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where x
0
is given. Recall that the error is defined by
e
k
D
x
k
x
;
(4.208)
where x
is a root of f . Our aim is now to show that, under proper conditions,
j
e
kC1
j
ce
k
:
(4.209)
Let us assume that the
f
x
k
g
generated by (
4.207
) are all located in a bounded interval
I . We assume also that f is a smooth function defined on I ,andthat
ˇ
ˇ
f
0
.x/
ˇ
ˇ
˛>0;
(4.210)
and
ˇ
ˇ
f
00
.x/
ˇ
ˇ
ˇ<
1
;
(4.211)
for all x
2
I .
(l) Use (
4.207
)and(
4.208
) to show that
e
k
f
0
.x
k
/
f.x
k
/
f
0
.x
k
/
e
kC1
D
:
(4.212)
(m) Use the Taylor series (
4.205
) to show that
1
2
e
k
f.x
/
D
f.x
k
/
e
k
f
0
.x
k
/
C
f
00
./;
(4.213)
where is a number in the interval bounded by x
k
and x
.
(n) Show that
1
2
e
k
e
k
f
0
.x
k
/
f.x
k
/
D
f
00
./:
(4.214)
(o) Combine (
4.212
)and(
4.214
) to show that
f
00
./
2f
0
.x
k
/
e
k
e
kC1
D
:
(4.215)
(p) Use (
4.210
)and(
4.211
) to conclude that
ˇ
2˛
e
k
j
e
kC1
j
:
(4.216)
