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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.