Newton

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ESTÁCIO

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Camila Barata Cavalcanti

30.08.2015

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Roots of a Nonlinear Equation
Topic: Newton-Raphson Method
Major: General Engineering
http://numericalmethods.eng.usf.edu
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Newton-Raphson Method
http://numericalmethods.eng.usf.edu
 f(x)
 f(xi)
 f(xi-1)
 
� EMBED Equation.3 ���
 
 
xi+2
xi+1
xi
 X
_1075143848.unknown
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Derivation
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 f(x)
 f(xi)
 A
 C
 B
xi+1

xi
 X
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Algorithm for Newton-Raphson Method
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Step 1
Evaluate f(x) symbolically
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Step 2
Calculate the next estimate of the root
Find the absolute relative approximate error
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Step 3
Find if the absolute relative approximate error is greater than the pre-specified relative error tolerance. 
If so, go back to step 2, else stop the algorithm.
Also check if the number of iterations has exceeded the maximum number of iterations.
http://numericalmethods.eng.usf.edu
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Example
You are working for ‘DOWN THE TOILET COMPANY’ that makes floats for ABC commodes. The ball has a specific gravity of 0.6 and has a radius of 5.5 cm. You are asked to find the distance to which the ball will get submerged when floating in water.
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Solution
The equation that gives the depth ‘x’ to which the ball is submerged under water is given by 
Use the Newton’s method of finding roots of equations to find the depth ‘x’ to which the ball is submerged under water. Conduct three iterations to estimate the root of the above equation. 
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Graph of function f(x)
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Iteration #1
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Iteration #2
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Iteration #3
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Advantages
Converges fast, if it converges
Requires only one guess
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Drawbacks
Inflection Point
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Chart1
		-64
		-54.872
		-46.656
		-39.304
		-32.768
		-27
		-21.952
		-17.576
		-13.824
		-10.648
		-8
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		-2.744
		-1.728
		-1
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		0.008
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		0.512
		1
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		2.744
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		5.832
		8
		
		0
		-8
		
		-8
		0
		
		0
		-2.37037037
		
		-2.37037037
		0
		
		0
		-0.702334595
		
		-0.702334595
		0
1
2
3
f(x)
x
Sheet1
		f(x)=(x-1)^3 inflection point pitfall in Newton_raphson Method
		
		
		-3		-64
		-2.8		-54.872
		-2.6		-46.656
		-2.4		-39.304
		-2.2		-32.768
		-2		-27
		-1.8		-21.952
		-1.6		-17.576
		-1.4		-13.824
		-1.2		-10.648
		-1		-8
		-0.8		-5.832
		-0.6		-4.096
		-0.4		-2.744
		-0.2		-1.728
		0		-1
		0.2		-0.512
		0.4		-0.216
		0.6		-0.064
		0.8		-0.008
		1		0
		1.2		0.008
		1.4		0.064
		1.6		0.216
		1.8		0.512
		2		1
		2.2		1.728
		2.4		2.744
		2.6		4.096
		2.8		5.832
		3		8
		
		-1		0
		-1		-8
		
		-1		-8
		-0.3333333333		0
		
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Sheet1
		
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Sheet3
		
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Drawbacks (continued)
Division by zero
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Chart10
		
		
		
		
		
		
		
		-0.0001976
		-0.0001096
		-0.0000516
		-0.0000176
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		0.0000024
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		0.0000184
		0.0000524
		
		-0.0000016
		-0.0000016
0.02
x
f(x)
Sheet1
		
		
		
		
		
		
		
		
		
		
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		-0.04		-0.0001096
		-0.03		-0.0000516
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		0.02		-0.0000016
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		0.05		0.0000524
		
		0.02		-0.0000016
		0		-0.0000016
		
		
		
		
		
		
		
		
		
		
		
		-0.63069		-0.5897021566
		0		0
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		2		0.9092974268
		3		0.1411200081
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		7		0.6569865987
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		9		0.4121184852
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		0.5226
		0
		
		0.5226
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		0
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		0
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7.53982
x
f(x)
Sheet2
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x
F(x)
Sheet3
		
		
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Drawbacks (continued)
Root Jumping
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Chart3
		-0.5897021566
		0
		0.8414709848
		0.9092974268
		0.1411200081
		-0.7568024953
		-0.9589242747
		-0.2794154982
		0.6569865987
		0.9893582466
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		0.5226
		0
		
		0.5226
		0
		
		0
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		0
		0.951
		
		0
		0.951
		
		-0.9688
		0
-0.063069
0.54990
4.462
7.53982
x
f(x)
Sheet1
		
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		-0.06		-0.0003216
		-0.05		-0.0001976
		-0.04		-0.0001096
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		0.03		0.0000024
		0.04		0.0000184
		0.05		0.0000524
		0.06		0.0001104
		0.07		0.0001984
		0.08		0.0003224
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		-0.63069		-0.5897021566
		0		0
		1		0.8414709848
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		0.5499		0.5226
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		0.5499		0.5226
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		4.462		0
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Sheet3
		
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Drawbacks (continued)
Oscillations near Local Maxima or Minima
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Chart1
		6
		5.24
		4.56
		3.96
		3.44
		3
		2.25
		2
		2.25
		3
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		4.56
		5.24
		6
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3
4
2
1
-1.75
-0.3040
0.5
3.142
f(x)
x
Sheet1
		
		
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