CAAM 553 Fall 2015
Homework 8
Due Oct. 23
(1) (10 points) A unit ball is dened as the set of x Cn such that x = 1. Consider
x R2 . Draw the unit ball in the 1, 2, and general p-norms.
(2) (30 points)
i. (10 points) Let be a matrix norm that is vector-indu

CAAM 553 Fall 2015
Homework 7
Due Oct. 16
1
(1) (20 points) Consider the function f (x) = 1+x
2 on [5, 5]. Construct two interpolation
polynomials of the function pn (x) and qn (x) where pn (x) uses equispaced interpolation
points and qn (x) uses the Cheb

CAAM 553 Fall 2015
Homework 9
Due Oct. 30
(1) (25 points)
Let x and y be real vectors of length n.
a. Show that if E = x y T is an outer product then E 2 = x 2 y 2 .
b. Let x be a unit vector ( x 2 = 1) such that A1 x 2 = A1 2 . Dene y to be
the vector
A1

CAAM 553 Fall 2015
Homework 10
Due Nov. 6
(1) i. (10 points) Show that if a matrix A is both triangular and unitary, then it is diagonal.
ii. (10 points) Prove that if A = LLT with L real and nonsingular, then A is symmetric
and positive denite.
iii. (20

CAAM 553 Fall 2015
Homework 11
Due Nov. 13
(1) (30 points) Let A be a real 50 5 matrix and suppose that the Singular Value Decomposition (short form SVD) of A is
A=U S V
T
where S = diag(5, 4, 3, 2, 1) and U R505 , V R55 have orthonormal columns
u 1 , u 2

CAAM 553 Fall 2015
Homework 11
Due Nov. 20
Helpful denition: The set of all eigenvalues of a square matrix A is called the spectrum of
A and is denoted (A).
(1) (25 points) Let A be an n by n complex matrix. Let AQ = Q R be a Schur decomposition of A with

CAAM 553
Homework 13
You do not have to hand in but please check your solutions!
(1) (25 points) The linear system
y = Ay,
y(0) = y0
where A is symmetric is solved by the explicit Eulers method. Let en = yn y(nh), n = 0, 1, . . .
and prove that
en 2 y0 2

CAAM 553
Homework 1 Solutions
Due Sept. 4
(1) (10 pts) How would you perform the following calculations to avoid cancellation? Justify
your answers.
i. Evaluate x + 1 1 for x 0.
Soln: For x 0, x + 1 1. Thus
x + 1 1 = 0.
x+1+1
However if we multiply by 1

CAAM 553 Fall 2015
Homework 4
Due Sept. 25
(1) i. (10 points) Given x0 = 0.2, x1 = 0 and x2 = 0.2 construct a second degree polynomial to approximate f (x) = ex via Newtons divided dierences.
Soln:
x0 = 0.2 f (x0 ) = e0.2
x1 = 0
x2 = 0.2
e0.2 1
0.2
f (x1

CAAM 553 Fall 2015
Homework 3
Due Sept. 18
(1) (15 pts) Which of the following iterations will converge to the indicated xed point x
(provided x0 is suciently close to x )? If it does converge, give the order of convergence;
for linear convergence, give t

CAAM 553 Fall 2015
Homework 2 Solutions
Due Sept. 11
(1) (30 pts) In laying water mains, utilities must be concerned with the possibility of freezing.
Although soil and weather conditions are complicated, reasonable approximations can
be made on the basis

CAAM 553 Fall 2015
Homework 5 Solutions
Due Oct. 2
(1) (10 points) Prove that an inner product space V over the field R equipped with the
induced norm is a normed linear space over R.
Soln: We need to show for f, g V
(a) kf k > 0 if f 6= 0 and kf k = 0 if

CAAM 553 Fall 2015
Homework 7
Due Oct. 16
1
(1) (20 points) Consider the function f (x) = 1+x2 on [5, 5]. Construct two interpolation
polynomials of the function pn (x) and qn (x) where pn (x) uses equispaced interpolation
points and qn (x) uses the Cheby

CAAM 553 Fall 2015
Homework 5
Due Oct. 2
(1) (10 points) Prove that an inner product space V over the eld R equipped with the
induced norm is a normed linear space over R.
(2) Suppose we have m data points cfw_(ti , yi )m , where the t-values all occur in

CAAM 553 Fall 2015
Homework 6
Due Oct. 9
(1) Consider approximations to x for x [0, 1].
i. (10 points) Find the line that best approximates x in the minimax (L ) sense, and
report the error. Hint: Consider the oscillation theorem, and the derivative of th

1
CAAM 453: Numerical Analysis I
Problem Set 5
Code Solutions to Problem 1
Code to set up least squares B-spline approximation to sine function
clear;
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% to sine(t) 0 .le. t .le. 2pi.
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CAAM 553
Homework 1
Due Sept. 4
(1) (10 pts) How would you perform the following calculations to avoid cancellation? Justify
your answers.
i. Evaluate x + 1 1 for x 0.
ii. Evaluate sin(x) sin(y) for x y.
iii. Evaluate 1cos(x) for x 0.
sin(x)
(2) (20 pts)

CAAM 553 Fall 2015
Homework 3
Due Sept. 18
(1) (15 pts) Which of the following iterations will converge to the indicated xed point x
(provided x0 is suciently close to x )? If it does converge, give the order of convergence;
for linear convergence, give t

CAAM 553 Fall 2015
Homework 4
Due Sept. 25
(1) i. (10 points) Given x0 = 0.2, x1 = 0 and x2 = 0.2 construct a second degree polynomial to approximate f (x) = ex via Newtons divided dierences.
ii. (10 points) Derive an error bound for p2 (x) when x [0.2, 0

CAAM 553 Fall 2015
Homework 2
Due Sept. 11
(1) (30 pts) In laying water mains, utilities must be concerned with the possibility of freezing.
Although soil and weather conditions are complicated, reasonable approximations can
be made on the basis of the as

CAAM 553 Fall 2015
Homework 6
Due Oct. 9
(1) Consider approximations to x for x [0, 1].
i. (10 points) Find the line that best approximates x in the minimax (L ) sense, and
report the error. Hint: Consider the oscillation theorem, and the derivative of th