Numerical Analysis –MTH603 VU
© Copyright Virtual University of Pakistan 1
R
RI
IC
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AR
RD
DS
SO
ON
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EX
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AT
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To improve the accuracy of the derivative of a function, which is computed by starting
with an arbitrarily selected value of h, Richardson’s extrapolation method is often
employed in practice, in the following manner:
S
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on
n,
,
t
th
he
en
n
w
we
e
h
ha
av
ve
e
()()
() 2
( )
T
T
yx h yx h
yx E
h
Fh E
+
−−
′
=
+
=+
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n
s
se
ee
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t
th
ha
at
t
246
123T
E chchch
=
+++
T
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a
o
of
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ic
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xt
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at
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om
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om
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pu
ut
te
ed
d
v
va
al
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ue
es
s
o
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()yx
′
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us
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ng
g
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e
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e
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et
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od
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ut
t
w
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it
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t
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if
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er
re
en
nt
t
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te
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si
iz
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es
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us
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ua
al
ll
ly
y
h
h
a
an
nd
d
h
h/
/2
2
t
to
o
y
yi
ie
el
ld
d
a
a
h
hi
ig
gh
he
er
r
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or
rd
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er
r
m
me
et
th
ho
od
d.
.
T
Th
hu
us
s,
,
w
we
e
h
ha
av
ve
e
24
12
() ()yx Fh ch ch
′=+++
And
24
12
() 2416
hhh
yx F c c
′=+++
H
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er
re
e,
,
c
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a
ar
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on
ns
st
ta
an
nt
ts
s,
,
i
in
nd
de
ep
pe
en
nd
de
en
nt
t
o
of
f
h
h,
,
a
an
nd
d
F
F(
(h
h)
)
a
an
nd
d
F
F(
(h
h/
/2
2)
)
r
re
ep
pr
re
es
se
en
nt
t
a
ap
pp
pr
ro
ox
xi
im
ma
at
te
e
v
va
al
lu
ue
es
s
o
of
f
d
de
er
ri
iv
va
at
ti
iv
ve
es
s.
.
E
El
li
im
mi
in
na
at
ti
in
ng
g
c
c1
1
f
fr
ro
om
m
t
th
he
e
a
ab
bo
ov
ve
e
p
pa
ai
ir
r
o
of
f
e
eq
qu
ua
at
ti
io
on
ns
s,
,
w
we
e
g
ge
et
t
46
1
4()
2
() ( )
3
h
FFh
yx dh Oh
−
′=++
N
No
ow
w
a
as
ss
su
um
mi
in
ng
g
t
th
ha
at
t
1
4()
2
23
h
FFh
h
F
−
=
E
Eq
qu
ua
at
ti
io
on
n
f
fo
or
r
y
y’
’(
(x
x)
)
a
ab
bo
ov
ve
e
r
re
ed
du
uc
ce
es
s
t
to
o
46
11
() ( )
2
h
yx F dh Oh
′=++
T
Th
hu
us
s,
,
w
we
e
h
ha
av
ve
e
o
ob
bt
ta
ai
in
ne
ed
d
a
a
f
fo
ou
ur
rt
th
h-
-o
or
rd
de
er
r
a
ac
cc
cu
ur
ra
at
te
e
d
di
if
ff
fe
er
re
en
nt
ti
ia
at
ti
io
on
n
f
fo
or
rm
mu
ul
la
a
b
by
y
c
co
om
mb
bi
in
ni
in
ng
g
t
tw
wo
o
r
re
es
su
ul
lt
ts
s
w
wh
hi
ic
ch
h
a
ar
re
e
o
of
f
s
se
ec
co
on
nd
d-
-o
or
rd
de
er
r
a
ac
cc
cu
ur
ra
at
te
e.
.
N
No
ow
w,
,
r
re
ep
pe
ea
at
ti
in
ng
g
t
th
he
e
a
ab
bo
ov
ve
e
a
ar
rg
gu
um
me
en
nt
t,
,
w
we
e
h
ha
av
ve
e
46
11
() ( )
2
h
yx F dh Oh
′=++
4
6
1
1
() ( )
416
dh
h
yx F Oh
′=++
E
El
li
im
mi
in
na
at
ti
in
ng
g
d
d1
1
f
fr
ro
om
m
t
th
he
e
a
ab
bo
ov
ve
e
p
pa
ai
ir
r
o
of
f
e
eq
qu
ua
at
ti
io
on
ns
s,
,
w
we
e
g
ge
et
t
a
a
b
be
et
tt
te
er
r
a
ap
pp
pr
ro
ox
xi
im
ma
at
ti
io
on
n
a
as
s
6
2
() ( )
4
h
yx F Oh
′=+
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