E19: Numerical Methods for Engineering Applications Spring 2016 - HOMEWORK 3

1. Condition number

Read the document at https://www.math.utk.edu/~ccollins/M577/Handouts/cond_stab.pdf. Then consider the function

f(x) =  (x - 5)(x - 2)/x + 2 = x² - 7x + 10/x + 2

Why is the function poorly conditioned at x = -1.99? Why is it also poorly conditioned at x = 1.99? Hint: the reasons are different in these two cases. Is f(x) well conditioned at x = 3.3? Explain.

FYI, next week, we will define and analyze the condition number of an n x n matrix A.

2. Engineering applications for linear systems

Identify a linear system of equations Ax = b which you solved in a previous Engineering course, and which involved real-world physical quantities, and then answer the following questions about it:

a. Present your system using a diagram or schematic, as well as in matrix form.

b. What are the units on A, x, and b?

c. How does the system arise from a model? What physical laws or principles govern the system of equations (e.g. conservation laws, steady-state assumptions, etc.)?

d. Is the underlying system actually linear, or have simplifying assumptions been made (e.g. small angle assumption) in order to induce linearity? How reasonable are the simplifying assumptions?

e. Aside from satisfying Ax = b, are there any other constraints on the unknowns x? Is there a fast method to eyeball the computed values to see if they make sense?

f. Is there any way your system could admit multiple solutions (i.e. if some of the underlying constants were changed)? What would the physical interpretation of the multiple solutions be?

3. Gaussian elimination with partial pivoting

Note: we will cover row pivoting on Tuesday 2/9.

Modify our implementation of Gaussian elimination from the course website to add partial (row) pivoting to the forward eliminating step. For each system below, have your program compute the solution to the system Ax = b. Also make sure your program prints out the corresponding upper-triangular system, before back-substitution is performed. Submit your code.

Attachment:- Assignment.zip