Thank you so much for these videos! I was wondering if Wegstein's method works in the general case of a system of nonlinear equations. If it does, do we have a vector of q values each corresponding to a variable or one common value?
It is a vector of q's. This was actually the main topic of Charles Guztler's thesis, which I used for background info on my root-finding Wegstein's Method video. Check out Chapter 4 of "An Iterative Method of Wegstein for Solving Simulteanous Nonlinear Equations" ir.library.oregonstate.edu/downloads/2r36v1962 for more.
@@OscarVeliz Oh god, thank you! I downloaded the thesis yesterday, but didn't have time to look at it yesterday. I am developing a program which finds an optimal distribution of chemical species in a system (gas phase, stoichiometric solids, liquid and solid solutions) at constant temperature and pressure (Gibbs energy constrained minimization). I derived a set of non-linear equations which can be solved for some cases using iterations but I found cases when the solution diverges. I checked yesterday Wegstein's formula for 1 variable (finding a solubility of a component in a liquid with strong interactions) and it worked as charm)) Hope it will work with several equations at once.
I just re-ran the Rust program I provided for that example and got the same numbers. Depending on the language and even the machine you're on you might get different results.
It is helpful to see them on the Cartesian plane but the Banach Fixed Point Theorem is meant to be applied on Banach spaces. This is why you can use the contraction mapping test.
@@OscarVeliz Using the fractal maps, I found out that: For G(X) = (x^2 - y + x - 1, x - y^2 + y + 1), the solution is (-1, 0); And for G(X) = (-x^2 + y + x + 1, -x + y^2 + y - 1), the solution is (0, -1).
You can find the program I provided on Fixed Point Iteration for Systems of Equations and Generalized Aitken-Steffensen in the GitHub repo for this channel (github.com/osveliz/numerical-veliz). You can also run the Rust code online by using the CodingGround link in the documentation wiki.
hello sir, thank you for your videos. i have a question i can't find an answer for it , how can we study of uniqueness of coupled nonlinear differential equations ?
I plan on covering existence and uniqueness of diff eq solutions although there a lot of topics in the queue ahead of it. I will suggest trying to visualize the system and I hope that helps.
I 've spend 6 hours reading about this topic , still dont understand, now I suddenly understand. I ve got adhd and dyslexia. THANKS! :)
th-cam.com/video/QXy_soGFi5Y/w-d-xo.html
Awesome video! Really nice valuable videos! Your videos are really quite superb and unique and in-depth
Thanks so much for this video, I was struggling with this but now I understand it, the explanation was super clear. thankyouthankyou~~
Thank you so much for these videos! I was wondering if Wegstein's method works in the general case of a system of nonlinear equations. If it does, do we have a vector of q values each corresponding to a variable or one common value?
It is a vector of q's. This was actually the main topic of Charles Guztler's thesis, which I used for background info on my root-finding Wegstein's Method video. Check out Chapter 4 of "An Iterative Method of Wegstein for Solving Simulteanous Nonlinear Equations" ir.library.oregonstate.edu/downloads/2r36v1962 for more.
@@OscarVeliz Oh god, thank you! I downloaded the thesis yesterday, but didn't have time to look at it yesterday. I am developing a program which finds an optimal distribution of chemical species in a system (gas phase, stoichiometric solids, liquid and solid solutions) at constant temperature and pressure (Gibbs energy constrained minimization). I derived a set of non-linear equations which can be solved for some cases using iterations but I found cases when the solution diverges. I checked yesterday Wegstein's formula for 1 variable (finding a solubility of a component in a liquid with strong interactions) and it worked as charm)) Hope it will work with several equations at once.
I noticed a typo at 4:36 -- should it be 1.24407?
I just re-ran the Rust program I provided for that example and got the same numbers. Depending on the language and even the machine you're on you might get different results.
@@OscarVeliz It said 124407 instead of 1.24407.
Oh I see. I was looking at the blue chart. Yes, in the orange chart there should be a decimal.
Thanks so much for this video , you are the best
At 3:23 why are you pointing out that it is not cartesian points but its Banach spaces? Whats the difference?
It is helpful to see them on the Cartesian plane but the Banach Fixed Point Theorem is meant to be applied on Banach spaces. This is why you can use the contraction mapping test.
@@OscarVeliz i see. Thanks
well done
You can turn this into a fractal map in the xy plane to see more clearly where our solutions are, and which points do converge toward it.
Certainly possible although there would be quite a few combinations of functions to check in order to determine the most interesting-looking fractals.
@@OscarVeliz Using the fractal maps, I found out that:
For G(X) = (x^2 - y + x - 1, x - y^2 + y + 1), the solution is (-1, 0);
And for G(X) = (-x^2 + y + x + 1, -x + y^2 + y - 1), the solution is (0, -1).
Sir do you have code for fixed iteration method for system of non-linear equations in MATLAB?
You can find the program I provided on Fixed Point Iteration for Systems of Equations and Generalized Aitken-Steffensen in the GitHub repo for this channel (github.com/osveliz/numerical-veliz). You can also run the Rust code online by using the CodingGround link in the documentation wiki.
Sir, Does it make sense to calculate spectral radius to prove convergence? If yes, How?
I'm not certain how something like Gershgorin Circle Theorem could be used on nonlinear systems. It would require some looking into.
hello sir, thank you for your videos. i have a question i can't find an answer for it , how can we study of uniqueness of coupled nonlinear differential equations ?
I plan on covering existence and uniqueness of diff eq solutions although there a lot of topics in the queue ahead of it. I will suggest trying to visualize the system and I hope that helps.
Nice