A strange bug in Solve

*To*: mathgroup at smc.vnet.net*Subject*: [mg24354] A strange bug in Solve*From*: Andrzej Kozlowski <andrzej at tuins.ac.jp>*Date*: Sun, 9 Jul 2000 04:52:57 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

I have long ago learned to be careful when making claims about kernel bugs in Mathematica, but this time I am pretty sure I have found a fairly serious one, even though it looks rather strange. I asked Mathematica to solve a system of two trigonometric equations: In[1]:= eqns = {2*Cos[2*t]*Cos[u/2] + Cos[t]*Sin[u/2] == 0, (Cos[u/2]*Sin[t])/2 - (Sin[2*t]*Sin[u/2])/2 == 0}; To my surprise Solve returned the empty list (I knew these equations do have solutions for geometric reasons): In[2]:= Solve[eqns, {u, t}] Out[2]= {} I then tried Reduce, and guess what, it gave the right answers: In[3]:= sols = Reduce[eqns, {u, t}] Reduce::ifun: Inverse functions are being used by Reduce, so some solutions may not be found. Out[3]= 1 1 t == 0 && u == 2 ArcCos[-(-------)] || t == 0 && u == -2 ArcCos[-------] || Sqrt[5] Sqrt[5] -Pi Pi t == --- && u == -Pi || t == -- && u == Pi || 2 2 3 -Sqrt[-] 2 3 t == ArcCos[--------] && u == 2 ArcCos[-Sqrt[-]] || 2 5 3 Sqrt[-] 2 3 t == ArcCos[-------] && u == 2 ArcCos[Sqrt[-]] 2 5 We can check them as follows: solrules = sols /. {Or -> List, And -> List, Equal -> Rule}; In[5]:= FullSimplify[Unevaluated[eqns /. solrules]] Out[5]= {{True, True}, {True, True}, {True, True}, {True, True}, {True, True}, {True, True}} Here one might ask: why did I need to define solrules? Why not use the options MakeRules->True in Reduce? The answer is: try and see what happens! This is peculiar and rather unpleasant. When Solve returns an ampty list it ought to mean that there are no generic solutions (although solutions might exist for some particular values of the parameters). Reduce is supposed to try to work out these special cases. However, the solutions found by Reduce in this case are perfectly generic. Moreover, the spectacular failure of Reduce[eqns, MakeRules->True] suggest than somehting else than the difficulty of finding the solutions that is causing the problem here. Andrzej Kozlowski -- Andrzej Kozlowski Toyama International University, JAPAN For Mathematica related links and resources try: <http://www.sstreams.com/Mathematica/>