Please help to solve the following equations by the Newton–Raphson method

Hi All,

Please help to solve the following equations by the Newton–Raphson method:

fx1 = (2*x1^2) + (x2^2) -10;
fx2 = (x1^2) - (x2^2) + (x1*x2) - 4;

Start with an initial guess of x1 = 1 and x2 = 1., y =0

My code as below:

clc;clear;close all;
syms x1 x2 fx1 fx2 y1 y2;

fx1 = (2*x1^2) + (x2^2) -10;
fx2 = (x1^2) - (x2^2) + (x1*x2) - 4;
Fx = [fx1;fx2]

J11 = diff(fx1,x1);
J12 = diff(fx1,x2);
J21 = diff(fx2,x1);
J22 = diff(fx2,x2); 

JJ = [J11 J12 ; J21 J22]
JJ1=[J11; J12]
JJ2=[J21 ; J22] 

y1 = 0;
y2 = 0;
YY=[y1;y2]

x1(1) = 1;
x2(1) = 1;
X = [1;1];

epsilon = 0.001;
n = 10; 

func1 = inline(Fx(1))
func2 = inline(Fx(2))


M = diff(sym(Fx(1)))
N = diff(sym(Fx(2)))

DF1 = inline(M)
DF2 = inline(N)



for i=1:4
   
    X(i+1)=X(i)+(YY-func1(X(i)))
    
end

Thanks in advance

[Walter Roberson]

Do not use inline(). Use matlabFunction() to convert symbolic expressions into anonymous functions.

Answer 1

[Divija Aleti]

Hi,

Have a look at the following code which solves the above mentioned equations by Newton-Raphson Method :

clc;clear;close all;
syms fx1(x1,x2) fx2(x1,x2) 
fx1(x1,x2) = (2*x1^2) + (x2^2) -10;
fx2(x1,x2) = (x1^2) - (x2^2) + (x1*x2) - 4;
Fx = [fx1;fx2]
J11 = diff(fx1,x1);
J12 = diff(fx1,x2);
J21 = diff(fx2,x1);
J22 = diff(fx2,x2); 
JJ = [J11 J12 ; J21 J22];
epsilon = 0.001;
x1_o = 1;
x2_o = 1;
x = [x1_o];
y = [x2_o];
for i=1:10
    h = det([-fx1(x(i),y(i)) J12(x(i),y(i)); -fx2(x(i),y(i)) J22(x(i),y(i))])/det(JJ(x(i),y(i)));
    k = det([J11(x(i),y(i)) -fx1(x(i),y(i)); J21(x(i),y(i)) -fx2(x(i),y(i))])/det(JJ(x(i),y(i)));
    x(i+1) = x(i) + h;
    y(i+1) = y(i) + k;
    if abs(x(i+1)-x(i)) <= epsilon && abs(y(i+1)-y(i)) <= epsilon
        break
    end
end
% The two roots are : 
x_1 = x(i+1)
x_2 = y(i+1)

For additional information on 'syms', refer to the following link:

Create symbolic variables and functions - MATLAB syms (mathworks.com)

Regards,

Divija