CAAM 454: Exam 2 (Spring 2013)
Due Date: Friday April 19, 6pm
Accepted Monday April 25, 6pm
This exam is take-home, close-book and close-note. The time limit is three (3) hours. On
this cover page, please ll in the date and time started and the date and t

CAAM 554: Mid-Term Exam (Spring 2011)
Due Date: Thursday, March 17, 2:30pm in class
This exam is take-home, close-book and close-note. The time limit is three (3) hours. On
this cover page, please ll in the date and time started and the date and time nish

CAAM 554: Mid-Term Exam (Spring 2013)
Due Date: Thursday, Feb. 14th, 10:00pm
This exam is take-home, close-book and close-note. The time limit is three (3) hours. On
this cover page, please ll in the date and time started and the date and time nished, pri

CAAM 454/554: Practice Exam 1 Solutions
1. Prove that the Jacobi method for Ax = b is convergent if A is column strictly diagonally
dominant, i.e.,
|ajj | >
|aij |, j cfw_1, 2, , m,
i=j
where the sum is over the running index i.
Solution: See the solution

CAAM 454/554: Practice Exam 1
Due Date: Tuesday, March 9, 2:30pm in class
This exam is take-home, close-book and close-note. The time limit is three (3) hours. On this
cover page, please ll in the date and time started and the date and time nished, print

CAAM 454/554: Practice Exam2 Solutions
1. (10 points) Let X Rn be a convex set and f : X Y R be a convex function, and let g : Y R
be convex and non-decreasing. Prove that the composite function h(x) = g (f (x) is a convex function
from X to R; that is, f

CAAM 454/554: Practice Exam 2
This exam is take-home, close-book and close-note. The time limit is three (3) hours. On this cover page,
please ll in the date and time started and the date and time nished, print your name and sign the honor
code pledge. Up

Solution Sketches for CAAM 454: Mid-Term Exam (Spring 2011)
Assume that A, M Rmm , b, c Rm and A is nonsingular. For W
is x W = xT W x. Let 2 (A) be the condition number of A in 2-norm.
1. (20 points) Let
0, the weighted 2-norm
be a norm in Rm .
a) Give t

Solution Sketches for CAAM 454: Exam 1 (2013s)
Assume that A, M Rmm , b, c Rm and A is nonsingular. For W
is x W = xT W x. Let 2 (A) be the condition number of A in 2-norm.
1. (20 points) Let
0, the weighted 2-norm
be a norm in Rm .
a) Give the denition o

CAAM 454: Exam 2 Solutions (Spring 2013)
1. (20 points) Let f : Rn R be twice continuously dierentiable.
(a) State the rst- and second-order necessary conditions for a point x Rn to be a
local minimizer of f .
(b) Give a set of sucient conditions for a po

Solution Sketches for CAAM 554: Mid-Term Exam (Spring 2011)
Assume that A, Q Rmm are nonsingular, and b Rm . Let Kn (A, b) be the n-th Krylov
subspace generated by A and b. For a real symmetric positive denite matrix W 0, the weighted
2-norm (by W ) is de

Solution Sketches for CAAM 554: Exam 1 (2013s)
Assume that A, Q Rmm are nonsingular, and b Rm . Let Kn (A, b) be the n-th Krylov
subspace generated by A and b. For a real symmetric positive denite matrix W 0, the weighted
2-norm (by W ) is dened by x W =

CAAM 554: Exam 2 Solutions (Spring 2013)
1. (25 points) Consider the gradient descent method xk+1 = xk k f (xk ) for minimizing a
dierentiable function f : Rn R, where at each iteration k is chosen to satisfy the two
Wolfe conditions. Assume that f is bou

CAAM 454: Mid-Term Exam (Spring 2013)
Due Date: Thursday, Feb. 14th, 10:00pm
This exam is take-home, close-book and close-note. The time limit is three (3) hours. On this
cover page, please ll in the date and time started and the date and time nished, pri