7
Math Homework 371, Fall 2011
Assigned: Saturday, October 28, 2011
Due: Thursday, November 3, 2011
Clearly label all plots using title, xlabel, ylabel, legend
Use the subplot command to compare multiple plots
Include printouts of all Matlab code, labeled with your name, date, and section.
Add 12 for page numbers in the international edition.
(1) (Condition numbers) For two n n matrices A and B , prove that (AB )1 = B 1 A1 ,
1
and (A)1 = A1 . Now do P. 187 #1, 2.
(2) (Residual errors vs. forward errors) As in the previous homework, let A be the n n
matrix of the form
2 1
1 2 1
..
.
1 2
.
..
..
. 1
.
1 2
Write a matlab function named [rerr, ferr, condnum] = ResidualFwdErrors(n)which
takes n as an argument, generates the matrix A as above, and then a vector x, and then
a vector b = Ax, and returns the relative residual error, relative forward error, and the
condition number.
Make a table with four columns corresponding to n and the three numbers returned by the
function, taking n to be 50, 100, 500, 1000, 2000, 4000.
Connect the table to the theorem on page 182 of the textbook. Which part of the theorem
does the table this illustrate?
For problem, generate A using the diag() command as shown in class, but do not use
sparse().
(3) (Cholesky Factorization) Consider the following matrix factorization:
a11 a21
a21 a22
=
l11
l21 l22
l11 l21
l22
.
Write this down as three equations for a11, a21 , a22 . Explain in what order and in what way
the equation can be solved for the entries of the L matrix.
(4) (Cholesky Factorization) Consider a matrix factorization as in the previous problem,
but now with 3 3 matrices. Write down 6 equations for the aij s. Explain how these
equations can be ordered and used to solve for the lij s.
(5) (Triangular Factorization) Consider the following matrix factorization A = LU :
l11
1 u12
a11 a12
a21 a22 a23
l21 l22
1 u23
=
.
a32 a33 a34
l32 l33
1 u34
1
a43 a44
l43 l44
Write this down as a set of 10 equations and explain how to solve for the lij s and uij s given
the aij s. Verify the operation count of 3n 3 given on p. 219.
1
2
Suppose you want to solve
b1
x1
x2 b2
A
x3 = b3
b4
x4
for x. Explain how to do that using the LU factorization. Verify the operation count of
5n 4 on p. 219.