Math 151A Homework #1 โ due Wednesday 10/11, in class
Show all your work!
1. On the importance of stopping criteria. (section 2.1, problem 16)
Let f(x) = (xโ1)10 (with a root at p= 1), and consider the sequence of approximations to
the root pgiven by pn= 1 + (1/n). Show the the stopping criterion |f(pn)|<10โ3is met
for nโฅ2, but |pโpn|<10โ3requires that n > 1000.
Answer:
The first stopping criterion requires that
|f(pn)|=|(pnโ1)10|=๎
๎
๎๎1
n๎10๎
๎
๎=๎1
n๎10
<10โ3
โ10 log(n)<โ3
n > 100.3โ1.995
which is satisfied when nโฅ2.
The second stopping criterion requires that
|pโpn|=|1/n|= 1/n < 10โ3
which is satisfied when n > 103.
2. On the importance of stopping criteria, II. (section2.1, problem 17)
Let pnbe the sequence defined by pn=Pn
k=1(1/k). Show that pndiverges even though
limnโโ(pnโpnโ1) = 0.
Answer:
pnโpnโ1=
n
X
k=1
๎1
k๎โ
nโ1
X
k=1
๎1
k๎
=1
n
So
lim
nโโ
(pnโpnโ1) = lim
nโโ
1
n= 0
so the difference between consectutive terms vanishes. But {pn}is the well-known
harmonic series, which diverges.
3. The sqare root of two.
Use the bisection method to approximate the value of โ2, given that it lies in the interval
[1,2]. Do this in two parts:
a. Calculate the maximum number of iterations required to estimate โ2 to within 10โ4,
starting with the interval [1,2].
