2019 - Fall
General Homework
page 79: 3.5 - 3.10
page 105: 4.1 - 4.6, 4.8 - 4.10
page 142: 5.1 - 5.3, 5.5, 5.7, 5.8
page 173: 6.1 - 6.7, 6.10, 6.11
page 216: 8.2, 8.5
page 276: 9.1 - 9.11
page 297: 10.2 - 10.5
page 316: 11.1, 11.2, 11.5, 11.9
page 487: 17.3, 17.4, 17.6, 17.7, 17.19
For interpolating polynomial, make up your own data and compute the polynomials(both Newton's and Lagrange). Also, consider the following problems:
page 574: 20.15, 20.16, 20.28, 20.39
page 630: 21.1 - 21.5, 21.11
page 752: 25.1, 25.2
Assignment 1
Due: Thursday, Sept. 5, 2019 at class time
For these problems, calculate all results to 6 significant figures.
Problem 1: xi+1 = (xi + A/xi)/2 is an iterative formulation that will calculate the square root of a number A. They taught it to kids in the 4th grade back in the '60s before calculators were invented. We .... I mean ... they called it 'divide and average'. So, take A to be 225, and let x0 be 100; this is your initial value. Use the formula to calculate x1, x2, and x3. For each, compute Et and et. Is it getting better? Give the results for the 10th iteration.
Problem 3.6 on page 79. Use only 6 terms, not 20. Compute the errors only for the final sum (6 terms for each method). Is there a difference? If so, why?
Problem 4.6 on page 108.
Problem 5.5 on page 142. For this problem, use both bisection and then again using false position.
Problem 6.10 parts a - c on page 174.
Assignment 2
Due: Tuesday, Oct. 1, 2019 at class time
For these problems, calculate all results to 6 significant figures unless otherwise specified.
Find the inverse of
using LU-decomposition.
Problem 9.10 parts c and d. For Gaussian, be sure to show the steps very clearly, noting the multiplying factors you use at each step. I reallly don't need to read what your calculator tells you.
Problem 11.3 on page 316. This problem requires you to use a method/algorithm I did NOT go over in class, but it's very important that you realize its importance. Be careful with the book's write-up; the algorithm can be a little confusing.
Problem 11.12 part a. Do 3 iterations and compute the ea's.
Assignment 3
Due: Tuesday, Oct. 22, 2019 at class time
Problem 17.9 on page 487. Disregard the instructions to plot.
Problem 17.18 on page 488. Disregard the instructions concerning standard error and correlation coefficient.
Problem 18.6 on page 524.
Find the natural spline that passes through the points {(xk , f(xk))} 3k=0, on the graph of f(x) = x + 3/x, using the nodes x0 = 1, x1 = 2, x2 = 3, x3 = 4. Check the accuracy of your interpolant at the value x = 3/2. Check it again at the value x = 7/2.
Assignment 4
Due: Tues. Nov. 19, 2019 at class time
In problems 1 and 2 below, refer to I given by
Compute an approximation to I using a single application of the Trapezoidal rule (h = 3), Simpson's rule (h = 3/2), Simpson's 3/8 rule (h = 1). Which is the best result? Choose an appropriate adjective to describe your first approximation and an appropriate adjective to describe your third approximation.
Now compute an approximation to I using the composite Trapezoidal rule by doubling the number of subintervals two times (final h = 3/4). The use Romberg integration to produce a really good approximation to I using these approximations. Compare this value to your best in #1.
Now, for the next two questions, let O.D.E. refer to y' = yt-2, y(1) = 2
For O.D.E., using h = 0.5, compute an approximation to y(1.5) using Euler's method, then Heun's method, then the Midpoint method.
Now use RK-4 with h = 0.5 to generate an approximation to the solution to O.D.E. at t = 1.5 and t = 2.0. Now calculate the percent relative error to the exact solution given by y = 2e(t-1)/t at t = 2.0. Was it good???! (ans: yes)
Special Assignment - OPTIONAL
Due: Monday, December 9, 2019, by noon in IN MY OFFICE (rm 325 F of CS)
Problem #26.6, page 780, but use Milne-Simpson method as I went over in class with h = 0.5 given all the "pre values" in that problem statement. Find and estimate to y(4.5). Iterate the corrector 3 times and ignore the text's demand to use Heun's and Adams. Also ignore their ea's.
Problem #13.3 on page 368.
Problem #13.5 on page 368.
Problem #23.5 on page 670.
NOTE: This assignment is optional - you can choose to do it or not do it. If you do it, I will add 5 pts for each correct answer to your hw points. That can mean the difference for some between one grade and the next. Please do not submit answers if you have no chance of improving your grade....like those of you with A's already in the bag.