MATH 4445, Homework 5 solutions
1. n = 2
0
1
T
5 10
2
,
A
A
=
10 30
3
4
1.6351
0.7698
T
T
A Aa = A y =
a=
1.0562
0.2214
1
1
A=
1
1
1
r = y Aa = [0.6698, 0.3931, 0.6823, 0.1355, 0.5410]T , |r|2 = 1.1746
n=3
0 0
5 10 30
1 1
T
2 4
, A A = 10 30 100
30 100
MATH 4445, 2016F, Homework 9
1. The function
f (x) = 3x3 2x2 6x + 4
has the following graph. Use the bisection method to find 2 roots of your choice. For
each root, (i) choose the appropriate interval [a, b], (ii) estimate the number of iterations require
MATH 4445, Homework 6 Solutions
1. In the following, two different approaches are provided to prove (a) and (b).
R
R
mm
(a) Proof: We consider A = Q
, where Q R
and
Rmn . In
0
0
the following we will use the theorem from linear algebra that rank(P A)
MATH 4445, Fall 2016, Homework 8
You may use Matlab in the following problems, unless otherwise specified. Keep you answers
accurate to at least 4 decimal places.
1. Consider matrix
1 1 1
A = 1 1 0
1 0 1
which has distinct eigenvalues |1 | > |2 | > |3 |.
MATH 4445, Homework 3 Solutions
1. (a) Algorithm: Gauss-Jordan
for k = 1 : n
for i = [1 : k 1, k + 1 : n]
lik = aik /akk
for j = k + 1 : n
aij = aij lik akj
end
bi = bi lik bk
end
end
for i = 1 : n
xi = bi /aii
end
P
( nk=1 )
(loop executed n 1 times)
(1
MATH 4445, Homework 4 solutions
1. (a) The matrix is not strictly diagonally dominant (but its diagonally dominant).
A is symmetric positive definite, because
the three leading principal submatrices
1 1
= 4 > 0, and det(A) = 36 > 0.
all have positive de