System of non-linear equations

[Gabriel]

Hi,
I tried to solve a system of non linear and non polynomial equations and I managed to solve it with the Solve App and in Advanced graph mode, looking for intersections in the grap or in num mode(using Trace when I was nearby). I wanted to know if there was a way to solve non-linear systems with something like solve-fsolve-nsolve(i think I got the syntax wrong), maybe specifing for example 1..7 for x and 2..4 for y rather than putting one number, like you do in the Solve App.

[Tim Wessman]

Please give your example problem.

Generally, it will look like this. fsolve([x^2+y-2,x+y^2-2],[x,y]) You can give it a range by doing 0..1 for example for your intial guess.

[Gabriel]

I tried your system(my original problem is a bit more complex, I guess its fine to try with simple systems), how do i give the range? Because we have both x and y, should I give 2 ranges in this way : fsolve([x^2+y-2,x+y^2-2],[x,y],[0..1,0..1])?

[Gabriel]

Wait I thought I saw y^(-2) while it was y^2, I need to solve non linear non polynomial systems then lets change with: fsolve([cos(x)-x,x-y],[x,y],[0..1,0..1]) which should give us x=0.739085133215 and y=0.739085133215

[Helge Gabert]

Try

[x=0 .. 1 , y=0 .. 1] as the second argument to fsolve.

[Gabriel]

Thanks a lot, it works for that system, but I tried it with my problem and even with an accurate range i cant get my solution :/
my system is:
39.8344274/(y-0.1495645628)-24.52151/(y^2)=
39.8344274/(x-0.1495645628)-24.52151/(x^2)
and
39.8344274/(y-0.1495645628)-24.52151/(y^2)=
39.8344274*ln((x-0.1495645628)/(y-0.1495645628))/(x-y)-24.52151/(x*y)

the solution i am aiming for is x=1.66649546538 and y=0.238068203637, which i can get in graph mode or with the solve app giving an inital guess value like 1.5-0.2, but with fsolve even if i put ranges like 1.665-1.667 and 0.22-0,24 or initial guess 1.5-0.2
I cant get it

[parisse]

It's slow but it works in Xcas, e.g.

Code:


fsolve([39.8344274/(y-0.1495645628)-24.52151/(y^2)=39.8344274/(x-0.1495645628)-24.52151/(x^2),39.8344274/(y-0.1495645628)-24.52151/(y^2)=39.8344274*ln((x-0.1495645628)/(y-0.1495645628))/(x-y)-24.52151/(x*y)],[x,y],[1.6,0.2])

The reason is probably because I compute symbolically the inverse of the jacobian, perhaps it would be faster if done at each Newton iteration step, once x and y have values.
For multi-dimensional systems, you can not provide intervals as guesses, this work only for 1-dimension equation where you can start a bisection.

[Gabriel]

Then why the intervals worked for fsolve([cos(x)-x,x-y],[x=0..1,y=0..1]?
Anyway I tried single values this way:
fsolve([39.8344274/(y-0.1495645628)-24.52151/(y^2)=39.8344274/(x-0.1495645628)-24.52151/(x^2),39.8344274/(y-0.1495645628)-24.52151/(y^2)=39.8344274*ln((x-0.1495645628)/(y-0.1495645628))/(x-y)-24.52151/(x*y)],[x,y],[0.8,0.25]) and I got my solution(why did you say it was slow, it was quite fast), and for some unknown reasons its important that i put an accurate y value, for example if i put 1.66 and 0.2 i doesnt work, while it works with 0.8 and 0.25

[parisse]

Didi you try on emulator or real calc? With inversion of jacobian after substitution (new code) it takes about 1/100s on the emulator.
The starting point is important for iterative methods: sometimes it converges, sometimes not...
For fsolve([cos(x)-x,x-y],[x=0..1,y=0..1]), the solver probably restarted from a random point, after an initial try with an invalid guess.

[Gabriel]

I tried with my real calc, if its 1/100 why did you say it was slow? its very fast(i can count seconds not 1/10 s or 1/100 s xD). You are right for the starting point, there must be some mathematical reason behind it.
One last thing, when I put these equations in advanced graph app for a graphic solution, it takes some time to plot the graph, like it plots a curved stripe/area and then it gets reduced till its a one dimension curve...its because they are complex non linear implicit functions ?

[parisse]

1/100s is on the emulator with the modified code (the original code takes about 100* more time). I don't know for the graph app.

[Gabriel]

Can i get the modified code for my real calc too? By the way thanks for the patience!!!

[parisse]

No, but you can test it in inside Xcas (test version 1.1.1) today (windows and linux 64 bits) or tomorrow (mac and linux 32 bits).