How to calculate jacobian matrix for R^n X R^n system in matlab

How to find the jacobian for R^n X R^n system

Function F(i,j) is a system nonliear functions, which constitutes n^2 equation.

i=1,2,....,n and j=1,2,......,n

Then how to find d F(i,j) / d(x(i,j))

where x(i,j) is matrix elements

[Torsten]

Define

f((i-1)*n+j ) = F(i,j) 
X((i-1)*n+j ) = x(i,j)

Then you have a system of n^2 equations in n^2 unknowns.

Now you can define the n^2 x n^2 Jacobian matrix as usual: df(i)/dX(j).

[Chandan Kumawat]

Thank you for your reply. Can you please send matlab code for this type of simple system of nonlinear equation for better understanding?

[Torsten]

What is your problem ?

Transforming your (NxN) matrix of functions into an (N^2,1) vector or writing code to calculate a numerical Jacobian for a system of n^2 equations or both ?

[Chandan Kumawat]

Both for 2-D burgers equations.

Answer 1

[Bjorn Gustavsson]

Maybe this illustration of how to reshape your variable x and function f from n-by-n to n^2-by-1:

syms x [2,2] % a small n-by-n set of independent variables
% and a 2-by-2 array of functions:
f = [sin(2*x(:).'*x(:)) cos(3*x(:).'*x(:));exp(-x(:).'*x(:)),tanh(x(:).'*x(:))];
% Make these into (2*2)-by-1 arrays
y = x(:)
 
x1_1
x2_1
x1_2
x2_2

F = f(:);
dfdx = jacobian(f(:),x(:))
 
dfdx =
 
[  4*x1_1*cos(2*x1_1^2 + 2*x1_2^2 + 2*x2_1^2 + 2*x2_2^2),   4*x2_1*cos(2*x1_1^2 + 2*x1_2^2 + 2*x2_1^2 + 2*x2_2^2),   4*x1_2*cos(2*x1_1^2 + 2*x1_2^2 + 2*x2_1^2 + 2*x2_2^2),   4*x2_2*cos(2*x1_1^2 + 2*x1_2^2 + 2*x2_1^2 + 2*x2_2^2)]
[       -2*x1_1*exp(- x1_1^2 - x1_2^2 - x2_1^2 - x2_2^2),        -2*x2_1*exp(- x1_1^2 - x1_2^2 - x2_1^2 - x2_2^2),        -2*x1_2*exp(- x1_1^2 - x1_2^2 - x2_1^2 - x2_2^2),        -2*x2_2*exp(- x1_1^2 - x1_2^2 - x2_1^2 - x2_2^2)]
[ -6*x1_1*sin(3*x1_1^2 + 3*x1_2^2 + 3*x2_1^2 + 3*x2_2^2),  -6*x2_1*sin(3*x1_1^2 + 3*x1_2^2 + 3*x2_1^2 + 3*x2_2^2),  -6*x1_2*sin(3*x1_1^2 + 3*x1_2^2 + 3*x2_1^2 + 3*x2_2^2),  -6*x2_2*sin(3*x1_1^2 + 3*x1_2^2 + 3*x2_1^2 + 3*x2_2^2)]
[-2*x1_1*(tanh(x1_1^2 + x1_2^2 + x2_1^2 + x2_2^2)^2 - 1), -2*x2_1*(tanh(x1_1^2 + x1_2^2 + x2_1^2 + x2_2^2)^2 - 1), -2*x1_2*(tanh(x1_1^2 + x1_2^2 + x2_1^2 + x2_2^2)^2 - 1), -2*x2_2*(tanh(x1_1^2 + x1_2^2 + x2_1^2 + x2_2^2)^2 - 1)]
 
>> dfdy = jacobian(F,y)
 
dfdy =
 
[  4*x1_1*cos(2*x1_1^2 + 2*x1_2^2 + 2*x2_1^2 + 2*x2_2^2),   4*x2_1*cos(2*x1_1^2 + 2*x1_2^2 + 2*x2_1^2 + 2*x2_2^2),   4*x1_2*cos(2*x1_1^2 + 2*x1_2^2 + 2*x2_1^2 + 2*x2_2^2),   4*x2_2*cos(2*x1_1^2 + 2*x1_2^2 + 2*x2_1^2 + 2*x2_2^2)]
[       -2*x1_1*exp(- x1_1^2 - x1_2^2 - x2_1^2 - x2_2^2),        -2*x2_1*exp(- x1_1^2 - x1_2^2 - x2_1^2 - x2_2^2),        -2*x1_2*exp(- x1_1^2 - x1_2^2 - x2_1^2 - x2_2^2),        -2*x2_2*exp(- x1_1^2 - x1_2^2 - x2_1^2 - x2_2^2)]
[ -6*x1_1*sin(3*x1_1^2 + 3*x1_2^2 + 3*x2_1^2 + 3*x2_2^2),  -6*x2_1*sin(3*x1_1^2 + 3*x1_2^2 + 3*x2_1^2 + 3*x2_2^2),  -6*x1_2*sin(3*x1_1^2 + 3*x1_2^2 + 3*x2_1^2 + 3*x2_2^2),  -6*x2_2*sin(3*x1_1^2 + 3*x1_2^2 + 3*x2_1^2 + 3*x2_2^2)]
[-2*x1_1*(tanh(x1_1^2 + x1_2^2 + x2_1^2 + x2_2^2)^2 - 1), -2*x2_1*(tanh(x1_1^2 + x1_2^2 + x2_1^2 + x2_2^2)^2 - 1), -2*x1_2*(tanh(x1_1^2 + x1_2^2 + x2_1^2 + x2_2^2)^2 - 1), -2*x2_2*(tanh(x1_1^2 + x1_2^2 + x2_1^2 + x2_2^2)^2 - 1)]

You can do very much the same with arbitrary numerical functions too, just keep track of where the different elements should go, then after integrating the solution you will just reshape the solution back into your n-by-n shape for each step of the solution. It is admittedly bit fidgety to get these things right - I have found it very helpful to check that I get this right by looking at small enough systems that I can manually inspect every matrix in the process. Even if I want it at 100-by-100 or more I check that the procedures work for 4-by-4, 10-by-10 etc.

HTH