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Metode Terbuka
(Open Methods)
Pertemuan - 3
Mata Kuliah : Analisis Numerik dan Pemrograman
Kode : TSP – 303
SKS : 3 SKS
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• TIU : Mahasiswa dapat mencari akar-akar suatu persamaan, menyelesaikan
persamaan aljabar linear, pencocokan kurva, dan membuat program
aplikasi analisis numeriknya
• TIK : Mahasiswa dapat menjelaskan Metode Terbuka (Open Methods)
untuk menyelesaikan permasalahan akar-akar persamaan
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• Sub Pokok Bahasan :
Simple Fixed-Point Iteration
Metode Newton-Raphson
Metode Secant
Kelemahan dan Kelebihan dari metode tersebut
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• For the bracketing methods, the root is located within an interval prescribed by a lower and an upper bound.
• Repeated application of these methods always results in closer estimates of the true value of the root.
• Such methods are said to be convergent because they move closer to the truth as the computation progresses
• In contrast, the open methods are based on formulas that require only a single starting value of x or two starting values that do not necessarily bracket the root
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• As such, they sometimes diverge or move away from the true root as the computation progresses (Fig. b).
• However, when the open methods converge (Fig. c), they usually do so much more quickly than the bracketing methods.
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Simple Fixed-Point Iteration
• Open methods employ a formula to predict the root.
• Such a formula can be developed for simple fixed-point iteration (or, as it is also called, one-point iteration or successive substitution) by rearranging the function f(x)=0 so that x is on the left-hand side of the equation:
x=g(x) (1)
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• This transformation can be accomplished
either by algebraic manipulation or by simply
adding x to both sides of the original
equation.
• For example,
0
0322
xsin
xx
xxsinx
xx
2
32
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• The utility of Eq. (1) is that it provides a formula to predict a
new value of x as a function of an old value of x.
• Thus, given an initial guess at the root xi, Eq. (1) can be used
to compute a new estimate xi+1 as expressed by the iterative
formula
xi+1= g(xi) (2)
• the approximate error for this equation can be determined
using the error estimator
%
x
xx
i
iia 100
1
1
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Example 1 :
Use simple fixed-point iteration to locate the
root of f(x)=e−x− x
The function can be separated directly and expressed
in the form
xi+1=e−xi
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• Starting with an initial guess of x0=0, this
iterative equation can be applied to compute
Iteration xi g(xi) a (%) t (%)
0 0 1
1 1 0,36788 100
2
3
4
5
6
7
8
9
10
the true value of
the root: 0.56714329.
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The Newton-Raphson Method
• Perhaps the most widely used of all root-locating formulas is the Newton-Raphson equation.
• If the initial guess at the root is xi, a tangent can be extended from the point [xi, f(xi)].
• The point where this tangent crosses the x axis usually represents an improved estimate of the root.
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• the first derivative at x is equivalent to the
slope
• Which can be rearranged to yield Newton-
Raphson formula :
1
0
ii
ii
xx
xfxf
i
iii
xf
xfxx
1 (3)
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Example 2 :
Use the Newton-Raphson method to estimate
the root of f(x)=e−x− x, employing an initial
guess of x0 = 0
The first derivative of the function can be evaluated as
f(x = −e− − , hi h can be substituted along with the
original function into Eq. (3) to give
11
i
i
x
ix
iie
xexx
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• Starting with an initial guess of x0=0, this iterative
equation can be applied to compute
Iterasi x i f(x i ) f /(x i ) a (%) t (%)
1 0 1 -2
2 0,5 0,1065 -1,607
3 0,5663 0,0013 -1,568
4
5
the true value of
the root: 0.56714329.
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• Although the Newton-Raphson method is
often very efficient, there are situations
where it performs poorly.
• Fig. a depicts the case where an inflection
point [that is, f’’ = ] o urs in the i init of a root.
• Fig. b illustrates the tendency of the
Newton-Raphson technique to oscillate
around a local maximum or minimum.
• Fig. c shows how an initial guess that is
close to one root can jump to a location
several roots away.
• O iousl , a zero slope [f’ = ] is trul a disaster because it causes division by zero
in the Newton-Raphson formula (see Fig d)
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The Secant Method
• A potential problem in implementing the Newton-Raphson method is the evaluation of the derivative.
• Although this is not inconvenient for polynomials and many other functions, there are certain functions whose derivatives may be extremely difficult or inconvenient to evaluate.
• For these cases, the derivative can be approximated by a backward finite divided difference, as in the following figure
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ii
iii
xx
xfxfxf
1
1
This approximation can be
substituted into Eq. (3) to
yield the following iterative
equation:
ii
iiiii
xfxf
xxxfxx
1
1
1(4)
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• Equation (4) is the formula for the secant
method.
• Notice that the approach requires two initial
estimates of x.
• However, because f(x) is not required to
change signs between the estimates, it is not
classified as a bracketing method.
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Example 3 :
Use the Secant method to estimate the root of
f(x)=e−x− x, employing an initial guess of x0 = 0
Start with initial estimates of x−1 =0 and x0= 1.0.
Iteration x i-1 x i x i+1 f(x i-1 ) f(x i ) a (%)
0 0 1 0,6127 1 -0,632
1 1 0,6127 0,5638 -0,632 -0,071 -8,666
2
3
4
5
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The Difference Between the Secant and False-Position Methods
• The Eqs. Above are identical on a term-by-term basis.
• Both use two initial estimates to compute an approximation of the slope of the function that is used to project to the x axis for a new estimate of the root.
• However, a critical difference between the methods is how one of the initial values is replaced by the new estimate.
ii
iiiii
xfxf
xxxfxx
1
1
1
ul
uluur
xfxf
xxxfxx
False-Position Formula Secant Formula
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• Recall that in the false-position method the
latest estimate of the root replaces
whichever of the original values yielded a
function value with the same sign as f(xr).
• Consequently, the two estimates always
bracket the root.
• Therefore, the method always converges
• In contrast, the secant method replaces the
values in strict sequence, with the new
value xi+1 replacing xi and xi replacing xi− .
• As a result, the two values can sometimes
lie on the same side of the root.
• For certain cases, this can lead to
divergence.
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Homework
Use : Simple Fixed-Point Iteration Method (x = g(x))
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Homework
Use Secant Method
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Homework
Use Newton Raphson Method