Re: question about Solve

*To*: mathgroup at smc.vnet.net*Subject*: [mg91708] Re: question about Solve*From*: Bill Rowe <readnews at sbcglobal.net>*Date*: Sat, 6 Sep 2008 02:06:35 -0400 (EDT)

On 9/5/08 at 7:14 AM, Florian.Jaccard at he-arc.ch (Jaccard Florian) wrote: >I don't understand the following behaviour of Solve. >Consider the following system : >Solve[{x*y==(a+2*b)/(c+2*d),1/Sqrt[2]==Sqrt[(e*y)/(z*f*g*h)],2*Pi*= i= >=0.9/(z*f)},{x,y,z}] >Everything fine, I obtain : >{{z -> 0.1432394487827058/(f*i), >x -> (1.419827298426263*^-9*(9.83403688*^9*a + >1.966807376*^10*b)*e*i)/((c + 2.*d)*g*h), y -> >(0.0716197243913529*g*h)/(e*i)}} >But if I ask for the answer of almost the same (only a 4 in the >denominator of the second equation), Solve isn't abble anymore to >manage without using inverse functions... why? >Solve[{x*y == (a + 2*b)/(c + 2*d), >1/Sqrt[2] == Sqrt[(e*y)/(4*z*f*g*h)], 2*Pi*i == 0.9/(z*f)}, {x, y, >z}] >Worse: >If I have numerical values for a, b, c, d, e, f, g, h, i: a = >65/10^6; b = 1/10^3; c = 1.9; d = 0.19; e = 1/(2.5/10^3); v = 18= ; w >= 8; g = (2*v)/((c + 2*d)*w); i = 3000; h = 0.2; >Then Solve isn't able anymore! Mathematica thinks there is no >solution. I believe all of issues you are seeing here has to do with round off as result of using machine precision numbers. I can do: In[39]:= Solve[{x*y == (a + 2*b)/(c + 2*d), 1/Sqrt[2] == Sqrt[(e*y)/(z*f*g*h)], 2*Pi*i == 9/10/(z*f)}, {x, y, z}] getting: Out[39]= {{x -> (3304*Pi)/27, z -> 3/(20000*f*Pi), y -> 9/(1216000*Pi)}} with no messages generated. Similarly if I do: a = 65/10^6; b = 1/10^3; c = 19/10; d = 19/100; e = 1/(25/10^2); v = 18; w = 8; g = (2*v)/((c + 2*d)*w); i = 3000; h = 2/10; then Solve is able to find In[37]:= Solve[{x*y == (a + 2*b)/(c + 2*d), 1/Sqrt[2] == Sqrt[(e*y)/(4*z*f*g*h)], 2*Pi*i == 9/10/(z*f)}, {x, y, z}] Out[37]= {{x -> (826*Pi)/27, z -> 3/(20000*f*Pi), y -> 9/(304000*Pi)}} with no difficulty. As a general rule you will get better, more consistent results with one of the functions in Mathematica that begin with N when using machine precision numbers in the input. When using functions like Solve that are designed to use algebraic methods, entering the coefficients as exact numbers works better.