MATH 128A
Homework #2
Due (in class) Tuesday, June 29.
All numbered exercises are from the textbook Numerical Analysis by Bur-
den and Faires, 7th ed.
a) Section 1.3 6(a,b),7(d),8,10,15,16
b) Section 2.1 10,12,14,16
If xn→x∗then let us define a new seqence pn=|xn+1−x∗|
|xn−x∗|α.
For α= 1 and pn→λ < 1, we say that xn→x∗linearly; if λ= 1, the
convergence is sublinear. (What happens if λ > 1?)
For α≥2 if pn→λ(notice, no restriction on λ), we call this convergence
of order α.
For the following sequences perform the asymptotic error analysis, i.e.
numerically compute pnfor n= 1, . . . , 1000:
a) xn= (1 + 1
n)n; here x∗=eand α= 1
b) xn+1 =xn−x2
n
x2
n+x2
n−1
;x0= 20, x1= 15; here x∗= 0 and α= 1 and
c) xn+1 = 1/2xn+1
xn,x0= 2; here x∗=√2, try α= 1,2
Plot pnthat you are getting and guess what the limit is in each case.
For the last sequence, experiment with various starting values and see for
what range of starting values the original sequence xnwill still converge to
√2.