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## Homework 4 (due 2/18) CSC 431

Midterm will be next week, 2/11 for in-class students; it'll  be 90 minutes, open book, open notes with class afterwards. Online students can sign up at col.

This week we finished the material on solving linear systems of equations, from chapter 2 of Heath's book (2.3, 2.4.5, 2.4.8). After the midterm we will talk about the linear least squares method (Chapter 3).

During class we saw two of Heath's simulations:

and we discussed my solution to solving quadratic equations from the homework.

Submission: you can submit the homework by hardcopy in class or by sending it to me as an email.

For some of the problems below you will need an implementation of Gaussian elimination. If you are using Python, you can use the linalg package that comes with numpy. E.g. here's the code for solving a simple 2x2 problem:

from numpy import *

A = array([[3,4],[2,1]]);
b = array([4,5]);
print linalg.cond(A);   #computes the condition number
x = linalg.solve(A,b);
print("(%r, %r)"%(x,x));

1. [Condition numbers and vector norms, 10pt] Solve question 2.61 (page 95).

2. [Determinant versus condition number, 10pt] The condition number measures how badly a solution to the linear system Ax = b is affected by small changes to A and b. Matrices that are close to being singular are particularly bad, since very small changes to A can turn a solvable system into an unsolvable one. Traditionally singularity is tested by computing the determinant det(A). Show that for our purposes this is not a useful criterion: Exhibit a matrix A, for which det(A) is very close to 0, but so that A is invertible (indeed, has condition number 1).

3. [Computing the Inverse, 10pt] Use Gauss-Jordan elimination with (partial) pivoting to compute the inverse of

 1 4 6 1 2 2 -2 4 6

4. [Growth factor, 15pt] When performing Gaussian elimination, entries in the matrix can increase, indeed significantly so; fortunately, this doesn't typically happen, however, there are matrices where it does. Do Exercise 2.7 (page 102) to analyze one bad case.

5. [Ill-conditioned matrices, 15pt] Do Exercise 2.10 (page 102) of the book.

6. [Extra Credit] Do Problem 2.4 (page 101).

Marcus Schaefer
Last updated: February 5th, 2010.