Last Updated on June 15, 2025 by Maged kamel
Solved problems for Newton-Raphson method.
The first problem of the two Solved problems for the Newton-Raphson method
Another solved problem for the Newton-Raphson method for root extraction: find the roots of x^3-3x-5=0.
1- starting by the initial point x0=3.00.
First, we try to find that root value by plotting the curve by plugging different values of x,, for example, starting from 0 to 3.0 and negative values from x= -0.50 to -4.00.
We are getting the corresponding values of y accordingly. To find roots, we find that the function value between one positive and one negative review lies between x=2 and x =2.50.
1-So, our first iteration is selecting x0 value =3.
2-Get f(x0=3) and f'(x0=3) f (x) is =x^3-3x-5=0 and f'(x) = 3*x^2-3.
At x0=3 then f(3) = 3^3-3*3-5 =13 and f'(x0=3.00)=3*(3)^2-3=24.00.
3- X1 =3-(13/24)=2.45833.
For the second iteration
4—Get the values of f(2.45833) and f'(2.45833). The Excel sheet displays a new value, x2 = 2.2943, along with the corresponding f(x) for each iteration, as shown in the next slide.
We continue making iterations until we achieve x4 = 2.279; the error value is small.
The second Problem of the two Solved problems for the Newton-Raphson method.
This is the second solved problem of the two Solved problems for the Newton-Raphson method, which is problem number 6: Use the Newton method for root extraction to find the roots of this function.
Problem number 6: Use the Newton method for root extraction to find the roots of this function.
This is the first derivative of the function.
Solution: He did not give us a starting point, so it is good to make a casing for the root range by substituting
1-We make the table, we put different values of X, and we find the corresponding values of our function. 2-We’ll find that when we plug X=0 then (0*e^0)-2, we get -2
3-For the next point where x=0.25, we get (-1.678994).
4-For x=0.50 we get -1.175639.
5- For x=0.750, we get. (-0.41225).
6- For x=1, we get +0.7182818.
Between x=0 and x=1.0, forx=0 gives a minus, and for x=1 gives a positive value.
So, our root should exist between 0 and 1. You have the choice either to start point x=0 or X =1. We put X=0 f(0)=(0)*e^0-2=-2.00, for the slope value, check the next relation, it will be=1.
x1=x0-f(x0)/f'(x0)=0-(-2/1)=+2.00.
For f(2), we have the value of 12.778 and f'(2)=22.16716. So X2 will be =+2-f(x1=2)/f'(x1=2).X2 will be =+2-(12.778/22.16716)=1.4325.
After substitution, we get the value of the function and the derivative value. We use the expression of X3=x2- f(x2)/f'(x2), we get x3=1.034936.
It is better to calculate using an Excel sheet. This sheet includes a starting point with a selected value followed by a column that represents the function value f(x), another new column for the value of f'(x), a column for the numerator, which I f(x), and the denominator, which is ‘(x), and then a column for the numerator/ denominator.
We have two starting points: x0 = 0 and x0 = 1. This is a function value f'(x) =4.34575, and we go on until we find that f(x) is coming closer to (0) at x value=0.8526.
When X =0.853003, the f(x) is zero if our starting point x0=1.
The next post is about the Modified-Newton-Raphson method, for which the Newton-Raphson method is modified.
This is a useful link for a numerical analysis calculator.