Unrecognized function or variable 'x'.

I am having a difficult time understanding how to define "x" in this scenario.

Here's the Problem:

Here's my code:

%%%%% Problem 3.13 %%%%%
xi = 1; % given initial x value
yi = 1; % given initial y value
e = 0.001; % error/tolerance
i_max = 5; % max number of iterations

f1 = @(x,y) -2*x^3 + 3*y^2 + 42; % function 1
f2 = @(x,y) 5*x^2 + 3*y^3 - 69; % function 2

d_f1x = diff(f1,x);        %partial derivative of f1 wrt x
d_f1y = diff(f1,y);        %partial derivative of f1 wrt y
d_f2x = diff(f2,x);        %partial derivative of f2 wrt x
d_f2y = diff(f2,y);        %partial derivative of f2 wrt y

%Jacobian
J = @(x,y) -60*x*y - 54*x^2*y^2;

for i = 1:i_max
    Ji = J(xi, yi);
    del_x = (-f1(xi,yi)*d_f2y(yi) + f2(xi,yi)*d_f1y(yi))/Ji; %delta x
    del_y = (-f2(xi,yi)*d_f1x(xi) + f1(xi,yi)*d_f2x(yi))/Ji; %delta y
    x1 = xi + del_x;
    y1 = yi + del_y;
    e_x = abs((x1-xi)/xi);
    e_y = abs((y1-yi)/yi);
    
    if e_x < e & e_y < e
        break
    else
        xi = x1;
        yi = y1;
    end
end
%x_Newton = []
%y_Newton = []

I get the error :

Unrecognized function or variable 'x'.

Error in solution (line 51)

d_f1x = diff(f1,x); %partial derivative of f1 wrt x

Answer 1

[Alan Stevens]

To avoid your error message simply specify the derivatives directly (they are easy to obtain from polynomials). With both initial guesses at 1 the system diverges (perhaps that's why you are told only to do 5 iterations!). Setting y = -1 allows the system to converge). Try the following (modify as you see fit):

%%%%% Problem 3.13 %%%%%
xi = 1; % given initial x value
yi = -1; % given initial y value  (diverges if both x and y = 1)
i_max = 5; % max number of iterations (use 7 iterations to get f1, f2 = 0)

f1 = @(x,y) -2*x^3 + 3*y^2 + 42; % function 1
f2 = @(x,y) 5*x^2 + 3*y^3 - 69; % function 2

d_f1x = @(x) -6*x.^2;       %partial derivative of f1 wrt x
d_f1y = @(y) 6*y;           %partial derivative of f1 wrt y
d_f2x = @(x) 10*x;          %partial derivative of f2 wrt x
d_f2y = @(y) 9*y.^2;        %partial derivative of f2 wrt y

%Jacobian
J = @(x,y) [d_f1x(x) d_f1y(y);
            d_f2x(x) d_f2y(y)];
F = @(x,y) [f1(x,y);
            f2(x,y)];
        
for i = 1:i_max
    Ji = J(xi, yi);
    Fi = F(xi, yi);
    del = Ji\Fi;
    xy = [xi;yi] - del;
    xi = xy(1);
    yi = xy(2);
end
disp([xi yi])
disp([f1(xi,yi) f2(xi,yi)])