MACM 316

MACM 316 Homework, Practice Problems, and Test Information

Problems are from the 8th Edition.

Solutions may be submitted to me electronically.

Here's some information about MAPLE and MATLAB .

You can download all the algorithms in the text (MAPLE and MATLAB) from the Author's website; http://www.as.ysu.edu/~faires/Numerical-Analysis/DiskMaterial/programs/ (Download these text files and open with MAPLE to run the procedures.)

Late penalty : 15% per day, 20% per weekend.

Homework #1. Due: Thursday, Jan 25 in class or via email.

Section 1.2 : 8, 12, 16c (here, use MAPLE with 45 digit accuracy " >Digits:=45; " to express your answer in decimal form to 45 digits), 19a, 20a

Problem A: Let p(x) = 3.1x⁵-2.31x³+4.7x-2.01. Use nesting and three-digit rounding to evaluate p(e). Compare this to the exact value of p(e).

Problem B: What is the nearest 64 bit machine number to the square root of 2?

Problem C: Which is more accurate, rounding or truncating? Answer in terms of absolute error and relative error.

Section 1.3 : 8, 10

Section 2.1 : 8

End of Homework #1

MAPLE worksheet from tutorial #1.
MAPLE worksheet from tutorial #2.

Practice problems :

Section 1.2 : 3, 5, 7, 13, 15,
Section 1.3 : 3, 9
Section 2.1 : 7

Homework #2. Due: Tuesday Feb 13.

Section 2.2 : 6, 8, 14

Section 2.3 : 6b, 6e, 8b, 8e

Problem A : Let f(x) = (sin(x) - x)/x.

(a) Show that x=0 is a root of f(x) of multiplicity 2 (see Section 2.4).
(b) Starting with p₀=1, compute approximations to this root;

(b1) the next 12 iterates (up to p₁₂) using Newton's method (equation (2.5)). Verify that this sequence is NOT converging to the root quadratically.
(b2) the next 5 iterates (up to p₅) using the Modified Newton's method (equation (2.11)). You will have to use a high number of digits of precision here. Verify that this sequence IS converging to the root quadratically.

(c) Use Steffensen's method (Section 2.5) with Newton's method and verify that the convergence is faster than with only Newton's method (compute up to p₁₂).

Here's some Maple commands that you may need for this

Tutorial #3 worksheet
Section 3.1 : 2c, 6c Construct the analogous table as Table 3.4, 10c, 18
Problem B : Use Maple to find the Lagrange interpolation polynomial P(x) to g(x)=sin(x^3/2) on the interval [0,6] using 11 equally spaced data points (so, x=0, 0.6, 1.2, . . . , 5.4, 6.0).
- (a) Calculate the theoretical error (equation (3.3)).
- (b) Find the actual error on the interval [0,6]. (Plot the difference g(x)-P(x). Where does this graph cross the x-axis?)
- (c) Find the actual error on the interval [1,5].
- (d) Add two more data points, one near 0 and one near 6, in the interval [0,6] so that the resulting Lagrange polynomial is a better approximation to g(x). What is the actual error now?
  
  Submit your plots with your answer.
Section 3.2 : 2b (Use eq. 3.10 - not 3.17!).
Section 3.3 : 2c, 6a (plot)

End of Homework #2

Practice problems :

Section 2.2 : 5, 7, 13
Section 2.3 : 5, 7
Section 2.5 : 3
Section 3.1 : 1, 5, 7, 17
Section 3.2 : 1
Section 3.3 : 1, 3, 7

Splines plots
Spline MAPLE worksheet
Bezier curve

Homework #3. Due: Tuesday Feb 20.

Section 3.4 : 4c, 6c, 7c (Plot your curves for 4c, 7c)
Section 3.5 : 1c, 3c (plot cuve for 3c)

End of homework #3

Some MAPLE commands for this homework

Plots of solutions

MAPLE worksheet for plotting Bezier curves from the midterm test.

Homework #4. Due: Thursday March 15 (New due date).

Section 4.1 : 2, 4, 6b, 20, 22, 24

Section 4.2 : 2, 4d, 6, 10

Section 4.3 : 2d, 4d, 6d, 8d, 16

Section 4.4 : 14, 19 only parts a,b for 14 and 19

Section 4.5 : 2d, 4d

Section 4.7 : 2d, 4d, 6

Section 5.2 : 6a, 8a, 10

Section 5.3 : 2b, 4b, 10

Section 5.4 : 2b, 10b (do these by hand)

End of Homework #4

MAPLE worksheet for calculating some solutions.

Practice problems :

Section 4.1 : 1, 5, 19
Section 4.2 : 1, 3, 5, 7, 9, 13, 15, 17, 19
Section 4.3 : 1, 5, 19
Section 4.4 : 1, 3, 7ab, 9, 11, 13ab, 19
Section 4.5 : 1, 3, 7
Section 4.7 : 1, 5
Section 5.2 : 1, 3
Section 5.3 : 1

Homework #5. Due: Thursday April 5.

Section 6.1 : 6b, 12 (modify Algorithm 6.1), 15
Section 6.2 : 4c+d, 6c+d, 10b (see Example 4), 16b, 28b
Section 6.3 : 2a+c, 8e
Section 6.4 :8
Section 6.5 :4d, 6a, 6c, 8c, 9c (Check 8c and 9c using MAPLE; include output with your solutions)
Section 7.1 : 4, 10 (Use Theorem 7.3)
Section 7.3 : 2c, 4c (do these two by hand), 6c (Use the MAPLE program alg071.mws), 10d

End of Homework #5

Last updated: April 2