Handout CIV 208 Analisis Numerik CIV 208 P2 3

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Akar Persamaan

(Roots of Equations)

Pertemuan – 2, 3

Mata Kuliah : Analisis Numerik Kode : CIV - 208

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• _{Sub Pokok Bahasan :}

 _{Metode Biseksi}

 _{Metode Regula Falsi}

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Years ago, we learned to use

the quadratic formula

to solve

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.

(1

)

₍₂

)

a

ac b

b x

2

4 2 _

 



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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 :}

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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}

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Example 1 :

Use the graphical approach to determine the real

root of :

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

 

x 4x3 6x2 7x 2,3

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These points are plotted in

Figure.

The x value as a horizontal

axes, and f(x) as a vertical

axes

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•

_{The validity of the graphical estimate can}

be checked by substituting it

•

_{which is close to zero, and x = 0,45 said}

as a root of



0,45



4



0,45



3 6



0,45



2 7



0,45



2,3 0,0005

f     

 

x 4x3 6x2 7x 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 x

l

to x

u

and f(x

l

) and f(x

u

)

have opposite signs, that is,

f(x

l

)f(x

u

)<0

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Termination Criteria and Error Estimates

•

an approximate percent relative error ε

a

can be

calculated, as in

•

where x

rnew

is the root for the present iteration and

x

rold

is the root from the previous iteration.

•

_{The absolute value is used because we are usually}

concerned with the magnitude of ε

a

rather than with

its sign.

•

When ε

a

becomes less than a prespecified stopping

criterion ε

s

, the computation is terminated.

%

x

x

x

new r

old r new

r

a



100





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Example 2 :

Use bisection to solve the same problem

approached graphically in Example 1, until

the approximate error falls below a stopping

criterion of

ε

_s

=0.5%.

Root of equation

 

x 4x3 6x2 7x 2,3

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The False-Position Method

• A shortcoming of the bisection method is that, in dividing the interval from xl to xu into equal halves, no account is

taken of the magnitudes of f(xl) and f(xu).

• For example, if f(xl) is much closer to zero than f(xu), it is

likely that the root is closer to xl than to xu.

• An alternative method that exploits this graphical insight is to join f(xl) and f(xu) by a straight line.

• The intersection of this line with the x axis represents an improved estimate of the root.

• The fact that the replacement of the curve by a straight line gives a “false position” of the root is the origin of the name, method of false position, or in Latin, regula falsi.

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•

_{Using similar}

triangles, the

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Example 3 :

Use the false-position method to

determine the root of the same

equation investigated in Example 1.

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Home work (Group)

Kerjakan soal dari buku Chapra Soal

no.

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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

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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}

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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:

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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,}

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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 x

i

, Eq. (1)

can be used to compute a new estimate x

i+1

as

expressed by the iterative formula

x

i+1

=

g

(

x

i

)

(2)

•

_{the approximate error for this equation can be}

determined using the error estimator

%

x

x

x

i

i i

a

100

1

1



_



 

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Example 1 :

Use simple fixed-point iteration to

locate the root of

The function can be separated directly and

expressed in the form

 

x 4x3 6x2 7x 2,3

f    

7

3 2 6

4 3 2

1

, x

x

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•

Starting with an initial guess of x

₀

=0,

this iterative equation can be

applied to compute

Root of equation

 

x 4x3 6x2 7x 2,3

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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}

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•

_{the first derivative at x is equivalent}

to the slope

•

_{Which can be rearranged to yield}

Newton-Raphson formula :

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Example 2 :

Use the Newton-Raphson method to

estimate the root of

employing an initial guess of x

0

= 0

The first derivative of the function can be

evaluated as

f’

(

x

)= 12x

2

– 12x + 7, which

can be substituted along with the original

function into Eq. (3) to give

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•

Starting with an initial guess of x

0

=0, this

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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’’(x)=0] occurs in the vicinity 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.

• _{Obviously, a zero slope [f’(x)=0] is}

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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}

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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

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Example 3 :

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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.

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(x_r).

• 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 x_i+1 replacing x_i and x_i replacing x_i−1.

• As a result, the two values can sometimes lie on the same side of the root.

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Method x_{root iteration} e_a(%)

Bisection 0,4512 10 0,4329 False Position 0,4506 4 0,3612 x=g(x) 0,44856 7 0,47396 Newton

Raphson 0,4501 6 0,001 Secant 0,4501 3 0,402

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Homework 1

Frekuensi alami dari getaran bebas (free vibration) balok

uniform yang terjepit pada salah satu ujungnya dan bebas pada ujung yang lain dapat dicari dari persamaan berikut

cos(L)cosh(L) = 1 Dengan :

L = panjang elemen balok = 2 meter

 = berat jenis elemen balok

n = frekuensi alami balok (rad/dt) EI = kekakuan lentur balok

Tetapkan 3 buah nilai _n dari persamaan di atas. kemudian gunakan nilai  untuk menentukan frekuensi alami balok.

a

n n



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Homework 2

HItunglah panjang kurva (L), dari suatu lengkung / busur jalan dengan parameter – parameter :

PC = titik kurvatur, PT = titik tangensial, PI = titik perpotongan. Kurva mempunyai 7 elemen :

1. Radius kurva, R 2. Sudut lengkung, I 3. Jarak tangensial, T 4. Panjang kurva, L 5. Panjang busur, Lc 6. Ordinat tengah, M 7. Jarak luar, E

Hubungan antara kelengkungan dan jari – jari : Diketahui nilai :

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Homework 3

Hubungan antara debit air Q penampang saluran terbuka

berbentuk trapesium terhadap parameter geometri penampang adalah : m3_{/dt. Hitung besarnya y!}