Re: equations

*To*: mathgroup at smc.vnet.net*Subject*: [mg31995] Re: equations*From*: Andrzej Kozlowski <andrzej at tuins.ac.jp>*Date*: Sun, 16 Dec 2001 03:44:30 -0500 (EST)*Sender*: owner-wri-mathgroup at wolfram.com

Dear Fred and Daniel I clearly should read the messages I am responding to more carefully. Fred's point is in fact one that I have written about before on this list. In general, if you give a set of algebraic equations with symbolic coefficients, I do not think it even makes sense to speak of a "general solution". There will be a lot of "branching" going on and it is not at all clear to me that one can give a "formula" for the roots, in any conventional sense anyway, that will Work for all values of the parameters. This is true even for a single equation: if the coefficients are numeric one can isolate the roots, but when they are not it seems to me that the concept of a general formula does not even make sense (for degrees greater than 5). In the case of systems of equations it gets even more complicated. Daniel knows a lot more about this than me, but I am not at all sure if the concept of "a general solutions" (that works for all values of the parameters) makes any mathematical sense. It also clearly does not make any practical sense, although questions about such formulas are probably the most common amongst the postings to this list. Andrzej On Sunday, December 16, 2001, at 02:36 AM, Fred Simons wrote: > Dear Andrzej and Daniel, > > I think I formulated my comment a little bit too short; the postings of > Daniel and mine are not mutually contradictory, as Andrzej wrote. As > Daniel > did, I drew the conclusion that the set of equations must have four > solutions. I my posting I referred to another problem. Knowing the > general > values of the unknowns does not imply that we know the general > solution. Let > me explain that more clearly. > > We do have (symbolic) expressions for the four possible values of each > of > the four unknowns R1, R2, R3 and R4: > > R1 may have one of the values v11, v12, v13, v14, > R2 may have one of the values v21, v22, v23, v24, > R3 may have one of the values v31, v32, v33, v34, > R4 may have one of the values v41, v42, v43, v44. > > Nevertheless, we have not yet found any solution of the system of > equations. > Such a solution much have the form > > {R1->value1, R2->value2, R3->value3, R4->value4}, > > where we expect that value(i) will be one of the four values for R(i). > > The testing I did was the following. For a given set of parameters we > can > solve the set of equations and therefore we can combine the symbolic > values > in a unique way such that say > > {R1 -> v11, R2 -> v24, R3 -> v31, R4 -> v44} > {R1 -> v12, R2 -> v22, R3 -> v32, R4-> v42} > {R1 -> v13, R2 -> v21, R3 -> v34, R4 -> v43} > {R1 -> v14, R2 -> v23, R3 -> v33, R4 -> v41} > > is the solution of the set of equations for that choice of the > parameters. > However, this solution is definitely NOT the general solution (i.e. for > all > choices of the parameters). It is easy to find values of the parameters > for > which substituting in the equations results in one or more outcomes > False. > So the general solution must be more complicated (a combination of > roots of > a polynomial of degree 16, in the worst case?). Maybe this explains > why I > did not manage to solve the set of equations using Solve within 14 > hours, > while finding the values v11, ..., v44 is pretty fast. > > By the way, playing with this example I also found that > > Solve[ Eliminate[eqs, {R1, R2, R3}], R4] is done within 3 seconds, while > > Solve[eqs, R4, {R1, R2, R3}] > > takes about 2 minutes. I thought these two commands were equivalent. > > > Fred Simons > Eindhoven University of Technology > > > > > > > > Andrzej Kozlowski Toyama International University JAPAN http://platon.c.u-tokyo.ac.jp/andrzej/