I have a system of non-linear multivariate equations. I am only interested in the roots of the system in an interval of each variable.
For example,
$$ \begin{align} \frac{1}{10} \sin \left( \frac{x y}{100} + \frac{y z}{5} \right) &+ \frac{1}{10} \sin \left( \frac{y z}{10} \right) \\ &+ \frac{1}{100} \left( y \cos \left( \frac{y z}{10} \right) \left(\frac{x}{10} + z \right) \right) \\ &+ \frac{1}{50} \left( y \cos \left( \frac{x y}{100} + \frac{y z}{5} \right) \left( \frac{x}{5} + z \right) \right) = 0 \end{align} $$ (z*cos((y*z)/10)*(x/10 + z))/100 + (cos((x*y)/100 + (y*z)/5)(x/5 + z)(x/100 + z/5))/10 = 0
sin((x*y)/100 + (y*z)/5)/50 + sin((y*z)/10)/100 + (y*cos((x*y)/100 + (y*z)/5)*(x/5 + z))/1000 = 0
and I am only interested in the interval of x in [0-10] and y in [-10 10 ] and z in [-5 , 10].
My questions are:
- Is there any method that can find all the roots in these intervals?
- Is there any solution that can tell me only number of roots in these intervals?
- If there is no general solution for questions 1 and 2 (which I think there is not), then under what condition I can find all the roots (e.g. linear equations or polynomial are clearly solvable, but I want something more general such as trigonometric equations).
Thank you very much for your help.