How can i take the Jacobian of a function

Hi there, I have a problem that I need to be solved quickly. I need the Jacobian of a function with respect to one of its input values and afterwards to fill in the values of the input. These are the input values and jacobian function:

if true
x = [5 7]
b = [2 3]
h = jacobian(qn(x,b),x)
end

This is the function itself:

if true
  function demand = qn(x,b)
  z1=b(1)-b(2)*x(1)+b(1)*x(2);
  z2=b(1)-b(2)*x(2)+b(1)*x(1);
  demand = [z1;z2];
end

However, I receive an error message saying:

"Undefined function 'jacobian' for input arguments of type 'double'.

Error in Algorithm3 (line 21) h = jacobian(qn(x,b),x);"

To recap, I need the Jacobian of the function qn with respect to x(1) and x(2), so that it gives a 2 by 2 matrix back and then I want to 'input' the vectors x & b. Thanks in advance!

[Adam]

Are you expecting it to find a function called jacobian? There is one in the symbolic Toolbox, but otherwise unless you have a 3rd party one or one you have written yourself the error is understandable.

[Martijn Mouw]

Well I have the symbolic Toolbox and for other problems the command 'jacobian' works fine, however when I try to find the jacobian matrix of a function specified with input variables ('qn' in this case) I don't know how to tell MATLAB that I want the jacobian matrix of the output of this function with respect to x(1) and x(2)

Answer 1

[Torsten]

function main
x = [5 7]
b = [2 3]
h = numerical_jacobian(@qn,x,b)
end

function df=numerical_jacobian(f,x,b) 
n=length(x); 
E=speye(n); 
e=eps^(1/3); 
for i=1:n 
 df(:,i)=(f(x+e*E(:,i),b)-f(x-e*E(:,i),b))/(2*e); % zentraler Differenzenquotient 
end 
end

function demand = qn(x,b)
  z1=b(1)-b(2)*x(1)+b(1)*x(2);
  z2=b(1)-b(2)*x(2)+b(1)*x(1);
  demand = [z1;z2];
end

[Martijn Mouw]

This works. Many thanks! Though, just for my own interest, it isn't possible to fix this problem inside the jacobian command of MATLAB itself?

[Martijn Mouw]

And would you be willing to share the intuition behind e=eps^(1/3) with me and using that in the central difference quotient (cdq)? I understand the cdq but this eps is new for me

[Torsten]

1. As far as I know, "jacobian" only works with symbolic variables.

2. Everybody uses other values as optimal h for numerical differentiation. Just test for your case which order of magnitude fits your needs.

Here is a link that derives h ~ eps^(1/3) for the central difference quotient:

https://math.stackexchange.com/questions/815113/is-there-a-general-formula-for-estimating-the-step-size-h-in-numerical-different

Best wishes

Torsten.

[Martijn Mouw]

Ah okay, I see. Thanks for the help!