Algorithm for solving system of nonlinear equations

[mooshasta]

I'm trying to find an algorithm to solve a 4 variable system of nonlinear equations.. the variables are named w,x,y,z and a,b,c,d are constants:

a = x - y + z
b = w + x
c = y * z
d = x * y / w


Can anyone offer any advice? Much appreciated...

 

[Hurkyl]

Gröbner basis algorithms could be used to, for example, eliminate all but one variable.

 

[tacman]

There are several algorithms that can be used to solve a system of nonlinear equations, such as the Newton-Raphson method, the Broyden's method, or the Levenberg-Marquardt algorithm. However, for a system of only 4 variables, a simpler approach can also be used.

One possible algorithm for solving this system of nonlinear equations is the Gauss-Newton method. This method involves iteratively updating the values of the variables until a solution that satisfies all the equations is found.

Here are the steps for implementing the Gauss-Newton method for this particular system of equations:

1. Start by guessing initial values for the variables w, x, y, and z.

2. Use these initial values to calculate the values of a, b, c, and d.

3. Calculate the error for each equation by subtracting the calculated value from the given constant value. For example, the error for the first equation would be a - (x - y + z).

4. Use the errors to construct a Jacobian matrix, which is a matrix of partial derivatives of the equations with respect to each variable. In this case, the Jacobian matrix would be:

J = [1, -1, 1, 0;
1, 1, 0, 0;
0, z, y, 0;
-y/w, 1, x/w, -x/w^2]

5. Calculate the pseudo-inverse of the Jacobian matrix using a method such as the Singular Value Decomposition (SVD) method.

6. Multiply the pseudo-inverse by the error vector to get a vector of updates for the variables.

7. Update the values of the variables by adding the update vector to the initial values.

8. Repeat steps 2-7 until the error for each equation is below a desired tolerance level.

This algorithm may require some additional fine-tuning and adjustments depending on the specific system of equations being solved, but it is a good starting point for solving a system of nonlinear equations with a small number of variables.