2021 - Spring
General Homework
Chapter 1:
page 12: 1, 2, 8a, 9a, 11a,b, 12b,c 13, 14a
page 23: 1a, 2a, 3a,b, 4a, 6, 7, 8b, 13b,
page 37: 1a,b, 2, 3a, 5a,b,c, 6a, 12
page 50: 1b, 2, 3a,b, 4, 11
page 61: 3, 4, 5, 9, 10, 11
page 85: 1, 2, 3, 6 - 11, 13 - 15, 22
page 119: 1, 2, 4, 5, 8, 10, 13, 15
page 124: 1, 23, 4b, 7
page 137: 1, 2, 5, 9, 11, 14a, 15
page 153: 1, 3, 4, 6
page 165: 1, 8
page 181: 3, 5, 6
page 196: 1, 3 - 6, 8, 9
page 205: 1a,b,d,
page 217: 1a, b, c, 2a, 3a, 6
page 228: 1, 2, 5, 8
section 4.5: I will assign no explicit problems to do, though many of them are good problems. But I will expect you to know the principle idea behind the use of Chebyshev polynomials.
page 259: 1a, 2a, 3a, 4, 5, 8, 10a
page 275: 1, 3, 6a, some of 8 - 15, 17, 18, 19
page 295: 5, 9b
page 334: 1a,b, 2a, 3a
page 349: 1, 4
page 362: 1c, 2c, 3
page 374: 1a, 8b,c, 9b
page 389: at least 3 of the problems in #1
page 463: part a only for 1, 3, and 5
page 472: 1 - 6, 8, 9
page 480: 1 - 5
page 485: 5
page 502: 5
Assignment 1
Due: Thursday, February 4, 2021 at class time
For these problems, calculate all results to 8 significant figures. Be sure to follow the RULES on homework submissions.
Problem 1:
i) page 24 #13a find the percent relative error
ii) Determine a base ten fractional expression (you know, like 3/65 or 88/153 or 5/9 or...) equal to 0.110110110110110...
Problem 2: page 37 #6a
Problem 3:
i) page 50 #3d
ii) For fixed-point iteration, explain the reason that it is advantageous for g'(P)≈0.
Problem 4: page 61 #5
Problem 5: page 87 #22a, except use A = 16 and p0 = 16. Include in your work all pertinent steps in the derivation of g(x) for f(x). Be careful because it's easy to screw up your algebra.
Assignment 2
Due: Thursday, Feb. 18, 2021 at class time
Problem 1:
i) pg. 137 #4 doing Gaussian elimination any way you desire to solve the system.
ii) pg. 137 #3 using partial pivoting to solve the system(ignore the second format).
Problem 2:
i) Perform LU-decomposition on the matrix
ii) pg. 153 #2
Problem 3:
i) pg. 165 #8b
Problem 4: Use the equations given in #6 on page 182 to rewrite as 0 = f1(x,y) and 0 = f2(x,y). Using p0 = (-0.3, -1.3), use Newton's method to generate 2 iterations.
Assignment 3
Due: Tuesday, March 30, 2021 at class time
For these problems, calculate all results to 6 significant figures. Be sure to follow the RULES on homework submissions.
Problem 1: Let y = sin(x). Use nodes x0 = 0, x1 = π/2, x2 = 3π/2, x3 = 2π. Compute the Lagrange form of the 3rd degree interpolating polynomial, P3(x). Using this, calculate P3(π). What is the absolute error.
Problem 2: page 229 #8
Problem 3: page 261 #9b
Problem 4: page 276 #6
Problem 5: page 295 #9b
Assignment 4
Due: Tuesday, May 4, 2021 at class time
For these problems, calculate all results to 6 significant figures. Be sure to follow the RULES on homework submissions.
Problem 1: For this problem, you will solve ∫142xdx on the interval [0.5, 1.5].
i) analytically...i.e. solve it as you would have in calc II for a numeric answer.
ii) Using a single application of the trapezoidal rule.
iii) Using a single application of the Simpson's 1/3 rule.
iv) Using a single application of Boole's rule.
And the concluding and obvious question is: which was best? So, compare the errors of (ii), (iii), and (iv) using your answer to (i).
Problem 2: For this problem, you will solve ∫142xdx on the interval [0.5, 1.5].
Use Romberg integration, starting with T(0) (this should be exactly what you did in part (i) above). Compute T(1), T(2), and T(3) and then S(1), S(2), and S(4), and then B(2) and B(3), and finally using these last two for one final refinement. Pretty much, this is exactly what I did in class. Use theorem 7.4 for computing successive trapezoidal approximations (that's the formula I developed in class).
Problem 3: For this problem, the IVP is y' = 3y + 3t, y(0) = 1. Note: the true solution is: y(t) = (4/3)e3t - t - 1/3
Use Euler's and Huens methods on the IVP with the step sizes of 0.2 and then 0.1 (2 iterations and then 4 iterations). Compare your answers with the true values computed with the solution provided.
Problem 4: Solve the same problem as in #3 above, but using the classical RK-4 method. Use step size h = 0.1. Compare the value obtained from this method at t = .2 with those obtained in #3 above.