       Re(2): Re: Eigenvalue Problem

• To: mathgroup at smc.vnet.net
• Subject: [mg17058] Re(2): [mg16991] Re: Eigenvalue Problem
• From: Andrzej Kozlowski <andrzej at tuins.ac.jp>
• Date: Wed, 14 Apr 1999 02:12:05 -0400
• Sender: owner-wri-mathgroup at wolfram.com

```On Tue, Apr 13, 1999, Paul Abbott <paul at physics.uwa.edu.au> wrote:

>At 1:17 PM +0800 13/4/99, Andrzej Kozlowski wrote:
>
>>I do not think this problem is essentially to do with roots of equations.
>>Or rather, there is a deeper underlying problem. It is Mathematica's
>>inability to recognize certain complex expressions as real.
>
>Perhaps ...
>
>But to me, the most elegant and general solution to problems of this type
>is to work with Root objects anyway and, except for trivial problems, an
>explicit formula in terms of radicals is no more enlightening.
>
>In ftp://ftp.physics.uwa.edu.au/pub/Mathematica/MathGroup/RealRoots.nb I
>have included a bit more on using Roots (following on from Dan) which I
>hope makes this clearer.
>
>Cheers,
>	Paul
>
>
>
>
I generally agree, though I still think ComplexExpand should be
improved. Mathematica's inability to recognize even very simple algebraic
expresions as positive or negative when all the variables are assumed to
be real, and to deal effectively with conjugates of expressions
containing variables reduces the usefulness (for what it's worth) of
ComplexExpand in algebraic manipulations .

In:=
ComplexExpand[a+b*I+Conjugate[a+b*I]]
Out=
2 a
In:=
ComplexExpand[a+Sqrt[-b^2]+Conjugate[a+Sqrt[-b^2]]]
Out=
2 a
So far so good. But

In:=
ComplexExpand[a+Sqrt[-b^2-1]+Conjugate[a+Sqrt[-b^2-1]]]
Out=
2      1           2
2 a + 2 Sqrt[1 + b ] Cos[- Arg[-1 - b ]]
2
which is hardly an elegant way to write 2a.

But it gets worse:

In:=
ComplexExpand[Sqrt[a + b I] + Sqrt[Conjugate[a + b I]],

TargetFunctions -> {Im, Re}]
Out=
2    2 1/4     1
(a  + b )    Cos[- ArcTan[a, -b]] +
2

2    2 1/4     1
(a  + b )    Cos[- ArcTan[a, b]] +
2

2    2 1/4     1
I (a  + b )    Sin[- ArcTan[a, -b]] +
2

2    2 1/4     1
I (a  + b )    Sin[- ArcTan[a, b]]
2

As an answer this leaves a lot to be desired. But now try evaluating

ComplexExpand[Sqrt[a+Sqrt[-1-b^2]]+Sqrt[Conjugate[a+Sqrt[-1-b^2]]],
TargetFunctions->{Im,Re}]

I am not sure if ComplexExpand in this type of situations has any serious
uses (I have never found one myself) but on the principle that if
anything is worth doing at all it is worth doing well I think some
improvements should be made in the future.

P.S. I have looked at your Roots.nb notebook. Near the end of it you need
to show that a  certain expresion is always non-positive.  One can save
oneself a little effort by letting Mathematica do the job:

In:=
<<Algebra`AlgebraicInequalities`
In:=
SemialgebraicComponents[1/27*
(-419904*A^6 - 16848*B^2*A^4 - 42768*C^2*A^4 -
225*B^4*A^2 + 423*C^4*A^2 - 774*B^2*C^2*A^2 - B^6 -
C^6 - 3*B^2*C^4 - 3*B^4*C^2) > 0, {A, B, C}]
Out=
{}

Of course, if one does that, one has to face the problem of whether
having a computer program do this  can even be taken to constitute a proof.

Andrzej Kozlowski
Toyama International University
JAPAN
http://sigma.tuins.ac.jp/
http://eri2.tuins.ac.jp/

```

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