Re: FindInstance puzzler

*To*: mathgroup at smc.vnet.net*Subject*: [mg83712] Re: FindInstance puzzler*From*: m.r at inbox.ru*Date*: Wed, 28 Nov 2007 05:50:25 -0500 (EST)*References*: <figur5$fpn$1@smc.vnet.net>

On Nov 27, 5:25 am, Tom Burton <n... at brahea.com> wrote: > A diagnosis seems easy enough, but so far a cure eludes me. Observe > that your conditions contain the radical Sqrt[z1^2+4z2]. Your > invocation of FindInstance asks it to assume that, not only the > variables but also all function values are real, in particular Sqrt, > implying that z1^2+4z2>0. Indeed, all points found satisfy this > condition. No puzzler here. Unfortunately, when I try the suggested > variation to relax this assumption, > > RegionPlot[conds,{z1,-.75,2.25},{z2,-1.25,1.25},PlotPoints->90, > Epilog->Point[{z1,z2}/. > FindInstance[conds&&Element[{z1,z2},Reals],{z1,z2},10^3] > ]] > > the kernel goes out to lunch. I hope someone has a better idea. > > Tom > > When responding, please replace news with my first initial and full > last name, as one word. > > Tom Burton > > > ... > > RegionPlot[conds, {z1, -.75, 2.25}, {z2, -1.25, 1.25}, PlotPoints - > > > 90, > > Epilog -> Point[{z1, z2} /. FindInstance[conds, {z1, z2}, Reals, > > 10^3]]] > > ... Hitting conds a couple of times with a hammer helps: In[2]:= Reduce[ComplexExpand[conds /. Sqrt[z1^2 + 4 z2] -> r, r] && Element[{z1, z2}, Reals] /. a_/z2^2 >= b_ :> a >= b z2^2 && z2 != 0 /. r -> Sqrt[z1^2 + 4 z2], {z1, z2}] Out[2]= (z1 == -1/2 && z2 == 1/2) || (-1/2 < z1 < 0 && -z1 <= z2 <= 1 + z1) || (0 <= z1 < 1 && 0 < z2 <= 1 - z1) || (0 < z1 <= 1 && (-z1^2/4 < z2 < 0 || -z1 <= z2 <= -z1^2/4)) || (1 < z1 < 2 && (-z1^2/4 < z2 <= 1 - z1 || -1 <= z2 <= -z1^2/4)) || (z1 == 2 && z2 == -1) RegionPlot[conds, {z1, -.75, 2.25}, {z2, -1.25, 1.25}, PlotPoints -> 90, Epilog -> Point[{z1, z2} /. FindInstance[%, {z1, z2}, 10^3]]] Maxim Rytin m.r at inbox.ru