Finding Jacobian matrix for a similar set of equations

Hello ,

I want to form a jacobian matrix for a case where i have multiple equations of the same form, e.g. 10 functions F1i of form xi - yi - 1 and similarly 10 of form F2i = xi^2 + yi^2 -1 .

I ultimately want to solve multiple such sets of similar equations using the newton raphson method.

Edit :

I would potentially like to use loops for the equations from say i = 1 : n

[Sam Chak]

The following equations are created using LaTeX. Could you please review them and confirm if they are correct?

[Parthsarthi]

Thank you for the response. I am actually looking as to how I could club all the i's from 1 to n , in one vector and them find a jacobian based on it.

[Sam Chak]

Now that you have the Jacobian, could you please demonstrate how you intend to solve this simple problem using the Newton-Raphson method in MATLAB code?

syms x y
F1  = x   - y   - 1;
F2  = x^2 + y^2 - 1;
eqn = [F1; F2];
J   = jacobian(eqn, [x, y])

J = 

Answer 1

[prabhat kumar sharma]

Hello Parthsarthi,

I understand that you're looking to solve your system of nonlinear equations using the Newton-Raphson method. For illustrative purposes, let's assume you're dealing with two equations, though this method can be extended to handle more equations and variables.

First, let's define the system of equations and their Jacobian matrix as you have begun. To solve the system using the Newton-Raphson method, you will need an initial guess for the values of x and y. Then, iteratively update these values using the Newton-Raphson formula until convergence is achieved.

For a clearer understanding and potential solution to your issue, you can refer to this MATLAB answer:

Solving a Nonlinear Equation using Newton-Raphson Method - MATLAB Answers - MATLAB Central (mathworks.com)

I hope this helps!

Answer 2

[Nipun]

Hi Parthsarthi,

I understand that you want to generalize a system of equations with different orders in a matrix to find the roots using an iterative Newton-Ralphson method. Based on the shared information, I see that you want to first calculate the Jacobian of the generalized matrix and then use the result to iteratively solve for roots.

Bases on the shared information, I assume all coefficients to be 1 and each "x_i" (respectively "y_i") represents a different variable. Since the coefficients are same, a solution to "F11" is same as solution to "F1n" (similar case holds for "F2i").

You can create the Jacobian as follows in MATLAB:

syms x y

F1 = x - y - 1;

F2 = x^2 + y^2 - 1;

eqn = [F1; F2];

J = jacobian(eqn, [x, y])

J =

For more information on "solving equations using symbolic math" in MATLAB, refer to the following MathWorks documentation:

https://in.mathworks.com/help/symbolic/equation-solving.html

If you intend to create a column vector of "F1i" and a separate column vector for "F2i" with varying "i", use the following MATLAB script:

n = 10;

xi = sym('x', [1 10])

yi = sym('y', [1 10])

F1i = xi - yi - 1;

F2i = xi.^2 + yi.^2 - 1;

J1i = jacobian(F1i, [xi, yi])

J2i = jacobian(F2i, [xi, yi])

xi =

yi =

J1i =

J2i =

Hope this helps.

Regards,

Nipun