Problems with Mathematica 8.0 Solve
leigh pascoe
Like many of you I have recently installed Mathematica 8.0 and have been
exploring its new features. I have encountered the following problem
using the Solve Function:
Firstly in Mathematica 7.0
I try to solve the following two simultaneous equations
In[35]
eq1 = (b + d + f)/x - (a + b)/(1 + x) - 2*(c + d + e)/(1 + 2*x + y) -
(f + g)/(x + y) == 0
eq2 = (e + g)/y - (c + d + e)/(1 + 2*x + y) - (f + g)/(x + y) == 0
In[36}
Timing[sols = Solve[{eq1, eq2}, {x, y}];]
Out[36]
{52.437, Null}
A bit slow but it gives me the result I want. I then Simplify the
solutions and evaluate them with a set of vectors of the constants
{a,b,c,d,e,f,g}.
subs = {{a -> 45, b -> 79, c -> 4, d -> 11, e -> 8, f -> 0, g -> 2},
{a -> 95, b -> 85, c -> 9, d -> 13, e -> 6, f -> 0, g -> 1},
{a -> 18, b -> 45, c -> 0, d -> 1, e -> 2, f -> 0, g -> 0}, {a -> 50,
b -> 41, c -> 0, d -> 1, e -> 0, f -> 0, g -> 0},
{a -> 17, b -> 61, c -> 1, d -> 9, e -> 14, f -> 0, g -> 1}, {a ->
68, b -> 60, c -> 10, d -> 7, e -> 3, f -> 2, g -> 0},
{a -> 42, b -> 90, c -> 5, d -> 29, e -> 49, f -> 6, g -> 17}, {a ->
140, b -> 108, c -> 23, d -> 32, e -> 19, f -> 16, g -> 8}}
nsols = Simplify[RootReduce[sols /. subs]]
N[%]
giving me the answers I want
{{{y -> -2.18473, x -> 2.02599}, {y -> -0.0457403, x -> -0.53058}, {y ->
2.72299, x -> 1.70867}}, {{y -> -1.05434, x -> 1.00657}, {y ->
-0.0244319, x -> -0.536953},
{y -> 0.83139, x -> 0.871734}}, {{y -> -2.66667, x -> 2.66667}, {y ->
-0.0183209, x -> -0.50458}, {y -> 12.1294, x -> 2.53236}}, {{y -> -0.84,
x -> 0.84},
{y -> 0., x -> -0.503749}, {y -> 0., x -> 0.833749}}, {{y -> -4.83381,
x -> 4.69898}, {y -> -0.100097, x -> -0.530885}, {y -> 12.0561, x ->
3.63746}}, {{y -> -1.05985,
x -> 0.917926}, {y -> -0.014617, x -> -0.542283}, {y -> 0.351798, x
-> 0.797433}}, {{y -> -4.52487, x -> 3.85637}, {y -> -0.240225, x ->
-0.548865}, {y -> 7.61935, x -> 2.24569}},
{{y -> -1.46526, x -> 1.07904}, {y -> -0.0612374, x -> -0.561339}, {y ->
0.684794, x -> 0.770645}}}
In Mathematica 8.0 this process hangs at the Solve command. I was forced
to abort the evaluation after more than an hour with no result. The
program seems to use only 25% of the available cpu (i.e. 1 of the 4
available in the quadcore) and never arrives at a result. I thought that
equation solving was much improved in the new version, so what is going on?
Perhaps there is a much better way to get the results I want. I need to
evaluate the solutions to the above equations for large numbers of data
vectors. I would appreciate any suggestions. Thanks in advance.
LP
Daniel Lichtblau
We are not exactly sure why this worked better in earlier versions (we know where it gets into trouble in version 8). First are some suggestions from Adam Strzebonski, regarding your use of parametrized Solve.
"[...] mention that
for the (hopefully rare) cases when the new Solve method turns out
to be slower than the old one we still have the old method available.
In[3]:= Timing[sols = Solve[{eq1, eq2}, {x, y}, Method->"Legacy"];]
Out[3]= {15.4497, Null}
Another possibility
parallelSolve[args__] :=
ParallelTry[#[args] &, {Solve, Solve[##, Method -> "Legacy"] &}]
In[10]:= AbsoluteTiming[sol = parallelSolve[{eq1, eq2}, {x, y}];]
Out[10]= {30.6728358, Null}
Adam"
Let me point out some other issues. First is that you will not save significant time if your goal is to get "nice" results. Specifically, you can solve relatively fast once you plug in those numeric values.
In[5]:= Timing[solns = Map[Solve[(exprs /. #) == 0] &, Take[subs, 4]];]
Out[5]= {0.889, Null}
But the reduction is relatively slow.
In[6]:= Timing[RootReduce[solns];]
Out[6]= {161.009, Null}
Also in version 8 you can readily suppress the radical solutions and then it will RootReduce quote quickly.
In[6]:= Timing[
solns = Map[Solve[(exprs /. #) == 0, Cubics -> False] &,
Take[subs, 4]];]
Out[6]= {0.063, Null}
In[9]:= Timing[RootReduce[solns];]
Out[9]= {0.032, Null}
If your end goal is the numeric solutions, I generally recommend NSolve.
In[10]:= Timing[Map[NSolve[exprs /. #] &, subs]]
Out[10]= {0.28, {{{y -> -2.18473, x -> 2.02599}, {y -> 2.72299,
x -> 1.70867}, {y -> -0.0457403, x -> -0.53058}}, {{y -> -1.05434,
x -> 1.00657}, {y -> -0.0244319, x -> -0.536953}, {y -> 0.83139,
x -> 0.871734}}, {{y -> 12.1294, x -> 2.53236}, {y -> -0.0183209,
x -> -0.50458}}, {}, {{y -> 12.0561,
x -> 3.63746}, {y -> -4.83381, x -> 4.69898}, {y -> -0.100097,
x -> -0.530885}}, {{y -> -1.05985,
x -> 0.917926}, {y -> -0.014617, x -> -0.542283}, {y -> 0.351798,
x -> 0.797433}}, {{y -> -4.52487, x -> 3.85637}, {y -> 7.61935,
x -> 2.24569}, {y -> -0.240225, x -> -0.548865}}, {{y -> -1.46526,
x -> 1.07904}, {y -> -0.0612374, x -> -0.561339}, {y -> 0.684794,
x -> 0.770645}}}}
Daniel Lichtblau
Wolfram Research
Chris Chiasson
> ----- Original Message -----
> > From: "leigh pascoe" <le...@evry.inserm.fr>
> > To: mathgr...@smc.vnet.net
> > Sent: Friday, November 26, 2010 4:32:06 AM
> > Subject: Problems with Mathematica 8.0 Solve
> > Dear Mathgroup,
>
> > Like many of you I have recently installed Mathematica 8.0 and have
> > been
> > exploring its new features. I have encountered the following problem
> > using the Solve Function:
>
> > Firstly in Mathematica 7.0
>
> > I try to solve the following two simultaneous equations
> > In[35]
> > eq1 = (b + d + f)/x - (a + b)/(1 + x) - 2*(c + d + e)/(1 + 2*x + y) -
> > (f + g)/(x + y) == 0
ow where it gets into trouble in version 8). First are some suggestions fro=
m Adam Strzebonski, regarding your use of parametrized Solve.
>
> "[...] mention that
> for the (hopefully rare) cases when the new Solve method turns out
> to be slower than the old one we still have the old method available.
>
> In[3]:= Timing[sols = Solve[{eq1, eq2}, {x, y}, Method->"Legacy"];]
> Out[3]= {15.4497, Null}
>
> Another possibility
>
> parallelSolve[args__] :=
> ParallelTry[#[args] &, {Solve, Solve[##, Method -> "Legacy"] &}]
>
> In[10]:= AbsoluteTiming[sol = parallelSolve[{eq1, eq2}, {x, y}];]
> Out[10]= {30.6728358, Null}
>
> Adam"
>
ficant time if your goal is to get "nice" results. Specifically, you can so=
lve relatively fast once you plug in those numeric values.
>
4]];]
> Out[5]= {0.889, Null}
>
> But the reduction is relatively slow.
>
> In[6]:= Timing[RootReduce[solns];]
> Out[6]= {161.009, Null}
>
it will RootReduce quote quickly.
>
> In[6]:= Timing[
> solns = Map[Solve[(exprs /. #) == 0, Cubics -> False] &,
> Take[subs, 4]];]
> Out[6]= {0.063, Null}
>
> In[9]:= Timing[RootReduce[solns];]
> Out[9]= {0.032, Null}
>
> If your end goal is the numeric solutions, I generally recommend NSolve.
>
> In[10]:= Timing[Map[NSolve[exprs /. #] &, subs]]
>
> Out[10]= {0.28, {{{y -> -2.18473, x -> 2.02599}, {y -> 2.72299,
,
> x -> 1.00657}, {y -> -0.0244319, x -> -0.536953}, {y -> 0.8313=
9,
> x -> 0.871734}}, {{y -> 12.1294, x -> 2.53236}, {y -> -0.0183209,
> x -> -0.50458}}, {}, {{y -> 12.0561,
> x -> 3.63746}, {y -> -4.83381, x -> 4.69898}, {y -> -0.100097,
> x -> -0.530885}}, {{y -> -1.05985,
> x -> 0.917926}, {y -> -0.014617, x -> -0.542283}, {y -> 0.351798,
> x -> 0.797433}}, {{y -> -4.52487, x -> 3.85637}, {y -> 7.61935,
,
> x -> 1.07904}, {y -> -0.0612374, x -> -0.561339}, {y -> 0.6847=
94,
> x -> 0.770645}}}}
>
> Daniel Lichtblau
> Wolfram Research
Thanks for posting about the Method->"Legacy" option. I called
technical support a few days ago and asked how to use the old Solve
method, but they never got back to me. Here is an example of an
equation with a few branches where the new default method falls over.
Solve[Max[coefficient Cv, low\[UnderBracket]limit,Min[static\
[UnderBracket]limit, z/(SWL x + y)]] == EBSTNLL/(Mf SWL + WDead), SWL]
Obviously, the Legacy method doesn't provide the conditions under
which the solutions are valid, but at least it does go through, take
each branch, and quickly solve it for the SWL rather than requiring me
to do that manually. Something needs to be done to make the new
default Solver more robust in the face of simple equations like the
one above.