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Chapter 6
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"
Open MethodsChapter 6
Open methods
are based on
formulas that
require only asingle starting
value of x or two
starting values
that do notnecessarily
bracket the root.
Figure 6.1
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#
imple !ixed"point #teration
...$%&%k%given'('()'(
& ==
==
okk xxgxxxgxf
*racketing methods are +convergent,.
!ixed"point methods may sometime
+diverge,% depending on the stating point
(initial guess' and how the function behaves.
-earrange the function so that x is on the
left side of the equation
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$
xxg
or
xxgor
xxg
xxxxf
$&'(
$'(
$'(
)$'(
$
$
+=
+=
=
=
/xample
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%
Convergence
x0g(x' can be expressed
as a pair of equations
y&0x
y$0g(x' (component
equations'
1lot them separately.
Figure 6.2
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&
Conclusion
!ixed"point iteration converges if
x'f(x'linetheof(slope&'( = xg
2hen the method converges% the error is
roughly proportional to or less than the error of
the previous step% therefore it is called +linearlyconvergent.,
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'
3ewton"-aphson Method
Most widely used method.
*ased on 4aylor series expansion
'(
'(
'()
g%-earrangin
)'f(xwhenxofvaluetheisroot4he
5$'('('('(
&
&
&i&i
$
&
i
iii
iiii
iiii
xf
xfxx
xx)(xf)f(x
xOx
xfxxfxfxf
=
+=
=
+
++=
+
+
++
+
3ewton"-aphson formula
olve for
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(
7 convenient method for
functions whose
derivatives can be
evaluated analytically. #t
may not be convenientfor functions whose
derivatives cannot be
evaluated analytically.
Fig. 6.5
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)
Fig. 6.6
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!*
4he ecant Method
7 slight variation of 3ewton8s method forfunctions whose derivatives are difficult toevaluate. !or these cases the derivative can beapproximated by a backward finite divided
difference.
%%$%&'('(
'(
'('(
'(
&
&&
&
&
=
=
+
ixfxf
xxxfxx
xfxf
xxxf
ii
iiiii
ii
iii
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!!
-equires two initial
estimates of x % e.g% xo%
x&. 9owever% becausef(x' is not required to
change signs between
estimates% it is not
classified as a
+bracketing, method.
4he scant method has the
same properties as
3ewton8s method.
Convergence is not
guaranteed for all xo%
f(x'.
Fig. 6.7
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!"
Fig. 6.8
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!#
Multiple -oots
3one of the methods deal with multiple roots
efficiently% however% one way to deal with problems
is as follows
'(
'(&xfind4hen
'('('(et
i
i
i
i
ii
xu
xu
xfxfxu
+
= 4his function has
roots at all the same
locations as the
original function
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!$
Fig. 6.13
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!%
+Multiple root, corresponds to a point where a
function is tangent to the x axis. :ifficulties
;!unction does not change sign at the multiple root%
therefore% cannot use bracketing methods.;*oth f(x' and f
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!&
ystems of >inear /quations
)'%%%%(
)'%%%%(
)'%%%%(
$&
$&$
$&&
=
=
=
nn
n
n
xxxxf
xxxxf
xxxxf
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!'
4aylor series expansion of a function of more than
one variable
'('(
'('(
&&&&&
&&&&&
iii
ii
iii
iii
ii
iii
yyyvxx
xvvv
yyy
uxx
x
uuu
+
+=
+
+=
+++
+++
4he root of the equation occurs at the value of xand y where ui?&and vi?&equal to =ero.
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!(
y
vy
x
vxvy
y
vx
x
vy
uy
x
uxuy
y
ux
x
u
ii
iiii
ii
i
ii
iiii
ii
i
+
+=
+
+
+=
+
++
++
&&
&&
7 set of two linear equations with two
unknowns that can be solved for.
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!)
x
v
y
u
y
v
x
ux
uvx
vu
yy
x
v
y
u
y
v
x
uy
uv
y
vu
xx
iiii
iiii
ii
iiii
ii
ii
ii
=
=
+
+
&
&
:eterminant of
the Jacobianof
the system.