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Newton-Raphson Method Computer Engineering Majors Authors: Autar Kaw, Jai Paul http://numericalmethods.eng.u sf.edu Transforming Numerical Methods Education for STEM Undergraduates 05/08/22 1 http:// numericalmethods.eng.usf.edu

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Newton-Raphson Method Figure 1 Geometrical illustration of the Newton-Raphson method. 3

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Page 1: Newton-Raphson Method Computer Engineering Majors Authors: Autar Kaw, Jai Paul  Transforming Numerical Methods Education

Newton-Raphson MethodComputer Engineering Majors

Authors: Autar Kaw, Jai Paul

http://numericalmethods.eng.usf.eduTransforming Numerical Methods Education for STEM

Undergraduates05/04/23 1

http://numericalmethods.eng.usf.edu

Page 2: Newton-Raphson Method Computer Engineering Majors Authors: Autar Kaw, Jai Paul  Transforming Numerical Methods Education

Newton-Raphson Method

http://numericalmethods.eng.usf.edu

Page 3: Newton-Raphson Method Computer Engineering Majors Authors: Autar Kaw, Jai Paul  Transforming Numerical Methods Education

Newton-Raphson Method

)(xf)f(x - = xx

i

iii 1

f ( x )

f ( x i )

f ( x i - 1 )

x i + 2 x i + 1 x i X

ii xfx ,

Figure 1 Geometrical illustration of the Newton-Raphson method. http://numericalmethods.eng.usf.edu3

Page 4: Newton-Raphson Method Computer Engineering Majors Authors: Autar Kaw, Jai Paul  Transforming Numerical Methods Education

Derivation

f(x)

f(xi)

xi+1 xi X

B

C A

)()(

1i

iii xf

xfxx

1

)()('

ii

ii

xxxfxf

ACAB

tan(

Figure 2 Derivation of the Newton-Raphson method.4 http://numericalmethods.eng.usf.edu

Page 5: Newton-Raphson Method Computer Engineering Majors Authors: Autar Kaw, Jai Paul  Transforming Numerical Methods Education

Algorithm for Newton-Raphson Method

5 http://numericalmethods.eng.usf.edu

Page 6: Newton-Raphson Method Computer Engineering Majors Authors: Autar Kaw, Jai Paul  Transforming Numerical Methods Education

Step 1

)(xf Evaluate symbolically.

http://numericalmethods.eng.usf.edu6

Page 7: Newton-Raphson Method Computer Engineering Majors Authors: Autar Kaw, Jai Paul  Transforming Numerical Methods Education

Step 2

i

iii xf

xf - = xx1

Use an initial guess of the root, , to estimate the new value of the root, , as

ix1ix

http://numericalmethods.eng.usf.edu7

Page 8: Newton-Raphson Method Computer Engineering Majors Authors: Autar Kaw, Jai Paul  Transforming Numerical Methods Education

Step 3

0101

1 x

- xx = i

iia

Find the absolute relative approximate error asa

http://numericalmethods.eng.usf.edu8

Page 9: Newton-Raphson Method Computer Engineering Majors Authors: Autar Kaw, Jai Paul  Transforming Numerical Methods Education

Step 4Compare the absolute relative approximate error with the pre-specified relative error tolerance .

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.

s

Is ?

Yes

No

Go to Step 2 using new estimate of the

root.

Stop the algorithm

sa

http://numericalmethods.eng.usf.edu9

Page 10: Newton-Raphson Method Computer Engineering Majors Authors: Autar Kaw, Jai Paul  Transforming Numerical Methods Education

Use the Newton-Raphson method of finding roots of equations to

a) Find the inverse of a = 2.5. 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

c) The number of significant digits at least correct at the end of each iteration.

http://numericalmethods.eng.usf.edu10

Example 1To find the inverse of a number ‘a’, one can use the equation

01)( x

axf

where x is the inverse of ‘a’.

Page 11: Newton-Raphson Method Computer Engineering Majors Authors: Autar Kaw, Jai Paul  Transforming Numerical Methods Education

0 0.25 0.5 0.75 1100

80

60

40

20

0

20

f(x)

1.5

97.5

0

f x( )

10.01 x

http://numericalmethods.eng.usf.edu11

Example 1 Cont.

01)( x

axf

axxx

xaxx

xaxx

x

xa

x

xfxfxx

iii

iii

iii

i

ii

i

iii

21

2

2

2

1

2

1

1

1

Figure 3 Graph of the function f(x).

2

1)('

01)(

xxf

xaxf

Solution

Page 12: Newton-Raphson Method Computer Engineering Majors Authors: Autar Kaw, Jai Paul  Transforming Numerical Methods Education

http://numericalmethods.eng.usf.edu12

Example 1 Cont.

%33.33

1001

01

x

xxa

Entered function along given interval with current and next root and the tangent line of the curve at the current root

0 0.24 0.48 0.72 0.96 1.2

79.5

59

38.5

18

0

f(x)prev. guessnew guesstangent line

2.5

97.5

0

f x( )

f x( )

f x( )

tan x( )

1.20 x x 0 x 1 x

Initial guess: Iteration 1The estimate of the root is

5.00 x

0.375 5.2)5.0()5.0(2

22

2001

axxx

The absolute relative approximate error is

Figure 4 Graph of the estimated root after Iteration 1.

The number of significant digits at least correct is 0.

Page 13: Newton-Raphson Method Computer Engineering Majors Authors: Autar Kaw, Jai Paul  Transforming Numerical Methods Education

http://numericalmethods.eng.usf.edu13

Example 1 Cont.

0.39844

5.2375.0375.022

375.0

22

2112

1

xaxxx

x

Entered function along given interval with current and next root and the tangent line of the curve at the current root

0 0.25 0.5 0.75 1

79.14

58.29

37.43

16.58

0

f(x)prev. guessnew guesstangent

4.27778

97.5

0

f x( )

f x( )

f x( )

tan x( )

10 x x 1 x 2 x

Figure 5 Graph of the estimated root after Iteration 2.

Iteration 2The estimate of the root is

%8224.5

1002

12

x

xxa

The absolute relative approximate error is

The number of significant digits at least correct is 0.

Page 14: Newton-Raphson Method Computer Engineering Majors Authors: Autar Kaw, Jai Paul  Transforming Numerical Methods Education

http://numericalmethods.eng.usf.edu14

Example 1 Cont.

0.39999

5.239844.039844.022

39844.0

2

2223

2

axxxx

0 0.25 0.5 0.75 1

79.24

58.49

37.73

16.98

0

f(x)prev. guessnew guesstangent

3.77951

97.5

0

f x( )

f x( )

f x( )

tan x( )

10 x x 2 x 3 x

Iteration 3The estimate of the root is

The absolute relative approximate error is

The number of significant digits at least correct is 2.

Figure 6 Graph of the estimated root after Iteration 3.

%38911.0

1003

23

x

xxa

Page 15: Newton-Raphson Method Computer Engineering Majors Authors: Autar Kaw, Jai Paul  Transforming Numerical Methods Education

Advantages and Drawbacks of Newton

Raphson Method

http://numericalmethods.eng.usf.edu

15 http://numericalmethods.eng.usf.edu

Page 16: Newton-Raphson Method Computer Engineering Majors Authors: Autar Kaw, Jai Paul  Transforming Numerical Methods Education

Advantages Converges fast (quadratic

convergence), if it converges. Requires only one guess

16 http://numericalmethods.eng.usf.edu

Page 17: Newton-Raphson Method Computer Engineering Majors Authors: Autar Kaw, Jai Paul  Transforming Numerical Methods Education

Drawbacks

17 http://numericalmethods.eng.usf.edu

1. Divergence at inflection pointsSelection of the initial guess or an iteration value of the root that is close to the inflection point of the function may start diverging away from the root in ther Newton-Raphson method.

For example, to find the root of the equation .

The Newton-Raphson method reduces to .

Table 1 shows the iterated values of the root of the equation.The root starts to diverge at Iteration 6 because the previous

estimate of 0.92589 is close to the inflection point of .

Eventually after 12 more iterations the root converges to the exact value of

xf

0512.01 3 xxf

2

33

1 13512.01

i

iii x

xxx

1x

.2.0x

Page 18: Newton-Raphson Method Computer Engineering Majors Authors: Autar Kaw, Jai Paul  Transforming Numerical Methods Education

Drawbacks – Inflection Points

Iteration Number

xi

0 5.00001 3.65602 2.74653 2.10844 1.60005 0.925896 −30.1197 −19.74618 0.2000 0512.01 3 xxf

18 http://numericalmethods.eng.usf.edu

Figure 8 Divergence at inflection point for

Table 1 Divergence near inflection point.

Page 19: Newton-Raphson Method Computer Engineering Majors Authors: Autar Kaw, Jai Paul  Transforming Numerical Methods Education

2. Division by zeroFor the equation

the Newton-Raphson method reduces to

For , the denominator will equal zero.

Drawbacks – Division by Zero

0104.203.0 623 xxxf

19 http://numericalmethods.eng.usf.edu

ii

iiii xx

xxxx06.03

104.203.02

623

1

02.0or 0 00 xx Figure 9 Pitfall of division by zero or near a zero number

Page 20: Newton-Raphson Method Computer Engineering Majors Authors: Autar Kaw, Jai Paul  Transforming Numerical Methods Education

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.For example for the equation has no real roots.

Drawbacks – Oscillations near local maximum and minimum

02 2 xxf

20 http://numericalmethods.eng.usf.edu

3. Oscillations near local maximum and minimum

Page 21: Newton-Raphson Method Computer Engineering Majors Authors: Autar Kaw, Jai Paul  Transforming Numerical Methods Education

Drawbacks – Oscillations near local maximum and minimum

21 http://numericalmethods.eng.usf.edu

-1

0

1

2

3

4

5

6

-2 -1 0 1 2 3

f(x)

x

3

4

2

1

-1.75 -0.3040 0.5 3.142

Figure 10 Oscillations around local minima for .

2 2 xxf

Iteration Number

0123456789

–1.0000 0.5–1.75–0.30357 3.1423 1.2529–0.17166 5.7395 2.6955 0.97678

3.002.255.063 2.09211.8743.5702.02934.9429.2662.954

300.00128.571 476.47109.66150.80829.88102.99112.93175.96

Table 3 Oscillations near local maxima and mimima in Newton-Raphson method.

ix ixf %a

Page 22: Newton-Raphson Method Computer Engineering Majors Authors: Autar Kaw, Jai Paul  Transforming Numerical Methods Education

4. Root JumpingIn 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.

For example

Choose

It will converge to instead of

-1.5

-1

-0.5

0

0.5

1

1.5

-2 0 2 4 6 8 10

x

f(x)

-0.06307 0.5499 4.461 7.539822

Drawbacks – Root Jumping

0 sin xxf

22 http://numericalmethods.eng.usf.edu

xf

539822.74.20 x

0x

2831853.62 x Figure 11 Root jumping from intended location of root for

. 0 sin xxf

Page 23: Newton-Raphson Method Computer Engineering Majors Authors: Autar Kaw, Jai Paul  Transforming Numerical Methods Education

Additional ResourcesFor 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_raphson.html

Page 24: Newton-Raphson Method Computer Engineering Majors Authors: Autar Kaw, Jai Paul  Transforming Numerical Methods Education

THE END

http://numericalmethods.eng.usf.edu