Plotting of solutions using Newton Raphson method for a nonlinear function

I need help while plotting a nonlinear equation using Newton Raphson method:

(x: solution to be calculated using NR)

plot(x,y)

a = 0:0.1:2;

fun = x^2 - 2*(a+x) + 4*a*x;

[Riccardo Scorretti]

Please clarify your question: Newton-Raphson method is used to solve (not to plot) a nonlinear equation. You can use the function plot to make a graphics of the function. By the way, you have a function with two variables (a and x). What do you want to do precisely?

[Supriya Khatoniar]

Need to plot the solution of the above function after solving it by applying Newton Raphson method into it, for the ranges of values "a". I want to plot the solution of the function for each values of "a".

This is what I tried, but not sure enough:

clc; clear all; close all;
        
        A=0:0.1:2;
        syms x;
        
        for i = 1:numel(A)
            a = A(i);
            
            fun = x^2 - 2*(a+x) + 4*a*x;
            x0 = 0; %intial value
            N = 100;
            step = 1;
            e= 0.00001;
            
            dfun = diff(fun,x);
            fa = eval(subs(fun,x,x0));
            
        while abs(fa)> e
            fa = eval(subs(fun,x,x0));
            ga = eval(subs(dfun,x,x0));
             
            b = x0 - fa/ga;
            fprintf('step=%d\ta=%f\tf(a)=%f\n',step,x0,fa);
            x0 = b;
            step = step + 1;
            X(i) = x0;
        end
        fprintf('Root is %f\n', x0);        
        end
        
        plot(A,X);

Answer 1

[KSSV]

Refer this: https://in.mathworks.com/matlabcentral/answers/107508-solving-a-nonlinear-equation-using-newton-raphson-method

SEarch in goole, you will get many more links.

Answer 2

[Sam Chak]

Hi @Supriya Khatoniar

Check if this script is helpful. You can adjust plot properties to suit your preferences.

function Demo_NR

close all

clc

x0 = 1;

epsilon = 1e-9;

n = 0:0.1:2;

x = zeros(1, length(n));

for a = 0:0.1:2

f = @(x) x^2 - 2*(a + x) + 4*a*x;

x = NewtonRaphson(f, x0, epsilon);

plot(a, x, '*', 'color', 'b')

hold on

end

grid on

xlabel('a');

ylabel('x');

title('Solutions via Newton–Raphson method')

print -f1 -dpng -r300 fig_Demo_NR.png

end

function [x, Iter] = NewtonRaphson(f, x0, epsilon)

Iter = 0;

df = @(x) (f(x+1.0e-6)-f(x))*1e6;

x1 = x0 - f(x0)/df(x0);

while abs(f(x1)) > epsilon

x0 = x1;

x1 = x0 - f(x0)/df(x0);

Iter = Iter + 1;

end

x = x1;

end

Result

Answer 3

[Nada Ahmed]

help me .. i need question soultion matlab

Assignment (1)

1. Problem Description

The equation of state developed by Beattie and Bridgeman is an extension of the ideal gas law and is given by the following equation: 𝑃=𝑅𝑇𝑉+𝑎𝑉2+𝑏𝑉3+𝑐𝑉4

Where 𝑃 is the gas pressure in [atm], 𝑇 is temperature in [K], 𝑉 is volume in [L/mol], 𝑅 is the ideal gas constant in [atm∙L/mol∙K] and 𝑎, 𝑏 and 𝑐 are constants. For the conditions and constants given in the table below, use MATLAB to determine the volume (𝑉). Note that Eqn. 1 has multiple roots but the one you are looking for will fall between 0.001 and 0.01 L/mol.

Solve this problem using MATLAB and each of the following numerical methods to an error tolerance of s  = 1∗10−5 %.

• Bisection method

• False position method

• Newton-Raphson method

Script File:

A main script file must also be created (Lab1.m) that does the following:

• Creates all the necessary variables (parameters in Table).

• Creates an anonymous function for the non-linear equation and any other anonymous functions required in the function files.

• Creates variables for the error tolerance and the maximum number of iterations.

• Creates variables for the initial guesses for each numerical method.

• Calls each of the three function files (Bisection.m, FalsePosition.m, NewtRaph.m) to solve for the volume 𝑉. Use different variable names for the resulting volume, approximate error and number of iterations from each method such that these values will not be overwritten when another function is called.

Results

Include the following results/discussions in your report:

• A table presenting the results from each of the three numerical methods. The table should include the following results for each method:

➢ the initial guess(es) o the final volume 𝑉

➢ the number of iterations required for the method to converge

➢ the approximate percent relative error after convergence is reached

➢ You may use any software to create this table (ex. Microsoft Excel)

𝑃 [atm]

𝑇 [K]

𝑅 [atm∙L/mol∙K]

𝑎

𝑏

𝑐

25

293

0.08206

−0.106

0.057

−0.0001

Submission Requirements

o Each student must submit the report as a pdf document and the four MATLAB m-files (Lab1.m, Bisection.m, FalsePosition.m, NewtRaph.m)

o Please combine all the files mentioned above into a folder and compress the folder into a zip file.