(ii) Use the Fixed-Point Iteration method to find an approximation pN of the fixed-point p of g(x) in [1, 2], the root of the polynomial f(x) in [1, 2], satisfying RE(pN = PN-1) < 10-6 by taking po = 1 as the initial approximation. All calculations are to be carried out in the FPA7. Present the results of your calculations in a standard output table for the method of the form RE(pn ~ Pn-1) n Pn-1 Pn
(ii) Use the Fixed-Point Iteration method to find an approximation pN of the fixed-point p of g(x) in [1, 2], the root of the polynomial f(x) in [1, 2], satisfying RE(pN = PN-1) < 10-6 by taking po = 1 as the initial approximation. All calculations are to be carried out in the FPA7. Present the results of your calculations in a standard output table for the method of the form RE(pn ~ Pn-1) n Pn-1 Pn
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Numerical Analysis & it’s applications
Q4 only part (ii) of the question needs answering. Part (i) is answered and the answer is included here. Thanks.
![4. (Fired-Point Iteration Method;
Consider the problem of finding the root of the polynomial
f (x) = x* - 0.89x - 1.08
in [1, 2].
(i) Show that
f (x)
f'(x)
f(x) = 0 = x = x -
on [1, 2], thereby reducing the root-finding problem to the fixed-point problem for the function
f(x)
f'(x)
g(x) = x -
on [1,2].
(ii) Use the Fixed-Point Iteration method to find an approximation pN of the fixed-point p of g(x) in [1, 2], the root of the polynomial f(x) in
[1, 2], satisfying RE(pN PN-1) < 10-6 by taking po = 1 as the initial approximation. All calculations are to be carried out in the FPA7. Present
the results of your calculations in a standard output table for the method of the form
Pn-1
Pn RE(pn = Pn-1)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdc92fb0c-7643-4ce5-8ef2-0330ed27f016%2F6fe53747-88f8-48dd-a4d9-dfa9da85cab3%2F9motrv5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4. (Fired-Point Iteration Method;
Consider the problem of finding the root of the polynomial
f (x) = x* - 0.89x - 1.08
in [1, 2].
(i) Show that
f (x)
f'(x)
f(x) = 0 = x = x -
on [1, 2], thereby reducing the root-finding problem to the fixed-point problem for the function
f(x)
f'(x)
g(x) = x -
on [1,2].
(ii) Use the Fixed-Point Iteration method to find an approximation pN of the fixed-point p of g(x) in [1, 2], the root of the polynomial f(x) in
[1, 2], satisfying RE(pN PN-1) < 10-6 by taking po = 1 as the initial approximation. All calculations are to be carried out in the FPA7. Present
the results of your calculations in a standard output table for the method of the form
Pn-1
Pn RE(pn = Pn-1)
![Step1
a)
let un eapress
Step3
c)
fen=o s a
x approximale fro by
eguahin win Ihe neghbourhood of
forat degre e equahan.
(0.89*1.06)"s
中ca)
prh degure
Ihe
Iheri
may
wrile
frm= aox+ a, = 6-0
Step2
Ashere a.t0. and a, ave arbihom
pavamelas. a be
appropnale Cnditans
delnmınid by preseribed
on f ) and ils denvaluis
b)
tiwo
ocx ) = (os9x + os )
on Substiluting ao and a, from O sin and represtenting
hi approximalue volues a x by xes ocue
obtain
%3D
Frax)
he problem of tinding rat of he eg Ois egenvaled
at lhe point (**itx) wilk Ihi
The
evaluahen'. fe, te' each eleration
(o-59)
let
X0=1+2=1-5
Lwi
d'ns) =
Repsomilhd reuirg
Newton
Suppore x* s the exact
8olution for= 0
At he 4 N-R. ite vaten,
defne lhe erm (Ei:
whuch es dhaving cğuadrahc
Convergina - EurCCEE, cs Canaloint
fro is molle and f'ram)fo
and 1x-x* |l fmook endugh
Ysdoal Comurrgence
%3D
0-2391
く1
%3D
=1-2466
, (a ): Co$9+ 1-08)""= (6 89 C1·2u6o)t 18)"-2164
$3c*)= (o89*2+ •08) "4 = Co•84 (1.21b4) +1•08)"4=t2167
fuca)= (o$9x3+1.08)4 = (089 (1:2167) +1 es) "4 = 1-2
f5107= Cofs ry t1-08)"4= (05s c1iz123)+18); 1-2122
The rool by ed point levaħau
Critenm
4 212と
=) 1= $(2)
1/4
x =1.21 22 E(!,2]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdc92fb0c-7643-4ce5-8ef2-0330ed27f016%2F6fe53747-88f8-48dd-a4d9-dfa9da85cab3%2Fgkzytfi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Step1
a)
let un eapress
Step3
c)
fen=o s a
x approximale fro by
eguahin win Ihe neghbourhood of
forat degre e equahan.
(0.89*1.06)"s
中ca)
prh degure
Ihe
Iheri
may
wrile
frm= aox+ a, = 6-0
Step2
Ashere a.t0. and a, ave arbihom
pavamelas. a be
appropnale Cnditans
delnmınid by preseribed
on f ) and ils denvaluis
b)
tiwo
ocx ) = (os9x + os )
on Substiluting ao and a, from O sin and represtenting
hi approximalue volues a x by xes ocue
obtain
%3D
Frax)
he problem of tinding rat of he eg Ois egenvaled
at lhe point (**itx) wilk Ihi
The
evaluahen'. fe, te' each eleration
(o-59)
let
X0=1+2=1-5
Lwi
d'ns) =
Repsomilhd reuirg
Newton
Suppore x* s the exact
8olution for= 0
At he 4 N-R. ite vaten,
defne lhe erm (Ei:
whuch es dhaving cğuadrahc
Convergina - EurCCEE, cs Canaloint
fro is molle and f'ram)fo
and 1x-x* |l fmook endugh
Ysdoal Comurrgence
%3D
0-2391
く1
%3D
=1-2466
, (a ): Co$9+ 1-08)""= (6 89 C1·2u6o)t 18)"-2164
$3c*)= (o89*2+ •08) "4 = Co•84 (1.21b4) +1•08)"4=t2167
fuca)= (o$9x3+1.08)4 = (089 (1:2167) +1 es) "4 = 1-2
f5107= Cofs ry t1-08)"4= (05s c1iz123)+18); 1-2122
The rool by ed point levaħau
Critenm
4 212と
=) 1= $(2)
1/4
x =1.21 22 E(!,2]
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