Download Homework #5 - Douglas Weathers

Homework #5: Sequences, matrices, and algorithms
MATH 174
Be aware that this document is two pages long.
1. (a) Give a closed-form rule for the sequence {xn } = 1, 3, 7, 15, 31, . . .
(b) Give a recurrence relation for the sequence given by the rule yk = 3 + 5k.
(c) Write out the first five elements of the sequence given by the recurrence relation fm =
fm−1 + fm−2 , where f0 = 1 and f1 = 1.
2. Let {xn }, {yk }, and {fm } be the same sequences they were in (1).
(a) Compute
4
X
xn .
n=0
(b) Compute
(c) Compute
4
X
1
.
f
m=0 m
3
2 X
X
(yk − xn ).
k=0 n=1
3. (a) Compute
(b) Compute
(c) Compute
(d) Compute
3
0
2
4
−1
5
3
0
2
4
−1
5
3
0
2
4
−1
5
0
1
1
1
0
0
+
6
1

6
 −2
−1

1
 1
0
6
1
0 −2
−1
3
0 −2
−1
3
or say it does not exist.
or say it does not exist.

0
1  or say it does not exist.
3

1
0  or say it does not exist.
1
4. (a) Describe an algorithm that computes the sum
n
X
ak of a finite sequence a0 , a1 , a2 , . . . , an .
k=0
(b) Describe an algorithm that computes the sum of two k × n matrices A = [ai,j ] and
B = [bi,j ].
http://douglasweathers.nfshost.com/s17math174.html
5. Let



A=

1
2
1
4
0
1
2
1
2
1
2
0
1
4
1
2






256





p =  256 


0
(a) Consider the matrix sequence1 p, Ap, A2 p, A3 p . . . Write a recurrence relation for this
sequence. (The notation An , as you may have guessed, is the square matrix A multiplied
by itself n times.)
(b) Using your recurrence relation in (a), write down the first five terms of this sequence.
(c) Describe an algorithm that produces the first N elements of this sequence.
(d) Does it look as though the matrices in the sequence are getting “closer and closer”
(whatever that means2 ) to a single matrix? If so, what is that matrix? (Credit for completion.)
1 If you must know, p is a population of 512 bean plants. Half (256) of them are dominant (YY) yellow;
half are hybrid (Yy) yellow; none are recessive (yy) green. Left-multiplying p by A models the change in
successive populations when they are bred with a hybrid plant.
2 MATH 161 students, it means: What is lim An p as n → ∞? We’d really need to define what it means
for two matrices to be close together, but you get the idea. Question (5d) is hence asking, if we keep
crossing our population with a hybrid plant, does it eventually level out to some distribution? What is that
distribution?