Re: Solve problems
- To: mathgroup at smc.vnet.net
- Subject: [mg64248] Re: [mg64233] Solve problems
- From: Pratik Desai <pdesai1 at umbc.edu>
- Date: Wed, 8 Feb 2006 03:53:35 -0500 (EST)
- References: <200602070836.DAA29884@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Joerg Schaber wrote:
>Hi,
>
>I have a system of polynomial equations where Solve cannot find the
>right solutions. Any hints? By the way, actually I just want to find the
>steady states of a differential equation system. If there is another
>clever way, please let me know.
>There are also coonstraints that all variable must be >=0, but including
>those constraints and using Reduce also does not yield a valid solution.
>There exisits a valid solution. I checked this solving the differential
>equation system with NDSolve and let it run into the steady state. But
>this is not very elegant and principally the steady states can be
>calculated directly.
>
>eqns1={0 == 0. c1 + 0.001 c3 - 721.9 c1 c4 + 0.001 c2 c8,
>0 == 0.001 c3 - 721.9 c1 c4 + 0.001 c5 - 346.09 c4 c6,
>0 == 0.001 c5 - 346.09 c4 c6 - 989.77 c6 c7 + 0.001 c8,
>0 == -989.77 c6 c7 + 0.001 c8, c1 + c2 + c3 == 5.7,
>c3 + c4 + c5 == 19.3,
>c5 + c6 + c8 == 4.,
>c7 + c8 == 1.};
>
>sol = Solve[eqns1, {c1, c2, c3, c4, c5, c6, c7, c8}, VerifySolutions ->
>True];
>
>\!\({{c2 -> 0.`, c3 ->
> 5.69999921807596`, c5 -> 3.502051416006559`, c1 -> \
>7.819240400365867`*^-7, c7 -> 0.5020524180816763`, c8 ->
>0.4979475819183236`, \
>c6 -> 1.0020751177226038`*^-6, c4 -> 10.097949365917481`}, {c2 -> 0.`,
>c3 -> \
>5.699999640315395`,
> c5 -> \(-8.352092810365317`\), c1 -> 3.5968460447115093`*^-7,
> c7 -> \(-11.352093909700997`\), c8 -> 12.352093909700997`, c6 -> \
>\(-1.0993356796497231`*^-6\), c4 -> 21.95209317004992`}, {c2 -> 0.`,
> c3 -> 8.687577015258206`, c5 -> 10.612427012862872`,
> c1 -> \(-2.987577015258207`\), c7 -> \(-1.3272190975834564`*^-7\),
> c8 -> 1.0000001327219097`, c6 -> \(-7.612427145584781`\),
> c4 -> \(-4.028121078693263`*^-6\)}, {c2 -> 0.`, c3 -> \
>19.300001049304143`,
> c5 -> 9.165049916617935`*^-7, c1 -> \(-13.600001049304144`\),
>c7 -> \
>\(-3.0000004306092727`\), c8 -> 4.000000430609273`,
> c6 -> \(-1.3471142644128086`*^-6\), c4 -> \
>\(-1.965809135229149`*^-6\)}, {c2 -> 0.`,
> c3 -> 24.324407481766347`, c5 -> \(-5.024405672582244`\), c1 -> \
>\(-18.624407481766347`\), c7 -> 1.259078407457304`*^-7, c8 -> \
>0.9999998740921593`, c6 -> 8.024405798490085`,
> c4 -> \(-1.8091841042296682`*^-6\)}}\)
>
>
>Einsetzen ergibt:
>
>eqns1 /. sol
>
>{{True, False, False, False, True, True, True, True}, {True, True,
> False, False, True, True, True, True}, {True, False, False, True,
> True, True, True, True}, {True, False, True, True, True, True,
> True, True}, {False, False, False, True, True, True, True, True}}
>
>
>best wishes,
>
>joerg
>
>
>
I think that this is a precision issue, have look at
eqns1[[3]]/.sol[[3]]//Trace
the last lines will have something like
HoldForm[0 == 3.469446951953614*^-18], HoldForm[False]}
Workaround is to increase the precision or use exact arithmatic (see
Rationalize)
In[77]:=
Clear[eqns1, sol1]
eqns1=SetPrecision[{0 == 0. c1 + 0.001 c3 - 721.9 c1 c4 + 0.001 c2 c8,
0 == 0.001 c3 - 721.9 c1 c4 + 0.001 c5 - 346.09 c4 c6,
0 == 0.001 c5 - 346.09 c4 c6 - 989.77 c6 c7 + 0.001 c8,
0 == -989.77 c6 c7 + 0.001 c8, c1 + c2 + c3 == 5.7,
c3 + c4 + c5 == 19.3,
c5 + c6 + c8 == 4.,
c7 + c8 == 1.},40];
sol1=Solve[eqns1//Simplify, {c1, c2, c3, c4, c5, c6, c7,
c8},VerifySolutions\[Rule]True];
eqns1/.sol2
Out[80]=
{{True,True,True,True,True,True,True,True},{True,True,True,True,True,True,
True,True},{
True,True,True,True,True,True,True,True},{True,True,True,True,True,True,
True,True},{True,True,True,True,True,True,True,True}}
- References:
- Solve problems
- From: Joerg Schaber <schaber@molgen.mpg.de>
- Solve problems