jacobian of a function in matlab

To find the Jacobian of a function in Matlab, one can use the jacobian command from the Symbolic Toolbox.

Here is an example of how to find the Jacobian of a function:

main.m
syms x y z    % define symbolic variables
f = [x*y^2*z^3;
    x*y*z^2;
    sin(x)+cos(y)+exp(z)];   % define function
J = jacobian(f, [x y z])    % calculate Jacobian
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In this example, we define three symbolic variables x, y, and z, and we define the function f as a 3x1 vector. Then we use the jacobian command to calculate the Jacobian of f with respect to x, y, and z. The output is a 3x3 matrix representing the Jacobian matrix.

Note that the jacobian command only works with symbolic inputs, so if your function involves numerical values, you will need to approximate the Jacobian using numerical differentiation.

main.m
X = [1 2 3];
Y = [4 5 6];
Z = [7 8 9];
f = @(x,y,z) [x*y^2*z^3; x*y*z^2; sin(x)+cos(y)+exp(z)];
delta = 1e-6;
J(1,:) = (f(X+delta,Y,Z) - f(X,Y,Z))./delta;
J(2,:) = (f(X,Y+delta,Z) - f(X,Y,Z))./delta;
J(3,:) = (f(X,Y,Z+delta) - f(X,Y,Z))./delta;
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This code will approximate the Jacobian of f using numerical differentiation with a step size of delta and the forward difference formula. The J matrix will represent the approximate Jacobian matrix.