, pg 255. #3, 5, 7e.
(#7e spoiler)
Friday, Mar 1
◊ Investigate the different pivoting methods.

Week 8

Monday, Mar 4
§7.2, pg 272. #3, 4, 6, Look at 18.
Wednesday, Mar 6
❄ No class today — all classes are cancelled again! ➃
¤ Use your Google-Fu on
- NumPy
- SciPy
- SymPy
Friday, Mar 8
➥ Read Key Moments in the History of Numerical Analysis by Benzi
◊ Since we lost Wednesday, the "Gaussian Elimination with Partial Pivoting" Python program is now due Monday, 3/18.

— Spring Break — “Let's be careful out there!”

Monday, Mar 11
Wednesday, Mar 13
Friday, Mar 15

Week 9

Monday, Mar 18
§10.1, pg 436. #1ab, 2a, 3a, 8.
Wednesday, Mar 20
◊ Project: (due Wednesday, March 27)
1. Work the Euler's Method project.
2. Redo the Euler's Method project using the second order Taylor's method.
Friday, Mar 22
◊ Project Day!

Week 10

Monday, Mar 25
◊ Project Day!
Wednesday, Mar 27
§10.2, pg 445. #2, 6, 8.
Friday, Mar 29
§10.2, pg 447. #2, 5.

Week 11

Monday, Apr 1
No classes — «State holiday»
Wednesday, Apr 3
¤ Write Maple code implementing an adaptive RK45 DE numeric solver.
Friday, Apr 5
- Finish the Maple version of the adaptive RK45 DE numeric solver.
- Test your code with the first-order IVPs below and compare your values with the exact solution:
  1. IVP: $\left\{y^\prime + y = 2,\; y(0)=1 \right\}$; exact solution: $y(x)=2-e^{-x}$
  2. Stiff IVP: $\left\{y^\prime =-100y,\; y(0)=10 \right\}$; exact solution: $y(x)=10\,e^{-100x}$
  3. IVP: $\left\{y^\prime = \sin(x\,y)+\frac12 x - y,\; y(0)=1 \right\}$; exact solution: not this time
    Use:
    DEtools[DEplot](y'=sin(x$\cdot$y)+x/2-y, [y(x)], x=0..10, y(x)=0..4, [[y(0)=1]], linecolor=blue)
    to see Maple's numeric solution.

Week 12

Monday, Apr 8
§10.3, pg 460. #9, 11.
Wednesday, Apr 10
§11.1, pg 475. #1.
Friday, Apr 12
◊ Predator-Prey with Euler's method and RK4 Maple worksheet.
¤ Change the Lotka-Volterra system in the predator-prey worksheet to a "Competitve Hunter" model \[ \begin{cases} x^\prime = 0.2x - 0.001 x y \\ y^\prime = 0.3 y - 0.002 x y \end{cases} \] with $\langle x_0,y_0\rangle = \langle 100,100\rangle$ where $x$: owls and $y$: hawks.
1. Produce plots similar to those for our predator-prey system. (You'll have to adjust plot ranges.)
2. Find the equilibrium point(s).
3. Describe long-term behaviour.

Week 13

Monday, Apr 15
◊ Investigate the nonlinear pendulum with the Maple worksheet.
Wednesday, Apr 17
§11.2, pg 483. #3.
Friday, Apr 19
◊ ¡ ☛ TEST ☚ today!
¤ Your ☛ TEST ☚ solutions are due Wednesday, Apr 24.

Week 14

Monday, Apr 22
§17.1, pg 666. #4.
Wednesday, Apr 24
¤ Finish working your test and turn it in!
Friday, Apr 26
◊ Set up the initial simplex tableaux for the housing project problem from class:
A developer has 60 acres for a new subdivision of townhouses, single-story houses, and two-story houses. On one acre he can put 6 townhouses, 4 single-story houses, or 2 two-story houses. It costs $40,000 to build a townhouse, $50,000 for a single-story house, and $60,000 for a two-story house. He makes a profit of $15,000 on a townhouse, $18,000 on a single-story house, and $20,000 on a two-story house. He has $2,880,000 of capital available. Townhouses require 2,500 hours of labor, single-story houses require 3,000 hours of labor, and two-story houses require 4,000 hours of labor. He has 240,000 hours of labor available.

Week 15 ♞ (This Week)

Monday, Apr 29
¤ Try the Lego simplex problem.
Wednesday, May 1
¤ Set up the Power Company LP problem.
¤ Set up the dual for the Power Company problem.
Friday, May 3
◊ The ☛ Final Exam ☚

Week 16 — Final Exams

Thursday, May 9, Noon to 2:30 pm
◊ The exam is...

“On two occasions I have been asked, ‘Pray, Mr. Babbage, if you put into the machine wrong figures, will the right answers come out?’ ... I am not able rightly to apprehend the kind of confusion of ideas that could provoke such a question.”

— Charles Babbage in Passages from the Life of a Philosopher

Last modified: Wednesday, 01-Feb-2023 08:32:07 EST