l&l < Ii < 1, vr ⬠(o,b) s\,ceo
1) (10 points)
g(r) - 2 sin(zr ,) lnPUse the theorer4llelpw to determine the interval [a, b] and to show that g(x) has a unique fixed point on.[a, SgUs" fixed point iteration, with initial guess g5!$, to find an approximation to ih-e fixed point that is accurate to within lJJ UFiEe-corollary below to estimate the number of iterations required to achieve Ia%,curacy. Ur" ebgglulg€1ggr to stop iterations. Illustrate on the plot how the Consider the function
lF-?o\t L
approximations are obtained
@ilty.
Theorem:If9eClo,b]@)4,b]foraIlr€[o,b],theng.hasafixedpointin [o,b). Suppose, in addition, that g'(r) exists on (a, b) and that a positive .constant k exists
with
x.
3
=
'511 + c'llsrlf , -z lo,S^lq-0'to5-l \'t1 XC?
c€et-}} a rrx
cr(
L <X
^lo
S.b3-t
Qr+=
3
-L
t
Io't35t l*€-
C)r
L
* 0 '(ob-l
\ u't'3'l-O'Soz\
I ll. Z co$ rsA
g.b \rled
V
q i\ecs"fion 5 ?o -) ?s
-
\.
zl5
-o.z \
= 9,b
2) (10 points)
a) Using Taylor's series for / about rs, derive an O(h2) three-point formula to approximate f"(ro) that uses /(zo - h), f (*o), and f (*o + h).
b) A particularly important
subject in the study of numerical differentiation is the effect roundoff and truncation errors play in the approximation. Suppose that in evaluating f ("0+h), f ("0) and /(zs-h) the computed values are l@s+h), i@d and, i@s-h) with roundoff errors e(ro + h),, e(no) and e(r6 - h), respectively. Assume that 1",1*s
- h)|,1"(ro)1,1"(ro + h)l S .,
l/(n)(€) I S
U.
r^
Find a bound to the total error of the approximation you obtained. Discuss the effect of roundoff and truncation errors in terms of the step size h. Find the optimal step size h that minimize the error.
o.) fi*.'+t-1 , {l*o, t !.f tt"r +
h.f, "cxoJ
+
f,v"-ht' f.u"t - Lf['v", + 5] f"a*", L
f".uor'
7?tu.)
*
h' l.
*
f",*or ' f,*".h\ -
€, Lt" *r'l
h?-
A,
7A
@
Ir-
\r\* /\J A.
'z---
h"
+ Yf,'t1) lL ,*-4
-
- t"-lJ , r[[ror - \c*o-*] le
. 5] f]^t"f c",)
f.-.-*rl + \t ['"t [1,.,)
i'fw,),-- { (*or e txo) rt.'-h') {tto - xl = ia*"- k\ - €-(
t'r'a
/',*.',
t t*' vs)
b) L*" r\ \ -- {tx"*s)
{t'.*o,
z\
kt f " (v,) + f,ut( 1 .,.) -C IL
It,-"+h) - l{cr')
#
..-f $t*t(.{.*l) b
lo
L
\D+@ : f.*. +h) + Ctr"-b')
h' f,')to".,
--
= l-etxo+t..t[ !
h"
hl-
t'trr s t-\ .r' o*1 , te cv")\,\e c vo-b.\ \ ( t ,, 4L + h,zH
g(r) - 2 sin(zr ,) lnPUse the theorer4llelpw to determine the interval [a, b] and to show that g(x) has a unique fixed point on.[a, SgUs" fixed point iteration, with initial guess g5!$, to find an approximation to ih-e fixed point that is accurate to within lJJ UFiEe-corollary below to estimate the number of iterations required to achieve Ia%,curacy. Ur" ebgglulg€1ggr to stop iterations. Illustrate on the plot how the Consider the function
lF-?o\t L
approximations are obtained
@ilty.
Theorem:If9eClo,b]@)4,b]foraIlr€[o,b],theng.hasafixedpointin [o,b). Suppose, in addition, that g'(r) exists on (a, b) and that a positive .constant k exists
with
x.
3
=
'511 + c'llsrlf , -z lo,S^lq-0'to5-l \'t1 XC?
c€et-}} a rrx
cr(
L <X
^lo
S.b3-t
Qr+=
3
-L
t
Io't35t l*€-
C)r
L
* 0 '(ob-l
\ u't'3'l-O'Soz\
I ll. Z co$ rsA
g.b \rled
V
q i\ecs"fion 5 ?o -) ?s
-
\.
zl5
-o.z \
= 9,b
2) (10 points)
a) Using Taylor's series for / about rs, derive an O(h2) three-point formula to approximate f"(ro) that uses /(zo - h), f (*o), and f (*o + h).
b) A particularly important
subject in the study of numerical differentiation is the effect roundoff and truncation errors play in the approximation. Suppose that in evaluating f ("0+h), f ("0) and /(zs-h) the computed values are l@s+h), i@d and, i@s-h) with roundoff errors e(ro + h),, e(no) and e(r6 - h), respectively. Assume that 1",1*s
- h)|,1"(ro)1,1"(ro + h)l S .,
l/(n)(€) I S
U.
r^
Find a bound to the total error of the approximation you obtained. Discuss the effect of roundoff and truncation errors in terms of the step size h. Find the optimal step size h that minimize the error.
o.) fi*.'+t-1 , {l*o, t !.f tt"r +
h.f, "cxoJ
+
f,v"-ht' f.u"t - Lf['v", + 5] f"a*", L
f".uor'
7?tu.)
*
h' l.
*
f",*or ' f,*".h\ -
€, Lt" *r'l
h?-
A,
7A
@
Ir-
\r\* /\J A.
'z---
h"
+ Yf,'t1) lL ,*-4
-
- t"-lJ , r[[ror - \c*o-*] le
. 5] f]^t"f c",)
f.-.-*rl + \t ['"t [1,.,)
i'fw,),-- { (*or e txo) rt.'-h') {tto - xl = ia*"- k\ - €-(
t'r'a
/',*.',
t t*' vs)
b) L*" r\ \ -- {tx"*s)
{t'.*o,
z\
kt f " (v,) + f,ut( 1 .,.) -C IL
It,-"+h) - l{cr')
#
..-f $t*t(.{.*l) b
lo
L
\D+@ : f.*. +h) + Ctr"-b')
h' f,')to".,
--
= l-etxo+t..t[ !
h"
hl-
t'trr s t-\ .r' o*1 , te cv")\,\e c vo-b.\ \ ( t ,, 4L + h,zH
