newton Raphson method multivariable with single function

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Thirusabaresaan P
Thirusabaresaan P 2021-6-15
回答： vidyesh 2023-10-13
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

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回答（1 个）

vidyesh
vidyesh 2023-10-13
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.

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