newton Raphson method multivariable with single function

im trying to minimize a function by newtons method with 3 variables but its showing error while solving

the code is

%code 2
 % Newton's Method 2-dimension
% Written by thirusabaresaan
% Date: May 14, 2021
clc
clear
format long
% Function Definition (Enter your Function here):
syms X Y Z
f =(3/2)*X^2+(3/2)*Y^2+Z^2-X*Y-10*(X+Y+Z)+75 % equation
% Initial Guess (Choose Initial Guesses):
X0(1) = -5
Y0(1) = 5
Z0(1) = 0
u = 10^(-2) % Convergence Criteria
i = 1 % Iteration Counter
% Gradient and Hessian Computation: 
J = subs(gradient(f),[X Y Z],[X0(1) Y0(1) Z0(1)]) % Gradient
H =  subs(hessian(f),[X Y Z],[X0(1) Y0(1) Z0(1)]) % Hessian
S = inv(H) % Search Direction
% Optimization Condition:
while (norm(J) > u)
    I = [X0(i),Y0(i),Z0(i)]'
    x0(i+1) = I(1)-S(1,:)*J
    Y0(i+1) = I(2)-S(2,:)*J
    Z0(i+1) = I(3)-S(3,:)*J
    i = i+1
    J = subs(gradient(f),[X Y Z],[X0(i) Y0(i) Z0(i)]) % Gradient updated
H =  subs(hessian(f),[X Y Z],[X0(i) Y0(i) Z0(i)]) %% Hessian updated
    S = inv(H) % New Search Direction
end
% Result Table:`
Iter = 1:i;
X_coordinate = X0';
Y_coordinate = Y0';
Z_coordinate = Z0';
Iterations = Iter';
T = table(Iterations,X_coordinate,Y_coordinate,Z_coordinate = Z0');
% Output:
fprintf('Initial Objective Function Value: %d\n\n',subs(f,[X,Y,Z], [X0(1),Y0(1),Z0(1)]))
if (norm(J) < u)
    fprintf('Minimum succesfully obtained...\n\n')
end
fprintf('Number of Iterations for Convergence: %d\n\n', i)
 X0(i)
 Y0(i)
 Z(i)
 subs(f,[X,Y,Z], [X0(i),Y0(i),Z0(i)])
disp(T)

I got error over here

J = subs(jacobian(f),[X Y Z],[X0(i) Y0(i) Z0(i)])

can any one help me to solve this

Answer 1

[vidyesh]

Hi Thirusabaresaan P,

I understand that you want to minimize a function using Newton-Raphson method, but the code is generating an error while updating the gradient value.

By making the following changes to your code, it should run smoothly without encountering any errors:

• In the line "x0(i+1) = I(1)-S(1,:)*J," change 'x0' to 'X0':

Original:

x0(i+1) = I(1)-S(1,:)*J  

Revised:

X0(i+1) = I(1)-S(1,:)*J 

• In the line "T = table(Iterations,X_coordinate,Y_coordinate,Z_coordinate = Z0');" replace "Z_coordinate = Z0'" with "Z_coordinate":

Original:

T = table(Iterations,X_coordinate,Y_coordinate,Z_coordinate = Z0');  

Revised:

 T = table(Iterations,X_coordinate,Y_coordinate,Z_coordinate); 

• In the third last line of code, change "Z(i)" to "Z0(i)":

Hope this answer helps.