Re: How to find solutions for conditioned equations?
- To: mathgroup at smc.vnet.net
- Subject: [mg19909] Re: [mg19861] How to find solutions for conditioned equations?
- From: "Wolf, Hartmut" <hwolf at debis.com>
- Date: Tue, 21 Sep 1999 02:22:45 -0400
- Organization: debis Systemhaus
- References: <199909190520.BAA10200@smc.vnet.net.>
- Sender: owner-wri-mathgroup at wolfram.com
Wen-Feng Hsiao schrieb: > > Suppose I have an equation > > A(x)= (x+1)/2, for -1<x<=1 > = (3-x)/2, for 1<x<=3 > = 0 , otherwise > > Now if I want to find solutions for A(x)==a, how can I otain the > solutions simultaneously (i.e., represented as (2a-1, 3-2a))? I always > use the stupid method to solve them separately, but I found it's laborous > when I have B(x), C(x) needed to be solved together. (I am trying to use > alpha-cut to proceed the interval operations) Any suggestions? Dear Hsiao, what to do best, depends on your intents. If you continually use the triangle-functions (e.g. as fuzzy sets) then an idea would be to define your own appropriate funtional calculus, e.g. as such: A[x_, x0_, width_] := (x - x0 + width)/width UnitStep[x - x0 + width] UnitStep[-(x - x0)] - (x - x0 - width)/width UnitStep[-(x - x0 - width)] UnitStep[x - x0] (width is the width at half height), so you have a function you can integrate, differentiate etc., or e.g. In[128]:= Plot[A[x, 1, 2], {x, -2, 4}] To find the inverse, calculate once (by hand): Ainvers[a_, x0_, width_] := If[0 <= a <= 1, width {-1, 1}(1 - a) + x0,{}] In[134]:= Simplify[Ainvers[a, 1, 2], 0 <= a <= 1] Out[134]= {-1 + 2 a, 3 - 2 a} Of course if you want to deal with the case a==0 you may include Interval[{-Infinity, -width + x0}, {width + x0, Infinity}] or not, that depends on what you want, perhaps that could be something more like In[147]:= chi[A, a_, x0_, width_] := If[0 <= a <= 1, Interval[width {-1, 1}(1 - a) + x0] ] In[149]:= chi[A, 1/2, 1, 2] Out[149]= Interval[{0, 2}] In[150]:= chi[A, 1, 1, 2] Out[150]= Interval[{1, 1}] In[152]:= chi[A, 0, 1, 2] Out[152]= Interval[{-1, 3}] Kind regards, hw
- References:
- How to find solutions for conditioned equations?
- From: d8442803@student.nsysu.edu.tw (Wen-Feng Hsiao)
- How to find solutions for conditioned equations?