4/17/2020
Electrical Engineering Majors
Authors: Autar Kaw, Jai Paul http://numericalmethods.eng.usf.edu
Transforming Numerical Methods Education for STEM
Undergraduates http://numericalmethods.eng.usf.edu
1

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f(x)
f(x i
)
 x i , f
  i
 x i

1
= x i
- f f(x i

(x i
)
)
f(x i-1
)
 x i+2 x i+1 x i
X
3
Figure 1 Geometrical illustration of the Newton-Raphson method.
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f(x)
4
f(x i
) B tan(
  
AB
AC f ' ( x i
)
 x i f

( x i x i
)

1
C
 x i+1
A x i
X x i

1
 x i
 f f (

( x i x i
)
)
Figure 2 Derivation of the Newton-Raphson method.
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5
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6
Evaluate f

( x ) symbolically.
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7
x i

1
= x i
- f f x x i

  i
  i http://numericalmethods.eng.usf.edu

8
 a
= x i

1
- x i x i

1

10 0
 http://numericalmethods.eng.usf.edu

9
Compare the absolute relative approximate error
 s
Yes
Go to Step 2 using new estimate of the root.
Is ?
a s
No Stop the algorithm
Also, check if the number of iterations has exceeded the maximum number of iterations allowed. If so, one needs to terminate the algorithm and notify the user.
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Thermistors are temperature-measuring devices based on the principle that the thermistor material exhibits a change in electrical resistance with a change in temperature. By measuring the resistance of the thermistor material, one can then determine the temperature.
Thermally conductive epoxy coating
For a 10K3A Betatherm thermistor, the relationship between the resistance,
R , of the thermistor and the temperature is given by
Tin plated copper alloy lead wires
Figure 3 A typical thermistor.
1

1 .
129241

10

3 
2 .
341077

10

4 ln
T where
T is in Kelvin and
R is in ohms.

8 .
775468

10

8
 ln
  
3 http://numericalmethods.eng.usf.edu

11
For the thermistor, error of no more than ±0.01
o C is acceptable.
To find the range of the resistance that is within this acceptable limit at 19 o C, we need to solve
19 .
01

1
273 .
15

1 .
129241

10

3 
2 .
341077

10

4 ln and
1
18 .
99

273 .
15

1 .
129241

10

3 
2 .
341077

10

4 ln


8 .
775468

10

8
 ln
  
3
8 .
775468

10

8
 ln
  
3
Use the Newton-Raphson method of finding roots of equations to find the resistance
R at 18.99
o C. a)
Conduct three iterations to estimate the root of the above equation. b)
Find the absolute relative approximate error at the end of each iteration and the number of significant digits at least correct at the end of each iteration.
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Entered function on given interval
0.00002
11000 12000 13000 14000 15000
0.00002
0.00004
0.00006
Figure 4 Graph of the function f(R).
12 f ( R )

2 .
341077

10

4 ln

8 .
775468

10

8
 ln
  
3 
2 .
293775

10

3 http://numericalmethods.eng.usf.edu

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Entered function on given interval with guess and estimated root Initial guess: R
0

15000
0.0001
0.00005
Iteration 1
The estimate of the root is
R
1

R
0
 f f
/
 
0
 
0
12000 14000 16000 18000 20000
R
1

15000

3 .
5383

10

5
1 .
7230

10

8

12946
0.00005
Figure 5 Graph of the estimate of the root after Iteration 1.
The absolute relative approximate error is
 a

12946

15000
12946

15.862%
The number of significant digits at least correct is 0.

100
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14
Entered function on given interval with guess and estimated root
0.0001
0.00005
0.00005
12000 14000 16000 18000 20000
Figure 6 Graph of the estimate of the root after Iteration 2.
Iteration 2
The estimate of the root is
R
2

R
1
 f f
/
 
1
 
1
R
2

12946


2 .
6140

10

6
1 .
9906

10

8

13078
The absolute relative approximate error is
 a

13078

12946
13078

100

1 .
0041 %
The number of significant digits at least correct is 1.
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15
Entered function on given interval with guess and estimated root
0.0001
0.00005
0.00005
12000 14000 16000 18000 20000
Figure 7 Graph of the estimate of the root after Iteration 3.
Iteration 2
The estimate of the root is
R
3

R
2
 f f
/
 
2
 
2

13078


1 .
2914

10

8
1 .
9710

10

8

13078
The absolute relative approximate error is
 a

13078

13078
13078

100

0 .
0050097 %
The number of significant digits at least correct is 3.
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16 http://numericalmethods.eng.usf.edu
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17


Converges fast (quadratic convergence), if it converges.
Requires only one guess http://numericalmethods.eng.usf.edu

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1. Divergence at inflection points
Selection of the initial guess or an iteration value of the root that f
  diverging away from the root in ther Newton-Raphson method.
x f

   x

1
 x


0 .
512
 x
3 
3

1
 x i
3 

1
0 .
512

2

0
Table 1 shows the iterated values of the root of the equation.
The root starts to diverge at Iteration 6 because the previous estimate x

1
Eventually after 12 more iterations the root converges to the exact value of x

0 .
2 .
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Table 1 Divergence near inflection point.
Iteration
Number
0
1
6
7
18
4
5
2
3 x i
5.0000
3.6560
2.7465
2.1084
1.6000
0.92589
−30.119
−19.746
0.2000
Figure 8 Divergence at inflection point for f
   x

1

3 
0 .
512

0 http://numericalmethods.eng.usf.edu

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Drawbacks – Division by Zero
2. Division by zero
For the equation f
 x
3 
0 .
03 x
2 
2 .
4

10

6 
0 the Newton-Raphson method reduces to x i

1
 x i
 x i
3 
0 .
03
3 x i
2 x i
2


2
0 .
06
.
4 x i

10

6 x

0 or x

0 .
02 denominator will equal zero.
Figure 9 Pitfall of division by zero or near a zero number http://numericalmethods.eng.usf.edu

21
Drawbacks – Oscillations near local maximum and minimum
3. Oscillations near local maximum and minimum
Results obtained from the Newton-Raphson method may oscillate about the local maximum or minimum without converging on a root but converging on the local maximum or minimum.
Eventually, it may lead to division by a number close to zero and may diverge.
roots.
f
 
 x
2 
2

0 http://numericalmethods.eng.usf.edu

Drawbacks – Oscillations near local maximum and minimum
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Table 3 Oscillations near local maxima and mimima in Newton-Raphson method.
Iteration
Number
0
1
2
3
4
5
6
7
8
9 x i
–1.0000
0.5
–1.75
–0.30357
3.1423
1.2529
–0.17166
5.7395
2.6955
0.97678
f
  i
3.00
2.25
5.063
2.092
11.874
3.570
2.029
34.942
9.266
2.954
 a
%
300.00
128.571
476.47
109.66
150.80
829.88
102.99
112.93
175.96
4
5
6 f(x)
3
3
2
2
1
1
4
-2
-1.75
-1
-0.3040
0
0
-1
0.5
1 2 3
3.142
Figure 10 Oscillations around local
  x http://numericalmethods.eng.usf.edu

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4. Root Jumping f
  of roots, one may choose an initial guess close to a root. However, the guesses may jump and converge to some other root.
1.5
f(x)
For example f
 
 sin x

0
1
0.5
Choose x
0

2 .
4
 
7 .
539822
It will converge to x

0
-2
0
-0.06307
0
-0.5
instead of x

2
 
6 .
2831853
-1
2
0.5499
4 6
4.461
8
7.539822
x
10
-1.5
Figure 11 Root jumping from intended location of root for f
 
 sin x

0 http://numericalmethods.eng.usf.edu

For all resources on this topic such as digital audiovisual lectures, primers, textbook chapters, multiple-choice tests, worksheets in MATLAB, MATHEMATICA, MathCad and MAPLE, blogs, related physical problems, please visit http://numericalmethods.eng.usf.edu/topics/newton_ra phson.html

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