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Re: Finding a continuous solution of a cubic

  • To: mathgroup at smc.vnet.net
  • Subject: [mg85936] Re: [mg85899] Finding a continuous solution of a cubic
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Thu, 28 Feb 2008 02:48:31 -0500 (EST)
  • References: <200802270925.EAA16050@smc.vnet.net>

On 27 Feb 2008, at 10:25, Hugh Goyder wrote:

> I am investigating the maxima and minima of the expression, e, below.
> The expression is a quartic in x and has two parameters a and b which,
> for my problem, are restricted to the range -1 < a < 1 and 0 < b (the
> interesting part is 0<b<1).
>
> I start with a simple approach and take the derivative with respect to
> x and solve to find the three turning points. The solution is
> complicated and not easy to examine. I work out the discriminant to
> see when I will have three real roots or one and plot the region. I
> work out three functions for the three roots r1, r2 and r3, and three
> functions for the value of e at the roots, f1, f2, and f3. I am
> expecting three real functions when within the region defined by the
> discriminant and one outside this region. I plot the three roots and
> the three functions (there are some artifacts where the functions turn
> from real to complex, this is not a problem).
>
> Now my problem. The function f3 is continuous in the region where
> there are three solutions and this I like. Functions f1 and f2 are not
> continuous but contain holes. If functions f1 and f2 are plotted
> together then they each fill in the others holes. The holes makes my
> life difficult because I do not have continuous functions.
>
> Is it possible to make Solve find roots that will give rise to three
> functions one corresponding to the maxima and two corresponding to
> each minima without the functions crossing over and jumping between
> the turning points?
>
> I also notice, from the plots when shown together,  that my desired
> functions are even with respect to the parameter a. Is this correct
> and how would I show this if I can't assemble them as continuous
> functions?
>
> Thanks
>
> Hugh Goyder
>
> e = ((1 - x)^2 + b*(-1 + a)^2)*((1 + x)^2 +
>          b*(1 + a)^2);
>
> d = Simplify[D[e, x]];
>
> dis = Discriminant[d, x];
>
> ContourPlot[dis, {a, -1, 1}, {b, 0, 1},
>   ContourShading -> False, Contours -> {0},
>   FrameLabel -> {"a", "b"}]
>
> sol = Solve[d == 0, x];
>
> ClearAll[x1, x2, x3, f1, f2, f3];
> x1[a_, b_] := Evaluate[x /. sol[[1]]];
> x2[a_, b_] := Evaluate[x /. sol[[2]]];
> x3[a_, b_] := Evaluate[x /. sol[[3]]];
> f1[a_, b_] := Evaluate[e /. sol[[1]]];
> f2[a_, b_] := Evaluate[e /. sol[[2]]];
> f3[a_, b_] := Evaluate[e /. sol[[3]]];
>
> r1 = Plot3D[x1[a, b], {a, -1, 1}, {b, 0, 1}]
>
> r2 = Plot3D[x2[a, b], {a, -1, 1}, {b, 0, 1}]
>
> r3 = Plot3D[x3[a, b], {a, -1, 1}, {b, 0, 1}]
>
> Show[r1, r2, r3, PlotRange -> All]
>
> s1 = Plot3D[f1[a, b], {a, -1, 1}, {b, 0, 1}]
>
> s2 = Plot3D[f2[a, b], {a, -1, 1}, {b, 0, 1}]
>
> s3 = Plot3D[f3[a, b], {a, -1, 1}, {b, 0, 2}]
>
> Show[s1, s2]
>
>
>

Mathematica will certainly not give you such a formula. I think in  
your case it will be quite hard to obtian one, and I can't at the  
moment suggest a good way to do so, but it may be helpful to first  
consider the the cubic geometrically and understand how Mathematica  
deals with the ordering of its roots. In general a cubic (with real  
coefficients) can have either 1 or all 3 real roots.  If the cubic ir  
represented by a polynomial poly (in your case poly = d) in the  
variable x  then these roots can are given by the Mathematica  
functions Root[poly,x,1], Root[poly,x,2] and Root[poly,x,3].
For example:

In[31]:= Root[x^3 + x + 1, x, 1]
Out[31]= Root[#1^3 + #1 + 1 &, 1]

In[32]:= ToRadicals[%]
Out[32]= -(2/(3*(-9 + Sqrt[93])))^(1/3) + ((1/2)*(-9 +  
Sqrt[93]))^(1/3)/3^(2/3)



Root[poly,x,1],Root[poly,x,2] and Root[poly,x,3] are your functions  
f1,f2, and f3.

What Mathematica counts as the first root of a cubic, as mentioned  
above, is always  real. So suppose you define your real root function  
as Root[poly,x,1] and also suppose that we "start" in a situation (a  
set of parameter values) where we have  one real root and two  
conjugate non-real ones. The real root is our Root[poly,x,1]. Now,  
however, as we vary the parameters, the two non-real roots Root[poly,x, 
2] and Root[poly,x,3] will at some point coalesce to form a double  
real root, which may be smaller than your original root. At this point  
the roots are "renamed", in other words these two roots become  
Root[poly,x,1] and Root[poly,x,2] and the original root then becomes  
Root[poly,x,3]. So if we stick to the function Root[poly,x,1] we will  
have a discontinuous jump. In order to avoid it, we should take  
Root[poly,x,3] as the value of our function after the jump point.

So in fact, to construct a continuous function you only need to  
consider Root[poly,x,1] and Rot[poly,x,3] for the root will never jump  
from the first to second place, because whenever two complex roots  
coalesce, two new real of the same magnitude roots are created and so  
what used to be root number one will become root number 3.

As you noted, all the jumps take place on the curve  
dis=Discriminant[d, x] = 0, because the choice of a smallest root can  
only be changed once a double root appears. However, not every  
appearance of a double root will result in a jump, because the two  
real roots that appear due to the merging of two complex ones may be  
larger than the real root we had before their merger.

Now, in the case of a cubic with a single parameter, the equation dis  
can actually be solved numerically so we can find all the jump points  
and use the function Piecewise to combine them to get a continuous  
function of the parameter. This is easy and has been considered more  
than once on this list in the past. Your two parameter case seems much  
more complicated (however, I have not considered it very carefully!)  
since the discriminant is now a real curve of degree 9 in the  
parameter space so at this time I have no good idea to suggest. Still,  
I hope you can find something useful in the above remarks.

Andrzej Kozlowski


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