the solution of non-linier equation
TRANSCRIPT
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The Solution of Non-Linier Equation
Nur Laila Hamidah
Ref : Erwin Kreyszig-Advanced Engineering Mathematics http://numericalmethods.eng.usf.edu
Matematika Rekayasa 1
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CapaianPembelajaran:
• Mampu menyelesaikan akar akar
persamaan non linier dengan
menggunakan metode regula falsi, biseksi,
newton raphson dan secant
• Mampu membandingkan kelebihan dan kekurangan keempat metode tersebut
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False Position Methods
Secant Methods
Newton-Raphson
Bisection02
0504
03
01 Error of Numerical Result
Solution of equation by iteration
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Error in Numerical Result
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Didalam setiap metode numerik harus terdapat perhitungan error. Jika tidak
terdapat formulasi eror, maka hasil numerik tersebut akan menjadi complicated
Prinsip dasar error
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The False-Position Method
(Regula-Falsi)
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We can approximate the solution by doing a linear interpolationbetween f(xu) and f(xl)
Find xr such that l(xr)=0, where l(x) is the linear approximation of f(x)between xl and xu
Derive xr using similar triangles
Basis of False Position Method
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( ) ( )
( )( )( ) ( )
)()().(
afbf
abbfbc
xfxf
xxxfxx
xx
xf
xx
xf
ul
uluur
ur
u
lr
l
−
−−=
−
−−=
−=
−
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Step 1
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Step 2
( ) ( )
( )( )( ) ( )
)()().(
afbf
abbfbc
xfxf
xxxfxx
xx
xf
xx
xf
ul
uluur
ur
u
lr
l
−
−−=
−
−−=
−=
−
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Step 3
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Step 4( ) ( )
( )( )( ) ( )
)()().(
afbf
abbfbc
xfxf
xxxfxx
xx
xf
xx
xf
ul
uluur
ur
u
lr
l
−
−−=
−
−−=
−=
−
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Step 5
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Bisection Method
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Theorem
x
f(x)
xu x
An equation f(x)=0, where f(x) is a real continuous function, has at least one root between xl and xu if f(xl) f(xu) < 0.
Figure 1 At least one root exists between the two points if the function is real, continuous, and changes sign.
Basis of Bisection Method
http://numericalmethods.eng.usf.edu
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Basis of Bisection Method
x
f(x)
xu x
Figure 2 If function does not change sign between two points, roots of the equation may still exist between the two points.
( )xf
( ) 0=xf
http://numericalmethods.eng.usf.edu
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Basis of Bisection Method
x
f(x)
xu x
Figure 3 If the function does not change sign between two points, there may not be any roots for the equation between the two points.
x
f(x)
xu
x
( )xf
( ) 0=xf
http://numericalmethods.eng.usf.edu
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Basis of Bisection Method
x
f(x)
xu x
http://numericalmethods.eng.usf.edu
Figure 4 If the function changes sign between two points, more than one root for the equation may exist between the two points.
( )xf
( ) 0=xf
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Step 1
http://numericalmethods.eng.usf.edu
Choose x and xu as two guesses for the root such that f(x) f(xu) < 0, or in other words, f(x) changes sign between x and xu. This was demonstrated in Figure 1.
x
f(x)
xu x
Figure 1
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Step 2
http://numericalmethods.eng.usf.edu
x
f(x)
xu x
xm
Estimate the root, xm of the equation f (x) = 0 as the midpoint between x and xu as
xx
m = xu +
2
Figure 5 Estimate of xm
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Step 3
http://numericalmethods.eng.usf.edu
Now check the following
a) If , then the root lies between x and xm;
then x = x ; xu = xm.
b) If , then the root lies between xm and xu;
then x = xm; xu = xu.
c) If ; then the root is xm. Stop the algorithm
if this is true.
( ) ( ) 0ml xfxf
( ) ( ) 0ml xfxf
( ) ( ) 0=ml xfxf
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Step 4
http://numericalmethods.eng.usf.edu
xx
m = xu +
2
100−
=new
m
old
m
new
ax
xxm
root of estimatecurrent =new
mx
root of estimate previous=old
mx
Find the new estimate of the root
Find the absolute relative approximate error
where
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Step 5
http://numericalmethods.eng.usf.edu
Is ?
Yes
No
Go to Step 2 using new upper and lower
guesses.
Stop the algorithm
Compare the absolute relative approximate error with the pre-specified error tolerance .
a
s
sa
Note one should also check whether the number of iterations is more than the maximum number of iterations allowed. If so, one needs to terminate the algorithm and notify the user about it.
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Tugas Penyelesaian Persamaan Non Linier
1. 2x3-2x-5 = 0, interval [1,2]
4. (x-2)2- lnx = 0, interval [1,2]
Cari akar akar persamaan non linier dibawah dengan menggunakan
metode regula falsi dan biseksi
0 ,1 ,0
0sin3)(
1010 ==
=−+=
ffxx
exxxf x2.
3. 5sin2x - 8 cos5x = 0, interval [0.5, 1.5]