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

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]

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] 

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 (x_n) and value of function at x = x_nis 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 (x_n) and value of function f(x) =x⁴ – 1at atx=x_n

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 (x_n) and value of function at x = x_nis 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 (x_n) and value of function f(x) =x⁴ – 2at atx=x_n

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

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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