brief illustration for -Post 7A-modified Newton-Raphson method

7A-Solved problem-8 by Modified Newton-Raphson method

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Solved problem-8 by Modified Newton-Raphson method.

For the given Solved problem-8 by the Modified Newton-Raphson method. The function f(x)=X^3-5*X^2+7x-3 with the first choice of x0=0.

Again I make a graph by plugging different values and getting the corresponding values of f(x), I have started from x=0 to x=3.50.

From the graph, as we can see from the next slide image, the roots are three roots x1=3 & x2=1 and x3=1 as shown in the Excel sheet for the Solved problem-8 by Modified Newton-Raphson method. the x=0 will give a negative value of -3.

Pict 5 num 7 numerical posts

1-We start to use the modified Newton-Raphson method, consider the initial point x0 equals 0, and we start to find the expressions for f(x),f'(x) , f’^2(x), and f”(x) .
2- Substitute at x=0 and get the values for f (0), f'(0) & f’^2(0), and f”(0). The value of f(0)=-3. The value of f'(0)=7.

The value of f”(0)=-10. Substitute in the equation of the modified Newton-Raphson method, we get the value of x1, which will be =1.105. Please refer to the next slide image for a detailed estimation of the various parameters.

The detailed calculation of how to get the value of x1 for solved problem #8.

We plug in with the value of x=1.105263 and get the corresponding values of f(1.105263),f'(1.105263), and”(1.105263).

The detailed calculation of how to get the value of x2 for the solved problem #8 from an esmated x1 value.

4- Substitute in the modified Newton-Raphson method. We get a new point with x2=1.00. and again continue to estimate f(1.00) and f'(1.00) and then apply them in the equation to get a new point that will be point x3.

The detailed calculation of how to get the value of x2 for solved problem #8.

5-Continue the process until x converges to 1.00. The Excel sheet for the various values of x, based on the modified Newton-Raphson method from x0=0 till x4=1.00 is shown.  

An excel sheet to show the values of xi till we will get the f(1.00)=0.

This table shows how many iterations and the corresponding f(x),f'(x), and f”(x) for each case. 

   

Solved problem-8 by Modified Newton-Raphson method using x0=4

Now if we consider the starting point as x0=4.00, and proceed to get the x value for f(x)& f'(x) , f’^2(x) and f”(x) for x0=4.00. We have f(4)=9,f'(4)=15 and f”(4)=14. The value of x1 is found to be=2.6363.      

Starting as the second choice by letting x0=4.00 and get the x1 value.

2- Substitute at x1=2.6363 and get the values for f (2.6363), f'(2.6363)  & f”(2.6363). the values are shown in the next slide image.

Substitute by the value of x1=2.6363 in the solved problem #8.

3-Plug in with previous calculation to the Modified Newton Raphson equation and d out the value of X2 Substitute at x2=2.8202 and get the values for f (2.6363), f'(2.8202)  & f”^2(2.8202)  and f”(2.8202) and get the value of x3, it will be=2.9617.

Get the value of x2 by the modified newton-raphson method.

This is the Excel sheet for the calculation based on the Modified Newton Raphson Method starting from x0=4 till x5=3.00.

Excel table for the different values of x for solved problem #8.

Solved problem-8 by Newton Raphson method with x0=0.

This is a comparison by the Newton Raphson method without modification for the same problem-8 but based on the Newton formula. Starting as before with x0=0 and using the equation of xi=x0-(F(x0)/f'(x0), we can get the x1 value 0.4285, then substitute and get the x2 value=0.6857.

Resolve problem #8 by the Newton method.

This is an Excel sheet for the points that are obtained by using the Newton-Raphson method for the Solved problem-8

The initial point is (0) the table lists all the values of the function till point x4. That point will be equal to 0.95578.

Excel table for the different values of x for solved problem #8.

If we start with x0=4.00 to get the other root but based on the Newton-Raphson method. The Excel sheet with more details shows the different values of x.

The steps to estimate x1 and x2 from starting x0=4.

This is an Excel sheet for the points obtained while considering the initial point x0=4.00 ending with point x6 with a value of 3.

Excel table for the different values of x for the solved problem #8 for x0=4.00.

This is a comparison between the Newton Raphson and Modified Newton Raphson methods for the same solved problem-8 using an Excel sheet.

Excel table for the different values of x for solved problem #8.

This is the PDF file used for the illustration of this post.

The next post is Structural analysis numerically by the Newton-Raphson method.

This is a useful link for a numerical analysis calculator.

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