MAT 401 - FALL 2013
Homework 2
Ryan Szypowski
Due October 24, 2013
1. Consider nding a root of the function
f (x) = x3 x.
(a) Use the IVT to justify that f has a root in the interval [2, 3].
(b) Using bisection, compute p4 starting with the interval [2, 3
MAT 401 FALL 2011
Name:
Test #2 Solutions
Please show all of your work. Answers without justifaction may be worth 0.
Please make your answers easy to read. This means it should be clear what you are doing
from one step to the next and your work should be
MAT 401 FALL 2011
Name:
Test #1
ID #:
Please show all of your work. Answers without justifaction may be worth 0.
Please make your answers easy to read. This means it should be clear what you are doing
from one step to the next and your work should be legi
MAT 401 - FALL 2011
Homework 3
Ryan Szypowski
Due November 15, 2011
1. Chapter 3.4 # 12 from the textbook.
2. The Frobenius norm of an n n matrix is dened as
A
F
n
1
2
n
a2 .
i,j
=
i=1 j=1
This is not a natural norm associated with any vector norm. Prove
MAT 401 - FALL 2011
Computer Homework 1
Ryan Szypowski
Due October 27, 2011
I have provided some basic Matlab code to perform most of the rootnding methods
discussed in class as well as a test driver to show you how to call the routines. Each
one returns
MAT 401 - FALL 2011
Homework 4
Ryan Szypowski
Due December 1, 2011
1. Let
x1 x2 + 1
x3 + x3
2
1
F(x) =
.
Perform 3 iterations of Newtons method to solve F(x) = 0 using
x(0) =
1
0
.
Perform 3 iterations of Broydens method to solve F(x) = 0 using
x(0) =
1
0
MAT 401 - FALL 2011
Homework 2 Partial Solutions
Ryan Szypowski
Due October 13, 2011
1. Justify the approximation
g (p)
pn pn1
pn1 pn2
used in the stopping criterion for xed point iteration when it is converging only linearly. This should not require muc
MAT 401 - FALL 2011
Homework 1
Ryan Szypowski
Due October 6, 2011
1. Recall the following fact about alternating series: the absolute error made
in truncating a convergent alternating series is bounded by the magnitude
of the following term. That is,
n1
(
MAT 401 - FALL 2011
Homework 3 Solutions
Ryan Szypowski
Due November 15, 2011
1. Chapter 3.4 # 12 from the textbook.
2. The Frobenius norm of an n n matrix is dened as
A
=
F
1
2
n
n
a2
i,j
.
i=1 j=1
This is not a natural norm associated with any vector n
MAT 401 - FALL 2011
Homework 2
Ryan Szypowski
Due October 13, 2011
1. Justify the approximation
g (p)
pn pn1
pn1 pn2
used in the stopping criterion for xed point iteration when it is converging only linearly. This should not require much work.
2. Show th
MAT 401 - FALL 2011
Homework 1 Partial Solutions
Ryan Szypowski
Due October 6, 2011
1. Recall the following fact about alternating series: the absolute error made in truncating
a convergent alternating series is bounded by the magnitude of the following t
MAT 401 - FALL 2013
Homework 4 Solutions
Ryan Szypowski
Due December 5, 2013
1. Let A be a matrix with eigenvalues 1 , . . . , n and associated eigenvectors v1 , . . . , vn .
Prove the following:
1
(a) The eigenvalues of A1 are 1 , . . . , 1n and associat
MAT 401 - FALL 2013
Homework 1
Ryan Szypowski
Due October 10, 2013
1. Consider computing the quantity
n
n
xi y j
i=1 j=1
where xi and yj are given sequences.
(a) Being naive, write an algorithm to compute this as written above and determine
the cost of th
MAT 401 - FALL 2013
Homework 1
Ryan Szypowski
Due October 10, 2013
1. Consider computing the quantity
n
n
xi y j
i=1 j=1
where xi and yj are given sequences.
(a) Being naive, write an algorithm to compute this as written above
and determine the cost of th
MAT 401 - FALL 2013
Homework 4
Ryan Szypowski
Due December 5, 2013
1. Let A be a matrix with eigenvalues 1 , . . . , n and associated eigenvectors v1 , . . . , vn .
Prove the following:
(a) The eigenvalues of A1 are
1
, . . . , 1n
1
and associated eigenve
MAT 401 - FALL 2013
Homework 2 Solutions
Ryan Szypowski
Due October 24, 2013
1. Consider nding a root of the function
f (x) = x3 x.
(a) Use the IVT to justify that f has a root in the interval [2, 3].
Solution: First, notice that f is continuous. Then, co
MAT 401 - FALL 2013
Homework 3
Ryan Szypowski
Due November 26, 2013
1. Prove that
x
A
=
xT Ax
is a proper vector norm provided A is a positive denite matrix. Hint: Write A in terms
of its eigenvalues and eigenvectors. You may need to do some research on p
MAT 401 - FALL 2013
Homework 3
Ryan Szypowski
Due November 26, 2013
1. Prove that
x
A
=
xT Ax
is a proper vector norm provided A is a positive denite matrix. Hint: Write A in terms
of its eigenvalues and eigenvectors. You may need to do some research on p
MAT 401 - FALL 2011
Homework 4
Ryan Szypowski
Due December 1, 2011
1. Let
x1 x2 + 1
x3 + x3
2
1
F(x) =
.
Perform 3 iterations of Newtons method to solve F(x) = 0 using
x(0) =
1
0
.
Perform 3 iterations of Broydens method to solve F(x) = 0 using
x(0) =
1
0