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Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
1
Chapter 6
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Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
2
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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3
Simple Fixed-point Iteration
...2,1,k,given)()(0)(
1
okk xxgxxxgxf
Bracketing methods are convergent.
Fixed-point methods may sometime
diverge, depending on the stating point
(initial guess) and how the function behaves.
Rearrange the function so that x is on the
left side of the equation:
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4
xxg
or
xxgor
xxg
xxxxf
21)(
2)(
2)(
02)(
2
2
Example:
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5
Convergence
x=g(x) can be expressed
as a pair of equations:
y1=x
y2=g(x) (component
equations)
Plot them separately.
Figure 6.2
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6
Conclusion
Fixed-point iteration converges if
x)f(x)linetheof(slope1)( xg
When 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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Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
7
Newton-Raphson Method
Most widely used method.
Based on Taylor series expansion:
)(
)(
)(0
g,Rearrangin
0)f(xwhenxofvaluetheisrootThe
!2)()()()(
1
1
1i1i
3
2
1
i
iii
iiii
iiii
xf
xfxx
xx)(xf)f(x
xOx
xfxxfxfxf
Newton-Raphson formula
Solve for
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8
A convenient method for
functions whose
derivatives can be
evaluated analytically. It
may not be convenientfor functions whose
derivatives cannot be
evaluated analytically.
Fig. 6.5
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9
Fig. 6.6
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10/19Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
10
The Secant Method
A slight variation of Newtons method forfunctions whose derivatives are difficult toevaluate. For these cases the derivative can beapproximated by a backward finite divided
difference.
,3,2,1)()(
)(
)()(
)(
1
11
1
1
ixfxf
xxxfxx
xfxf
xxxf
ii
iiiii
ii
iii
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11
Requires two initial
estimates of x , e.g, xo,
x1. However, becausef(x) is not required to
change signs between
estimates, it is not
classified as a
bracketing method.
The scant method has the
same properties as
Newtons method.
Convergence is not
guaranteed for all xo,
f(x).
Fig. 6.7
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12
Fig. 6.8
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13
Multiple Roots
None of the methods deal with multiple roots
efficiently, however, one way to deal with problems
is as follows:
)(
)(1xfindThen
)()()(Set
i
i
i
i
ii
xu
xu
xfxfxu
This function has
roots at all the same
locations as the
original function
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14
Fig. 6.13
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15
Multiple root corresponds to a point where a
function is tangent to the x axis. Difficulties
Function does not change sign at the multiple root,
therefore, cannot use bracketing methods. Both f(x) and f(x)=0, division by zero with
Newtons and Secant methods.
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16
Systems of Linear Equations
0),,,,(
0),,,,(
0),,,,(
321
3212
3211
nn
n
n
xxxxf
xxxxf
xxxxf
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17
Taylor series expansion of a function of more than
one variable
)()(
)()(
11111
11111
iii
ii
iii
iii
ii
iii
yyyvxx
xvvv
yyy
uxx
x
uuu
The root of the equation occurs at the value of xand y where ui+1and vi+1equal to zero.
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18
y
vy
x
vxvy
y
vx
x
vy
uy
x
uxuy
y
ux
x
u
ii
iiii
ii
i
ii
iiii
ii
i
11
11
A 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
1
1
Determinant of
the Jacobianof
the system.