Hi, everyone, I want to solve a nonlinear equation system with multiple solutions by quasi-newton broyden method, because it seems impossible to derive its jacobian matrix, but the equations are not polynomial, and it is very difficult to transform it into a polynomial system. Thank you very much...
Good day
I think this link will be helpfull for you.
Article Multiple Solutions for a System of Nonlinear Equations
Sir Armand Augustin Fondjo thank you very much, i will read it and hope it helpfull.
You may use numerical Jacobian in the solution of your nonlinear equations. In many practical applications it is not possible to derive the analytical Jacobian matrix. You may simply use forward, backward difference formulas to determine the partial derivatives in the Jacobian matrix by using a small increment. These formulas are O(h). If you want more accuracy in the Jacobian matrix you may use higher order formulas. For instance, central difference formula will be O(h^2). However, higher order formulas will require more computational effort.
For multiple solutions, Newton type solvers will require a good initial guess for each one of the roots. Therefore, you need to estimate the roots by other means. If you are seeking the the intersection points of the figure you've given above then you can discretize your domain first. Then calculate your functions and subtract them from each other and look for the points which are close to zero in absolute sense. These points will be good initial guesses to start your solution. Not all of them are the solutions, however, some of them are not. You can eliminate them by inspecting the figure if the case is like above like you can plot. Otherwise you can decrease the step sizes used in the discretization (only around the possible root locations) and check for a second time. Anything does not result in a convergent root at vicinity of the initial guess with sufficiently small discretization is not an actual root then.
If you have all the equations, you can solve them using a genetic algorithm approach.