The basics of fsolve

In this screencast, we discuss the basics of Matlab's function fsolve, which can be used to solve systems of non-linear equations.

[@vjdipchan6436]

Thank you for your in-depth demonstration of the fsolve function. Much appreciated!

[@danielgladstone1789]

Interesting haha

[@LaraxWarrior3]

Thank you

[@danielt.plazas5401]

Thank you Dr Gregory Reeves! Do you know if there is any way to find all the roots of a system in which there is, for example, an equation with two unknowns? Of course in a finite range.

[@drgregoryreeves]

Sorry for the slow reply. In general, there is no way to do that, as far as I know. For some kinds of equations, you can be sure about all the roots (if the equations are polynomials, for example). But non-linear equations might not have any roots, and you wouldn't know that. Also, within a finite range that you specify, you might not be in the right range. For one equation with two unknowns, though, then you have an underspecified system. You'd need either another equation, or you'd need to specify one of your unknowns. If you have enough computing power, and you want to be very sure, you can always plot the equation for many values of your unknown within your finite range, and see how many times it crosses the x-axis. You can usually be pretty sure your equation is "smooth", so that you won't be missing any roots as long as you have a high enough density of points. This is a very brute force method, so it's very slow.

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[@sefaarslan1488]

I have a system with 10 equations and 8 unknowns, how can I solve these equation sets?

[@GregReeves1520]

If you have an overdetermined system, fsolve can still work with it, but it will try to do some sort of optimization to get the best fit values of your unknown that minimizes your 10 equations in some way. I am not 100% sure how that works, since I've only tried to use fsolve with square systems. Are you trying to do an optimization problem?

[@sefaarslan1488]

@@GregReeves1520 no, I'm dealing with trim analysis of an aircraft in forward flight and trying to find the unknowns from dynamic equations. Thanks for your reply I will try f solve first

[@saddamhussainsoomro2810]

Sir can you help me for 12 non linear equations with 12 variables, or guide anything please. Thanks

[@GregReeves1520]

Hi, happy to help. Can you clarify? Are you running into problems that are not helped by this video? fsolve can, in principle, handle a 12x12 system, but it may be difficult and you might need to use a very good initial guess. Have you already tried to program a script and function to use fsolve on your system? What did you find? Was there a particular error?

[@saddamhussainsoomro2810]

@@GregReeves1520 yes sir I tried using fsolve. That shows error. The equation not solved! I also tried Newton raphson method but the roots are coming to be same as my initial guesses.

[@GregReeves1520]

@@saddamhussainsoomro2810 If you have already taken the time to write a Newton's method code to solve the problem, I would focus on that. So you wrote a while loop that quits when either "f" or "delta_x" (or both) become small (say, 1e-6). In each iteration of the loop, you need to evaluate "f", calculate the Jacobian "J", then take a newton step of "delta_x = -J \ f; x = x + delta_x;" Correct? How many iterations are happening in the while loop? If your solution is the same as your initial guess, then my guess is that either you are not entering the while loop correctly, or within the while loop, you are not updating your value of x. Thoughts?

[@saddamhussainsoomro2810]

@@GregReeves1520 sir can you send me email I will send you my code

[@drgregoryreeves]

@@saddamhussainsoomro2810 I am not planning on posting my email publicly here, but you can google me, and you can probably find me on an NC State website.

[@criccenter9619]

How can I plot (X,Y) When X=-3:0.1:3; Y=X² + 3*Y - exp(X-2*Y);

[@drgregoryreeves]

You can convert your equation to the form f(X) = 0 by moving the lone Y on the LHS to the RHS. Then, in that form, you can run a for loop to use fzero to solve for Y for every X in your mesh. Alternatively, you could use fsolve once instead of fzero in a for loop...that might work too. Let me know if you have other questions. Once you have a value of Y for every value of X, you can plot(X,Y).

[@criccenter9619]

@@drgregoryreeves x=[-3:0.05:3]; n = size(x,1); yy=zeros(n,1); for i=1:n yy(i)=fsolve(@(y) fun(x(i),y),0); end plot(x,yy) function S=fun(x,y) S=X²+3*Y-exp(X-2*Y) end I used still unable to get answer please suggest your code more precisely

[@drgregoryreeves]

​@@criccenter9619, when you say "unable to get answer", does that mean you can't get any kind of solution? Or is it that you are not getting the solution you expect? Because your code looks good, syntax-wise. The only thing is that in your first post, you had a "Y" on the LHS of your equation, but that "Y" is gone in your second post. If the first post is correct, you should move that "Y" from the LHS to the RHS, so it appears as a "-Y" in the RHS of the "S" equation in your function. Did that help? Let me know!

[@criccenter9619]

@@drgregoryreeves I am sorry I made mistake in editing there S=Y-(X² + 3*Y - exp(X-2*Y)); I wasn't able to get anything on plot .Nothing except the axis. Please provide your solution if u don't mind.

[@drgregoryreeves]

@@criccenter9619 It sounds like your equation didn't converge, and you got a bunch of NaN's for your elements in the "yy" vector. Can you double check that?