P: ISSN No. 0976-8602 RNI No.  UPENG/2012/42622 VOL.- XII , ISSUE- IV October  - 2023
E: ISSN No. 2349-9443 Asian Resonance

Convergence of the Newton-Raphsons Method
Paper Id :  17870   Submission Date :  28/09/2023   Acceptance Date :  15/10/2023   Publication Date :  25/10/2023
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Lavkush Pandey
Assistant Professor
Mathematics
M. L. K. Post Graduate College
Balrampur,Uttar Pradesh, India
Abstract
With the help of Newton-Raphson method, the author has found out the fourth roots of the natural numbers from 1 to 30. These values have been compared with the actual values. The minimum error and minimum percentage error (both are zero) has been obtained in the determination of fourth roots of 1 and 16. The average value in the error has been found to be0.000000094927. The maximum error 0.000001154116 and maximum percentage error 0.000097049224have been obtained in the determination of fourth roots of 2. The average value of percentage error is 0.000006303776. Generally, numerical rate of convergence in the determination of the fourth root of numbers from 1 to 30 by Newton-Raphson method decreases as the number increases.
Keywords Convergence, Newton-Raphson method, Numerical accuracy, Iteration, Stopping tolerance, Approximation.
Introduction

In this paper researcher is going to discuss the convergence of Newton-Raphson method.The Newton-Raphson method, or Newton Method, is a powerful technique for solving equations numerically. Like so much of the differential calculus,it is based on the simple idea of linear approximation.
Aim of study

The main motive of this research paper is to find out the fourth roots of natural numbers with the help of Newton-Raphson method.
Review of Literature

Two research papers entitled “Convergence of bisection method” and “convergence of the method of false position” has been studied.[1,2] The Newton-Raphson method finds the slope (the tangent line) of the function at the current point and uses the zero of the tangent line as the next reference point. The process is repeated until the root is found. The Newton-Raphson method is much more efficient than other "simple" methods such as the Bisection method. However, the Newton-Raphson method requires the calculation of the derivative of a function at the reference point, which is not always easy. Furthermore, the tangent line often shoots wildly and might occasionally be trapped in a loop. The promised efficiency is then unfortunately too good to be true. It is recommended to monitor the step obtained by the Newton-Raphson method. When the step is too large or the value is oscillating, other more conservative methods should take over the case.[18-24]
Methodology

The Newton-Raphson method finds the slope (the tangent line) of the function at the current point and uses the zero of the tangent line as the next reference point. The process is repeated until the root is found.


The Newton-Raphson method is much more efficient than other "simple" methods such as the Bisection method. However, the Newton-Raphson method requires the calculation of the derivative of a function at the reference point, which is not always easy. Furthermore, the tangent line often shoots wildly and might occasionally be trapped in a loop. The promised efficiency is then unfortunately too good to be true. It is recommended to monitor the step obtained by the Newton-Raphson method. When the step is too large or the value is oscillating, other more conservative methods should take over the case.[18-24] 
Result and Discussion

Calculation of fourth root of 1 by the Newton-Raphson method

 

The Newton-Raphson method has been used to calculate the root of equation

f(x) = x4 – 1 = 0

with initial guess of x0 = 0.1 by using C++ computer program. Number of iterations, root guessed by the Newton-Raphson method in each iteration (xn) and value of function at x = xnis given in Table-1. Root guessed by the Newton-Raphson method after each iterationis shown in Graph-1.

Table-1: Root guessed by the Newton-Raphson method in each iteration (xn) and value of function f(x) =x4 – 1at atx=xn

No. of iteration

Root guessed by the

Newton-Raphson method  (xn)

Value of function at x=xn

i. e. f(xn)

1

250.074981689453

3910938463.357570000000

2

187.556243896484

1237445573.834880000000

3

140.667190551758

391535597.855772000000

4

105.500396728516

123884327.493161000000

5

79.125297546387

39197774.812289400000

6

59.343975067139

12402421.847590400000

7

44.507984161377

3924204.113129460000

8

33.380989074707

1241642.165968030000

9

25.035749435425

392863.136867540000

10

18.776828765869

124304.110246735000

11

14.082659721375

39330.339948690000

12

10.562084197998

12444.103161413400

13

7.921775341034

3937.130351975160

14

5.941834449768

1245.471031092210

15

4.457567691803

393.813312559357

16

3.345998287201

124.343800101109

17

2.516172409058

39.083125938481

18

1.902822732925

12.109717254544

19

1.463403582573

3.586236597327

20

1.177324175835

0.921251628497

21

1.036190629005

0.152812405653

22

1.001852273941

0.007429706708

23

1.000005125999

0.000020504155

24

1.000000000000

0.000000000000

Actual value of fourth root of 1

1.000000000000

Calculated value of fourth root of 1 by Newton-Raphson method

1.000000000000

Difference between actual and calculated values of fourth root of 1  by Newton-Raphson method

0.000000000000

Percentage error in the value of fourth root of 1 calculated by Newton-Raphson method

0.000000000000

Numerical rate of convergence of  Newton-Raphson method in the calculation of fourth root of 1

1.388888888889

Graph-1: Root guessed by the Newton-Raphson method in the equation x4 – 1 =0


Calculation of fourth root of 2 by the Newton-Raphson method

The Newton-Raphson method has been used to calculate the root of equation

f(x) = x4 – 2 = 0

with initial guess of x0 = 0.1 by using C++ computer program. Number of iterations, root guessed by the Newton-Raphson method in each iteration (xn) and value of function at x = xnis given in Table-2. Root guessed by the Newton-Raphson method after each iterationis shown in Graph-2.

Table-2: Root guessed by the Newton-Raphson method in each iteration (xn) and value of function f(x) =x4 – 2at atx=xn

No. of iteration

Root guessed by the Newton-

Raphson method  (xn)

Value of function at x=xn

i. e. f(xn)

1

500.074981689453

62537499276.950300000000

2

375.056243896484

19787257239.279900000000

3

281.292175292969

6260811180.259070000000

4

210.969131469727

1980959786.136660000000

5

158.226852416992

626788116.410148000000

6

118.670143127441

198319701.590961000000

7

89.002609252930

62749597.093287200000

8

66.751960754395

19854367.876608000000

9

50.063972473145

6282045.676109990000

10

37.547985076904

1987678.359152300000

11

28.160997390747

628913.255375148000

12

21.120769500732

198991.526190695000

13

15.840630531311

62961.644504997500

14

11.880599021912

19920.936631488300

15

8.910747528076

6302.585737668240

16

6.683767318726

1993.654196839160

17

5.014500141144

630.281669704365

18

3.764840602875

198.902978040374

19

2.833000183105

62.414910893650

20

2.146740436554

19.238221574373

21

1.660594820976

5.604220760033

22

1.354635119438

1.367358247330

23

1.217118501663

0.194479140750

24

1.190152645111

0.006368333123

25

1.189208269119

0.000007763951

Actual value of fourth root of 2

1.189207115003

Calculated value of fourth root of 2 by Newton-Raphson method

1.189208269119

Difference between actual and calculated values of fourth root of 2  by Newton-Raphson method

0.000001154116

Percentage error in the value of fourth root of 2 calculated by Newton-Raphson method

0.000097049224

Numerical rate of convergence of  Newton-Raphson method in the calculation of fourth root of 2

1.333333333333

Graph-2: Root guessed by the Newton-Raphson method in the equation x4 – 2 =0

 

Consolidated analysis of the fourth roots of numbers from 1 to 30 calculated by Newton-Raphson method

The value of fourth root, error in the determination of fourth root, percentage error and numerical rate of convergence in the Newton-Raphson method are shown in Table-3(a) and Table-3(b). The actual value of fourth root and the value of fourth root calculated by Newton-Raphson method are shown in Graph-3. Error in the value of fourth root calculated by Newton-Raphson methodis given in Graph-4. Percentage error in the values of fourth root calculated by Newton-Raphson methodis given in Graph-5. Numerical rate of convergence in the determination ofthe fourth roots by Newton-Raphson methodis given in Graph-6.

Table-3(a): Actual value of fourth root, value of fourth root calculated by Newton-Raphson method and error in the determination of fourth root by Newton-Raphson method in findingthe roots of equations

f(x) = x4 – n = 0; n = 1, 2, …., 30







Table-3(b): Actual value of fourth root, percentage error in the calculation of fourth root and numerical rate of convergence of Newton-Raphson method in the determination of roots of equations f(x) = x4 – n = 0; n = 1, 2, …., 30


Graph-3: Actual value of fourth root and the value of root calculated by Newton-Raphson method in the equations f(x) = x4 – n=0; n=1, 2, …., 30






Graph-4: Error in the value of fourth root calculated by Newton-Raphson method in the equations f(x) = x4 – n=0; n=1, 2, …., 30


Graph-5: Percentage error in the value of fourth root calculated by Newton-Raphson method in the equations f(x) = x4 – n=0; n=1, 2, …., 30


Graph-6: Numerical rate of convergence in the determination of the fourth root by Newton-Raphson method in the equations f(x) = x4 – n=0; n=1, 2, …., 30
Conclusion

Fourth roots of the natural numbers from 1 to 30 have been found by Newton-Raphson method and these values have been compared with the actual values. The minimum error zero and minimum percentage error zero has been obtained in the determination of fourth roots of 1 and 16. The average value in the error is 0.000000094927. The maximum error 0.000001154116 and maximum percentage error 0.000097049224have been obtained in the determination of fourth roots of 2. The average value of percentage error is 0.000006303776. The normal trend in the fourth roots as obtained by the method of false position is that the error and percentage error in the roots increase as the number increase. Generally, numerical rate of convergence in the determination of the fourth root of numbers from 1 to 30 by Newton-Raphson method decreases as the number increases. Minimum, Maximum and average values of the numerical rate of convergence are 1.010101010101, 1.388888888889 and 1.104698523392 respectively.
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