The Newton Raphson Method - 2017
The Newton Raphson Method - 2017
The Newton Raphson Method - 2017
Method
Major: Chemical Engineering
Subject: Chemical Engineering
Mathematics 2
Author: Andrew KUMORO
Department of Chemical
Engineering
Diponegoro University
2017
Competences to be achieved:
2
ax bxc 0
Why? Lets find the roots of
Trial and errors through factorial
Analytical solution
b b 2 4ac
roots
2a
sin x x 0 x ?
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Thermodynamics application
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Unit Operations application
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METHODS FOR FINDING
ROOTS of NON LINEAR
EQUATIONS
1. Successive Substitution Method
2. Wegstein Method
3. Method of Linear Interpolation (False
Position)
4. Bisection Method
5. Secant Method
6. Newton-Raphson Method
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Method of Linear Interpolation
(False Position)
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The Bisection Method
For the arbitrary equation of one variable, f(x)=0
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If f(xl). f[(xl+xu)/2]>0, root
lies in the upper interval,
then xl= [(xl+xu)/2, go to
step 2.
xl xu
xl
2
100%
If f(xl). f[(xl+xu)/2]=0, xl xu
then root is (xl+xu)/2 and 2
terminate. or
xl xu
xu
2
4. Compare es with ea 100%
xl xu
2
5. If ea< es, stop. Otherwise
repeat the process.
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Evaluation of Method
Pros Contras
Easy Slow
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ff/(xi)xii/(xi)1
difference
The Secant Method
The derivative
is replaced by a
backward finite divided
xi 1 xi f ( xi ).
( xi 1 xi )
f ( xi 1 ) f ( xi )
Comparison of convergence of False
Position and Secant Methods
Use the false-position and secant method to find the
root of f(x)=lnx. Start computation with xl= xi-
1=0.5,
1.
xFalse
u=xi =position
5. method
Iter xl xu xr
1 0.5 5.0 1.8546
2 0.5 1.8546 1.2163
3 0.5 1.2163 1.0585
2. Secant method
Iter xi-1 xi xi+1
1 0.5 5.0 1.8546
2 5 1.8546 -0.10438
Introductio
n
Methods such as the bisection method and the
false position method of finding roots of a
nonlinear equation f (x) = 0 require bracketing
of the root by two guesses. Such methods are
called bracketing methods. These methods are
always convergent since they are based on
reducing the interval between the two guesses
so as to zero in on the root of the equation.
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Introductio
In the
n
Newton-Raphson method, the root is not
bracketed. In fact, only one initial guess of the
root is needed to get the iterative process started
to find the root of an equation. The method hence
falls in the category of open methods.
Convergence in open methods is not guaranteed
but if the method does converge, it does so much
faster than the bracketing methods.
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Open Methods
Bracketing methods are based on assuming
an interval of the function which brackets
the root.
The bracketing methods always converge to
the root.
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.
These method sometimes diverge from the
true root.
Open Methods-
Convergence and Divergence
Concepts
f(x) f(x)
x xi xi+1 x
xi xi+1
x
2
f ( xi 1 ) f ( xi ) f ( xi )x f ( xi ) ...
2!
The root is the value of xi+1 when f (xi+1) = 0
Newton-Raphson Method
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Newton-Raphson Method
0 f(xi ) f (xi )( xi 1 xi )
f ( xi )
xi 1 xi Newton-Raphson
f ( xi ) formula
f(xi)
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f(x)
Newton-Raphson Method
Root
f(x)
C
xi+1
Raphson method.
A
xi
B
Slope f /(xi)
x
/i
Figure 1. Geometrical illustration of the Newton-
()xfx()x
fxf(x
i1ii/i
1
tan(
0
)
AB
AC
A tangent to f(x) at the initial point xi is
extended till it meets the x-axis at the
improved estimate of the root xi+1.
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Step 1
Evaluate f (x) symbolically.
Step 2
Use an initial guess of the root,xi , to estimate the
new value of thexi root, , as f xi
1
xi 1 = xi -
f xi
Step 3
a
Find the absolute relative approximate error as
xi 1- xi
a = 100
xi 1
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Step 4
Compare the absolute relative approximate error
with the pre-specified relative error tolerance .
s
Go to Step 2 using
Yes
new estimate of the
Is a s ? root.
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1
while
a >s & False
i <maxi
i=1
or Stop
xn=0
True
x0=xn
of
fx(x)ex1e1x
0
1
2
3
4
i1i/i i
i x x
The Newton Raphson Method
Use the Newton-Raphson method to find the roots
0
F(x) = e-x-x
f(x) = e-x-x and f`(x)= -e-x-1; thus
Iter. xi
0.5
0.566311003
0.567143165
0.567143290
t%
100
11.8
0.147
0.00002
<10-8
Example 1
You are working for DOWN THE TOILET COMPANY
that makes floats for ABC commodes. The floating
ball has a specific gravity of 0.6 and has a radius of
5.5 cm. You are asked to find the depth to which
the ball is submerged when floating in water.
Radius = R
Density =
f x x -0.165 x +3.993 10
3 2 -4
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Example 1 Cont.
Solve for f ' x
f x x 3-0.165 x 2+3.993 10- 4
f ' x 3x 2 -0.33x
Let us assume the initial guess of the root off x 0
x0is 0.05m . . This is a
x0
reasonable 0.11(discuss
xguess m why
and are not good choices) as the
extreme values of the depth x would be 0 and the
diameter (0.11 m) of the ball.
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Example 1 Cont.
Iteration 1
The estimate of the root is
f x0
x1 x0
f ' x0
0.05
0.05 0.165 0.05 3.993 10 4
3 2
1.118 10 4
0.05
9 10 3
0.05 0.01242
0.06242
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Example 1 Cont.
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Example 1 Cont.
Iteration 2
The estimate of the root is
f x1
x2 x1
f ' x1
0.06242
0.06242 0.165 0.06242 3.993 10 4
3 2
3.97781 10 7
0.06242
8.90973 10 3
0.06242 4.4646 10 5
0.06238
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Example 1 Cont.
2 m
The maximum value of m for whicha 0.5 10 is
2.844. Hence, the number of significant digits at least
correct in the answer is 2.
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Example 1 Cont.
Iteration 3
The estimate of the root is
f x2
x3 x2
f ' x2
0.06238
0.06238 0.165 0.06238 3.993 10 4
3 2
4.44 10 11
0.06238
8.91171 10 3
0.06238 4.9822 10 9
0.06238
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Example 1 Cont.
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f ( x) x 2 x 2 x 0
g ( x) x 2
2
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Advantages and Drawbacks
Newton Raphson Method
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Advantages
Converges fast (quadratic convergence), if it
converges.
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Drawbacks
1. Divergence at inflection points
Selection of the initial guess or an iteration value of the
f x
root that is close to the inflection point of the function
may start diverging away from the root in ther Newton-
Raphson method.
f x x 1 0.512 0
3
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Drawbacks Division by
Zero
2. Division by zero
For the equation
f x x 3 0.03 x 2 2.4 10 6 0
the Newton-Raphson
method reduces to
xi3 0.03 xi2 2.4 10 6
xi 1 xi
3xi2 0.06 xi
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Drawbacks Oscillations near
local maximum and minimum
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Drawbacks Oscillations near
local maximum and minimum
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Drawbacks Root
Jumping
4. Root Jumping
f x
In some cases where the function is oscillating and has a
number of roots, one may choose an initial guess close to a root.
However, the guesses may jump and converge to some other
root. f(x)
1.5
For example
f x sin x 0 0.5
x
0
Choose
x0 2.4 7.539822
-2 0 2 4 6 8 10
-0.06307 0.5499 4.461 7.539822
-0.5
x0 -1
It will converge to
-1.5
xiiu/v(11x/i i
x) u
/f/v
/i 2( x)
xi/)fi([x2i)/
/( i
The Modified Newton Raphson Method
fi/]2(xii)/(ixi)
This function has
roots at all the
same locations as
the
function
original
/
Another u(x) is introduced such that u(x)=f(x)/f /(x);
Getting the roots of u(x) using Newton Raphson technique:
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fxi1xii1(i3xi2i1/03i/f((x25xii)i)2/7iif3/3x(5xi33i)527i21100x7i3x(i67)i310
Modified Newton Raphson Method:
Example
Using the Newton Raphson and Modified Newton
Raphson evaluate the multiple roots of
f(x)= x3-5x2+7x-3 with an initial guess of x0=0
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2
Z2
D,
Le
1
Z1 W
V, Q
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Example First order reaction, in adiabatic CSTR
The developed
equation:
B X
X Da 1 X exp 0 f(x) = 0
1 B
X
Da 5
25
B 0.1
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THE END