Handout CIV 208 Analisis Numerik CIV 208 P2 3

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Mata Kuliah : Analisis Numerik
Kode
: CIV - 208
SKS
: 3 SKS

Akar Persamaan
(Roots of Equations)
Pertemuan – 2, 3

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• Sub Pokok Bahasan :
 Metode Biseksi
 Metode Regula Falsi
 Simple Fixed-Point Iteration
 Metode Newton-Raphson
 Metode Secant

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Years ago, we learned to use
the quadratic formula
 b  b 2  4ac
x
2a

(1
to solve f  x  ax 2 )bx  c 0 (2
The values calculated with )
Eq. (1) are called the “roots”
of Eq. (2).
They represent the values of
x that make Eq. (2) equal to
zero.

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Bracketing Methods
• This topic on roots of equations deals with methods that
exploit the fact that a function typically changes sign in
the vicinity of a root.
• These techniques are called bracketing methods because
two initial guesses for the root are required.
• As the name implies, these guesses must “bracket,” or be
on either side of, the root.
• The particular methods described herein employ different
strategies to systematically reduce the width of the
bracket and, hence, home in on the correct answer
• Three methods classified as Bracketing Methods are :
graphical method, bisection method and the falseposition method

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Graphical Method
• A simple method for obtaining an
estimate of the root of the equation

f(x)=0 is to make a plot of the
function and observe where it
crosses the x axis.
• This point, which represents the xvalue for which f(x)=0, provides a
rough approximation of the root.

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Example 1 :
Use the graphical approach to determine the real
root of
f  x: 4 x 3  6 x 2  7 x  2 ,3
Various values of x can be substituted
into the right-hand side of this
equation to compute f(x),
for example if x = 0, then
f(x) = 4(0)3 – 6(0)2 + 7(0) – 2,3 = − 2,3
Try for another value of x, and
tabulated the results

x

f(x)

0

-2,3

0,1

-1,656

0,2

-1,108

0,3

-0,632

0,4

-0,204

0,5

0,2

0,6

0,604

0,7

1,032

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These points are plotted in
Figure.
The x value as a horizontal
axes, and f(x) as a vertical
axes
Resulting curve crosses
the x axis between 0,4 and
0,5.
Visual inspection of the
plot provides a rough
estimate of the root of
0,45.

1.5

root
location

1
0.5

0
0

0.1

0.2

0.3

0.4

f(x) -0.5
-1
-1.5
-2
-2.5
x

0.5

0.6

0.7

0.8

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• The validity of the graphical estimate can
be checked by substituting it
f  0 ,45 4 0,45 3  6 0,45 2  7 0,45  2,3  0,0005
• which is close to zero, and x = 0,45 said
as af root
of
x  4 x 3  6 x 2  7 x  2 ,3

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Bisection Method
• When applying the graphical technique in
Example 1, we have observed that f(x)
changed sign on opposite sides of the root.
• In general, if f(x) is real and continuous in
the interval from xl to xu and f(xl) and f(xu)
have opposite signs, that is,
f(xl)f(xu)