How do I find a point in every region defined by a system of linear equations?
I have a system of eight linear equations of the form
Ai and Bi are numerical values.
If I plot all eight equations on one graph then I get numerous bounded regions defined by three or more of the equations and numerous unbounded regions defined by two or more of the equations.
(It is possible – though unlikely - for two of the solutions to be parallel or collinear. )
My aim is to find one point – any point does - within every bounded region and one in every unbounded region. It is easy to do this by inspection.
However, I need to do this automatically. The system is dynamic so the Ai and Bi values change and I do not know in advance how many regions there are.
I can see that it is straightforward to find all – (8-1)! I think - solutions of pairs of equations (intersecting points) and I can see that it will be possible to write an algorithm to find the regions but before I do two questions:
1) Is there a particular branch of linear programming which deals with this sort of problem?
2) Or even better are there commands in maple to directly deal with this?
Any suggestions most welcome.